Download thesis - Curtin University

Department of Spatial Sciences
Improvement of Geodetic Parameter Estimation in Coastal Regions
from Satellite Radar Altimetry
Xiaoli Deng
This thesis is presented for the Degree of
Doctor of Philosophy
of
Curtin University of Technology
August 2003

i
Declaration
This thesis contains no material which has been accepted for the award of any other degree
or diploma in any university.
To the best of my knowledge and belief this thesis contains no material previously published
by any other person except where due acknowledgment has been made.
Signature:
………………………………………….
Date:
………………………...

ii
ABSTRACT
Altimeter-derived sea surface heights (SSH) are most probably in error in coastal
regions due, in part, to the complex nature of echoes returned from rapidly varying
coastal topographic surfaces (both land and sea). This dissertation presents improved
altimeter-derived SSH results in coastal regions using the waveform retracking
technique, which reprocesses the waveform data through a ‘coastal retracking
system’.
The system, based upon a systematic and comprehensive analysis of satellite radar
altimeter waveforms in the Australian coast, provides an efficient means of
improving altimeter-derived SSH data, not from a single retracker, but instead from
several retrackers depending on the altimeter waveform characteristics. Central to the
system is the use of two retracking techniques: the modified iterative least squares
fitting and the threshold retracking algorithms. To overcome the ‘noise’ in
waveforms caused by fading noise, the fitting algorithm has been developed to
include a weighted iterative scheme. The retrackers adopted in the system include
five fitting models and the threshold method with varying threshold levels. A
waveform classification procedure has also been developed, which enables the
waveform to be sorted and retracked by an appropriate retracker.
Twenty-Hertz waveform data from the ERS-2 and POSEIDON satellites have been
used to estimate a broad contaminated distance of ~10 km offshore the Australian
coast. Two cycles of ERS-2 waveform data are, then, reprocessed using the coastal
retracking system to obtain the improved SSH estimates. Using the AUSGeoid98
geoid grid as a partly independent ground reference, the coastal retracking system
has been able to reduce the contaminated distance to ~5 km. Improved ERS-1 SSH
data from waveform retracking has also been obtained in another coastal region of
Taiwan. Two altimeter-based gravity anomaly grids were created near Taiwan using
the SSH data before and after retracking. Compared with ship-track gravity data,
results show that the accuracy of the gravity anomalies has been improved from
±13.9 mgal before retracking to ±9.9 mgal after retracking.

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ACKNOWLEDGEMENTS
I wish to extend my warmest and sincerest thanks to my supervisor, Professor Will
Featherstone. Will has always been patient, helpful and supportive. I have learned a
lot about the research methodology and geodesy from him. To him, I am indebted
not just for his scientific supervision, but also for his understanding, help and
friendship.
I am most grateful to my associate supervisor, Professor Cheinway Hwang (National
Chiao Tung University, Taiwan), for giving me the opportunity to work in his Space
Geodesy Laboratory for a month, and for his help with many aspects of this research.
I am grateful also to my other associate supervisors, Dr Jon Kirby and Professor
Phillipa Berry (De Montfort University, UK), for their guidance during much of this
research.
My thanks go to all of the friendly team members, both staff and students, in the
Western Australian Centre for Geodesy for their help and advice, particularly to
people who worked with me in the lab: Troy Forward, Simon Holmes, Minghai Jia,
Sten Claessens and Ireneusz Baran. To all the staff in the Department office, both
past and present, I also extend my thanks.
My thanks also go to Dr Michael Kuhn, Dr Mark Stewart and Dr Ruey-Gang Chang
for their helpful comments towards this dissertation, and to Dr Graham Quartly
(Southampton Oceanography Centre, UK), Dr Ronald Brooks (NASA, USA) and Dr
George Hayne (NASA, USA) for providing some references and useful discussions.
This research was funded by the International Postgraduate Research Scholarship
(IPRS) and Curtin University Postgraduate Scholarship (CUPS). In-kind support was
received from AVISO and ESA, by many of kindly providing POSEIDON and ERS-
1/2 waveform data, respectively.
Finally, I would like to thank my family for the constant love and support, in
particular, my daughter, Lulu, with whom I shared happiness in the past two and half
years. Without this love, patience and support, I am quite sure that this dissertation
would never have been finished.

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TABLE OF CONTENTS
ABSTRACT.................................................................................................................ii
ACKNOWLEDGEMENTS ........................................................................................iii
TABLE OF CONTENTS............................................................................................ iv
LIST OF FIGURES .................................................................................................... ix
LIST OF TABLES ...................................................................................................xvii
ACRONYMS AND ABBREVIATIONS ................................................................. xix
1. INTRODUCTION ............................................................................................... 1
1. 1 Applications of Satellite Radar Waveforms................................................. 1
1.1.1 Description of the Satellite Radar Altimeter Waveforms .................... 1
1.1.2 General Applications of Waveform Data............................................. 4
1. 2 Problems in Coastal Areas ........................................................................... 6
1.2.1 Topographic Effects on Waveforms .................................................... 7
1.2.2 Topographic Effects on Geophysical Corrections ............................. 10
1. 3 Justifications for This Study ...................................................................... 11
1.3.1 Previous Work.................................................................................... 11
1.3.2 Significance of the Improvement of Altimeter Data in Coastal
Regions ............................................................................................................ 12
1. 4 The Study Areas......................................................................................... 15
1. 5 Thesis Outline ............................................................................................ 16
2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES..... 18
2. 1 Introduction................................................................................................ 18
2. 2 General Information on Construction of Altimeter Measurements ........... 19
2.2.1 Fundamental Radar Measurement Principles..................................... 19
2.2.2 The Returned Waveform.................................................................... 21
2.2.3 The Range and Sea Surface Height.................................................... 25
2. 3 The Altimeter Waveform Data Used ......................................................... 26
2.3.1 ERS-1 Waveform Data ...................................................................... 26
2.3.2 ERS-2 Waveform Data ...................................................................... 27
2.3.3 POSEIDON Waveform Data ............................................................. 28
2. 4 External Input Data .................................................................................... 29
2.4.1 The AUSGeoid98 Geoid Model of Australia..................................... 30

v
2.4.2 Sea Surface Topography .................................................................... 31
2.4.3 Marine Gravity Data around Taiwan ................................................. 34
2.4.4 The Australian Digital Elevation Model (DEM) ............................... 35
2. 5 Summary .................................................................................................... 36
3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS............... 37
3. 1 Introduction................................................................................................ 37
3. 2 Convolutional Representation of Ocean-Returned Waveforms................. 38
3.2.1 The Brown Model .............................................................................. 38
3.2.2 Average Flat Surface Impulse Response Function ............................ 40
3.2.3 Surface Elevation PTR of the Scattering Distribution ....................... 41
3.2.4 Radar System PTR ............................................................................. 42
3.2.5 Solution of the Radar Returns............................................................ 42
3. 3 Modified Convolutional Representation of the Ocean Return Waveform. 43
3.3.1 Effects of the Earth’s Curvature......................................................... 44
3.3.2 Simplification of the Bessel Function................................................ 44
3.3.3 Approximate Expression of the Non-Gaussian Scattering Surface ... 46
3.3.4 Approximate Expression of the System PTR .................................... 49
3.3.5 A Modified Solution of the Radar Returns Developed in This Study50
3. 4 Ocean Waveform Retracking..................................................................... 55
3.4.1 Least Squares Fitting Procedure ........................................................ 56
3.4.2 Deconvolution Method ...................................................................... 58
3. 5 Ice-Sheet Waveform Retracking................................................................ 60
3.5.1 Fitting Algorithm - β-Parameter Retracking...................................... 62
3.5.2 Off-Centre of Gravity (OCOG) Retracking Algorithm ..................... 65
3.5.3 Threshold Retracking Algorithm ....................................................... 66
3.5.4 Surface/Volume Scattering Retracking Algorithm ............................ 67
3. 6 Land Waveform Retracking....................................................................... 69
3. 7 Brief Introduction to Coastal Waveform Retracking................................. 71
3. 8 Summary .................................................................................................... 72
4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS .. 74
4. 1 Introduction................................................................................................ 74
4. 2 Background ................................................................................................ 75
4.2.1 ERS-2 and POSEIDON Altimeter Footprints.................................... 76

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4.2.2 ERS-2 and POSEIDON Altimeter Waveforms.................................. 76
4. 3 Processing Methodology for Determination of Contamination Area ........ 78
4.3.1 Threshold Retracking......................................................................... 78
4.3.2 Mean Power of the Waveform ........................................................... 80
4. 4 Data and Editing......................................................................................... 81
4.4.1 ERS-2 ................................................................................................. 81
4.4.2 POSEIDON........................................................................................ 83
4. 5 Determination of Contamination Area....................................................... 84
4.5.1 Using 50% of Threshold Retracking Points of ERS-2....................... 84
4.5.2 Using the Mean Waveform and its Standard Deviation..................... 89
4.5.3 Using Statistics of the 50% Retracking Point .................................... 94
4. 6 Waveform Contamination Analysis........................................................... 97
4.6.1 Waveform Contamination for Water-to-Land Ground Tracks .......... 97
4.6.2 Waveform Contamination for Land-to-Water Ground Tracks ........ 101
4.6.3 Contamination Caused by the Altimeter’s Operation ...................... 107
4.6.4 Summary of Waveform Contamination in Coasts ........................... 109
4. 7 An Experiment of the Land Effects on Waveforms Using the DEM Data....
.................................................................................................................. 110
4.7.1 Concept of Land Effects Using the DEM Model............................. 111
4.7.2 A Preliminary Test of Land Effects Using the DEM Model ........... 114
4. 8 Summary .................................................................................................. 116
5. COASTAL WAVEFORM RETRACKING SYSTEM: DESIGN AND
IMPLEMENTATION.............................................................................................. 119
5. 1 Introduction.............................................................................................. 119
5. 2 Coastal Waveform Retracking System Design........................................ 120
5. 3 Extraction of the Geodetic Parameter: the Sea Surface Height ............... 122
5. 4 Analysis of Waveform Shapes................................................................. 123
5.4.1 Determination of the Peaks in the Waveform.................................. 123
5.4.2 Waveform Classification.................................................................. 124
5. 5 Fitting Functions ...................................................................................... 127
5.5.1 The Ocean Model............................................................................. 127
5.5.2 Five- and Nine-β-Parameter Models................................................ 131
5.5.3 Comparison between the Ocean and Five-β-Parameter Models...... 131

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5. 6 Iterative Nonlinear Fitting Procedure ...................................................... 136
5.6.1 Solutions of the Waveform Retracking Fitting Function................. 137
5.6.2 Linearisation of the Waveform Retracking Fitting Function........... 138
5.6.3 Initial Estimates of the Unknown Model Parameters ...................... 139
5.6.4 Data Weight and Outlier Detection Scheme .................................... 142
5.6.5 Iterative Procedure ........................................................................... 144
5. 7 Threshold Retracking............................................................................... 145
5.7.1 Threshold Retracking Algorithm ..................................................... 145
5.7.2 Selection of the Threshold Level ..................................................... 146
5.7.3 Results of Selecting the Threshold Levels....................................... 149
5. 8 An Assessment of Biases among Waveform Retracking Algorithms ..... 155
5.8.1 Analysis Methods and Data ............................................................. 155
5.8.2 Retracking Results and Analysis...................................................... 157
5.8.3 Biases among Retracking Algorithms.............................................. 160
5.8.4 Biases between the SSH Data before and after Retracking ............. 164
5. 9 Summary .................................................................................................. 167
6. WAVEFORM RETRACKING APPLICATION TO THE AUSTRALIAN
COAST..................................................................................................................... 169
6. 1 Introduction.............................................................................................. 169
6. 2 Waveform Classification in Australian Coastal Regions......................... 170
6.2.1 Typical Waveform Shapes ............................................................... 170
6.2.2 Results of Waveform Categories ..................................................... 176
6. 3 Waveform Retracking Results and Discussion........................................ 177
6.3.1 Data Editing ..................................................................................... 177
6.3.2 Waveform Retracking Using Different Algorithms......................... 177
6.3.3 Waveform Retracking Using Different Fitting Functions ............... 179
6. 4 Analysis of the Single-Track Retracked Results...................................... 182
6. 5 Analysis of the SSH Data Before and After Retracking.......................... 185
6. 6 Collinear Analysis.................................................................................... 187
6.6.1 Methodology .................................................................................... 187
6.6.2 Discussion of Results ....................................................................... 188
6. 7 Evaluation of SSH Data Using Geoid Heights ........................................ 190
6.7.1 Comparison with the AUSGeoid98 Geoid Model ........................... 190

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6.7.2 Results of the Comparison between SSH Data and Geoid Heights. 191
6. 8 Summary .................................................................................................. 193
7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES NEAR
TAIWAN FROM WAVEFORM RETRACKING .................................................. 196
7. 1 Introduction.............................................................................................. 196
7. 2 Data and Area........................................................................................... 198
7. 3 Waveform Retracking .............................................................................. 200
7.3.1 Retracking Scheme........................................................................... 201
7.3.2 Examples of Waveform Retracking Results .................................... 202
7. 4 The SSH Extracted from Waveforms ...................................................... 205
7.4.1 Corrections ....................................................................................... 205
7.4.2 The Weight Function ....................................................................... 211
7.4.3 Data Filtering and Editing................................................................ 211
7. 5 Marine Gravity Anomalies from Satellite Altimetry ............................... 212
7.5.1 Along-track Deflections of the Vertical........................................... 216
7.5.2 Averaging Gradients ........................................................................ 216
7.5.3 Gravity Anomaly Computation........................................................ 219
7.5.4 Comparisons between Marine and Ship Gravity Anomalies ........... 219
7. 6 Summary .................................................................................................. 224
8. CONCLUSIONS AND RECOMMENDATIONS .......................................... 226
8. 1 Summary of Dissertation.......................................................................... 226
8. 2 Specific Conclusions................................................................................ 228
8.2.1 The Quantitative Estimation of the Contaminated Distance............ 228
8.2.2 Practical Implementation of the Coastal Waveform Retracking System
.......................................................................................................... 230
8.2.3 Development and Validation of a Coastal Retracking System ........ 232
8.2.4 Other Conclusions............................................................................ 235
8. 3 Recommendations for Future Work......................................................... 235
8.3.1 Extraction of Ocean Returns from the Mixed Waveform Shape ..... 236
8.3.2 Improvement of the Correction Algorithms near Land.................... 236
8.3.3 Improvement of the Altimeter Operation......................................... 237
8.3.4 Geodetic Application ....................................................................... 237
REFERENCES......................................................................................................... 238

ix
LIST OF FIGURES
Figure 1.1 Schematic altimeter mean return waveform over sea surfaces................... 3
Figure 1.2 An observed ERS-2 waveform over deep oceans. (1 bin/gate = 0.45
m and tracking gate = 32.5)....................................................................... 3
Figure 1.3 Altimeter simultaneously illuminates land and ocean near the
coastline..................................................................................................... 8
Figure 1.4 Differences between the ERS-2 SSH and AUSGeiod98 geoid height
along four collinear repeat tracks from cycles 42 and 43 at the
Southern Australian coast, showing the metre-level uncertainty of
SSHs closer to the coastline. ..................................................................... 9
Figure 2.1 Basic measurement principle of the altimeter-derived SSH (From
http://www.jason.oceanobs.com/html/portail/galerie
/banque_img_welcome_uk.php3). .......................................................... 18
Figure 2.2 A schematic geometrical description of the interaction of a pulse and
the scattering surface, and the build up of a return waveform over
the duration of a pulse. Note that, an average returned waveform
from 50 typical individual returns is depicted, while a single typical
return shows more noise. (PLF is the pulse-limited footprint). .............. 22
Figure 2.3 Quasi-time-independent SST (contour interval is 10 cm) from Levitus
et al. (1997). ............................................................................................ 33
Figure 2.4 Coverage of ship-track gravity observations around Taiwan................... 34
Figure 3.1 Values of d ( ξ = 0 ), showing that d

x
Figure 3.6 Nine- β -parameter model, fitting a double-ramp function to two
leading edges of the waveform samples.................................................. 64
Figure 3.7 Schematic description of the OCOG method applied to an observed
waveform. Using the OCOG method to calculate the amplitude (A)
and width (W) of the rectangle, and the position of the 50% of
amplitude interpolated at the leading edge as the estimate of
expected tracking gate............................................................................. 66
Figure 4.1 An example of the observed POSEIDON waveform over deep oceans
(tracking gate = 31.5). ............................................................................. 77
Figure 4.2 ERS-2 ground tracks (35-day repeat orbit, from March to April 1999)
to a distance of 350 km from Australian shoreline (Lambert
projection). .............................................................................................. 82
Figure 4.3 The ocean-ice-mode flag in ERS-2 data supplied close to the
Australian shoreline compared to the Wessel (2000) coastline,
showing that most flags around Australia are on the land (Lambert
projection). .............................................................................................. 82
Figure 4.4 POSEIDON ground tracks (10-day repeat orbit, from January 1998 to
January 1999) to a distance of 350 km from Australian shoreline
(Lambert projection). .............................................................................. 83
Figure 4.5 One cycle of ERS-2 contaminated waveforms (highlighted by circles)
along ground tracks in the southeast coasts of Australia. ....................... 86
Figure 4.6 Distributions of 50% retracking points plotted versus DS759.2 ocean
depths for contaminated waveforms in areas of 105°≤λ

xi
Figure 4.9 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for ERS-2 in five 10 km-wide bands from
the Australian coastline........................................................................... 91
Figure 4.10 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for POSEIDON cycle 197 in five 10 kmwide
bands from the Australian coastline. .............................................. 92
Figure 4.11 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for ERS-2 in five 2 km-wide bands from the
Australian coastline................................................................................. 93
Figure 4.12 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for POSEIDON cycle 197 in five 2 km-wide
bands from the Australian coastline........................................................ 94
Figure 4.13 Fifty percent retracking point frequency distributions for 5 cycles of
Poseidon waveform data (left) and 1 cycle of ERS-2 waveform data
(right). They are shown as the percentage of observations in each bin
for five 2 km-wide bands around the Australian coastline. .................... 95
Figure 4.14 The 20 Hz SSH profile of pass 21792 (ascending track, cycle 43 of
ERS-2), approaching the eastern Australian shoreline from the
Tasman Sea. The number beside the SSH samples corresponds to the
waveform in Figure 4.15. ........................................................................ 98
Figure 4.15 20 Hz waveforms of pass 21792 (ascending track, cycle 43 of ERS-
2), approaching Australian eastern shoreline in the Tasman Sea. The
number in the top left-hand corner is related to that in Figure 4.14........ 98
Figure 4.16 The 20 Hz SSH profile of pass 21450 (ascending track, cycle 43 of
ERS-2), approaching the western Australian shoreline from the
Indian Ocean. The number beside the SSH samples corresponds to
the waveform in Figure 4.17. ................................................................ 100
Figure 4.17 The 20 Hz waveforms of pass 21450 (ascending track, cycle 43 of
ERS-2), approaching Australian western shoreline in the Indian
Ocean. The number is related to that in Figure 4.16............................. 100
Figure 4.18 The 20 Hz SSH profile of pass 21636 (ascending track, cycle 43 of
ERS-2), receding from the northwestern Australian shoreline to the
Indian Ocean. ........................................................................................ 103

Figure 4.19 20 Hz waveforms of pass 21636 at the 1st second (~0-7 km alongtrack
distance) from the shoreline (ascending track, cycle 43 of ERS-
2), receding from the northwestern Australian shoreline to the Indian
Ocean..................................................................................................... 103
Figure 4.20 20 Hz waveforms of pass 21636 at the 2nd second (~7-14 km alongtrack
distance) from the shoreline (ascending track, cycle 43 of ERS-
2), receding from northwestern Australian shoreline to the Indian
Ocean..................................................................................................... 104
Figure 4.21 20 Hz waveforms of pass 21636 at the 3rd second (~14-21 km
along-track distance) from the shoreline (ascending track, cycle 43
of ERS-2), receding from the northwestern Australian shoreline to
the Indian Ocean. .................................................................................. 104
Figure 4.22 The 20 Hz SSH profile of pass 21886 (descending track, cycle 43 of
ERS-2), receding from the southern Australian shoreline to the
Southern Ocean. .................................................................................... 105
Figure 4.23 20 Hz waveforms of pass 21886 at the 1st second (0-7 km alongtrack
distance) from the shoreline (descending track, cycle 43 of
ERS-2), receding from the southern Australian shoreline to the
Southern Ocean. .................................................................................... 106
Figure 4.24 20 Hz waveforms of pass 21886 at the 2nd second from the
shoreline (ascending track, cycle 43 of ERS-2), receding Australian
southern shoreline to the Southern Ocean............................................. 106
Figure 4.25 20 Hz waveforms of pass 21886 at the 3rd second from the
shoreline (ascending track, cycle 43 of ERS-2), receding Australian
southern shoreline to the Southern Ocean............................................. 107
Figure 4.26 A schematic geometrical description of the radar return from a point
on the land surface at off-nadir angle ξ and associated colatitude
angle φ where the land elevation above the mean sea level is H .
The distance from the altimeter to the land surface is
xii
'
R
φ
. ................... 113
Figure 4.27 Retracking gate estimates corresponding to the land elevation. The
search radius is 2.7 km (top) and 3.5 km (bottom). .............................. 115
Figure 5.1 Block diagram of the coastal retracking system..................................... 121
Figure 5.2 Typical ERS-2 waveform categories in Australian coasts ..................... 126

Figure 5.3 Variation of the leading edge’s slope with varying SWH (0 - 20 m),
xiii
modelling waveforms using the ocean model ( ξ = 0 °). ....................... 129
Figure 5.4 Effect of the off-nadir angle (in degrees) on the amplitude, leading
edge, and especially the slope of the trailing edge, modelling
waveforms using the ocean model (SWH = 4 m). ................................ 130
Figure 5.5 Variations of the trailing edge’s slope with varying parameter β 5 ,
modelling waveforms using the five-β-parameter parameter function
(linear trailing edge, SWH = 4 m, β
21
= 500 counts, β 5 = β
51)........... 134
Figure 5.6 Variations of the trailing edge’s slope with varying parameter β 5 ,
modelling waveforms using the five-β-parameter parameter function
(exponential decayed trailing edge, SWH = 4 m, β
21
= 500 counts,
β 5 = β 51
). ............................................................................................. 135
Figure 5.7 Mean waveforms and the standard deviation in two 5 km-wide bands
of 0-5 km (top) and 5-10 km (bottom) from the Australian coastline.. 148
Figure 5.8 Ten 15º×15º coastal areas around Australia, showing the area number
(Lambert projection). ............................................................................ 150
Figure 5.9 Mean differences between the SSH data after retracking and
AUSGeoid98 geoid heights in ten areas 5 km from the Australian
coastline................................................................................................. 152
Figure 5.10 Standard deviations of the mean differences in ten areas 5 km from
the Australian coastline......................................................................... 152
Figure 5.11 Mean differences between the SSH data after retracking and
AUSGeoid98 geoid heights in ten areas 5-10 km from the Australian
coastline................................................................................................. 154
Figure 5.12 Standard deviations of the mean differences in ten areas 5-10 km
from the Australian coastline. ............................................................... 154
Figure 5.13 Geoid height and SSH profiles along ground tracks 21085 (top) and
21364 (bottom) (ERS-2, cycle 42) before and after retracking using
the ocean model, the five-parameter model and the 50% threshold
level in coastal regions.......................................................................... 158
Figure 5.14 Retracked and unretracked SSH profiles over open oceans, showing
the biases among them. ......................................................................... 161

xiv
Figure 5.15 Histogram of the differences between the ocean model and fiveparameter
model.................................................................................... 162
Figure 5.16 Histograms of the differences between the 50% threshold retracking
technique and the ocean model as well as the five-parameter model. .. 162
Figure 5.17 Histograms of the differences between unretracked SSH data and
retracked SSH data using the ocean model and the five-parameter
model..................................................................................................... 164
Figure 5.18 Histogram of the differences between untracked SSH data and
retracked SSH dada using the 50% threshold retracking technique...... 165
Figure 6.1 Part of ERS-2 ground tracks off the northwest Australian coast,
showing the orbit number of each ground track. The highlighted
diamonds along tracks will be shown in detail waveform shapes later
in this Section........................................................................................ 171
Figure 6.2 Typical ocean waveforms from pass 21815 (cf. Figure 6.1). ................. 173
Figure 6.3 Waveforms recorded when the ground track 21815 is approaching an
island at latitude ~ -20.5º and leaving the island to water at ~ -20.9º.
The data gap is due to the island (cf. Figure 6.1).................................. 173
Figure 6.4 Waveforms leaving land to water (pass 21636 in Figure 6.1). The
high-peaked specular responses are probably due to off-nadir
brighter calm water. The waveform shifting (~ -22.25°

xv
Figure 6.9 STD (positive values) of the collinear 20 Hz SSH differences of
cycles 42 and 43 before and after retracking in six 5 km wide
distance bands, showing improvement from SSH data after
retracking............................................................................................... 189
Figure 6.10 Mean differences of geoid heights (AUSGeoid98) and SSH data
before and after retracking in six 5 km wide distance bands (cycle
42). ........................................................................................................ 191
Figure 6.11 STD (positive values) of the mean difference before and after
retracking in six 5 km wide distance bands, showing improvement
on SSH data after retracking (cycle 42). ............................................... 192
Figure 6.12 Mean differences between geoid heights (AUSGeoid98) and SSH
data before and after retracking in six 5 km wide distance bands
(cycle 43)............................................................................................... 192
Figure 6.13 STD (positive values) of the mean difference before and after
retracking in six 5 km wide distance bands, showing improvement
on SSH data after retracking (cycle 43). ............................................... 192
Figure 7.1 Distribution of ERS-1 ground tracks ...................................................... 199
Figure 7.2 The histogram of dh (the SSH difference between the five-parameter
model and the ocean model (mean -0.7 cm; STD 13.5 cm; 3070
observations). ........................................................................................ 202
Figure 7.3 Three-ground tracks in the study area .................................................... 203
Figure 7.4 SSHs from ground track A1 (a), D1 (b), and D2 (c) in Figure 7.3
before and after retracking using the retrackers of the five-parameter
and ocean models. ................................................................................. 204
Figure 7.5 SSH profiles and parts of corrections along the track A1. ..................... 208
Figure 7.6 SSH profiles and parts of corrections along the track D1. ..................... 209
Figure 7.7 SSH profiles and parts of corrections along the track D2. ..................... 210
Figure 7.8 Untracked SSH profiles, showing the filtered SSH data (dark curves)
and unfiltered SSH data (light curves) along the tracks A1 (a), D1 (b)
and D2 (c) in Figure 7.3. ....................................................................... 213
Figure 7.9 Retracked SSH profiles, showing filtered SSH data (dark curves) and
unfiltered SSH data (light curves) along the tracks A1 (a), D1 (b)
and D2 (c) in Figure 7.3. ....................................................................... 214

xvi
Figure 7.10 Flow diagram for the SSH extraction from waveforms and recovery
of gravity anomalies.............................................................................. 220
Figure 7.11 Gravity anomalies from altimeter data before retracking (units in
mgal) ..................................................................................................... 221
Figure 7.12 Gravity anomalies from altimeter data after retracking (units in mgal)
............................................................................................................... 221
Figure 7.13 Differences between altimeter-derived (before retracking) and shiptrack
gravity anomalies ......................................................................... 223
Figure 7.14 Differences between altimeter-derived (after retracking) and shiptrack
gravity anomalies ......................................................................... 223

xvii
LIST OF TABLES
Table 2.1 Altimeter Operating Parameters................................................................. 20
Table 2.2 Corrections applied to ERS-1 range measurements and their likely
error......................................................................................................... 25
Table 2.3 The repeat cycle characteristics of ERS-1. ................................................ 26
Table 2.4 The reference ellipsoid characteristics of ERS-1/2 and T/P. ..................... 29
Table 4.1 Contaminated 1 Hz ocean depths and 20 Hz distances from the nearest
shoreline in different ............................................................................... 85
Table 4.2 Differences between 50% retracking points and the tracking gate (bin
31.5) for 5 cycles (January 1998 to January 1999) of Poseidon
waveforms (each bin has the range distance of ~0.469 m). .................... 96
Table 4.3 Differences between 50% retracking points and the tracker point (bin
32.5) for 1 cycle (March to April 1999) of ERS-2 waveforms (each
bin with the range width of ~0.454m)..................................................... 96
Table 4.4 Descriptive statistics of the differences of retracking gate estimates
before and after shifting rightward waveform samples in the range
window (20 Hz waveforms from pass 21886 at the 2nd second from
the coastline, 1 gate≈0.4542 m) ............................................................ 108
Table 5.1 ασ² Values for ERS-1/2 (ocean mode) and TOPEX ............................... 132
Table 5.2 Criteria of the absolute differences between two iterations (the ocean
model for ERS-1/2) ............................................................................... 145
Table 5.3 Along-track geoid and SSH gradients (track 21085)............................... 159
Table 5.4 Along-track geoid and SSH gradients (track 21364)............................... 160
Table 5.5 Biases between different retracking techniques from fitting a quadratic
function to the ERS-2 SSH (cycle 42) differences over an ocean area
(-30°S

xviii
Table 5.8 Biases between the ERS-2 (cycle 43) SSHs before and after retracking
over an ocean area (-30°S

xix
ACRONYMS AND ABBREVIATIONS
AGC
Automatic gain control
AGSO Australian Geological Survey Organisation, now Geoscience
Australia (GA)
ALT.WAP
AVISO
AWF
DEM
ECMWF
EMB
EODC
ESA
FFT
GDR
GPS
IAG
MLE
OCOG
PLF
PTF
PTR
RMS
SKB
SSB
SSH
Altimeter waveform products
Archiving, Validation and Interpretation of Satellite Oceanographic
team
AIDA waveform format
Digital elevation model
European Centre for Medium-Range Weather Forecasts
Electromagnetic bias
Earth Observation Data Centre, UK
European Space Agency
Fast Fourier transform
Geophysical data record
Global Positioning System
International Association of Geodesy
Maximum likelihood estimation
Off centre of gravity method
Pulse-limited footprint
Probability density function
Radar system point target response
Root-mean-square
Skewness bias
Sea state bias
Sea surface height

xx
SST
STD
SWH
T/P
UK-PAF
Sea surface topography
Standard deviation
Significant wave height
TOPEX/POSEIDON
United Kingdom Processing and Archiving Facility

Chapter 1. INTRODUCTION 1
1. INTRODUCTION
This dissertation is concerned with the intersection of two topics of geodetic satellite
radar altimetry in coastal regions: satellite radar altimeter waveform retracking; and
some applications of the waveform retracking. This Chapter will introduce both the
background and the justification for the dissertation.
1. 1 Applications of Satellite Radar Waveforms
Radar altimeters have been flown on a number of satellites (GEOS-3, SEASAT,
GEOSAT, TOPEX/POSEIDON (T/P), ERS-1 and 2, JASON-1, and ENVISAT) and
been found to give useful information about the oceans, such as the large-scale
movement of water and their total mass and volume (e.g., Marth et al., 1993; Chelton
et al., 2001). The basic altimetric measurement is the vertical distance or range
between the satellite and the at-nadir instantaneous sea surface. This range
measurement, when combined with precise knowledge of the satellite orbit,
propagation effects, and the Earth’s geopotential field, allows the determination of
sea surface slopes and hence surface currents. The altimeter range measurements are
determined from the time interval between the time that the radar pulse is transmitted
and the time that the return reflected from the (assumed at-nadir) mean sea surface is
received. Satellite radar altimeters transmit pulses at a certain frequency (e.g.,
1,020 Hz for ERS-2 and 1,700 Hz for POSEIDON). The on-satellite processor
averages the reflected radar returns to give the waveform that relates to the shape of
the sea surface, which in turn contains information on currents, tides and ocean
bottom features related to oceanography and solid Earth geophysics (e.g., Tokmakian
et al., 1994)
1.1.1 Description of the Satellite Radar Altimeter Waveforms
Pulsed-limited radar altimeters produce a measurement of power as a function of
time (e.g., Marth et al., 1993; Quartly and Srokosz, 2001; Chelton, 2001). From this
temporal profile of received power, which is referred to as the altimeter waveform
(described in detail in Section 2.2.2), the range to the at-nadir sea surface can be
determined over oceans. In addition to the range, the waveform also indicates the

Chapter 1. INTRODUCTION 2
reflectivity (estimated from the amount of power in the reflected pulse) and the largescale
roughness of the scattering surface (estimated from the slope of the leading
edge of the waveform). Of these, only the range will be considered in this
dissertation as a geodetic parameter with a view to developing an improved range
determination in coastal regions.
The range is measured by the altimeter indirectly through a direct measurement of
the two-way travel time between the transmission and reception of a radar pulse. The
observed time delay can be converted into a range measurement as long as the
propagation velocity is known. A determination of this velocity, with corrections for
the instrument (e.g., the Doppler shift, antenna gain pattern, and automatic gain
control (AGC) attenuation), the atmospheric refraction (e.g., wet and dry troposphere,
the ionosphere), the sea-state bias, and geophysical adjustments (e.g., ocean tides and
atmospheric pressure loading), is an essential part of radar altimetry (Chelton et al.,
2001).
The profile of the backscattered power (i.e., the waveform) is a function of range
measured by the altimeter. It is the convolution of three terms based on specular
reflection from the sea surface: the system’s point target response (PTR), the impulse
response of the smooth spherical scattering surface, and the ocean-surface height
distribution (e.g., Brown, 1977; Hayne, 1980; Rodriguez, 1988; Jensen, 1999). If the
ocean waves on the sea surface are assumed to be linear, the corresponding statistics
of surface elevation and slopes are Gaussian. This means that the ocean-surface
height distribution is assumed symmetric about some mean value. Therefore, the
altimeter waveform is an odd function relative to the midpoint (i.e., half-power point)
on the leading edge of the waveform, and the range to the at-nadir sea surface
corresponds to this midpoint of the leading edge. The Brown (1977) model is thus
the general function used by the on-satellite data processor (Chelton et al., 2001).
Figures 1.1 and 1.2 show a schematic altimeter waveform based on the above
theoretical description and an observed ERS-2 waveform, respectively. As can be
seen, the waveform consists mainly of three parts (Figure 1.1): the thermal noise,
leading edge and the trailing edge (see Section 2.2.2 for the details), and recorded in
64 range bins or gates (Figure 1.2). From Figures 1.1 and 1.2, there are differences

Chapter 1. INTRODUCTION 3
between the modelled and actual waveforms. The observed waveform shows wraparound
noise at the first and the last five bins, and noise along the trailing edge.
Noise also affects the whole waveform. Due to these differences, it is hard to define a
tracker point that is related to the mid-point on the leading edge of the waveform
tracked by an altimeter. Therefore, it is replaced by a tracking gate, which is
designed before the satellite is launched and located at bin 32.5 for ERS-1/2. The
onboard tracking algorithm (e.g., Brown model) tries to centre the leading edge at the
position of this gate. The ERS leading edge usually contains 3-4 range bins (Figure
1.2).
Returned Power
Thermal Noise
Leading Edge
Trailing Edge
Time or Bins
Figure 1.1 Schematic altimeter mean return waveform over sea surfaces.
1400
1200
1000
Power (counts)
800
600
400
200
0
0 8 16 24 32 40 48 56 64
Bins or Gates
Figure 1.2 An observed ERS-2 waveform over deep oceans.
(1 bin/gate = 0.45 m and tracking gate = 32.5).

Chapter 1. INTRODUCTION 4
1.1.2 General Applications of Waveform Data
The above-mentioned satellite radar altimetric tracking theory has allowed good
estimates of geophysical and oceanographic parameters to be obtained over open
ocean surfaces (e.g., Chelton et al., 2001; Hayne, 1980). The waveform shape based
on the convolution form (e.g., Brown, 1977), especially the sharply rising leading
edge of the waveform (see Figure 1.2), is the basis for the precise range estimation.
Analysing the waveforms provides estimates of oceanographic parameters, which
include the two-way travel time of the pulse (to give the range), the slope of the
leading edge of the returned waveform (to give the significant wave height), and the
backscattered power σ 0 (to relate empirically to the wind speed). These estimates
are obtained by the on-satellite processing of the radar return based on the tracking
algorithm of the Brown (1977) ocean statistics model (e.g., Tokmakian et al., 1994;
Marth et al., 1993; Fernandes, 2003; Chelton et al., 2001). Thus, waveforms are the
fundamental measurements of a radar altimeter.
Although the waveform recorded in 20 Hz format is a basic measurement of the
radar altimetry, the sea surface height (SSH) derived from the range and satellite
altitude above an ellipsoid reference is generally used in geodetic and oceanographic
investigations. Usually, the SSH is averaged to 1 Hz for a typical oceanographic
application (e.g., ocean circulation) and 2 Hz (or a higher rate) for geodetic and
geophysical applications (e.g., gravity anomalies) over an along-track distance per
second (e.g., ~7 km for T/P). However, the 20 Hz waveform data have also found
important applications both in oceanography and geophysics by applying a data postprocessing
technique known as waveform retracking, which extracts the geophysical
parameters from the waveform data using empirical, semi-empirical or physically
based algorithms (e.g., Quartly and Srokosz, 2001; Chelton et al., 2001).
Over oceans, the modified convolutional representation allows for non-linear wave
parameters, such as skewness of the surface distribution, to be derived from the
shape of the waveform (e.g., Barrick and Lipa, 1985; Hayne, 1980; Rodriguez, 1988;
Tokmakian et al., 1994). The effects of atmospheric liquid water (both clouds and
rain) on the waveform has been investigated using the waveform data over open
oceans by Walsh et al. (1984), Quarlty et al. (1998), and Tournadre (1998).

Chapter 1. INTRODUCTION 5
Satellite radar altimeters have principally been launched to study ocean variability
and the marine geoid using measurements from repeated orbits, and map ocean
gravity using measurements from geodetic missions (e.g., GEOSAT and ERS-1
and 2). However, since Robin (1966, cited in Wingham et al., 1993) first recognised
the potential of satellite altimetry for ice-sheet topography mapping, it has also been
shown that many applications exist over non-ocean surfaces, such as digital terrain
modelling and polar ice sheet topographic mapping in the field of ice dynamics
(Berry et al., 1998; Bamber et al., 1997; Bamber, 1994; Brenner et al., 1997; Davis,
1995; Partington et al., 1991; Remy et al., 1989; Remy and Minster, 1993; Frey and
Brenner, 1990). Over non-ocean surfaces, the situation is more complex, less
predictable, and the nature of return echoes varies widely as a function of the surface
type. The data post processing involves range estimate refinement procedures (i.e.,
waveform retracking) and a slope correction to provide the correct range for the
corresponding ground location.
Waveform retracking over non-ocean surfaces aims to determine the offset between
the tracking gate derived from the on-satellite software and a known, fixed,
instrument-independent position on the leading edge of the waveform and correct the
satellite range measurement accordingly (e.g., Martin, et al., 1983; Davis, 1995;
Bamber, 1994; Zwally, 1996). The slope-induced error correction aims to reduce the
altimeter range between satellite and the closest point to that at nadir (e.g., Brenner et
al., 1983; Cooper, 1989). Therefore, several non-linear range estimation algorithms
have been developed mainly over ice since 1983, such as the β-parameter retracking
algorithm (Martin et al., 1983), the threshold retracking method (Wingham et al.,
1986), and the surface/volume scattering-retracking algorithm (Davis, 1993).
These algorithms provide an important monitoring ‘tool’ for accurate estimates of
growth or shrinkage of the ice sheets (e.g., the Greenland and Antarctic ice sheets).
Wingham et al. (1998) indicate that the elevation of the Antarctic ice sheet fell by
0.9±0.5 cm/year between 1992 and 1996. Information on the characteristics and
changes of the firn (compacted snow), including surface roughness and volume
scattering (e.g., Partington et al., 1989; Davis, 1993; Legresy and Remy, 1998), and
information on the ice-sheet topography, including the surface melt streams and

Chapter 1. INTRODUCTION 6
slopes (e.g., Phillips, 1998, cited in Zwally and Brenner, 2001) can also be derived
from radar altimetry.
Altimetry over land is even more complicated owing to rapidly varying topography.
The echo waveform over land comprises reflections from an essentially static, nonhomogeneous
surface. It contains various types of cover and each shows different
characteristics. Some of the methods used over ice surfaces could be applied after
proper modification. For example, Bamber et al. (1997) use a modified threshold
method, with an optimal power threshold. Berry et al. (1998) develop an expert
system to retrack waveforms over land from the ERS-1 mission by selecting one of
the retrackers, which were designed to optimise the determination of the individual
range correction. The ‘ice-mode’ operation of the ERS-1/2 enables the track to
maintain lock over land on ~80% of surfaces (Berry, 2000a). Using a variety of
different tracker algorithms, Berry (2000b) has determined land elevations, achieving
a good repeatability between different passes. This has in turn revealed the
inconsistencies in various digital elevation models (DEMs) obtained from
conventional mapping techniques (e.g., Hilton et al., 2002).
1. 2 Problems in Coastal Areas
Modern satellite radar altimeters can measure the instantaneous SSH to a precision of
approximately 5 cm in the open oceans (e.g., Shum et al., 1998; Chelton et al., 2001).
However, in coastal regions, the waveform measurements are affected by the noisier
radar returns from the (generally rougher) coastal sea states and simultaneous returns
from the land (e.g., Brooks and Lockwood, 1990; Brooks et al., 1997; Nerem, 1995;
Mantripp, 1996), and by less reliable geophysical, wet delay, orbit and instrument
corrections (e.g., Shum, 1998; Andersen and Knudsen, 2000; Chelton et al., 2001;
Fernandes et al., 2003; Quartly and Srokosz, 2001). Poorly modelled tides also affect
the data owing to the shallow depth of the ocean and the irregular shape of the
shoreline. This prevents the altimeter data from providing valuable information on
the geoid shape, tides, the wind field, current characteristics, and dynamic the sea
surface topography in coastal regions.

Chapter 1. INTRODUCTION 7
1.2.1 Topographic Effects on Waveforms
The shape of the radar return waveform can also be significantly affected in coastal
regions by varying coastal topography, such as cliffs, shallow water tides, or the
solid ground that is illuminated together with the sea surface. It is well known that
the altimeter range may be estimated poorly by on-board tracking software in coastal
regions (e.g., Nerem, 1995; Deng et al., 2002; Mantripp, 1996). As stated, oceanic
waveform models are given under assumptions of scattering surface statistical
homogeneity. However, this spatial uniformity of the topographic statistics and
microwave properties of the surface cannot be met in coastal regions (Mantripp,
1996; Quartly and Srokosz, 2001).
In the proximity of the coastline, the altimeter simultaneously views both scattering
from water and land surface (Figure 1.3). These two scattering surfaces may have
different elevations, but the shortest range to the altimeter depends on the distance to
the shoreline and the slope and reflectivity of the land (Brooks, et al., 1997; Brooks,
2002). The nadir range between the altimeter and ocean surfaces may be the same as
the slant range from land to the altimeter within the altimeter footprint. But as the
satellite gets closer to the coastline, the slant range to the altimeter reflected from
higher (or brighter) land may be shorter than that from the nadir oceans. Since the
altimeter measures the shortest range to the reflecting surfaces, the land return within
the footprint can contribute to the received power by the altimeter which then affects
the returned waveform. The received power from different reflecting surfaces is
dependent upon their respective backscattering coefficient and the angle at which
they are being illuminated. These waveform features can be difficult to interpret
using the Brown (1977) model. As a result, waveforms will be distorted by land
return effects in coastal regions (Chapter 4).
In addition to the land return effects, calm (usually brighter) water surfaces (both
enclosed sea and inland water surfaces) near the coastline, such as bays and estuaries,
make the returned power higher and narrower, thus contaminating the altimeter
waveform as well. It will be shown in this research that land topography affects the
waveform though the radar altimeter’s operation when crossing the coastline
(Section 4.6). The altimeter changes its tracking mode when flying land-to-water or

Chapter 1. INTRODUCTION 8
water-to-land, but data or signal reacquisition may take a second or more of time. As
a result, land return effects on waveforms will last longer near the coastline. The
effects of this performance on the waveform should also be considered when
processing altimeter waveform data in coastal regions.
Land
Ocean
Figure 1.3 Altimeter simultaneously illuminates land and ocean near the coastline
The altimeter-derived SSHs are affected by the above factors. The difference
between the SSH after applying all geophysical and environmental corrections and
AUSGeoid98 geoid height along four ERS-2 collinear ground tracks from cycles 42
and 43 is plotted in Figure 1.4. It shows that there is a large uncertainty in the
altimeter-derived SSH data with respect to the geoid when approaching the coastline.
The magnitude of the difference increases closer to the coastline. Since altimeterderived
SSHs should provide the same geoid structure, but different noise
components from the repeat observations, this large data scatter with respect to the
geoid clearly shows the general problem that exists with the altimeter range data

Chapter 1.INTRODUCTION 9
6
5
Cycle42
Cycle43
SSH -Geoid (m)
4
3
2
1
0
0.06 3.07 6.37 9.42 11.62 14.25 16.76 19.30 22.04 24.89 27.81
-1
Distance from the coastline (km)
15
Cycle43
Cycle42
10
SSH - Geoid (m)
5
0
0.19 3.40 6.60 9.82 12.76 15.56 18.47 21.48 24.57 27.70
-5
-10
Distance from the coastline (km)
6
5
Cycle42
Cycle43
SSH - Geoid (m)
4
3
2
1
0
10.60 13.89 16.25 17.65 19.51 21.71 24.15 26.77 29.53
Distance from the coastline (km)
15
10
Cycle42
Cycle43
SSH - Geoid (m)
5
0
0.07 3.17 6.38 9.64 12.38 15.22 18.16 21.20 24.30 27.45
-5
-10
Distance from the coastline (km)
Figure 1.4 Differences between the ERS-2 SSH and AUSGeiod98 geoid height along
four collinear repeat tracks from cycles 42 and 43 at the Southern Australian coast,
showing the metre-level uncertainty of SSHs closer to the coastline.

Chapter 1.INTRODUCTION 10
from the on-satellite tracking algorithm in coastal regions. Therefore, waveforms can
be contaminated by the land returns and sea states in coastal regions. This is the
definition of the term contamination used in this dissertation.
1.2.2 Topographic Effects on Geophysical Corrections
In the vicinity of the coastline, land topography also affects the modelling of some of
the geophysical corrections applied to the range measurement (e.g., Anderson and
Knudsen, 2000; Shum et al., 1998; Fernandes et al., 2003). Of all corrections, the
effects of wet tropospheric delay and the ocean tides on the altimeter data are most
critical (e.g., Anzenhofer et al., 2000; Chelton, 2001). The result is that the accuracy
of the altimeter-derived SSH will also degrade with the degradation of these
geophysical corrections.
For ERS-1/2 altimeters, the wet tropospheric corrections are derived from the actual
measurements made by the onboard microwave radiometer (e.g., Anzenhofer et al.,
2000; Fernandes et al., 2003). The corrections contained in the waveform data
products are also calculated using an estimate of the total integrated water vapour
and surface air temperature at the sub-satellite point using data supplied by the global
forecast analysis in the European Centre for Medium-Range Weather Forecasts
(ECMWF) (NRSC, 1995). However, the radiometer does not work correctly near the
land and cannot switch on or off when the satellite flies across the coastline, causing
some useful data to be unavailable and errors in the range measurements in coastal
regions (e.g., Fernandes et al., 2003; Anzenhofer et al., 2000).
Due to the spatial complexity of tides in shallow water, the global ocean tide models
are still not accurate enough over shallow seas for detailed oceanographic studies.
The error resulting from using global ocean tidal models can be at the decimetre
level in coastal regions (Anzenhofer et al., 2000; Shum et al., 1998). For both
ERS-1/2 altimeter waveform data and geophysical data records (GDRs), the ocean
tide corrections are provided in a format of 1 Hz and based on the Schwiderski (1983,
cited in NRSC, 1995) model rather than on the recently improved tidal models
(NRSC, 1995). Since the ERS-1/2 satellite orbit is sun-synchronous and the choice of
35 days for its main repeat period leads to severe aliasing of the most important tidal
constituents, the use of the Schwiderski ocean tidal model is appropriate for the

Chapter 1.INTRODUCTION 11
ERS-1/2 data. The problem is that the model cannot provide an adequate resolution
for shallow water tidal constituents. This is because tides over shallow waters
become highly complicated and vary rapidly in amplitude and phase. In these areas,
tides are highly dependent on the bathymetry, the shape of the continental shelf, and
the regional tidal regime (Andersen, 1999). In addition, shorter wavelength tidal
features near sharply topographical changes are not represented in the current
dynamical and empirical global tidal models (Tierney et al., 1998).
Although relatively less research has been reported, the sea state bias (SSB)
correction, which is a function of the significant wave height (SWH), is
unfortunately affected by the land topography near the coastline. Land affecting the
SWH and thus the SSB correction for GEOSAT altimeter has been reported by
Fernandes et al. (2003). It will be shown in Chapter 7 that the erroneous SSB
corrections have also been found for ERS-1 range measurements in the Taiwan Strait
approaching the coastline.
1. 3 Justifications for This Study
1.3.1 Previous Work
As mentioned above, improved determination of the SSH data through waveform
retracking in coastal regions is essential because the coastal topography and the sea
states contaminate the altimeter waveform (cf. Section 3.2.1). However, the potential
of the altimeter waveform data in coastal regions has been the subject of
comparatively little study. So far, considering the existing retracking algorithms used
for coastal waveforms from the open literature, they are those developed for dealing
with the waveform data over ice-sheet surfaces (cf., Anzenhofer et al., 2000; Brooks
et al., 1997).
One earlier study by Brooks and Lockwood (1990) indicates that the presence of land
within a radar altimeter’s footprint affects the altimeter’s tracking performance and
thus contaminates the altimeter waveform data. They analysed GEOSAT waveform
data over the Tuamotu Archipelago in the southern Pacific Ocean to determine the
contaminated distance (1.0-6.2 km) from the islands. Their research pointed out that
the island can locally affect the SSH by more than ±4 m. An edit criterion based on a

Chapter 1.INTRODUCTION 12
combination of waveform data, automatic gain control (AGC) and SWH was
suggested, but waveform retracking was not involved. Brooks et al. (1997) retracked
TOPEX waveforms along 18 satellite ground passes in Pacific Rim coastal zones
using the threshold retracking technique (cf. Sections 3.5.2 and 3.7). Anzenhofer et al.
(2000) present a retracking system in coastal regions, which consists of only a single
retracker and a fitting algorithm is employed. The five-β-parameter fitting function
with either a linear trailing edge or exponential decay trailing edge (cf. Section 3.5.1)
is used by Anzenhofer et al. (ibid.) to retrack ERS-1 waveforms.
Using only a single retracker, however, limits the precision of the recovered SSH
data because of various levels of complexity of the waveform data in coastal regions
(see Sections 5.8.2 and 6.4). Therefore, developing an alternative approach to
improving the precision of recovered SSH data in these areas is necessary, which
must be able to cope with diverse waveforms and estimate precise SSHs from these
reprocessed data. This is one of the justifications for this dissertation.
1.3.2 Significance of the Improvement of Altimeter Data in Coastal Regions
Over open oceans, satellite altimetry is one of the most effective techniques that
provides global and accurate homogeneous coverage of the shape of the sea surface.
The shape of the sea surface is the only physical variable measured from space that is
directly and simply connected to large scale movement of water and the total mass
and volume of the ocean. In coastal regions, although there are generally other data
sources available, such as tide gauges, airborne and shipborne gravity data sets,
altimetry is still a significant technique that is able to provide reliable and uniform
surface information. Since the launch of the SEASAT satellite, there has been a
demand for combining the uniform altimetric coverage of data with other data
sources in order to investigate the oceanic and climatic environment. As stated,
however, the precision of the altimeter data is degraded because of the coastal
topography (both land and sea).
As new satellite missions, such as GFO-1, ENVISAT, JASON-1, and CRYOSAT
will contribute more to the existing altimeter data sets of SEASAT, GEOSAT,
ERS-1/2 and T/P, applications of altimetry in coastal regions will continue to grow.
The coastal areas represent eight percent of the ocean surface. Most importantly,

Chapter 1.INTRODUCTION 13
economic and engineering activities are mainly concentrated in these areas, such as
fisheries and offshore oil drilling (Menard et al., 2000). Altimeter data are also
important for understanding coastal tides, interactions between coastal currents and
open ocean circulation, mesoscale and coastal variability, geoid modelling, and
marine gravity anomaly determination. However, this could only be achieved by
specialised improvement of altimeter measurements (both the data themselves and
the corrections) in coastal regions and accurate data sampling through merging, in an
optimal way, different altimeter data sets such as T/P and ERS-1/2, or JASON-1 and
ENVISAT in the future.
After declassification of the GEOSAT geodetic mission, cross-track resolution of the
satellite altimeter profiles has been improved considerably. In particular, the crosstrack
resolution is improved further because of other satellite altimeter missions,
especially the ERS geodetic missions. Consequently, the cross-track resolution has a
trend exceeding the along-track resolution (~7 km). ERS and GEOSAT geodetic
missions, for instance, have cross track spacings of ~8 km and ~5 km at the equator,
respectively (Chelton et al., 2001). Thus, it becomes necessary to keep and improve
20 Hz data for refining the along-track resolution not only in open oceans, but also in
coastal regions.
When compared ERS-1 with GEOSAT altimeters, the ERS-1 range is poorly tracked
by the altimeter. This is because ERS-1 was not a geodetic satellite and had its
geodetic mission added at the end of its life. The prime use of its data was to monitor
the trailing edge of the waveform and then to monitor the air-sea interactions. The
number of the gates along the leading edge is usually 3-4 sample bins. This makes
difficult for the on-satellite tracking algorithm to precisely define the waveform
shape and thus the range to the surface. Instead, GEOSAT has twice the number of
gates along the leading edge to estimate the range more precisely (Fairhead et al.,
2001). Fairhead et al. (2001) conclude that the ERS-1 ranges have been poorly
estimated by the on-satellite tracking algorithm.
In addition, ocean surface waves are a fundamental limitation to the recovery of the
gravity field from altimeter SSH profiles. For instance, to recover the gravity field at
an accuracy of 1 mgal at 20-km full wavelength requires the ocean surface height

Chapter 1.INTRODUCTION 14
change over a 10-km horizontal distance to a precision of 1 cm (Sandwell, 2003;
Fairhead et al., 2001). However, standard on-board data processing of altimeter data
provides a SSH precision of only ~5 cm at this length scale (Chelton et al., 2001;
Sandwell, 2003). The results from Fairhead et al. (2001) show that reprocessing
ERS-1 waveforms significantly reduces the along-track noise particularly in the 50-
10 km wavelength range, thus improving spatial resolution. This also suggests the
improvement of geodetic parameter estimation from waveform retracking is
necessary over both coastal regions and open oceans.
The present theory of altimetry, developed to describe scattering from the ocean
surface or to retrack the return waveform from the non-ocean surface, does not deal
properly with the waveforms in coastal areas. Data post-processing over topographic
surfaces of ice and land have been attended to since 1983 (e.g., Martin et al., 1983;
Partington et al., 1989; Wingham et al., 1986; Davis, 1995; Zwally, 1996; Berry,
2000a), but there is still a comparable lack of post-processing in the coastal areas
(see Section 1.3.1 and Chapter 3). Therefore, in order to fully realise the potential of
the altimetry in coastal areas, it is necessary to improve the altimeter measurements
and obtain a better understanding of the SSH estimates derived from satellite radar
altimetry. Accurate measurements in coastal regions would quantify one important
source of the SSH and thus enable better understanding of the various geophysical
and oceanographical features in coastal regions. This requires more sophisticated
correction models and data processing techniques than in the open oceans and on
land.
Finally, the International Association of Geodesy (IAG) has created a special study
group (3.186) for studying altimetry data processing for gravity, geoid and sea
surface topography determination (http://space.cv.nctu.edu.tw/IAG/main.html). The
subject of this study has been listed as one of the research problems in the group. The
Australian Research Council International Researcher Exchange Scheme (grant
number: X00001267) also supported a project related to this study. Therefore, the
results will contribute significantly not only to altimeter applications, but also to the
altimetric data processing theory and practice. Essentially, satellite altimeter tracking
will realise its potential over the whole Earth’s surface that is covered by the

Chapter 1.INTRODUCTION 15
altimeter orbits, rather than ignoring eight percent of the Earth’s surface in coastal
regions.
1. 4 The Study Areas
The study areas used to demonstrate the retracking system are the Australian coastal
regions and the Taiwan Strait. They are associated with different coastal
characteristics.
As well as for geographical convenience to the author, Australia is chosen for this
study for the following reasons. Firstly, it is surrounded by part of three large,
interconnected ocean basins of the Southern Hemisphere: the Pacific, Indian and
Southern Oceans. Secondly, the Australian coastline (including Tasmania), is
approximately 36,700 kilometres in length (Zann, 1990, cited in Resource
Assessment Commission, 1993), which is one of the longest coastlines in the world
for a single country. Finally, Australia’s coastal zone exhibits the following broad
range of topographical features: coastal plains, hills and mountains, beaches, cliffs,
estuaries, oceans, coral reefs, and islands. Thus, it is assumed that the performance of
altimetry around Australia is reasonably representative of what can be expected in
other parts of the world.
Although the Australian coast provides an almost perfect area for the investigation of
waveform retracking, it is not an area for the application of recovery of gravity
anomalies. In 1992, the Australian Geological Survey Organisation (AGSO, now
Geoscience Australia) released a gravity database that includes denser ship-track
gravity observations around Australia bound by 108°E ≤ λ ≤ 162°E and
8°S ≤ ϕ ≤48°S. However, these AGSO marine gravity data were not provided in a
format such that the crossover adjustment can be conducted to reduce some serious
biases among overlapping tracks (cf. Featherstone, 2003). They were also not
crossover adjusted before supply (ibid.). Therefore, they cannot offer any definitive
indication on the improvements of SSH data through the accuracy of the satellite
altimeter-derived gravity anomalies in the Australian coastal region.
Instead, another study area chosen is the narrow Taiwan Strait (roughly ~110 km
wide), which links the East China Sea to the South China Sea. This area is chosen

Chapter 1.INTRODUCTION 16
because firstly there are some on-going research activities, detailed and reliable shiptrack
gravity anomalies, and bathymetric data available in the area (e.g., Hwang and
Wang, 2002). Secondly, it is a shallow water region with high tidal amplitudes,
which allows the evaluation of the retracking system in a different coastal area.
Thirdly, ~2.5 years of ERS-1 waveform data including all phases provide denser
measurement coverage over the area. This will benefit the recovery of the gravity
field there.
1. 5 Thesis Outline
Chapters 1 and 2 introduce the background of this dissertation and the geodetic
principles of satellite altimetry. The altimeter data products, waveform data sets and
some external input data sources used in this study will be summarised and described
briefly in Chapter 2. In Chapter 3, the existing algorithms developed to retrack
altimeter waveforms over ocean surfaces and non-ocean surfaces will be reviewed
and discussed in the context of coastal retracking. The ocean model, which will be
used in this study, will be deduced, and a detailed qualitative analysis of the model
will be carried out.
Chapter 4 quantifies the contaminated distance around Australian coast using 20 Hz
waveform data of ERS-2 (one cycle, 35-day repeated orbit) and POSEIDON (five
cycles, 10-day repeated orbit). Several examples from ERS-2 waveforms along
satellite ground tracks near the coastline are analysed in detail to understand how
waveforms are contaminated and what the characteristics of contaminated
waveforms are at the Australian coast (Section 4.6). A preliminary test that estimates
the land effects on waveforms in the vicinity of land will be given in Section 4.7.
Chapter 5 develops a coastal retracking system that is based upon a detailed preanalysis
of the coastal waveforms given in Chapter 4. The retracking algorithms used
will be the fitting and threshold methods. Issues involving the fitting algorithms are
fitting functions of an ocean and β-parameter models, the least squares iterative
procedure, linearisation of the parameters, determination of the initial estimates of
parameters, and the weight scheme. A method will be introduced that uses iterative
weights to detect the outliers in the waveforms ensuring the effective convergence of
the fitting procedure. For the threshold retracking algorithm, a method will be

Chapter 1.INTRODUCTION 17
described that selects an appropriate threshold level for coastal waveforms. Finally, a
coastal retracking system will be developed, which makes it possible to improve the
SSH precision from reprocessing the altimeter waveforms, correcting the range
measurements, and extracting the precise SSH data from corrected ranges.
In Chapter 6, the coastal retracking system developed in Chapter 5 is applied to the
Australian coast (0-30 km from the coastline) using two cycles (42 and 43, March to
May 1999) of ERS-2 20 Hz waveform data. Results of waveform classification and
waveform retracking will be presented and discussed in Sections 6.2 and 6.3,
respectively. The internal validation of the quality of the retracked altimeter data will
be conducted by analysis of the single track SSH data in Section 6.4, comparisons
between SSH data before and after retracking in Section 6.5, and a collinear analysis
in Section 6.6. Using an external (partly) independent reference of the AUSGeoid98
geoid model, the results of retracking and comparisons with the unretracked and
external control data will be discussed in Sections 6.4 and 6.7 to demonstrate the
improvements from waveform retracking.
Chapter 7 investigates the improved accuracy of the altimeter-derived coastal marine
gravity field from SSH data after waveform retracking in the coastal region of the
Taiwan Strait. Using ~2.5 years of ERS-1 20 Hz waveform data (

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 18
2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES
2. 1 Introduction
Satellite altimetry employs a nadir-pointing, high-resolution radar to measure the
range to the ocean’s surface with an accuracy of a few centimetres. The range to the
at-nadir sea surface is measured by the altimeter directly through a measurement of
the time difference between the transmission of a radar pulse and the reception of the
echo. If the altitude of the satellite above a reference ellipsoid is known, the SSH
above the reference ellipsoid (Figure 2.1), which is the measurement expected from
the satellite altimetry and corresponds to the geophysical characteristics of the sea
surfaces, can be calculated by subtracting the range from the satellite altitude. In
order to precisely estimate the SSH, it is necessary to know the principle of the
altimeter’s operation, estimates of the range and then the SSH, and the error sources
for the determination of the SSH. These will be described next.
Figure 2.1 Basic measurement principle of the altimeter-derived SSH (From
http://www.jason.oceanobs.com/html/portail/galerie
/banque_img_welcome_uk.php3).

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 19
In addition, Section 2. 3 will briefly summarise the waveform data sets used in this
study. Some external input data sources will be described in Section 2. 4.
2. 2 General Information on Construction of Altimeter Measurements
The altimeter range measurement allows the determination of the SSH related to a
reference ellipsoid and hence oceanographic, geophysical and geodetic applications,
such as the determination of gravity anomalies, mesoscale eddies, Rossby waves and
the surface currents under an assumption of geostrophy (Chelton et al., 2001). Unlike
synthetic aperture radar, radar altimeters do not form an image observation, but make
point measurements at or close to the nadir of the satellite as it follows its orbit. The
coverage of the ground tracks is built up over time as the orbit arcs cover the Earth’s
surface. In addition, the shape of the return pulse has been found to give information
on SWH, while the backscattered power has been used to estimate the surface wind
speed (ibid.).
2.2.1 Fundamental Radar Measurement Principles
The altimeter-observed time delay can be converted into a range to the at-nadir sea
surface as long as the propagation velocity is known as
ct
R = (2.1)
2
where R is the range, c ≈ 3×10 8 m s -1 is the free-space speed of light and t is the
two-way travel time of the radar pulse. The range resolution is given by
c
∆ R ≥
τ
(2.2)
2
where τ is the pulse length. The resolution of one discrete measurement of range
depends on the altimeter’s operating parameters (Table 2.1). For instance, the
resolution is approximately 0.45 m for the ERS-1/2 altimeter operating in an ‘ocean
mode’ with a pulse length τ = 3.03 ns. In order to measure dynamic ocean signals,

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 20
Table 2.1 Altimeter Operating Parameters
Altimeter
Launch
date
Altitude
(km)
Inclination
(deg)
Beamwidth
(deg)
Frequency
(GHz)
PRF
(Hz)
No.
of
gate
s
Tracking
gate
Sampling time
(ns)
Waveform
frequency
(Hz)
No of
waveforms
in average
Seasat 27/06/1978 800 108 1.6 13.5 1020 60 30.5 3.125 10 100
Geosat 12/03/1985 800 108 2.1 13.5 1020 60 30.5 3.125 10 100
ERS-1 17/07/1991 784 98 1.3 13.8 1020 64 32.5 3.03 (ocean mode) 20 50
12.12 (ice mode)
ERS-2 21/04/1995 784 98 1.3 13.8 1020 64 32.5 3.03 (ocean mode) 20 50
12.12 (ice mode)
TOPEX (Ku) 10/08/1992 1334 66 1.0 13.6 4500 128 32.5 3.125 10 2×228
TOPEX (C) 10/08/1992 1334 66 2.7 5.3 1200 128 35.5 3.125 5 4×60
POSEIDON 10/08/1992 1334 66 1.1 13.65 1700 60 29.5 3.125 20 86
GFO 10/02/1998 800 108 1.6 13.5 1020 128 32.5 3.125 10 100
Jason-1 (Ku) 07/12/2001 1334 66 1.3 13.6 1800 104 32.5 3.125 20 90
Jason-1 (C) 07/12/2001 1334 66 3.4 5.3 300 104 32.5 3.125 20 15
Envisat (Ku) 01/03/2002 784 98 1.3 13.6 1800 128 3.125 18 100
Envisat (S) 01/03/2002 784 98 5.6 3.2 450 64 6.25 18 25

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 21
the range and timing precisions are required to a few centimetres (e.g., ~4 cm) and
less than 0.2 ns, respectively (Mantripp, 1996; Chelton et al., 2001). This is achieved
by averaging the estimates from several individual pulses (cf. the rightmost column
in Table 2.1).
2.2.2 The Returned Waveform
A schematic description of the interaction of a pulse-limited radar altimeter with a
diffuse, horizontal and planar sea surface is illustrated in Figure 2.2. Initially
( 0 t < t0
< ), a pulse of electromagnetic energy is transmitted from an on-satellite
altimeter antenna, propagating in a spherical wavefront. The area of pulse that can be
received on ground is defined by antenna beam-width ( θ
A
). When the wavefront
encounters the nearest crests of those ocean waves directly beneath the satellite (i.e.,
at nadir) at t = t0
, it illuminates a point and a reflected signal begins to return to the
altimeter. As time and the pulse progress, the wavefront reaches the surface at points
further from the nadir point. This increases the area of interaction between pulse and
surface, and the illuminated point spreads out rapidly to form a disc (assuming a
locally flat surface). Properly reflected facets within this disc scatter energy back to
the altimeter during the period of t
0
< t < t1
(Figure 2.2). Once the rear of the pulse
reaches the lowest trough at nadir, the region of interaction between the surface and
pulse forms an annular ring of increasing diameter, narrowing width, and the
constant mean area. The backscattered energy reaches its maximum at the point of
transition to an annular ring (i.e., t = t ). Thereafter ( t > t
1
1
), the backscattered energy
begins to decay owing to the finite antenna beam width and the fewer proper
reflected facets available at larger off-nadir angles. The returned power is recorded
over the duration of the pulse, building up a returned waveform (see the bottom in
Figure 2.2) with a rapidly rising leading edge and long decay of the trailing edge.
In order to reduce the fading noise arising from coherency among individual
waveforms, returned waveforms are actually an average of a number of pulses, 50 in
the case of ERS-1/2 (Mantripp, 1996; Quartly et al., 2001), forming theoretically a
average return altimeter waveform over ocean surface shown in Figure 1.1.

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 22
SIDE VIEW
transmitted
pulse
R
θ A
PLF
PLAN VIEW
(Illuminated area)_
t 0
t 1
t 2
RETURN WAVEFORM
leading edge
trailing edge
average return
typical return
t 0 t 1 t 2
time
Figure 2.2 A schematic geometrical description of the interaction of a pulse and the
scattering surface, and the build up of a return waveform over the duration of a pulse.
Note that, an average returned waveform from 50 typical individual returns is
depicted, while a single typical return shows more noise. (PLF is the pulse-limited
footprint).
This averaged returned waveform (Figure 1.1) is a time series of the mean returned
power recorded by a satellite altimeter, and is referred to as the convolution of three
terms in the time domain (e.g., Brown, 1977; Hayne, 1980; Chelton et al., 2001).
Over open oceans, it can be described by the Brown (1977) model (see Chapter 3). A
modelled or ideal average return altimeter waveform over ocean surface is shown in
Figure 1.1. It consists mainly of three parts:
(a)
The thermal noise contains the thermal noise power generated by the altimeter
prior to the first return of a signal from the scattering surfaces. Its effect is to
add a constant power level to the return waveform. It is to be noted that the

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 23
(b)
(c)
observed waveform of ERS-1/2 does not show the thermal noise because of the
operating feature of the altimeter (see Section 4.4).
The leading edge contains the return power from the scattering surfaces within
the pulse-limited footprint (see below for explanation), which involves in the
information about the SWH and range between the satellite altimeter and the
mean sea surface at the nadir.
The trailing edge consists of the return power from the scattering surface
outside the pulse-limited footprint, which can be approximated by a straight
line whose slope depends on the altimeter antenna pattern and the off-nadir
angle (see Chapter 5).
To date, all spaceborne radar altimeters use the pulse-limited (i.e., only the earliest
returns reflected from nadir are recorded) operation to measure the range to the
surface. The pulse-limited operation itself defines one of the diameters of the
measurement footprint (Martin et al., 1983; Quartly et al., 2001). The footprint is the
area of the reflecting surface which is illuminated by the altimeter, and its size
depends on the satellite altitude, the width of the range window and the features of
the reflecting surface (see Section 4.2.1). The pulse-limited footprint is the maximum
circular area from which backscatted power can be simultaneously received (Brooks
et al., 1978). The spreading of the pulse over the pulse-limited footprint corresponds
to the risetime of the leading edge of the return waveform, while the range to the
nadir sea surface measured by the altimeter corresponds to the midpoint at the
leading edge of the waveform.
In practice, a pulse compression technique (Marth et al., 1993; Chelton et al., 2001)
is used to obtain a high signal-to-noise ratio and an acceptable demand on the
satellite’s power system. The radar altimeter emits long-duration chirps, which are
linearly frequency-modulated pulses, then the received return signal is dechirped and
mixed with a deramping chirp. This mixed signal is then filtered and digitised in the
frequency domain where there is a one-to-one correspondence between frequency
and two-way travel time. Individual waveform samples result finally from a fast
Fourier transform (FFT) procedure. The resultant signal is a function of the
difference frequency, and its properties depend not only on the particular part of the
nadir scattering surface, but also the characteristics of the filter frequency used in the

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 24
chirp generation and the deramping processing. In this case, the power spectral
density estimates at discrete frequencies (64 bins for the ERS and POSEIDON) are
obtained and have a similar shape to that shown in Figure 1.1, with only the
difference for the two domains that the abscissa is frequency or two-way travel time
(Chelton et al., 2001). The further details on the actual implementation can be found
in Marth et al. (1993), Mantripp (1996), and Chelton et al. (2001).
Radar altimeters operate as closed-loop regulators in both the time-delay (range
tracking) and radiometric (AGC) domains (Marth et al., 1993; Jensen, 1999). Thus,
the on-satellite radar processor attempts to hold the reference position of the
altimeter waveform at a fixed position and with fixed amplitude so that successive
waveforms may be integrated or averaged without any loss of measurement
resolution. This is conducted by adjusting the timing of the data sampling and by the
attenuators within the receiver (ibid.). As a result, the processor must constantly
measure the varied range to the surface and reflectivity of the scattering surface. It
will be found in Section 5.4.1 (Figure 5.3) that the waveform’s leading edges are
centred on the midpoint, and this remains true regardless of the surface roughness. It
is this stable behaviour that makes the AGC and range tracking algorithm work.
The range to the surface can vary from several metres over ocean surfaces to several
kilometres over non-ocean surfaces within the diameter of the footprint (Mantripp,
1996; Chelton et al., 2001). However, in the vertical dimension the altimeter can
receive the returns only within a ‘range window’ specified by the width of the
frequency spectrum. For instance, it is ~28 m and ~112 m for ERS-1/2 in the ocean
mode (1 range bin equals ~0.45 m) and ice mode (1 range bin equals ~1.82 m),
respectively, and ~30 m for POSEIDON over oceans (1 range bin equals ~0.47 m).
The onboard tracker continuously adjusts the range window to keep the leading edge
of the waveform at a specified position at the centre of the range window. This
position is known as the ‘tracking gate’ (or tracking point), which is designed prior to
the launch of the satellite and is the range bins 32.5 for ERS-1/2 and 31.5 for
POSEIDON. When the surface elevation varies too rapidly for the tracker to respond,
the altimeter is said to have ‘lost lock’. Usually, the altimeter cannot track surfaces
where the mean slope is greater than about one degree (Barrick and Lipa, 1985;
Mantripp, 1996).

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 25
2.2.3 The Range and Sea Surface Height
If the altimeter tracker and AGC circuitry worked perfectly, the position of the
tracking gate would always be centred halfway up the return waveform leading edge
at the tracking gate. Therefore, the tracking gate would itself be an accurate measure
of the two-way travel time. In order to turn this measurement into the range from the
electromagnetic mean sea surface, and, ultimately, a surface elevation (e.g., SSH),
several corrections need to be applied to the data during ground processing. These
include the instrumental and satellite corrections (e.g., software processing delays
and Doppler shift), propagation corrections (e.g., dry and wet tropospheric delay and
ionospheric delay), and geophysical corrections (e.g., tides and inverse barometer
effect). Table 2.2 lists some corrections applied and the correction error to ERS-1
range measurements (cf. Cudlip et al., 1994).
Table 2.2 Corrections applied to ERS-1 range measurements and their likely error.
Error Source Correction magnitude Error
Doppler shift ±33 cm ±0.3 cm
Centre of gravity offset 0-90 cm ±0.1 cm
Ionospheric delay 0-100 cm ±5 cm
Tropospheric delay-dry 1.7-2.5 m ±1 cm
Tropospheric delay-wet 0-50 cm ±5 cm
Ocean Tide ±50 cm a ±10 cm
Earth Tide ±30 cm ±1 cm
Ocean Loading Tide ±10 cm

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 26
measurements, the SSH can be estimated by subtracting the corrected range from the
altitude.
2. 3 The Altimeter Waveform Data Used
Three types of waveform data products from ERS-1, ERS-2, and POSEIDON have
been used in this study. They are provided in the specific data format themselves by
the European Space Agency (ESA) for ERS-1/2 and the Archiving, Validation and
Interpretation of Satellite Oceanographic (AVISO) team for POSEIDON.
2.3.1 ERS-1 Waveform Data
The European Remote Sensing satellite ERS-1 was launched on 17 July 1991 by
ESA. The ERS-1 measurements effectively consist of a combination of several
different missions (Table 2.3).
Phase
Table 2.3 The repeat cycle characteristics of ERS-1.
Repeat cycle
(days)
Number of orbits
per cycle
Ground-track
spacing at the
equator (km)
ice 3 43 931
Multi-discipline 35 501 80
Geodetic 168 2411 16
The ERS-1 Ku-band radar altimeter (RA) is one of the remote sensors carried onboard
the satellite. It emits 1020 pulses per second and the on-board processor
averages the returns in groups of 50 to produce 20 Hz waveforms (e.g., Quartly et al.,
2001). The ERS-1 radar altimeter waveform is provided in a set of power signals
with respect to time at 64 sample bins (cf. Table 2.1 and Figure 1.2). It is noted that
ERS-1 geodetic mission contains two cycles of measurements and its second cycle of
orbit drifts away 8 km from the first cycle of orbit on the equator, thus making a
denser data coverage for geodetic applications.
ERS-1 altimeter waveform products (ALT.WAP) are supplied by the United
Kingdom Processing and Archiving Facility (UK-PAF) on behalf of ESA. The

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 27
ALT.WAP product contains all the information telemetered in the altimeter source
packets together with the corrections, calibration and orbit data required to further
process the data into the GDRs. The data are structured following the CEOS format,
which are unformatted, binary, direct access and with fixed record lengths (NRSC,
1995).
Because of the common format of the CEOS for all on-board remote sensing data
(Mansley, 1996, cited in Anzenhofer et al., 2000), the original ALT.WAP data in the
CEOS format are compressed to remove the un-used information for the altimeter,
generating compressed waveform products in the AIDA waveform format (AWF).
The intermediate ERS-1 waveform written in AWF contains only relevant
information to produce the altimeter GDRs, leading a data reduction from 1 GByte to
about 550 MByte for each 3-day data set (cf., Anzenhofer et al., 2000). They are
stored on the CD-ROM and contain 1 Hz of geophysical and environmental
0
corrections, and 20 Hz of observing time, range, location, orbit altitude, σ (radar
backscattering cross section per unit area), SWH, and waveform data in 64 sample
bins. The exact details of the AWF format can be found in, e.g., Anzenhofer et al.
(2000).
Nearly 2.5 years of ERS-1 waveform data are used in this research, which include
the data from all three phases. The data are supplied by the Ohio State University,
USA, but they were originally provided by UK-PAF. Unfortunately, the waveform
data used in this study are not provided in a format such that the orbit number can be
obtained to identify individual ground tracks. This presents a problem for averaging
repeated SSH to reduce the time variability and data noise when computing gravity
fields described later in Section 7.5.2. Therefore, a case-specific averaging method
will be developed to circumvent this problem.
2.3.2 ERS-2 Waveform Data
The ERS-2 satellite was launched on 21 April 1995 into a high inclination orbit
(98.5°), which provides a near-global coverage. Its mission consists of a combination
of the multi-discipline and tandem phases with both 35-day orbits. ERS-2 operates in
two tracking modes: ocean mode and ice mode. The on-satellite tracking algorithm

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 28
used for the ocean mode is known as the Sub-optimal Maximum Likelihood
Estimator (SMLE), while the off-centre of gravity (OCOG) algorithm is used by the
ice mode (Scott et al., 1994).
The ERS-2 waveform data used in this study are taken from CD-ROMs supplied by
De Montfort University, U.K., which are the ALT.WAP products supplied originally
by UK-PAF on behalf of ESA. The original data were read, compressed and stored
on the CD-ROM at De Montfort University. Each ERS-2 waveform record consists
of 1 Hz and 20 Hz data. Geophysical and environmental corrections are supplied at 1
Hz, and observations, such as the range, waveform longitude and latitude, altitude,
σ
0
, and 64 waveform samples, are supplied at 20 Hz. Two cycles of waveform data
(35-day repeat orbit, March to May 1999) are used in the study.
A problem found in the ERS-2 waveform data edited by De Montfort University is
that most of the Doppler range corrections exceed the minimum (-50 cm) and
maximum (50 cm) acceptable values (cf., NRSC, 1995). Therefore, this correction is
not applied to the altimeter range measurement. As stated, the returned signal is
analysed in the frequency domain. Thus, any change in the frequency of the returns
causes an error in the estimation of the range to the at-nadir sea surface. The Doppler
shift changes the relative velocity between the altimeter and the sea surface, thus
causing an error in the range estimate. The range correction for the Doppler shift is
typically ±33 cm for ERS data (cf., Mantripp, 1996). Without applying this
correction will cause the error to the range measurement, which is ~±0.3 cm from
Table 2.1. However, comparing with the required range accuracy of a few
centimetres (e.g., ~4 cm), the effect of this correction is small and can be neglected.
2.3.3 POSEIDON Waveform Data
The POSEIDON altimeter is a solid-state radar altimeter carried by the
TOPEX/POSEIDON (T/P) satellite, which was launched on 10 August 1992 into a
66-degree inclination orbit. It emits pulses at 1700 Hz and averages the returns in
groups of 86 every 53 ms. The waveform is sampled in 64 bins of 3.125 ns width,
with the tracking gate between bin 31 and 32. Waveform samples at the first two and
last two (aliased) bins have been removed from the data products provided by the

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 29
AVISO team. Since TOPEX and POSEIDON altimeters share an antenna on board
the satellite, the POSEIDON altimeter is operated for about 10 percent of the mission
time. The POSEIDON data used in this study comprise five cycles (January 1998 to
January 1999) of 20 Hz waveform data (ten-day repeat orbit) taken from the CD-
ROMs supplied by AVISO.
Table 2.4 The reference ellipsoid characteristics of ERS-1/2 and T/P.
Items ERS-1/2 T/P
Equatorial radius (m) 6378137 6378136.3
Flattening 1/298.257223563 1/298.257
The POSEIDON waveform product is structured as the classical T/P GDRs in terms
of cycle, pass files, headers and format. Each file is a fixed-length unformatted
record and contains a header. All file headers are ASCII and all other data are VAX
binary integers (AVISO/Altimeter, 1998). It consists of ten-day repeat cycles of data,
which were separated into ascending and descending passes. A POSEIDON
waveform record includes 1 Hz elementary data records of a pass-file and 20 Hz data
of the waveform measurements (ibid.). The data are referenced to different ellipsoids
(Table 2.4), so it is necessary to convert the data to a consistent reference (e.g.,
GRS80) if they are used together.
2. 4 External Input Data
The external input data sets used in this research for providing necessary information
for data editing and the result valuation include:
(1) GSHHS (0.2 km resolution) shoreline model (Wessel and Smith, 1996)
– provide information of the coastline’s location;
(2) The DS759.2 (5′×5′ resolution) ocean depth model (Dunbar, 2000) and the
Australian bathymetric model (30" resolution) (Buchanan, 1991) – identify
whether radar returns reflect from land or water, and for result analysis;
(3) Australian DEM (9"×9" resolution, version 2) (Hutchinson et al., 2001)
– provide land elevations above local mean sea level;

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 30
(4) A water/land grid (0.5′×0.5′ resolution) derived from the Generic Mapping
Tools (GMT) high-resolution shoreline data (Wessel, 2000) – combined with
the bathymetric models to flag data accurately;
(5) AUSGeoid98 geoid grid (2′×2′ resolution) (Featherstone et al., 2001) – provide
“ground truth” for validating the algorithms;
(6) Sea surface topography (SST) model (Levitus et al., 2002) – provide the SST
correction for estimating the gravity anomalies; and
(7) Marine gravity data around Taiwan (Hwang and Wang, 2002) – provide the
ground reference for the comparison between the gravity anomalies derived
from SSHs before and after retracking.
Some of these data will be described in detail below.
2.4.1 The AUSGeoid98 Geoid Model of Australia
The principal objective of the retracking system is to estimate the tracking error
caused by the coastal waveform contamination and correct the altimeter range
measurements accordingly. The analysis of the system discussed in Chapters 5, 6 and
7 has arisen because of the necessity of knowing how good the retracking system is.
To assess the retracking system, objective and independent ground references must
be used. In this study, the AUSGeoid98 gravimetric geoid model of Australia
(Featherstone et al., 2001) will be used as one of these independent ground
references. The altimeter-derived SSH before and after retracking will be compared
statistically with the geoid heights interpolated at the observation location from
AUSGeoid98 (Chapter 5 and 6). This model is chosen just because it is the most
recent and precise model readily available around Australia coastal regions at the
time of this study. It is acknowledged that other reference sources of local geoid
models (e.g., AUSGeoid91 and AUSGeoid93 models) or tide gauge data exist.
However, the AUSGeoid98 includes denser and more types of data than other global
or local models and uses arguably improved computational techniques (cf.,
Featherstone et al., 2001), thus precisely representing the geoid features in Australia.
The AUSGeoid98 geoid model was computed using data from the EGM96 global
geopotential model, the 1996 release of the Australian gravity database (both over

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 31
land and in marine areas), a national-wide digital elevation model (DEM), and
satellite altimeter-derived marine gravity anomalies. The geoid height has a
resolution of two minutes in longitude and latitude with respect to the GRS80
ellipsoid. Comparisons of AUSGeoid98 with GPS and Australian Height Datum
(AHD) heights across the continent indicate an RMS agreement of ±0.364 m (ibid.).
It is also acknowledged that the altimeter-derived gravity anomalies, the global two
minute grid of Sandwell et al. (1995), were used in the computation of the
AUSGeoid98 model. However, the ship-track gravity data are also combined with
the altimeter-derived gravity data using the ‘draping’ technique (cf., Kirby and
Forsberg, 1997; Featherstone et al., 2001). This technique involves least squares
collocation, as well as a 200 km high-pass filter. The satellite gravity grid is first
carried out by this filter and the difference between the filtered altimeter grid and the
ship-track gravity anomalies is calculated. Next, a difference grid is generated using
least squares collocation. Then, this grid is added back to the filtered altimeter grid
where no ship-track gravity data existed. Thus, because of the use of the ‘draping’
technique, the ship-track gravity data contribute mainly to the geoid signal in areas
which are close to the Australian coast. Therefore, the AUSGeoid98 can still be a
quasi-independent reference for the purpose of assessing the SSH data before and
after retracking.
The problem that must be noted is that many ship-track gravity data used in the
AUSGeoid98 model have not been cross-over analysed and adjusted (Featherstone et
al., 2001; Featherstone, 2002). Some serious biases among overlapping tracks exist.
Therefore, these ship-track gravity data should be used with some caution. This is
also the reason that the marine gravity data around Taiwan are used in this thesis (cf.
Section 2.4.3).
2.4.2 Sea Surface Topography
In order to estimate the gravity anomalies and geoid height from altimeter-derived
SSH, both the time-dependent and time-independent (or explicitly quasi-timeindependent)
sea surface topography (SST) must be removed (Hwang et al., 2002b;
Chelton et al., 2001). The SST is the dynamic sea surface height associated with
geostrophic surface currents (Chelton et al., 2001). The long-term mean SST from a

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 32
reference surface can be computed from a compilation of ship observations of
temperature and salinity profiles (ibid.). For instance, a 1°×1° mean SST grid related
to 3000 dB computed from 100 years of ship observations of temperature and salinity
profiles and smoothed by Levitus and Boyer (1994, cited in Chelton et al., 2001) can
be found in Chelton et al. (Figure 52, p.88, 2001). Another Levitus SST version
(1997) is provided by NOAA with the monthly averages on a 1°×1° grid for 12
months (cf, Hwang et al., 2002b), which is related to a reference sea surface of
1000 m depth level (Levitus et al., 2002). In this study, the quasi-time-independent
SST values adopted is from a 1°×1° grid averaged from these 12 month SST values
by Hwang et al. (2002b). Figure 2.3 shows the averaged Levitus (1997) SST, which
appears to be the similar profile to the version of Levitus (1994) SST, but the
different values due to the different references (cf, Chelton et al., Figure 52, p.88,
2001).
This SST grid (Figure 2.3) will be used to interpolate the required SST values at the
altimeter data point around the Taiwan Strait in Chapter 7. It is applied because of
the existence of the Kuroshio Current passing east of Taiwan and flowing along the
eastern boundary of the East China Sea near the research area (118°E ≤ λ ≤ 123°E
and 22°N ≤ ϕ ≤ 27°N). The Kuroshio Current is a strong western boundary surface
current (Le Traon and Morrow, 2001). It causes the larger SSH gradients (>1 mm/km
from Figure 2.3), so that the effect of SST cannot be neglected in the area around
Taiwan (Hwang, 2003).
However, this SST grid will not be applied to the altimeter data at the Australian
coast in this study. Firstly, from the SST data released by Levitus (1997) to create
this quasi-time-independent mean SST (Figure 2.3), there were no data sources near
the Australian coast. The SST values at this area are based on an interpolated
procedure from a 1°×1° grid (Hwang, 2003), thus causing additional errors to the
altimeter-derived SSH there. Secondly, the ocean currents near the Australian coast,
such as the Leeuwin Current, are weak when compared with the Kuroshio Current
(Le Traon and Morrow, 2001; Hwang, 2003), and do not cause large SSH gradients
(~0.2 mm/km). Thus, the SST effects on the SSH can be ignored at this area.

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 33
Figure 2.3 Quasi-time-independent SST (contour interval is 10 cm) from Levitus et
al. (1997).
Finally, it is known that the SSH can be expressed as (Chelton et al., 2001)
SSH h + h +
=
g d
C + ε
(2.3)
where h g is the geoid height related to the reference ellipsoid, h d is the SST height, C
is the range correction, and ε is the measurement error. However, it must be noted
that the absolute SST values cannot be obtained from an existing SST model,
because all existing SST versions are related to some reference surface (cf, Chelton
et al., 2001; Levitus et al., 1997). From Equation (2.3), if the altimeter-derived SSH
is used as an absolute measurement relative to the reference ellipsoid, nonuniqueness
will be caused by applying the SSTs from different references.
Due to these reasons, the SST was not applied to the altimeter-derived SSH data at
the Australian coast. The comparison between the SSH and the AUSGeoid98 geoid
height focuses on the observation of their profile variation rather than their difference.
The standard deviation of the difference between them will be as an indicator of the
SSH accuracy rather than the mean difference of them (Chapters 5 and 6).

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 34
2.4.3 Marine Gravity Data around Taiwan
To assess the accuracy of gravity anomalies derived from retracked satellite altimeter
waveform data, the ship-track gravity data provided by Hwang and Wang (2002a)
around Taiwan are used in this study (Figure 2.4). These marine gravity data contain
observations conducted from more than 50 marine geophysical cruises during the
period from 1965 to 1996. They are obtained from the gravity databases at Oxford
University, the National Geophysical Data Centre data set of the National Oceanic
and Atmospheric Administration (NOAA), and several overseas universities and
agencies (Hsu et al., 1998; Hwang and Wang, 2002a).
Figure 2.4 Coverage of ship-track gravity observations around Taiwan
Some basic data cleaning procedures had been applied to these ship-borne gravity
data, such as removing observations during turning the ship and crossover
adjustment (c.f. Hsu et al., 1998; Hwang and Wang, 2002a). Most gravity data near
Taiwan were provided by five geophysical cruises conducted in 1996 and GPS
navigation was used to obtain high-quality positions and Eötvös corrections. Other

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 35
ship-borne gravity data show lower precision because the Transit navigation system
was used. Therefore, the five cruises were used as control data and others were
adjusted with respect to them. The mean difference and standard deviation of the
crossover points of total shipboard gravimetric data are 6.3 mgal and ±11.2 mgal (see
Hsu et al., 1998). The free-air gravity anomalies were recomputed on the GRS80
ellipsoid.
Through the use of GPS positioning techniques and these careful data adjustments,
the gravity data around Taiwan form a ‘ground control’ on the accuracy of the
satellite altimeter-derived gravity anomalies. Gravity anomalies at 397 points along
the ship tracks are used in comparisons (Chapter 7), which are taken in the area
118°E ≤ λ ≤ 122°E and 23°N ≤ ϕ ≤ 26°N west of Taiwan to be consistent with the
altimeter data (Figure 7.1).
2.4.4 The Australian Digital Elevation Model (DEM)
The shape of the waveforms in coastal regions have been found to differ from those
recorded over open oceans from the previous studies (e.g., Brooks, 1997) and this
study (Chapters 4 and 6). One of the causes of this waveform contamination is that
the waveform may be mixed with the land and ocean returns when the altimeter
simultaneously illuminates land and water. The DEM can provide information about
the terrain elevation and topography on land within the altimeter footprint near the
coastal zone, and thus be used for analysis of the land effects on the waveforms
(Section 4.7).
A DEM is a representation of the terrain using averaged elevation information. The
recent upgraded (version 2) 9 second Australian DEM is a grid of elevation points
covering the whole of Australia with a grid spacing of 9 seconds in longitude and
latitude (approximately 250 m) in the Geocentric Datum of Australia 1994 (GDA94)
horizontal reference frame (Hutchinson et al., 2001). The source data used to create
the 9 second Australian DEM consists of:
(1) revised data sets (spot heights, linear watercourse features, Australian coastline,
and coastal inlets) from Australian GEODATA digital topography products

Chapter 2. SATELLITE ALTIMETER MEASUREMENTS AND DATA SOURCES 36
(including TOPO-250K Relief theme, Drainage layer of TOPO-250K, and
COAST-100K Coastline and State Borders);
(2) trigonometric data points (converted to the GDA94 coordinate system) from
the National Geodetic Data Base;
(3) radar altimeter point elevation data (300 observations) for Lake Eyre; and
(4) additional data sets (spot heights, stream line data, sink point data, cliff line
data and associated contour line data) from digital 1:100 000 scale mapping.
They are supplied by National Mapping Division of Geoscience Australia
(formerly AUSLIG) (ibid.).
Errors in the DEM depend mainly on the resolution of the DEM and the roughness of
the terrain (i.e., the slope). Theoretical estimates and tests of the 9 second DEM
against trigonometric data distributed evenly across the continent indicate that the
standard elevation error of the DEM varies from ~±7.5 m to ~±20 m for most of the
continent (ibid.). Errors are larger in highland areas with steep and complex terrain,
where the largest errors can exceed 200 m (ibid.).
2. 5 Summary
This Chapter has introduced the basic fundamental working principle of a radar
altimeter, the altimeter range measurements and the required sea surface information
(i.e., SSH), and data products. Some external input data sources used in this study,
such as the AUSGeoid98 geoid model, the Australian 9" DEM and the ship-track
gravity around Taiwan, were also described in this Chapter.

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 37
3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS
3. 1 Introduction
Radar altimeter return waveforms are the basic measurement for observing
geophysical parameters of the Earth surfaces, in particular, the ocean. They give the
range between the satellite and the mean sea surface at nadir (via pulse transit time),
the SWH (via return waveform shape characteristics), and the roughness of the sea
surface (via received energy, from which wind speed is empirically inferred).
Retracking is a procedure of waveform data post-processing to improve parameter
estimates over those given by GDRs. These parameters contain range corrections due
to the estimation algorithm and the limited computational time on-board the satellite,
and unmodelled ocean surface effects, such as the surface skewness. Waveform
retracking has different emphases for different applications depending on the
reflecting surfaces.
Over ocean surfaces, altimeter retracking algorithms aim to reduce the following
errors: short-wavelength random noise, and more importantly, potential longwavelength
errors caused by SWH biases, range biases, the altimeter antennamispointing
angle, or unmodelled non-Gaussian ocean surface parameters, such as
the surface skewness and kurtosis (e.g., Hayne and Hancock III, 1990; Hayne et al.,
1994; Hayne, 1980; Rodriguez, 1988; Rodriguez and Martin, 1994b). Over nonocean
surfaces, the on-board satellite altimeter tracker cannot follow the sharp
change of the surface topography (i.e., land/ice), causing an obvious departure
between the midpoint of the leading edge of the waveform and the predesigned
altimeter tracking gate. This leads to an error in the telemetered range measurements
made by the altimeter. Waveform retracking, therefore, aims to produce accurate
surface elevation measurements over ice sheets and land (e.g., Martin et al., 1983;
Ridley and Partington, 1988; Davis, 1993a; Berry et al., 1998). The algorithms
developed to retrack altimeter waveforms over ocean surfaces and non-ocean
surfaces will be discussed in Sections 3.4, 3.5 and 3.6.
In coastal regions, waveforms are contaminated by the land topography (see
Chapters 1, 2 and 4), which exhibits a variety of terrains including the coastline,

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 38
mountains, islands, lowlands, cliffs, reefs, and estuaries, as well as the sea states,
such as standing and very calm water. To the author’s knowledge, all previously
published retracking efforts involve retracking coastal waveforms using existing
algorithms developed for ice sheets (e.g., Anzenhofer et al., 2000; Brook et al., 1997).
A description of these methods will be given in Section 3.7.
3. 2 Convolutional Representation of Ocean-Returned Waveforms
The shape of the radar return waveform represents the observations measured by a
satellite radar altimeter as a function of two-way travel time, which is related to the
scattering characteristics of the reflecting surface. Moor and Williams (1957, cited in
Brown, 1977) first proposed the convolutional representation of the return waveform.
They demonstrated that for a ‘rough’ (i.e., incoherent) scattering surface, the average
return power (as a function of delay time) can be expressed as a convolution of the
transmitted power waveform envelope with a quantity involving
0
σ (radar
backscattering cross section per unit scattering area), the antenna gain, and the range
from the altimeter to any point on the Earth’s surface. Furthermore, when
considering the vertical distribution of the surface height and radar receiver effects,
the average return power (as a function of the delay time) is a convolution of three
terms: the smooth spherical surface, the probability density function of the height of
the specular points, and the radar system’s point target response (e.g., Brown, 1977;
Hayne, 1980; Rodriguez, 1988).
3.2.1 The Brown Model
The mathematical model used to derive the average altimeter returned waveform
from incoherent surface scattering has been based upon physical optics theory,
whereby the surface is treated as a set of specular facets with a given height and
slope probability density distribution. The time-series of the mean returned power
waveform P (t)
measured by a satellite altimeter is expressed as a convolution of
three terms in the time domain as (Brown, 1977)
P( t)
= P ( t)
∗ q ( t)
∗ P ( t)
(3.1)
fs
s
PTR

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 39
where t is the time measured at the satellite receiver such that t = 0 corresponds to
the range to the mean sea level at nadir, P fs
(t)
is the average impulse response from
a flat surface, q s
(t)
is the surface elevation probability density function (PDF) of
specular points within the altimeter footprint, and P PTR
(t)
is the radar system point
target response (PTR). Each of the three terms in Equation (3.1) will be presented
separately later in Sections 3.2.2, 3.2.3 and 3.2.4.
In order to give an analytical representation of the convolution in Equation (3.1),
Brown (1977) first presented a simplified expression of the terms for near-normal
incidence under the following assumptions, which are common to all satellite radar
altimeter systems. The basic assumptions that are inherent in the convolution model
of near-normal incidence rough surface backscatter (Barrick, 1972; Brown, 1977;
Fedor et al., 1979) are as follows:
(1) The root-mean-square (RMS) elevation variations are greater than the radar
wavelength;
(2) The ocean surface is statistically homogeneous and stationary over the number
of pulses needed to determine a mean waveform;
(3) The radar cross section is constant over the illuminated area;
(4) There is no backscatter of the electromagnetic energy from beneath the surface;
(5) The illuminated surface always contains a statistically significant number of
specular points;
(6) The surface elevation and surface slope density distributions are statistically
independent; and
(7) Multiple scattering among various parts of the surface is neglected.
These assumptions are generally valid for an altimeter operating over open ocean
surfaces. However, they are not generally the case over non-ocean surfaces, and
hence an alternative approach must be sought (e.g., Mantripp, 1996). In coastal
regions, areas of land can be simultaneously illuminated with water, thus causing
land returns to the altimeter. Therefore, it is not always the case that the above
assumptions are valid in the coastal zone (ibid.).

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 40
3.2.2 Average Flat Surface Impulse Response Function
Using a Gaussian approximation of the antenna gain, Brown (1977) presents P fs
(t)
over the illuminated area of the Earth’s surface as
where
P fs
1/ 2
( t)
= Aexp(
−α ' t)
I ( β ' t ) U ( )
(3.2)
0
t
4c
α'
= cos(2ξ
)
(3.3)
γ h
a
4
β ' =
γ
a
⎛ c
⎜
⎝ h
⎟
⎠
⎞
1/ 2
sin(2ξ
)
(3.4)
sin 2 θ
γ
a
=
ln 4
(3.5)
2
In Equation (3.2), U (t)
is a unit step function, I ( t
1/ ')
is the modified Bessel
0
β
function (see Section 3.3.2), A is an amplitude scaling term containing the off-nadir
pointing angle ξ and several other constants, such as the radar wavelength, the ocean
reflectivity and the radar antenna gain (Brown, 1977). In Equations (3.3) and (3.4),
h is the satellite altitude above the reference ellipsoid surface, and γ
a
is an antenna
beamwidth parameter defined by Brown (1977). In Equation (3.5), θ is the antenna
beamwidth.
The derivation of Equations (3.3) and (3.4) assumes that the altimeter antenna pattern
is approximately a Gaussian function, otherwise the closed form of Equation (3.2)
cannot be derived from an integration of P fs
(t)
(see Brown, 1977, Equation 2).
According to Brown (1977), this approximation is generally valid out to the point on
the antenna pattern for which there is no appreciable contribution to the
backscattered power.
In Equation (3.2), A can be divided into two parts as
A = A0
Aξ
, where 0
A is an
amplitude scaling term involving only constants of the radar parameters and the
ocean reflectivity, and A ξ
is a term related to the off-nadir pointing angle ξ . They
can be written as

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 41
where G
0
, λ and
A
0
2 2 0
0 λ cσ
( 0)
2 3
( 4π
) L h
G
= (3.6)
4
p
⎛ 4 ⎞
= ⎜ −
2
A
ξ
exp sin ξ ⎟
(3.7)
⎝ γ ⎠
L
p
are radar system parameters (Brown, 1977).
3.2.3 Surface Elevation PTR of the Scattering Distribution
The instantaneous illumination pattern of the altimeter depends on the specific wave
height field at time t within the footprint. Because of the random nature of the sea
surface elevation distribution, it is more instructive to consider the PTR of specular
points for the determination of the waveform. As a first-order approximation, the
distribution of the sea surface elevation is Gaussian (Chelton et al., 2001). The scale
of the wave-height distribution is described by the SWH, which is defined to be the
average crest-to-trough height of the 1/3 highest waves. Brown (1977) uses a
Gaussian specular point PDF to describe the statistics of the linear surface elevations.
It is given as
q s
( ζ )
2
1 ⎛ ζ ⎞
exp
⎜ −
⎟
1 / 2
(3.8)
(2π
) σ
s ⎝ 2σ
⎠
= s
where ζ is the surface elevation (positive upward) above the mean level of the
specular points, σ s
is the RMS surface elevation of the specular points related to
SWH by SWH = 4(2 / c)
σ . The PDF in Equation (3.8) is expressed in the space
s
domain in distance units (metres), while the ocean surface elevation density function
q s
(t) is written in the altimeter’s time domain in Equation (3.1). To coincide with
other two terms in Equation (3.1), a variable
t = −2ζ / c must be used to convert
q s(ζ ) from the space domain to the two-way range time domain in time units
(nanoseconds). The relevant standard deviation can be written as σ = 2σ c .
t s
/

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 42
3.2.4 Radar System PTR
The radar system PTR is primarily the transmitted radar pulse shape. For an idealised,
linear, frequency-modulated altimeter radar pulse, the radar system PTR is given by
(Ulaby et al., 1982, cited in Rodriguez, 1988)
P PTR
[(
at / 2)( T − t )]
2
sin
( t)
− T ≤ t ≤ T
(3.9)
=
2
( at / 2)
where T is the radar transmitted pulse duration (or the compressed pulse length if
pulse compression is employed) and a is a constant which depends on the radar
bandwidth. Thus, P PTR
(t)
is a symmetric expression of the impulse response about
t = 0 . However, using the PTR expressed by Equation (3.9), it is not possible, in
general, to obtain the analytic function of Equation (3.1). Since the width of the PTR
of the short pulse radar altimeters is of the order of 20 ns (e.g., Brown, 1977),
Equation (3.9) can be simplified to a Gaussian function as
P
PTR
2
⎡ ⎤
⎢
1 ⎛ ⎞
⎜
t
( t)
≈ η − ⎟ ⎥
p
PT
exp
⎢ 2
(3.10)
⎥
⎣ ⎝σ
p ⎠ ⎦
where η
P
is the pulse compression ratio, P T
is the peak transmitted power, and
is a measure of the pulse width
σ
P
σ
P
= 0. 425T
(3.11)
in which 0.425 is an approximation of the compressed pulse shape by a Gaussian
function (Marth et al., 1993).
3.2.5 Solution of the Radar Returns
Using Equations (3.2), (3.8) and (3.10), Brown (1977) shows that the convolution of
Equation (3.1) can be reduced to the following expression
P(
t)
≈ P
fs
( t)
∞ ∞
∫∫
0 −∞
⎛ c ⎞
⎜ ⎟P
⎝ 2 ⎠
PTR
⎛ c
( t −τ
) qs⎜
⎝ 2
( τ − ˆ τ ) dτ
d ˆ τ
⎟ ⎠
⎞
(3.12)

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 43
1 ⎡ ⎛ t ⎞ ⎤
≈ ηP
PT
Pfs(
t)
σ
P ⎢erf
⎜ ⎟ + 1
2
⎥
⎣ ⎝ 2σ
⎠ ⎦
where erf (x)
is the error function, and σ is the composite risetime given by
2 2 ⎛ 2
= σ p
+ ⎜ s
2
σ ⎞
⎟
⎝ c σ
⎠
(3.13)
In Equation (3.12), the first term, P fs
(t)
, includes the effects of the antenna
beamwidth and the off-nadir pointing angle. The flat surface emphasises that an
incoherent surface scattering process is assumed. The second term, P PTR
(t)
, contains
the effects of the receiver bandwidth on the transmitted pulse. The third term, q s
(t)
,
contains the sea-state effects. It is assumed that waves on the sea surface are linear
(i.e., an incoherent surface) and therefore the corresponding statistics of surface
elevations and slopes can be expressed by a Gaussian function.
Equation (3.12) is the fundamental model of the returned waveform over ocean
surfaces, named the Brown model. It is important because it clearly shows that the
simplified expression of the altimeter return waveform can be applied to extract
geophysical parameters over oceans from the observed return waveform. However, it
should be noted that some effects are neglected in the Brown model, such as the
curvature of the Earth’s surface, approximation of the Bessel function, and the
nonlinearity of the actual surface waves. These remain an open issue, which has been
addressed by other researchers, such as Rodriguez (1988), Hayne (1980), Challenor
and Srokosz (1989), and Rodriguez and Martin (1994a), and will be summarised in
the following Sections.
3. 3 Modified Convolutional Representation of the Ocean Return Waveform
The Brown model has been modified, e.g., by Hayne (1980) and Rodriguez (1988),
to present general expressions for the radar mean return waveform from non-
Gaussian surfaces. This has allowed more parameters to be estimated from the radar
return, such as the skewness of the sea surface (e.g., Lipa and Barrick, 1981;
Rodriguez, 1988; Rodriguez and Martin, 1994b; Challenor and Srokosz, 1989). This
Section will summarise the modifications made to the Brown model by these authors.

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 44
Then, an analytic expression for the radar ocean return will be derived from some of
these modified expressions for the purposes of this dissertation.
3.3.1 Effects of the Earth’s Curvature
The effects of the Earth’s curvature are neglected in Equations (3.3) and (3.4), but
the off-nadir angle for a given pulse delay time t is slightly smaller than that in the
flat-Earth approximation. Considering its effect, Rodriguez (1988) shows that the
corrected functions α and β related to Equations (3.3) and (3.4) are
4c
1
α = cos(2ξ
)
(3.14)
γ h (1 + h / R)
4 ⎛
β = ⎜
γ ⎝
c
h
1 ⎞
⎟
(1 + h / R)
⎠
1/ 2
sin(2ξ
)
(3.15)
where R ≈ 6371005 m is the spherical radius of the Earth (e.g., GRS80; Morita,
1980). The flat-Earth approximation in Equations (3.3) and (3.4) is not a realistic
case. For the ERS-2 and TOPEX orbit heights of about 785 km and 1336 km,
respectively, this flat-Earth approximation leads to a 5.8% and 9.1% reduction of the
illuminated area of the flat-Earth surface. Thus, the average radar impulse response
from a smooth spherical surface is given by (Rodriguez, 1988)
P s
1/ 2
( t)
= Aexp(
−α t)
I ( β t ) U ( )
(3.16)
0
t
Equation (3.16) includes effects due to the Earth’s curvature which, in general,
cannot be ignored (Hayne and Hancock III, 1990).
3.3.2 Simplification of the Bessel Function
The convolution in Equation (3.1) does not allow for easy analytical integration
because of the presence of the modified Bessel function (Equation 3.2). It can be
seen from Equations (3.2) and (3.4) that the off-nadir angle ξ and return time t are
two variables in the Bessel function. The off-nadir angle is ideally equal to zero, but
in practice the antenna does not always point along the nadir and the maximum
possible pointing error is 0.5 degrees (e.g., Challenor and Srokosz, 1989; Brown,

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 45
1977). In addition, the return time is also a small value compared to the leading edge
rise time (Rodriguez, 1988). For such small off-nadir angles and the case of shortpulse
radar altimeters, Hayne (1980) expands I 0
in the small-argument (assumed
convergent) series expansion (Jeffrey, 1995):
n
2
2
1
0
( ) ∑ ∞ ⎛ z ⎞ ⎛ ⎞
I z =
⎜ ⎜ ⎟
= 0 4
⎟
(3.17)
n ⎝ ⎠ ⎝ n!
⎠
which allows for the term-by-term integration in the convolution model in Equation
(3.1) (Hayne, 1980). Rodriguez (1988) expands the Bessel function presented in the
smooth surface impulse response using Equation (3.17), but keeps only terms of
order
2
β t and lower. Then, ( 1/ 2
)
I β is replaced by exp( 2 t / 4)
0
t
β to give a simpler
expression for analytical integration. Thus, considering the curvature of the Earth and
the above approximation for I
0
, Equation (3.2) can be expressed as the average radar
impulse response from a smooth sphere (Rodriguez, 1988):
P s
2
⎡ ⎛ β ⎞ ⎤
( t)
≈ Aexp⎢−
⎜α − t⎥U
( t)
⎣ 4
⎟
(3.18)
⎝ ⎠ ⎦
The error caused by this approximation was analysed by Rodriguez (1988) based
upon various off-nadir angles and return times, for SEASAT, GEOSAT and TOPEX.
The results show that the error is less than one percent for all times when the offnadir
angles is approximately less than half the altimeter beamwidth (see Table 2.1)
(Rodriguez, 1988).
The antenna off-nadir angle is also called the off-nadir pointing error, which
represents the displacement of the boresight axis of the altimeter antenna from the
nadir. Because of peak power limitations and the high operating altitude, spaceborne
radar altimeters generally use high gain antennas. A direct consequence of the high
gain requirement is a narrow beamwidth, typically less than 3° (Brown, 1977). With
current satellite attitude control systems, the beamwidth is 1.34° for ERS-1/2 and
1.1° for TOPEX (Ku-band, see Table 2.1). Thus, the off-nadir angle approximated as
half the antenna beamwidth cannot be larger than 1°. In addition, it is highly unlikely
that the pointing error could exceed 1° without causing loss of tracking (cf. Barrick

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 46
and Lipa, 1985). Therefore, Equation (3.18) is a reasonable approximation based on
the typical magnitude of the off-nadir angle.
3.3.3 Approximate Expression of the Non-Gaussian Scattering Surface
As stated, the shape of the returned waveform is determined by the distribution of
specular surface scatterers rather than by the actual sea surface height distribution
within the footprint. The differences between the distributions of the scatterers and
the sea surface height lead to biases between the altimeter-measured scattering
surface and the actual mean sea surface, which must be corrected for the altimeter
range measurements (Chelton et al., 2001; Rodriguez, 1988). The range to the sea
surface is estimated from the half-power point of the leading edge of the waveform,
which corresponds to the return from the median height of the specular scatterers
referred to as the electromagnetic (EM) sea level (ibid.).
The difference between the true mean sea level and the mean EM surface in the
footprint is called the sea state bias (SSB), which comprises the electromagnetic bias
(EMB) and the skewness bias (SKB). Owing to the non-Gaussian nature of the sea
surface, the wave troughs are brighter than the peaks for radar wavelengths, arising
in a greater backscattered power per unit surface area from wave troughs than from
wave crests. The result is that the mean EM surface measured by altimeters is biased
lower than the true mean sea surface toward wave troughs. This effect is known as
the EMB. From previous studies (cf. Chelton et al., 2001), the EMB has been
observed to increase monotonically with increasing the SWH. The SKB is the error
in tracking the mean height of the specular scatterers, which is the height difference
between the mean scattering surface and the median scattering surface (ibid.). The
SKB is generally proportional to the SWH, but it is not related in any simple manner
to the SWH or any other geophysical quantity that can be inferred from altimeter
data.
The specular point PDF q s
(t)
in Equation (3.1) represents the probability density
distribution of the specular scatterers within the altimeter footprint. It thus includes
the effects of the SSB. It is evident that the SSB in the surface elevation distribution,
through its effect on the specular point PDF, affects the ability of the on-satellite

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 47
tracking algorithm to determine the exact range to the at-nadir mean sea level. The
Brown model (Equation 3.12) uses a Gaussian specular point PDF to describe the
statistics of the ‘linear’ ocean wave heights. As stated, the ‘linear’ ocean surface is
defined as an incoherent surface (i.e., not including the SSB) in this study. To a firstorder
approximation, the height of the waves and slope of the waves are assumed
zero-mean, Gaussian, uncorrelated random variables. However, higher order effects,
including correlation between heights and slopes, as well as skewness, indicate that
the surface height and slope are not strictly Gaussian (Barrick and Lipa, 1985). For
example, the results obtained by Srokosz (1986) and Rodriguez (1988) regarding the
effects of the SKB on the range to the sea surface show that the SKB causes an error
of 0.8 cm for a skewness of 0.1 and the SWH of 2 m. This error increases to ~4 cm
for the SWH of 10 m.
Because of the effects from both the EMB and SKB on the shape of the leading edge
of the radar returned waveform, as well as the prospect of extracting information
about the sea state of a non-linear scattering surface, numerous studies have
attempted to estimate the SKB (e.g., Lipa and Barrick, 1981; Parsons, 1979;
Rodriguez, 1988; Barrick and Lipa, 1985; Hayne, 1980), and the EMB (e.g.,
Rodriguez, 1988) using a modified expression for the specular point PDF.
Longuet-Higgins (1966, cited in Challenor, 1989) obtained the PDF of surface height
using an approximation of the Gram-Charlier series (cf. Jeffrey, 1995) for the nonlinear
surface:
⎫ ⎛ ⎞
⎨
⎧ 2
1 λ
= + s
η
qs ( ζ ) 1 H
3( η) ⎬exp⎜
− ⎟
2π
σ ⎩ 6 ⎭ ⎝ 2
(3.19)
s
⎠
where
ξ
η = (3.20)
σ s
and
H
n
are the Hermite polynomials (e.g., Jeffrey, 1995) of degree n with

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 48
H
n
n
n 2 d
2
( x)
= ( −1)
exp( x ) exp( −x
)
(3.21)
n
dx
Equation (3.19) introduces the height skewness parameter ( λ
s
) of the sea surface,
which is a measure of the non-linearity of the wavefield, and is a dimensionless
quantity (Hayne and Hancock III, 1990). Barrick and Lipa (1985) and Srokosz (1986)
extend the result in Equation (3.19) to obtain the joint PDF of the elevation and slope,
which contains another non-linear parameter of the cross-skewness (δ ) as
2
1 ⎧ λ
⎛ ⎞
s
δ ⎫ η
qs ( ζ ) = ⎨1
+ H
3( η) − H1( η) ⎬exp⎜
− ⎟.
(3.22)
2π
σ ⎩ 6 2 ⎭ ⎝ 2
s
⎠
In addition, Hayne (1980) uses a four-term series as the expression of radar-observed
surface height PDF from the laboratory measurements of the surface elevation
density function for a wind-generated wavefield, which contains Hermite
polynomials of H
3
, H
4
, and H
6
(see Hayne, 1980, Equation 12). Two non-linear
parameters of skewness and kurtosis are introduced (discussed below).
The skewness parameter contributes significantly to the SSB (cf. AVISO/Altimeter,
1996, p.45; Chelton, 2001), while the cross-skewness term is thought to be related to
the EMB and thus cause a shift of the electromagnetic surface with respect to the true
mean sea surface (Rodriguez, 1988). A rewritten form of the PDF by Rodriguez
(1988) makes the effect of the EMB more apparent:
⎫ ⎛ ⎞
⎨
⎧ 2
1 = + s
( ) ⎬ ⎜ −
R
q ( ζ λ
η
s
) 1 H
3
ηR
exp ⎟
2π
σ ⎩ 6 ⎭ ⎝ 2
s
⎠
(3.23)
where
η
R
( ζ − ζ γσ / 2)
+ 0 s
= (3.24)
σ
s
where γ is the parameter related to the surface shift partly caused by the EMB and
ζ
0
is the altimeter tracker error. The expression in Equation (3.24) indicates that the
effect of γ is to shift the whole surface by the amount of − γσ / 2)
, which is the
(
s

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 49
magnitude of the EMB. This means that the total altimeter height estimation error,
i.e., the shift of the altimeter-observed sea surface with respect to the true mean sea
surface, consists of the altimeter tracker error ( ζ
0
) and the shift caused by the EMB.
Therefore, Equation (3.23) differs from Equations (3.19) and (3.22).
The SKB can be estimated from the above expressions of the specular point PDF
using the altimeter waveform data (e.g., Hayne, 1980; Rodriguez, 1988; Lipa and
Barrick, 1981; Barrick and Lipa, 1985). Using retracked TOPEX data after removing
the EMB, Rodriguez and Martin (1994) estimate that the dimensionless skewness is
very small and ranges from 0.01-0.12.
Theoretically, determining the EMB requires knowledge of the specular point PDF.
Unfortunately, the current theoretical understanding of the EMB is not sufficiently
well developed to estimate the EMB from a purely theoretical basis or from altimeter
data themselves without additional geophysical information or assumptions (Chelton,
2001; Rodriguez, 1988). There are two possible reasons for this. The first is because
two shifts of the EMB and SKB are introduced in the specular point PDF, and they
are not independent to all orders of approximation. Thus, they cannot be
distinguished. The second is that the EMB depends on the characteristics of the
wavefields, while only the SWH can be unambiguously extracted directly from the
altimeter waveform. Therefore, in practice, empirical estimation of the EMB is a
more appropriate method (Chelton, 2001).
3.3.4 Approximate Expression of the System PTR
The altimeter-transmitted pulses have a shape that is described by the radar system
PTR. However, the PTR also includes effects due to the bandwidth of the receiver
(Hayne, 1980). It is generally expressed by Equations (3.9) and (3.10). In the Brown
model (Equation 3.12), the PTR is expressed by a Gaussian function (Equation 3.10).
In reality, the PTRs are not exactly Gaussian (Hayne, 1980; Rodriguez, 1988).
Rodriguez (1988) compares measured PTRs of SEASAT and GEOSAT with an
idealised PTR ( sinc
2 x)
. The results (Rodriguez, 1988) show that the radar system
PTR presents a lack of symmetry and large side lobes, suggesting thus the PTR is not

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 50
2
sinc x or a Gaussian function. The effect of the Gaussian-approximated PTR will
change the shape of the return waveform, especially in the early parts of the leading
edge. To obtain realistic non-Gaussian ocean parameters, Rodriguez (1988)
concludes that the measured PTR must be used in the estimation process to obtain
realistic sea state parameters.
The modified PTR has been developed by Hayne (1980), and Rodriguez and Martin
(1994). Hayne (ibid.) expresses the PTR using a nearly Gaussian function (a fourterm
series). Rodriguez and Martin (1994) expand the PTR as a series of Gaussian
functions. They both appear to improve the estimates of the non-Gaussian ocean
surfaces (Chelton et al., 2001).
3.3.5 A Modified Solution of the Radar Returns Developed in This Study
Following Brown (1977), modified convolutional representations of the waveform
for non-Gaussian ocean surfaces have been given by Hayne (1980), Rodriguez
(1988), Rodriguez and Chapman (1989), Challenor and Srokosz (1989), and Barrick
and Lipa (1985). The main difference among the convolutional solutions is that
researchers take different modified expressions of the surface-height PDF, the radar
system PTR, and the Bessel function. Following this idea, and according to the
specific application of this study, a modified solution of the radar returns developed
by the author will be introduced in this Subsection.
The model introduced in this study is slightly different from the Brown model and
the modified models (e.g., the Hayne (1980) model). The main differences are as
follows:
(1) As stated, the current theory is not sufficiently well developed to accurately
estimate the EMB using only altimeter data. This term will be neglected from
the specular point PDF. Instead, Equation (3.19) is employed as the expression
of the specular point PDF.
(2) Equation (3.18) is used as the function describing the smooth surface response,
because it takes account of the effects of the Earth’s curvature. It is also
computationally simple, but without significant loss of accuracy (Rodriguez,

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 51
1988), provided that the Bessel function has been expanded appropriately
(ibid.).
(3) Since this study does not focus on estimating non-linear parameters of the non-
Gaussian ocean surface, the Gaussian function will still be assumed to be an
adequate expression of the radar system PTR for the purpose of this study.
Using a variable
t = −2ζ / c , Equation (3.19) can be rewritten in the time domain as:
q s
( t)
=
1 ⎧ λs
⎨1
+
2π
σ ⎩ 6
s
2
3 ⎫ ⎛ η ⎞
( − 3 ) exp⎜
−
t
η η
⎟
t
t
⎬
⎭
⎝
2
⎠
(3.25)
in which
c
ηt
= − t
(3.26)
2σ
s
Equation (3.25) is the surface height PDF in the time domain. It can be reduced to
the linear case when the skewness parameter λ
s
is omitted, i.e., by setting λ
s
=0.
The radar system PTR used is assumed to be a Gaussian function, given by
P
PTR
( t)
≈
1
⎡
1 ⎛ ⎞
⎢ ⎜
t
exp − ⎟
2π
σ ⎢ 2
p
⎣ ⎝σ
p ⎠
2
⎤
⎥
⎥
⎦
(3.27)
As stated, for typical short pulse radar altimeters, the width of the PTR is of the order
20 ns or less (Brown, 1977). This approximation of the PTR is found to be adequate
for these widths by Brown (1977).
From Equations (3.18), (3.25), and (3.27), describing the three terms in the
convolutional representation, a modification of Equation (3.1) will be given. The
convolution of the specular point PDF and the system point PTR can be written as
(Bracewell, 2000)

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 52
B(
t)
= qs(
t)
∗ PPTR(
t)
1 ⎡ 1 ⎛ t ⎞
= exp⎢−
⎜ ⎟
2π
σ ⎢⎣
2 ⎝σ
⎠
2
⎤⎡
1
⎥⎢1
+ λcsH
⎥⎦
⎣ 6
3
⎛ t ⎞⎤
⎜ ⎟⎥
⎝σ
⎠⎦
(3.28)
in which λ cs
and σ are the composite skewness and risetime (i.e., the information
relative to the sea surface roughness through the SWH) given by
2 2 ⎛ 2
= σ p
+ ⎜ s
2
σ ⎞
⎟
⎝ c σ
⎠
(3.29)
3
⎛σ
s ⎞
λcs = λs⎜
⎟
(3.30)
⎝ σ ⎠
Then, the waveform can be obtained as:
P(
t)
= P ( t)
∗ B(
t)
=
s
∫ ∞ −∞
= A
P ( x)
B(
t − x)
dx
A
s
exp
0 ξ
τ
[ − d(
+ d / 2) ](
C1
+ C2
) + PN
(3.31)
where
1 ⎡ ⎛ τ ⎞ ⎤
C
1
= ⎢erf
⎜ ⎟ + 1⎥
(3.32)
2 ⎣ ⎝ 2 ⎠ ⎦
3
2
λ ⎧
⎫ ⎛ ⎞
cs
⎡ ⎛ τ ⎞ ⎤ d 1
2
2 −τ
C2
= ⎨⎢erf
⎜ ⎟+ 1⎥
+ ( 1−
3d
− 3d
⋅τ
−τ
) ⎬exp⎜
⎟
(3.33)
6 ⎩⎣
⎝ 2 ⎠ ⎦ 2 2π
⎭ ⎝ 2 ⎠
with
t − t
τ = σ
0
− d
(3.34)
2
⎛ β ⎞
d = ⎜α
⎟
−
σ
(3.35)
⎝ 4 ⎠
where P
N
is an additive noise term that represents the altimeter’s thermal noise, the
time shift t 0
represents the mean level of the specular points on the ocean surface at

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 53
nadir, and σ represents the standard deviation (in delay time units) of the specular
point about this mean level. Equation (3.31) is an extension of the expression
obtained by Brown (1977) to include the effects of the skewness and Earth curvature.
It is also different from the Hayne (1980) model, which includes another non-linear
kurtosis term to estimate parameters for the non-Gaussian surface.
The effects of Equations (3.32) to (3.35) on the waveform can be discussed
qualitatively. The function C 1
is the predominant expression of the leading edge of
the return waveform, while the function C 2
represents the effects of the skewness
and the off-nadir angle on the leading edge. Because of small magnitude of the
skewness 0.01< λ

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 54
significantly affects the waveform in the leading edge when τ ≤ 2 . It has no effect
on the regions of the thermal noise and trailing edge of the waveform (Figure 3.3).
0.12
0.10
ERS
POSEIDON
TOPEX
Values of d
0.08
0.05
0.03
0.00
0.0 5.0 10.0 15.0 20.0
SWH (metres)
Figure 3.1 Values of d ( ξ = 0 ), showing that d

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 55
0.04
0.02
Values of C2
0.00
−0.02
−0.04
−0.06
ERS
POSEIDON
TOPEX
−0.08
0.0 20.0 40.0 60.0
Bin or Gate
Figure 3.3 Values of C
2
( d = 0 , λ
cs
= -1, and SWH = 4 m) showing the small
magnitude of effects on the leading edge of the waveform, and nearly overlapped
curves for ERS-1/2 and TOPEX.
As can be seen from Figure 3.3, for the times near the tracking gate when τ ≤ 1, the
effect of the skewness is to lower the power, and to raise it when the leading edge
starts to rise or near the plateau of the waveform.
Equation (3.31) can be considered as a fundamental model of the ocean return
waveforms. From the above qualitative and quantitative analysis, it is found that the
effects of non-Gaussian ocean surface through skewness λ cs
on the waveforms are
very small (~0.01-0.12). In this case, when retracking waveform to estimate the
range correction, the skewness term can be ignored without significantly affecting
the results in the complete convolutional form. In Chapter 5, Equation (3.31) will be
used to obtain a linear analytical function for retracking waveforms in coastal regions.
3. 4 Ocean Waveform Retracking
In general, a satellite radar altimeter’s on-board range estimation is limited by the onboard
computational capability and by real-time data processing constraints. As the

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 56
orbit errors are reduced by using geometric, dynamic and combined methods (e.g.,
T/P orbit determination), it becomes increasingly desirable to also reduce the errors
due to the on-board range estimation algorithm. In addition to the on-satellite range
estimates, the satellite also transmits the waveform measured by the radar altimeter
to the ground. This waveform contains information about the scattering surface, but
is retracked later to get improved ranges.
The convolutional representation of the radar return waveform, its shape, and the
precise position of the waveform in the range window allow for the retrieval of more
accurate estimates of the range and the SWH, as well as additional parameters such
as the antenna off-nadir angle and the skewness of the surface elevation distribution
using more sophisticated algorithms. Hence, it has long been recognised that
reprocessing altimeter waveform data will lead to improved geophysical estimates
using waveform retracking over ocean surfaces (Brenner et al., 1993; Barrick and
Lipa, 1985; Rodriguez and Martin, 1994a; Rodriguez, 1988).
Ocean waveform retracking is based upon the convolutional representation of the
waveform, such as the Brown model (Equation 3.12), or the modified model (e.g.,
Equation 3.31) for non-linear ocean surfaces. The retracking algorithms used to
estimate the geophysical parameters over ocean surfaces can mainly be categorised
into two classes of least squares estimation and deconvolution, as follows.
3.4.1 Least Squares Fitting Procedure
As stated, an analytic representation of the waveform can be obtained by formulating
a convolution of three terms. The most commonly used algorithm is to fit a
convolution model of the waveform (e.g., Equation 3.31) to the measured waveforms.
An iterative least squares fitting procedure (Hayne and Hancock III, 1990; Hayne
and Hancock III, 1982) or a maximum likelihood estimation (MLE) algorithm
(Challenor and Srokosz, 1989; Parsons, 1979; Rodriguez and Martin, 1994a;
Tokmakian et al., 1994) is performed to extract the parameter estimates.
Hayne and Hancock III (1990) use an iterative least squares fitting procedure to
retrack GEOSAT waveform data. In their algorithm, Equations (3.16), (3.19) and
(3.27) express three terms in the convolution Equation (3.1). The model of the

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 57
convolutional solution is fitted to the waveforms to produce six independent
estimates. These are:
(1) Waveform amplitude, A ;
(2) The time shift of the range-tracker-positioned waveform tracking gate relative
to the true tracking gate of the waveform, t 0
;
(3) SWH;
(4) Noise baseline, P
n
, (i.e., thermal noise);
(5) Skewness of the radar-observed surface elevation PDF, λ
cs
; and
(6) Off-nadir pointing angle, ξ .
The waveform has noise. It can be seen in the trailing edge, but is more difficult to
identify in the leading edge. The parameters affected by the noise in the tailing edge
will degrade the geodetic parameter estimates in the leading edge if the least squares
fitting procedure is applied to an individual waveform. The fitting method in this
case may appear to give statistically small error but does not provide a consistent
accuracy to the individual parameters. Therefore, to reduce the noise on the return
waveforms, effective fitting is usually carried out using waveform averages in
several seconds rather than individual waveforms (e.g., Challenor and Srokosz,
1989). In the procedure of Hayne and Hancock III (1990), ten-second average
waveforms are used. Since the closed form solutions of the theoretical waveform
cannot be deduced from realistic forms of the radar PTR, the fitting algorithm suffers
from the drawback that the fitting function is analytic only if the radar system PTR is
Gaussian (Brown, 1977; Challenor and Strkosz, 1989), near-Gaussian (Hayne, 1980),
or approximated by a set of Gaussians (Rodriguez and Martin, 1994a).
A typical example of using the MLE algorithm is from the UK Earth Observation
Data Centre (EODC). EODC retracks ERS-1 data to re-estimate all parameters
calculated by the on-board algorithm on the ground using the MLE algorithm (e.g.,
Tokmakian, 1994). The convolution function includes unknown parameters A , t 0
,
σ , ξ , and non-linear wave parameters λ cs
and δ . The MLE algorithm has been
tested by EODC with three forms of the convolution functions as follows:
(1) Non-linear model containing both non-linear wave parameters λ
cs
and δ .

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 58
(2) Non-linear model containing only one of non-linear wave parameters λ
cs
or δ .
(3) Linear model without λ
cs
and δ .
Their results show that the MLE algorithm is successful with the linear model for the
estimation of the geophysical parameters, but cannot accurately estimate all
parameters if the non-linear model is used. From EODC’s tests, it has been shown
that the 20 Hz individual waveform data does not produce as accurate results as
when averaging the waveforms to 1 Hz (ibid.). This means an averaging procedure is
essential as well.
Theoretically, the MLE algorithm is more accurate than least squares (Chelton et al.,
2001) and gives asymptotically minimum variance unbiased estimates of the
parameters (Challenor and Srokosz, 1989). However, in practice, because of
unreliability of the statistical weights due to unmodelled errors in the waveforms
(Chelton, 2001) and further complications introduced by numerics of the model
(Tokmakian et al., 1994), MLE estimates are not significantly better than those
estimated by least squares. Therefore, nonlinear least-squares fitting procedure
appears to still be an adequate method of waveform retracking.
3.4.2 Deconvolution Method
The second method estimates parameters from the specular point PDF obtained by
deconvolution, which was introduced by Priester and Miller (1979), Lipa and Barrick
(1981), and refined by Rodriguea and Chapman (1989). The deconvolution method
uses a fact that the average altimeter return waveform is the convolution of the ocean
surface specular point PDF, which is an unknown function, and two functions of the
radar PTR and the smooth surface response, which are known and depend only on
the radar design parameters. By taking the fast Fourier transform (FFT) of the slope
function, the specular point PDF can be obtained from dividing the Fourier transform
of the data by the Fourier transform of the convolution of the smooth surface
response and the radar PTR. The specular PDF will then allow estimation of three
independent sea-surface-state parameters (e.g., Lipa and Barrick, 1981; Barrick and
Lipa, 1985; Rodriguez, 1988; Rodriguez and Chapman, 1989). These are:
(1) the ocean surface standard deviation;

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 59
(2) ocean surface skewness; and
(3) shift of the altimeter tracking gate with respect to the mean electromagnetic
surface measured by an altimeter.
For the convenience of representation, altimeter return power Equation (3.1) can be
thought of as a matrix equation
y = Mx
(3.36)
where y is the vector of return power and M is the convolution given by
M = P ( t)
∗ P ( t)
(3.37)
fs
PTR
and x is the specular point PDF vector
x = q(t)
(3.38)
Thus, the specular point PDF can be obtained by inverting Equation (3.36).
In its most direct implementation, Lipa and Barrick (1981) suggest that
deconvolution can be performed on the slope of the leading edge of the waveform,
rather than the full waveform. This suggestion means that the radar antenna pattern
and the off-nadir angle can be ignored, and the slope of the mean waveform is given
by the convolution of the specular point PDF and the radar PTR. Thus, data can be
neglected in the trailing edge of the waveform, where the noise level is higher
(Section 3.3.5), and the returns depend weakly on the non-Gaussian ocean surface.
This algorithm has the advantages that it is computationally simple and fast, and it is
not necessary to assume any functional form for the radar specular point PDF.
However, it is impossible in general to ignore the radar antenna pattern and the offnadir
angle for altimeters, such as GEOSAT and TOPEX, because it introduces
errors which are of the same order as the quantities to be estimated (cf. Rodriguez,
1988). Another disadvantage of the slope deconvolution is that the algorithm is
highly sensitive to high-frequency noise in the waveform. This leads the
deconvolution of Lipa and Barrick (1981) being carried out only for waveforms that

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 60
had been averaged over a long time period (e.g., 10 seconds) for which an acceptably
low noise level was obtained.
In order to overcome these problems, Rodriguez (1988) and Rodriguez and Chapman
(1989) introduce a "corrected slope convolution model" as
2
1 ⎡ ⎛ β ⎞ ⎤
M = ⎢P′
( t)
+
⎜α
−
⎟P(
t)
⎥
(3.39)
A ⎣ ⎝ 4 ⎠ ⎦
where P′ (t)
is the derivative of the return power with respect to the return time. This
model differs from the expression derived by Lipa and Barrick (1981, Equation 3),
because the second term on the right-hand side represents the radar antenna pattern
and off-nadir angles. Since the return waveform is also a function of the thermal
noise and the off-nadir pointing angle, they must be estimated separately when using
Equation (3.39).
Both deconvolution algorithms presented by Lipa and Barrick (1981) and Rodriguez
and Chapman (1989) have advantages and disadvantages. Rodriguez and Chapman
(ibid.) investigate the use of both methods. They recommend that a spectral filtering
procedure be used to reduce the high-frequency noise in the estimated parameters.
3. 5 Ice-Sheet Waveform Retracking
Ice-sheet surfaces are characterised by the existence of topography, often with a
significant surface slope with respect to the ocean. Over such topographic surfaces,
the altimeter is generally unable to follow abrupt altitude changes and small-scale
slopes, thus losing signal lock. To maintain the satellite track over large surface
slopes, ERS-1/2 altimeters operate in ice mode over ice sheets at a reduced range
resolution or increased sampling time (12.12 ns) compared with the ocean mode
(3.03 ns). The form of the return is determined by the range to the different reflectors,
their scattering properties, and their position within the antenna beamwidth. This
results in a return waveform of generally unpredictable shape, which changes rapidly
and is offset from the tracking gate designed for the altimeter over oceans. The
present theory of altimetry, developed to describe scattering from the ocean surface
such as the Brown (1977) model, does not deal properly with the geometry of ice-

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 61
sheet surfaces. Therefore, onboard range measurements in altimeter records do not
always correspond to the reflecting surface at nadir.
retracking gate
tracking gate
Returned Power (counts)
offset
32.5
Time or Bins
Figure 3.4 The concept of the observed waveform retracking for ERS-1/2, in which
( retracking gate − tracking ) × 0. 45
offset = gate (m).
Estimation of the true ranges to the reflecting surfaces can be obtained using
waveform retracking techniques. Waveform retracking determines the offset
(Figure 3.4) of the actual tracking gate, which is related to the half-power point on
the leading edge of the waveform, from the pre-designed tracking gate, and corrects
the range calculated by the on-board algorithm accordingly. Retracking algorithms
developed over ice sheets can be categorised into four classes:
(1) Fitting algorithm: β -parameter retracking (Martin et al., 1983);
(2) The off centre of gravity (OCOG) technique (Wingham et al., 1986);
(3) Threshold retracking (e.g., Bamber, 1994; Davis, 1997; Ridley and Partington,
1988; Partington et al., 1989; Partington et al., 1991; Femenias et al., 1993;
Zwally, 1996; Remy and Minster, 1993); and
(4) A surface/volume scattering retracking (Davis, 1993a).

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 62
In addition, Brooks et al. (1978) assume that the actual tracking point on the leading
edge of the waveform is related to the half value of the maximum return power. This
retracking method is, in fact, the same as the 50% threshold retracking. Wingham
(1995b) develops a method for determining the average height of a large topographic
ice sheet from altimeter observations. This method is extended to cover the ice-sheet
geometry, as well as including the effects of the penetration of the surface by the
radar waves, and thus differs from all above methods. A Volterra type of integral
equation is derived (Wingham, 1995a), which allows the uniqueness of the solution
for the average ice-sheet height to be estimated simply. However, a discrete
representation is not given, so that practical implementation is not yet available.
3.5.1 Fitting Algorithm - β-Parameter Retracking
Martin et al. (1983) developed the first retracking algorithm for processing altimeter
return waveforms over continental ice sheets. The algorithm was used to retrack all
SEASAT radar altimeter waveforms to obtain corrected surface elevation estimates.
This algorithm fits a 5- or 9-parameter function to the altimeter waveform, which is
based partly on Brown’s surface scattering model (discussed in Section 5.4.3). The 5-
parameter function is used to fit single-ramp returns (Figure 3.5), while the 9-
parameter function is used to fit double-ramp returns (Figure 3.6) (Martin et al.,
1983). This retracking algorithm is also known as β-parameter retracking or the
NASA algorithm (e.g., Davis, 1995). It is a surface-scattering model that can deal
with complex waveforms reflected from one or two scattering surfaces over ice
sheets. The Ice Altimetry Group at NASA’s Goddard Space Flight Centre (GSFC)
has developed a series of retracking algorithms for ice-sheet waveforms based upon
Martin’s functions (Zwally, 1996). The general parameter function fitting the radar
returns is given as (Martin et al., 1983; Zwally, 1996):
n
⎛ t − β
3i
⎞
y(
t)
= β + ∑ +
⎜
⎟
1
β
2i
(1 β
5iQi
) P
(3.40)
i=
1 ⎝ β
4i
⎠
where

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 63
Q
i
⎧ 0
= ⎨
⎩t
− ( β
3i
+ 0.5β
)
4i
for t < β + 0.5β
3i
3i
4i
for t ≥ β + 0.5β
4i
(3.41)
⎛ −
∫ −∞
⎟ ⎞
= x 2
1 q
P(
x)
exp
⎜ dq
2π ⎝ 2 ⎠
(3.42)
where n = 1 or 2 is the number of the ramp in the waveform range window which
corresponds to single or double reflecting surfaces, respectively (Figures 3.5 and 3.6).
Double ramps indicate that two distinct, nearly equidistant surfaces are observed by
the altimeter. The unknown parameters are as follows.
(1) β
1: the thermal noise level of the return waveform.
(2) β
2i
: return signal amplitude.
(3) β
3i
: the mid-point on the leading edge of the waveform, which is chosen as the
correct range point in the retracking procedure.
(4) β
4i
: the return waveform risetime.
(5) β
5i
: the slope of the trailing edge.
When the linear trailing edge is replaced by an exponential decay term (cf., Zwally,
1996), Equation (3.40) is adapted to give
n
⎛ t − β
3i
⎞
y(
t)
= β + ∑ −
⎜
⎟
1
β
2i
exp( β
5iQi
) P
(3.43)
i=
1 ⎝ β
4i
⎠
where
Q
i
⎧0
= ⎨
⎩t
− ( β
3i
+ 0.5β
)
4i
for
for
t < β
3i
t ≥ β
3i
− 2β
− 2β
4i
4i
(3.44)
This function is used to fit the waveform with a fast-decaying trailing edge over seaice
or ice sheets, which are caused by the beam attenuation (cf. Zwally, 1996). It can
simulate the antennae attenuation as the pulse expands on the surface beyond the
pulse-limited footprint. This function is designed to fit the fast-decaying ice mode
return waveforms (ibid.). Another fitting function developed at GSFC is to fit a
simple linear trailing edge to the first ramp and the exponential decay-trailing edge to

Returned Power (counts)
Returned Power (counts)
Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 64
retracking gate
tracking gate
β
51
(trailing edge slope)
β
11
β
21
β
41
(risetime)
β
31
32.5
Time or Bins
Figure 3.5 Five- β -parameter model, fitting a single ramp function to a signal
leading edge of the waveform samples.
tracking
gate
1st retracking
gate
2nd retracking
gate
β
51
(1st trailing
edge slope)
β42
(2nd risetime)
β52
(2nd trailing
edge slope)
β
1
β
41
(1st risetime)
β
21
β
22
β
31
β
32
32.5
Time or Bins
Figure 3.6 Nine- β -parameter model, fitting a double-ramp function to two leading
edges of the waveform samples.

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 65
the second ramp, for which detailed information can be also found in Zwally (1996).
The β parameter retrackers will be employed in the coastal zone for this project
(Chapter 5).
3.5.2 Off-Centre of Gravity (OCOG) Retracking Algorithm
The OCOG retracking method was developed by Wingham et al. (1986), which uses
full waveform samples (excluding the samples in the bins at the beginning and end
due to the wrap-around error that occurs within the digital single processing). It is
based on the definition of a rectangle about the effective centre of gravity (COG) of
the waveform and the amplitude (A) and width (W); see Figure 3.7. The area of the
rectangle equals that of the waveform. The height of the COG is defined as half the
amplitude of the centre of gravity of the waveform. To reduce the effect of lowamplitude
samples in front of the leading edge, the squares of the sample values are
used in the computation (Wingham et al., 1986). The equation used to compute A is
given by
A =
64−n
a
∑
i=
1+
n
a
4
i
64−n
a
∑
P ( t)
P ( t)
(3.45)
i=
1+
n
a
2
i
where P i
(t)
is the value of waveform sample at the i th bin, n a
is the number of
aliased bins at the beginning and end of the waveform (Figure 3.7) depending on the
altimeter (typically n
a
= 4 for ERS-2 and n
a
= 0 for TOPEX/POSEIDON). The range
window is i = 1, 2, ···, 64 sample bins. The waveform width, W, is then obtained by
the relation
∑
AW = P
i
(t)
with the result
W
⎛
= ⎜
⎝
64−na
2
∑ Pi
i=
1+
na
⎞
( t)
⎟
⎠
2
64−na
4
∑ Pi
i=
1+
na
( t)
(3.46)
the location of the COG is formed by
COG =
64−n
a
∑
i=
1+
n
iP ( t)
a
2
i
64−n
a
∑
i=
1+
n
P ( t)
a
2
i
(3.47)

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 66
Finally, the location of the half-power point or mid-point on the leading edge of the
waveform, LEG , is given by
W
LEG = COG −
2
(3.48)
W
Returned Power
50% retracking point estimate
COG
A
Time or Bins
Figure 3.7 Schematic description of the OCOG method applied to an observed
waveform. Using the OCOG method to calculate the amplitude (A) and width (W) of
the rectangle, and the position of the 50% of amplitude interpolated at the leading
edge as the estimate of expected tracking gate.
3.5.3 Threshold Retracking Algorithm
The OCOG algorithm is simple to implement, though it is purely statistical and is not
based upon any physical model of the reflecting surfaces. Moreover, it is sensitive to
the waveform shape affected by surface undulations and off-nadir pointing, because
it uses the full samples in waveform bins. However, when dealing with the returns at
which the slope of the leading edge is smaller (i.e., longer pulse rise times), an
erroneous estimate is produced because the LEG cannot be precisely positioned at
the mid-point of the leading edge of the waveform (e.g., Partington et al., 1989).
To improve the estimation, an empirical method of threshold retracking was
developed and is used by ESA to process altimeter data from the ERS satellite
missions (Davis, 1995). It has also been adopted by NASA/GSFC as an alternative
retracking algorithm for the production of ice-sheet altimeter data sets (Davis, 1997).

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 67
This algorithm is based upon the dimensions of the rectangle (i.e., amplitude, width,
and hence the ‘centre of gravity’) computed using the OCOG method. The threshold
value is then referenced to the amplitude (Bamber, 1994) or the maximum waveform
sample (e.g., Davis, 1997; Zwally, 1996) estimate of the rectangle at 25%, 50%, and
75% of the waveform amplitude. The retracking gate estimate is determined by
linearly interpolating between adjacent samples of a threshold crossing at a steep part
of the leading edge slope of the waveform. This algorithm keeps the same
advantages as the OCOG, but it can determine a more accurate tracking gate position
than the OCOG does (Partington et al., 1989). A disadvantage is that, like the OCOG
method, it is not based on a physical model.
The selection of an optimum threshold level is important when applying this method
to waveforms, because the range to the surface is determined from it. Davis (1995;
1997) tests different threshold levels (10%, 20%, and 50% of the waveform
amplitude) for the purposes of measuring ice-sheet elevation change. The
comparison is performed on SEASAT, GEOSAT-GM, and GEOSAT-ERM
crossover data. It was found that a 10% threshold level can produce more repeatable
elevation estimates, a 20% threshold level is suitable for obtaining the true ice-sheet
elevation in only an average sense, and a 50% threshold level is appropriate only
when surface scattering dominates the return waveform shape.
Thus far, no account has been taken of the thermal noise effect (described in Section
3.4.1) in Equations (3.45), (3.46) and (3.47). However, since the magnitude of the
thermal noise varies with location and time in a given waveform dataset, these
variations cause the leading-edge retracking point to vary if the threshold level is
referenced only to the amplitude in which the thermal noise is included. To
overcome this problem, the mean thermal noise estimated by averaging over earlier
gates 6-10 should first be subtracted from the amplitude (Zwally, 1996).
3.5.4 Surface/Volume Scattering Retracking Algorithm
In general, the leading edge of altimeter waveforms is characterised by a rapid rise in
return power and continuing to increase to the plateau (Figures 2.3 or 3.7). Most
leading edges over ice sheets, however, show first a rapid rise in the return power.
Then, they continue to rise but at a reduced rate, indicating the influence from

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 68
subsurface volume scattering (e.g., Ridley and Partington, 1988; Davis, 1993a). The
reason is that the radar pulse can significantly penetrate the snow over most of the ice
sheet, resulting in elevation estimation errors of up to a few metres (e.g., Partington
et al., 1991) if seeking to determine the height of these surfaces. Therefore, when
retracking is carried out using the β-parameter retracking algorithm, there is an
implicit assumption that the ice sheet surface scatters the radiation in a similar
manner to the ocean surface.
To take into account radar penetration over ice sheets, Partington et al. (1991) use a
threshold retracking method but the position of the retracking gate is selected from
an average waveform derived from all SEASAT waveforms from the region of
interest. To a first order, this method overcomes the penetration problem (ibid.).
Using Gaussian approximations for the altimeter’s antenna pattern and transmitted
pulse shape (i.e., the radar system PTR), Davis and Moore (1993b) derive a closedform
analytical solution for the received power due to volume scattering. Because of
this closed-form, they combine the volume-scattering model with the Brown surfacescattering
model (Equation 3.12) to develop an algorithm capable of retracking
altimeter waveforms over ice sheets. The combined surface and volume-scattering
model is (Davis and Moore, 1993b)
Am
⎡ K ⎤
SV ( n)
= DC + ⎢S(
n)
+ V ( n)
⎥
S
2 ⎣ S1
⎦
(3.49)
in which DC is the thermal noise baseline, A m
is the maximum amplitude of the
model waveform, and S (n)
is an adjusted form of the Brown model (cf. Equation
3.12) given by
1 ⎡ ⎛ t − t ⎡−
⎤
0 ⎞⎤
4c
S( n)
= ⎢1
+ erf ⎜ ⎟⎥exp⎢
( t − t0)
2
⎥
⎣ ⎝ 2σ ⎠⎦
⎣ γ h ⎦
(3.50)
The derivation of Equation (3.50) from Equation (3.12) assumes that the off-nadir
angle ξ = 0 , and the modified Bessel function is expanded to order zero ( n = 0 in
Equation 3.17). The volume-scattering model, V (n)
, is given by

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 69
2 2 2
[ β k c − 2k
c ( t − t )]
2 2
⎛ c β c(
t t0
) ⎞
τ
−
V ( n)
= exp⎜
⎟
− − exp
2 4
2
e s e s 0
4h
c
h
τ
⎝ β β
c ⎠
(3.51)
where constants S 1
and S
2
are normalising factors that ensure the quantities V (n)
and S ( n)
+ V ( n)
range from zero to one, K represents the correct proportion of
volume scattering, c s
is the speed of light in snow, β c
is a constant related to the
antenna beamwidth, β τ
is a constant that determines the 3 dB width of the
transmitted pulse, and k e
is the extinction coefficient of snow. This is a fitting
function, and there are six unknown parameters in the model ( DC , σ , t 0
,
A
m
, K
and k
e
). This model will not be used for the coastal zone retracking in this study,
because there is no known penetration problem present at the coast.
3. 6 Land Waveform Retracking
In spite of the fact that the satellite radar altimeter was developed to measure the
ocean surface, and the on-board algorithms used to control the receiver gain and
height tracker were not designed to cope with terrain signals, substantial quantities of
seemingly useless data were obtained over land. However, SEASAT was the first
altimeter to show its mapping capability over land (and ice-sheet) surfaces. Over
about three months of its lifetime, it collected data over 34% of the Earth’s land
surface (Tapley et al., 1987, cited in Guzkowska et al., 1988). GEOSAT data were
also collected over land both in the 17-day ERM orbit and during period of the GM
(Brenner et al., 1990). Moreover, with the advent of ERS-1/2, the ice mode with a
longer sampling time on the altimeter and different orbit repeat phases provided far
better spatial coverage of data (~80% of the Earth’s surface) for topographic
purposes than previous satellite missions (Berry, 2000).
Waveform data from different satellite altimeters collected over land surfaces have
been analysed by Guzkowska et al. (1988), Brenner et al. (1990) and Berry (2000a).
Guzkowska et al. (1988) categorise the SEASAT waveform data into seven classes,
in which each type of data shows unique characteristics corresponding to the
reflecting surface. Berry (2000) also presents a selection of 10 classes of waveforms
with different characteristics for ERS-1 over land. These results indicate that specific

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 70
problems have been associated with the attempt to retrieve an accurate range to
surface over land. One of them, which is the same as that encountered over ice-sheet
surfaces, is that the on-board satellite processor cannot calculate an accurate range to
surface, because the altimeter is in general unable to centre the return reflected from
the rapidly varying topographic surface in the range window. Thus, in order to work
with these data, waveform retracking must be employed.
Land waveform retracking is mainly conducted via two approaches. The first is to
retrack waveforms using an expert system depending on the specific waveform
categories, which was developed by Berry et al. (1998), Berry (2000a) and Berry et
al. (2000b). ERS-1/2 geodetic missions, plus their operation in the ice mode, resulted
in the acquisition of large and effective volumes of radar returns over land surfaces.
Berry et al. (1998) generate orthometric heights from ERS-1 geodetic mission dataset
using retracked waveform data from an expert systems approach. Data from the
entire ERS-1 geodetic mission has been reprocessed to generate a Regional Altimeter
Result database, and the data are then used to generate orthometric heights.
Land topography, in particular topography with a non-uniform slope, and the nonuniform
reflectivity of the land are main courses that lead to the complicated return
waveform shapes over land. The expert system, on the other hand, contains several
retrackers (Berry, 2000a, 2001). Each retracker is specifically designed and
developed for a particular type of return waveform, for which ten retrackers have
been created for a land waveform-retracking system (Berry, 2001). Some of them are
(Berry et al., 1998):
(1) Spline retracker, which is more effective with pre-peaked waveforms by fitting
bicubic spline functions to the leading edge of the waveform;
(2) Narrow waveforms, for high power specular returns from inland;
(3) Bor retracker, for waveforms showing slope information;
(4) Bog retracker, for Bor-type waveforms; and
(5) Default retracker, i.e., threshold retracker, for waveforms that cannot be
retracked by other retrackers.
Two steps have been taken when reprocessing waveform data from land. The first
step is to sort the waveforms based on their shape and retain all waveforms which

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 71
show a leading edge regardless of complexity of the whole waveform shape. Next,
choosing the optimised retracker to estimate the range to the land surface at the nadir
based upon the individual waveform shapes. More detailed information about the
expert system of land waveform retracking can be found in Berry et al. (1998) and
Berry (2000).
The second approach to land waveform retracking is achieved by retracking all
waveforms using a single retracker, typically the OCOG technique (e.g., Bamber et
al, 1997, cited in Berry, 2000) or one optimised for double-peaked waveforms (eg.,
Brenner et al., 1997, cited in Berry, 2000). Before the retracker is applied to
waveform data, they are edited and filtered so that the single retracker can succeed
for a larger percentage of data. Shape corrections are then applied to individual
retracked ranges to surface and combined to derive mean or median height values.
Results from this approach represent the land topography with a spatial resolution of
15' or 5' (Berry, 2000).
The mathematical methods for these have not been presented because 1) they will not
be used in this dissertation, and 2) moreover, they are not presented by authors.
3. 7 Brief Introduction to Coastal Waveform Retracking
The potential of the altimeter waveform data to be used in coastal regions has been
the subject of comparatively little study (Anzenhofer et al., 2000; Brooks and
Lockwood, 1990; Brooks et al., 1997). So far, considering the existing retracking
algorithms, two approaches have been applied to retracking coastal waveforms,
which are the fitting procedure and threshold technique. The first is to fit an
analytical waveform model to the observed waveform data for determining the more
accurate range estimate. The fitting function used from open literature
(e.g., Anzenhofer et al., 2000) is the five-β-parameter function Equations (3.40) and
(3.43), choosing either the linear trailing edge or the exponential trailing edge. When
using the function fitting algorithm, usually n = 1 in Equations (3.40) and (3.43), an
iterative least squares procedure is used.
An alternative method used by Brooks et al. (1997) in coastal regions is a slightly
modified threshold retracking technique, which tracks the ocean return from the

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 72
hybrid waveforms of water and land close to the coastline. This retracking involves a
three-stage process:
(1) Returns are observed using a visual examination method to identify the ocean
and land returns.
(2) A threshold level is selected for each individual pass such that the threshold is
higher than both the thermal noise level and the minor waveform leakage in the
early gates, and is sufficiently low to intercept the ocean return part in the
waveform. Then, the actual tracking gate position of the ocean return is
obtained by converting the intercepted amplitude on the leading edge of the
waveform to the gate number.
(3) The position of the expected tracking gate is ascertained by averaging midpoint
positions on the leading edge of the waveforms from several seconds of openocean
waveforms in the same area, pass and date. The difference between this
expected tracking gate and the actual tracking gate is computed and then
converted to the range correction in metres. This range correction is applied to
the range measurement from the on-board satellite-tracking algorithm.
This method is appropriate for retracking the mixed waveforms in coastal regions.
However, it cannot be conducted automatically (Brooks, et al., 1997). The method
developed to select the threshold level by the author in Chapter 5 will solve this
problem.
3. 8 Summary
An outline of waveform retracking algorithms, both for ocean and non-ocean
surfaces, has been presented in this Chapter. Starting from the basic convolutional
representation of the waveform, the Brown model is introduced. Then, it is extended
to a form that includes additional corrections (i.e., skewness and Earth’s curvature)
so that it can be used to estimate more parameters over non-linear ocean surfaces.
Finally, a model which will be used in this study is deduced. A detailed qualitative,
and some quantitative, analysis is carried out on this model.
Over non-ocean surfaces, waveform-retracking algorithms have been developed for
obtaining reasonably accurate elevation estimates from altimeter data over ice-sheet

Chapter 3. RADAR ALTIMETER WAVEFORM RETRACKING METHODS 73
and land surfaces. Over ice-sheets, retracking algorithms are mainly based upon
either a linear surface scattering model (i.e., the Brown model) or the threshold
method. In addition, a volume and surface scattering model extends the Brown
model to deal with the returns from snow-covered ice sheets. Over land surfaces,
existing retracking algorithms are either those used over ice sheets or a retracking
system. Because a physically-based model is impossible over land due to the more
rugged and rapidly varying surface, an expert system that depends on the waveform
shapes is currently the only apparently effective retracking method.
It is important to note that improvement of the altimeter data in coastal regions has
been the subject of comparatively little study, although many previous studies have
reported the data are in error in these regions (e.g., Mantripp, 1996; Brooks and
Lockwood, 1990; Hwang, 1998; Featherstone, 2003). The existing retracking
algorithms are also either the β-parameter fitting procedure or the threshold
algorithm for ice-sheet surfaces. This suggests that it is necessary to investigate
further the coastal retracking algorithms, which is the objective of this dissertation.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 74
4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS
4. 1 Introduction
In order to avoid contaminated data in coastal regions, Nerem (1995) and Shum et al.
(1997) in their studies of analysis of global mean sea level and ocean tidal models
over open oceans, simply eliminate altimeter SSH data in areas with water depths
less than 200 m. Shum et al. (1998) show that it is necessary to improve correction
algorithms for data near coastal regions and semi-enclosed seas. Brooks et al. (1990;
1997) investigate land effects on GEOSAT and TOPEX radar altimeter
measurements in the Pacific Rim coastal zone, and provide a slightly modified
threshold retracking algorithm for the data user to rectify and recover the sea surface
topography in close proximity to the land. Although some of these efforts have been
reported previously in the open literature, empirical evidence is not yet published to
show how far away from the shoreline altimeter measurements are poor.
This Chapter uses two approaches of the threshold retracking (Section 3.5.3) and
mean waveform analysis of one cycle of ERS-2 and five cycles of POSEIDON 20
Hz waveform data to quantify the distance from the Australian shoreline that
altimeter measurements are contaminated. This distance is obtained by using the 50
percent threshold retracking point estimates and analysing mean waveform and its
standard deviation across a zone extending out to 350 km offshore from Australia. In
this Chapter, the modelling (and retracking) of the altimeter waveform will not be
addressed. Instead, only the distance from the coastline in which sea surface heights
are poorly estimated by the on-satellite tracking algorithm will be examined.
Quantified distance results will be presented in Section 4.5. In addition, several
examples from waveforms along satellite ground tracks near the coastline are taken
to analyse in detail how waveforms are contaminated and what the characteristics of
contaminated waveforms are in coastal regions (Section 4.6). A preliminary test that
estimates the land effects on waveforms in the vicinity of land will be given in
Section 4.7.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 75
4. 2 Background
When the altimeter pulse interacts with the Earth’s surface, the shape of a returned
waveform depends on the topography of the reflecting surface. Over open oceans,
the altimeter tracker operates as designed and thus uses the Brown (1977) ocean
statistics model (Section 3.2). The on-satellite tracking algorithm can well centre the
ocean return leading edge at a ‘tracking gate’, which is a pre-designed position at the
range window corresponding to the range to the nadir mean sea surface. Over land
and ice sheets, however, the reflection of a satellite radar altimeter signal differs from
the Brown model owing to the non-Gaussian-reflecting surface, where waveforms
are different to the expected/ideal waveform shape over oceans are generally
received (Sections 3.5 and 3.6). This leads to an error in the on-satellite range
estimates that must be corrected through retracking techniques. As stated, retracking
altimeter data aims to estimate the departure of the midpoint on the waveform’s
leading edge from the pre-designed tracking gate, and thus correct the on-satellite
range measurements accordingly (Martin et al., 1983; Bamber, 1994; Berry et al.,
1998).
In coastal regions, the situation differs from open oceans mainly because of the close
proximity to land and the complex spatial-temporal variation of coastal water
surfaces. As a satellite ground track approaches, recedes or runs parallel to the
coastline, even though the altimeter’s nadir point is over the sea, the altimeter may
also track the off-nadir returns reflected from the higher land and any brighter still
inland water. This causes a problem in that the altimeter cannot follow the apparent
accelerations in the surface shape. As a result, the waveforms recorded on board the
satellite will become misaligned with respect to the expected tracking gate (i.e.,
contaminated). Therefore, a postprocessing or retracking is necessary as well on the
on-satellite range calculation (Brooks and Lockwood, 1990; Mantripp, 1996; Quartly
et al., 2001), which will be covered in Chapters 5, 6 and 7. In this Chapter, the spatial
extent of this contamination is quantified so that efficiency can be achieved by
learning over what range to retrack coastal data.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 76
4.2.1 ERS-2 and POSEIDON Altimeter Footprints
The altimeter-illuminated ground region (i.e., along-track footprint) from which most
power returns for each 20 Hz measurement is typically oval in shape and its size
depends on the satellite altitude, surface slope and roughness, and the width of
waveform gates used to compute its diameter (e.g., Chelton et al., 2001, p.36). Two
types of altimeter along-track footprint are important, which are the middle gate and
the automatic gain control (AGC) along-track footprints. The middle gate is the
average of 2, 4, 8 or 16 waveform samples centred on the leading edge of the ocean
return, and the middle-gate along-track footprint is defined by the half-width of the
middle-gate after the tracking gate. For the sea surface with SWH=0 m (i.e., a flat sea
surface) and SWH=15 m, the size of the middle-gate along-track footprint varies
from 8.5 km to 15 km for ERS-2 and 8.5 km to 16 km for POSEIDON. The AGC
gate is the average of waveform samples 17 through 48 centred on the leading edge
of the ocean return. Similarly, the AGC along-track footprint is computed from 15.5
AGC gates after the tracking gate. Its diameter for POSEIDON is about 13.9 km for
a flat sea surface, increasing to about 16.9 km for SWH=15 m (Chelton et al., 2001,
p. 36 and Figure 22).
The on-satellite tracking algorithm compares the returned signal from the middle
gate to that from the AGC gate to estimate the range to nadir sea surface. Since the
AGC footprint is larger than the middle gate footprint, a sharp change in scattering
surface will be first sensed by the AGC gate before the middle gate is affected. In
addition, the footprint diameter estimated by all waveform gates after the tracking
gate is much larger than the above-mentioned sizes, such as 22 km for TOPEX
mission (Brooks et al., 1997). Therefore, it is necessary to consider a larger footprint
when investigating waveform contamination in coasts. However, the exact distance
from the coast at which this occurs is unknown. Hence, this is a secondary
justification for this Chapter.
4.2.2 ERS-2 and POSEIDON Altimeter Waveforms
Figures 1.2 and 4.2 show typical examples of observed ERS-2 and POSEIDON
waveforms over oceans, respectively. As can be seen, they are somewhat different
from the ideal waveform (Figure 1.1), due mainly to reasons of the ‘wrap-around’

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 77
200
150
Power (counts)
100
50
0
0 8 16 24 32 40 48 56 64
Bins or Gates
Figure 4.1 An example of the observed POSEIDON waveform over deep oceans
(tracking gate = 31.5).
error and ‘fading’ (Rayleigh) noise (cf. Partington et al., 1991; Quartly et al., 2001).
The former causes a reduction in power in the last few bins and a commensurate rise
at the beginning of the waveform, and the later causes an undulating trailing edge in
the waveform. The noise affects the leading edge as well, but it is more difficult to
identify in the leading edge and is clearly seen in the trailing edge. In addition,
Figure 1.2 does not show the thermal noise because the ERS-2 altimeter artificially
suppresses this noise (cf. Quartly et al., 2001).
Since altimeter-observed waveforms are different from the Brown model (Figure 1.1),
it is hard to correctly define the tracker point to be tracked by altimeter. Therefore, it
is displaced in the observed waveform by the tracking gate. After subtracting the
noise baseline (i.e., the thermal noise level) from each waveform gate, the tracking
gate is defined by a gate (or bin) number that corresponds to the midpoint on the
leading edge of the returned waveform. On-satellite algorithms try to centre the ramp
of the waveform at the tracking gate, but this generally succeeds only over open
oceans. The tracking gate is specified prior to launch as part of the overall design of
the radar system and corresponds to the mean surface range over the ocean (Brown,

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 78
1977). The tracking gate is chosen to align between waveform sample bins 31-32 for
POSEIDON and bins 32-33 for ERS-2 (e.g., Quartly et al., 2001). The parameters
linked to the reflecting surface can be estimated from the detailed shape of the
leading edge, trailing edge and precise position of the waveform in the sampled
window relative to the altimeter tracking gate location.
4. 3 Processing Methodology for Determination of Contamination Area
Whenever land is within the altimeter footprint, there is a likelihood that the land
return contaminates the waveforms, depending on the height and type of terrain
along the shoreline. In order to quantify the contaminated distance from the shoreline,
it is necessary to calculate waveform anomalies with respect to the ideal waveform
shape. Fitting an ocean-like function to each individual waveform then calculating
the standard deviation of the differences is a reasonable measure of determining if
the waveform contains any departure from the ocean-type response. However, it is
not a good way of calculating a better sea surface height in coastal regions because
the model does not fit the actual return. Therefore, two processing techniques are
used in this Chapter: threshold retracking and arithmetic averaging. The
contaminated area is determined by computing the tracking gate estimates and
analysing the shape of the mean waveform, instead of only focussing upon the
individual waveform shapes. Waveform retracking at the Australian coast will be
covered in Chapter 6.
4.3.1 Threshold Retracking
The threshold retracking technique (see Section 3.5.3) is employed in this Chapter
due to its perceived advantage of sensitivity to the surface topography (Partington et
al., 1989). To estimate a contaminated boundary for coastal waveforms, it is essential
to distinguish whether the returned waveforms are affected by coastal topography
(land and sea-states). Unlike the returned waveform over land or ice, where the
waveform shape is significantly different from that over the ocean, most waveforms
affected by land in coastal regions usually show a combination of water and land
(Brooks et al., 1997). Therefore, if the actual position of the mid-point on the leading

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 79
edge of the ocean waveform can be determined accurately, the purpose of estimating
a contaminated boundary for coastal waveforms will be achieved.
As stated, assuming that the reflecting surface approximates a uniform, spherically
smooth, and diffuse surface with a symmetrical distribution of small scale slopes, the
range from the satellite to the mean surface at nadir corresponds to the location of the
half-power point (i.e., the midpoint) on the leading edge of the waveform (Brown,
1977). These assumptions of statistical homogeneity are usually satisfied over open
ocean surfaces. Thus, the 50% threshold value can be used as the accurate estimate
of the mid-point on the leading edge of the ocean waveform. Since the 50% retracked
position should agree with the tracking gate for ocean waveforms, it may be different
from the tracking gate for the waveform affected by non-ocean surfaces. This can, in
turn, be used to assess the contamination of the waveform in coastal regions. This is
the principle on which the identification of contaminated altimeter data is based in
this study.
When applying the threshold-retracking algorithm, the formulae used to compute A,
W and COG are given by Equations (3.45), (3.46) and (3.47). The 50% threshold
value of the amplitude of the rectangle is used as the estimate of the mid-point on
leading edge of the waveform in this Chapter. It is linearly interpolated between the
bins adjacent to the threshold value at the leading edge of the waveform. Then, its
position at the range window (i.e., the bin or gate number) can be computed from this
estimate. This position is then compared with the expected tracking gate (bins 32-33
for ERS-2) to determine the contaminated distance.
For the ERS-2 altimeter, there is no need to account for the thermal noise effect.
However, attention must be paid to it for the POSEIDON altimeter waveform data.
To overcome this problem, the mean thermal noise estimated by averaging over gates
5-10 is subtracted from the amplitude; the threshold-retracking scheme adopted here
is referenced to 50% of the amplitude above the mean thermal noise level. This
approach has been applied to the POSEIDON data used in this study.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 80
4.3.2 Mean Power of the Waveform
In order to provide a statistically meaningful assessment of the contaminated area, an
averaging procedure was carried out around the Australian coast. This aims to
calculate a broad contaminated area by analysing shape variations between the mean
coastal waveform and the theoretical waveform due to the Brown model. It is likely
that the variation in scattering surfaces within the coastal footprint of the altimeter
creates the error in the individual returned waveforms. Averaging can reduce this
‘noise’ and leave a mean waveform, which should represent the typical shape of the
waveform over the reflecting surfaces (cf. Partington et al., 1989; Quartly et al.,
2001). The mean waveform generated by this averaging technique is then compared
with the theoretical or ideal waveform to estimate the general differences and hence
the contaminated distance from the coastline.
There are a number of possibilities for the variation of the mean waveform shape.
Firstly, the altimeter measurement is both a waveform and a gain measured by the
AGC. The signal processor maintains the value of the AGC gate at a specified value
by adjusting the attenuator in the receiver. Although the AGC applies this variable
attenuation to the return signal, the mean waveform shape will still vary with sharp
changes in the scattering surface. Secondly, the statistical homogeneity property of
the scattering surfaces within coastal areas is changed and probably invalid. Land
returns may contribute to the mean waveform, thus causing a variation. If this is the
case, the variation in different areas can provide the indication of the contaminated
distance from the coast. Finally, the SWH and the satellite off-nadir pointing angle
may cause the variations in the slope of the leading edge and in the trailing edge of
the mean waveform. However, mean waveforms with different slopes and trailing
edges should cross their leading edges at the same position as the tracking gate over
oceans (Brenner et al., 1993; Hayne, 1980). In this case, the contaminated distance
can be obtained from combining the position of the retracking point with the
variation of the mean waveform. Therefore, using the variation of the mean
waveform shape and its standard deviation in different coastal regions should give a
useful indication of the areas in which the waveform data may be contaminated.
For each waveform-sampled bin, the mean power of waveform M (i)
and its
standard deviation STD (i)
are calculated by

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 81
N
1
M
i
( t)
= ∑ Pij
( t)
(4.1)
N
j=
1
STD ( t)
=
i
n
∑
j=
1
( P ( t)
− M
ij
n −1
i
( t))
2
(4.2)
where n is the number of waveforms used in the averaging procedure. Averaging
was carried out over different distance bands with increasing distance from the
Australian coast. These distance bands were used to investigate broad variations in
waveform shape over the spatial-scale of the coastal region and over the time-scale
of a cycle.
4. 4 Data and Editing
4.4.1 ERS-2
One cycle (March to April 1999) of ERS-2 35-day repeat orbit individual waveforms
was used in this study (Section 2.3.2). The geographical distribution of the
observations out to 350 km around the Australian coast is displayed in Figure 4.2.
Each ERS-2 waveform record consists of 1 Hz and 20 Hz data. Geophysical and
environmental corrections are supplied at 1 Hz, and observations, such as the range,
0
location, σ , and waveforms, are supplied at 20 Hz (Section 2.4.2).
Since the purpose of this Chapter is to analyse waveform contamination over coastal
regions, all land returns must be exactly edited out when near the shoreline. For
ERS-2, a flag is set to record the working mode changes based on the location of the
sub-satellite point. When the satellite is over the land, it is flagged as in ice mode.
Therefore, it might be possible to estimate the location of shoreline by observing the
changes of the flag. However, the locations near the Australian coastline are usually
flagged incorrectly (Figure 4.3). This means the flag supplied with the data is not
appropriate for determining the location of the coastline.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 82
110˚
120˚
130˚
140˚
150˚
160˚
0˚
0˚
-10˚
-10˚
-20˚
-20˚
-30˚
-30˚
-40˚
-40˚
110˚
120˚
130˚
140˚
150˚
160˚
Figure 4.2 ERS-2 ground tracks (35-day repeat orbit, from March to April 1999) to a
distance of 350 km from Australian shoreline (Lambert projection).
0˚
110˚
120˚
130˚
140˚
150˚
160˚
0˚
-10˚
-10˚
-20˚
-20˚
-30˚
-30˚
-40˚
-40˚
110˚
120˚
130˚
140˚
150˚
160˚
Figure 4.3 The ocean-ice-mode flag in ERS-2 data supplied close to the Australian
shoreline compared to the Wessel (2000) coastline, showing that most flags around
Australia are on the land (Lambert projection).

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 83
Instead, to identify if the sub-satellite point is over ocean or land near the coastal
regions, the DS759.2 (5′×5′ resolution) ocean depth model (Dunbar, 2000) and a
water/land grid (0.5′×0.5′ resolution) derived from GMT high-resolution shoreline
data (Wessel, 2000) have been used together, instead of using the ERS-2 flags (see
Section 2.4).
4.4.2 POSEIDON
The geographical distribution of the POSEIDON coastal observations around
Australia’s coasts are shown in Figure 4.4. A POSEIDON waveform record consists
of 1 Hz and 20 Hz data. However, all data are supplied at 1 Hz except for 20 Hz
waveform measurements (AVISO/Altimeter, 1998). The land/water flag for
POSEIDON is provided by the AVISO in the waveform products. It is directly used
in this study due to its fairly good agreement with the actual shoreline (Section 2.4.3).
110˚
120˚
130˚
140˚
150˚
160˚
0˚
0˚
-10˚
-10˚
-20˚
-20˚
-30˚
-30˚
-40˚
-40˚
110˚
120˚
130˚
140˚
150˚
160˚
Figure 4.4 POSEIDON ground tracks (10-day repeat orbit, from January 1998 to
January 1999) to a distance of 350 km from Australian shoreline (Lambert
projection).

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 84
4. 5 Determination of Contamination Area
Before calculating the retracking point and averaging the waveforms, geophysical
and environmental corrections were applied to all altimeter range measurements. The
GSHHS shoreline model (Wessel and Smith, 1996) has been employed for
determining the position of the shoreline with the exception of the POSEIDON data.
The distance referred to below is the shortest distance (on the sphere) between the
point observed on the satellite ground track and the shoreline point from the GSHHS.
Since 1 Hz range measurement is usually taken by averaging 20 waveforms, the
whole 20 Hz data record is kept in the calculation, though some waveforms may be
returned from only the land in coastal regions. The purpose is to analyse how the 1
Hz range data averaged from 20 Hz data are contaminated in coastal regions, since
these 1 Hz data are used by most users from GDRs.
4.5.1 Using 50% of Threshold Retracking Points of ERS-2
The Australian coastal area is subdivided into 10 areas of 15°×15°, which will be
used in Sections 5.6 and 5.8. Within these areas, if the value of the 50% threshold
retracking point is between bins 31-33, the mid-point is considered to be at the
expected location, otherwise the data are considered to be contaminated by land or
coastal sea surface states. The criterion of bins 31-33 is referenced to the statistics of
50% retracking points over ocean areas (i.e., 50-350 km offshore Australia), where
96% of the values of 50% retracking points lie within bins 31-33. This proportion
agrees with that in open oceans (cf. Quartly, 2001). This criterion is used to analyse
the impact from land or coastal scattering surfaces on the individual waveform when
a satellite ground track is within a few kilometres of the shoreline. The calculations
were carried out using ERS-2 data and the results summarised in Table 4.1. Figure
4.5 shows contaminated waveforms along satellite ground tracks in a small, but
representative, coastal area (135° ≤ λ < 150°, -45° ≤ ϕ < -30°) based on these
calculations.
It can be seen from Table 4.1 that although the contaminated waveforms comprise a
small percentage (from 0.2% to 2.9%, depending on different sub-areas), they are all
in close proximity to the coastline. The more contaminated data and the longest
contaminated distances occur off the north Australian coast and off the south

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 85
Table 4.1 Contaminated 1 Hz ocean depths and 20 Hz distances from the nearest shoreline in different
Australian coastal regions for the ERS-2 waveform data (20Hz, March to April 1999).
a
Areas
Max.
Distance a
(km)
Min.
Distance
(km)
Mean
Distance
(km)
Max.
Ocean
Depth b
(m)
% of
Contaminated
Waveforms c
Coastal Types d
120°≤λ

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 86
Figure 4.5 One cycle of ERS-2 contaminated waveforms (highlighted by circles)
along ground tracks in the southeast coasts of Australia.
Australian coast, near Tasmania. There are two reasons for this. The first is the
complex shoreline in these coastal regions. The second is that there are more satellite
ground tracks in north and south of Australia because of the inclination of the
satellite orbit with respect to the Australia landmass. The eastern and western
Australian coasts show less contaminated data and a shorter contaminated distance.
Overall however, the shortest and the longest contaminated distances from the
coastline are 0.02 km and 21.76 km, respectively; and the mean contaminated
distance is 6.85 km.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 87
It can be seen from Figure 4.2 that some ERS-2 ground tracks are not available for
this cycle near the Australian coast. This is another (lesser) reason why a small
percentage of waveforms appears to be affected to the east and west. Thus, it is
necessary to estimate a broad boundary around Australia, which contains the most of
contaminated data, by using waveform data from both ERS-2 and POSEIDON
altimeters.
The ocean depths interpolated from an Australian 30" bathymetric model (Buchanan,
1991) for these contaminated data range from 12.2 m to 361.5 m (Table 4.1). From
Table 4.1, it is not evident that the contaminated distance correlates with the ocean
depth, while 50% retracking points show some correlation with ocean depths (Figure
4.6). Also, correlation with depth varies from one area to another. Waveforms over
areas where the ocean depth is less than 50 m (Figure 4.6a) and
100 m (Figure 4.6c) appear to be highly contaminated, while some contaminated data
are found in the deeper (up to ~300 m) ocean (Figure 4.6b). Therefore, if shallow
water data close to the shoreline with water depth less than 200 m are simply
removed (e.g., Nerem, 1995; Shum et al., 1998), some valuable data are probably
omitted. Moreover, some contaminated data may still be kept in the data set in
coastal regions while they should be removed.
The rightmost column in Table 4.1 describes the general nature of the coastal terrain
(cf. Thom, 1984; Kelleher et al., 2002). According to Brooks et al. (1997) and
Brooks (2002), the contaminated distance should be correlated with the coastal
topography, in which the correlation is not only related to the height of the coastal
terrain, but also the slope of the terrain.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 88
50% retracking points
64
7a
48
32
16
0
-400 -350 -300 -250 -200 -150 -100 -50 0
Ocean Depth (m)
50% Retracking Points
64
48
7b
32
16
0
-400 -350 -300 -250 -200 -150 -100 -50 0
Ocean Depth (m)
50% Retracking Points
64
7c
48
32
16
0
-400 -350 -300 -250 -200 -150 -100 -50 0
Ocean Depth (m)
Figure 4.6 Distributions of 50% retracking points plotted versus DS759.2 ocean
depths for contaminated waveforms in areas of 105°≤λ

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 89
4.5.2 Using the Mean Waveform and its Standard Deviation
The mean waveforms, and the standard deviation of these means, were computed
over three distance bands in coastal regions, which range in distance of 0-350 km, 0-
50 km, and 0-10 km off the Australian coast. Each has been divided into several
annuli with intervals of 50 km, 10 km and 2 km, respectively. The time span of the
data used to average is a cycle (35 days for ERS-2 and 10 days for POSEIDON). For
POSEIDON, similar results were obtained from five cycles. Therefore, only the
results from cycle 197, which are representative of all the data examined, are shown
here.
(a) 0-350 km off the Shoreline
Figures 4.7 and 4.8 show averaged waveforms and standard deviations, respectively,
in the distance-bands of 0-350 km with the interval of 50 km around the Australian
coast for ERS-2 and POSEIDON. It is important to note that the coastal effects on
the waveforms come from both Australia and Indonesia when the data are taken to
the north of Australia. Therefore, the data thought to be affected by land near
Indonesia have not been used to calculate or plot the mean waveform. Despite the
different number of measurements in each distance band, the mean waveforms are
generally in agreement with the theoretical model (Figure 1.1). The slope of the
leading edge of the ERS-2 and POSEIDON mean waveforms is similar to that of the
theoretical model. However, it can be seen that the trailing edges show some slight
departure feature from the theoretical model (Figure 1.1) caused by fading noise and
the wrap-around feature for both altimeters (cf. Figure 1.2 and Figure 4.1).
The standard deviations of individual waveforms with respect to the mean waveform
for ERS-2 show a peak in the waveform bins 38 and 39 and two ramps at the leading
edge. Since this occurs in all distance bands, it seems to be a systematic anomaly.
The reason for this should be investigated from more ERS-2 altimeter waveform data.
Compared with ERS-2, the standard deviation of the POSEIDON data shows a
smoother shape (Figure 4.8). Figures 4.7 and 4.8 show a large standard deviation in
the distance band of 0-50 km compared with other standard deviations, implying that
there is more contamination in this area. This will be studied next.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 90
Mean
Wavwform
(counts)
1200
900
600
300
0
64
55
46
37
Bins
28
19
10
1
0- 50 km
50-100 km
100-150 km
150-200 km
200-250 km
250-300 km
300-350 km
STD (counts)
1200
900
600
300
0
64
55
46
37
Bins
28
19
10
1
0- 50 km
50-100 km
100-150 km
150-200 km
200-250 km
250-300 km
300-350 km
Figure 4.7 The mean waveform (top) and the standard deviation of the mean
waveform (bottom) for 1 cycle of ERS-2 data in seven 50 km-wide
bands around the Australia’s coastline.
Mean
Waveform
(counts)
200
150
100
50
0
55
46
Bins
37
28
19
10
1
0- 50 km
50-100 km
100-150 km
150-200 km
200-250 km
250-300 km
300-350 km
STD (counts)
200
150
100
50
0
55
46
Bins
37
28
19
10
1
0- 50 km
50-100 km
100-150 km
150-200 km
200-250 km
250-300 km
300-350 km
Figure 4.8 The mean waveform (top) and the standard deviation of the mean
waveform (bottom) for POSEIDON cycle 197 in seven 50 km-wide
bands around the Australia’s coastline.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 91
(b) 0-50 km off the Shoreline
Mean and standard deviations in the distance bands of 0-50 km with an interval of 10
km are shown in Figures 4.10 and 4.11 for ERS-2 and POSEIDON, respectively. For
both altimeters, the leading edges of the mean waveforms in the bands of 10-20 km,
20-30 km, 30-40 km and 40-50 km have a similar slope of leading edge, but this
becomes less in band 0-10 km compared with the others.
From Figure 4.9, the ERS-2 standard deviations for 30-40 km and 40-50 km agree
well with these beyond 50 km (cf. Figures 4.8 and 4.9), while other standard
deviations also show similar shapes, even if there are some larger values around the
leading edge in other distance bands. The standard deviations for the 0-10 km band
are the largest, with a peak value of 1288 counts. From Figure 4.10 for POSEIDON,
the large standard deviation is evident in the 0-10 km band, especially close to the
tracking gate near the peak of the plot.
Mean Power of Waveform (counts)
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
Bins
STD (counts)
1600
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
Bins
0−10 km
N=9370
10−20 km
N=14202
20−30 km
N=14998
30−40 km
N=13995
40−50 km
N=12514
Figure 4.9 Mean power of the waveform (left) and the standard deviation of the mean
waveform (right) for ERS-2 in five 10 km-wide bands from the Australian coastline.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 92
Mean Power of Waveform (counts)
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
Bins
STD (counts)
60
40
20
0
0 10 20 30 40 50 60
60
40
20
0
0 10 20 30 40 50 60
60
40
20
0
0 10 20 30 40 50 60
60
40
20
0
0 10 20 30 40 50 60
60
40
20
0
0 10 20 30 40 50 60
Bins
0−10 km
N=4400
10−20 km
N=5160
20−30 km
N=3660
30−40 km
N=4620
40−50 km
N=4060
Figure 4.10 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for POSEIDON cycle 197 in five 10 km-wide bands from the
Australian coastline.
(c) 0-10 km off the Shoreline
Figures 4.11 and 4.12 show mean waveforms and their standard deviations 0-10 km
off the shoreline with an interval of 2 km for ERS-2 and POSEIDON, respectively. It
can be seen from Figure 4.11 that mean ERS-2 waveforms depart from the
theoretical waveform in distance bands of 0-2 and 2-4 km. In the band of 8-10 km,
the mean waveform agrees well with the theoretical or ocean waveform and with
others at greater distances (cf. Figures 4.7, 4.9 and 4.11). It is evident that the
standard deviation is large in gates for all distance bands. The variation of the
standard deviation in the 0-2 km and 2-4 km distance bands is very large, up to about
3000 counts in the peak.
From Figure 4.12, it is obvious that nearly the same conclusion for the mean
POSEIDON waveforms as for ERS-2 can be obtained. However, POSEIDON’s
standard deviation does not show the obvious large values that the ERS-2 does in the
4-6 km and 6-8 km bands. The largest standard deviation is also in the distance band

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 93
0-2 km and the next largest is in the 2-4 km band. The standard deviation of the
mean waveform increases with decreasing distance.
From Figures 4.11 and 4.12, it can be seen that the larger standard deviation from the
bands of 0-2 km and 2-4 km is dominant, which implies that this is the area of main
contamination of coastal waveforms, but some lesser contamination also occurs in
the other bands as well.
The above results suggest that it is possible to estimate a typical contaminated area
around coasts by simply analysing the variations of the mean waveform from the
ideal and the standard deviation of the mean. Using this approach around Australia
shows that the waveform data will mainly be contaminated when the ground track is
within a distance of about ~4 km from the coast, while there is some smaller level of
contamination to about 4-8 km for POSEIDON and 4-10 km for ERS-2.
Mean Power of Waveform (counts)
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
Bins
STD (counts)
3000
2000
1000
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
1200
800
400
0
0 10 20 30 40 50 60
Bins
0−2 km
N=1098
2−4 km
N=1495
4−6 km
N=1826
6−8 km
N=2323
8−10 km
N=2505
Figure 4.11 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for ERS-2 in five 2 km-wide bands from the Australian
coastline.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 94
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
0−2 km
N=560
Mean Power of Waveform (counts)
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
STD (counts)
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
150
100
50
0
0 10 20 30 40 50 60
2−4 km
N=740
4−6 km
N=840
6−8 km
N=1000
150
100
50
0
0 10 20 30 40 50 60
Bins
150
100
50
0
0 10 20 30 40 50 60
Bins
8−10 km
N=1260
Figure 4.12 Mean power of the waveform (left) and the standard deviation of the
mean waveform (right) for POSEIDON cycle 197 in five 2 km-wide bands from the
Australian coastline.
4.5.3 Using Statistics of the 50% Retracking Point
Statistics of the 50% threshold retracking points have been computed using both
POSEIDON and ERS-2 waveform data in a coastal area of 10 km from the
Australian shoreline. This distance is subdivided into 5 bands with an interval of
2 km. As stated, if there is no effect from land and coastal sea states on the returned
waveform, the threshold value of 50% retracking point should be very close to the
pre-designed tracking gate when it is over oceans. Five cycles of POSEIDON
waveform data and 1 cycle of ERS-2 waveform data are used to compute the 50%
threshold retracking point in the above bands.
Figure 4.13 shows the 50% retracking point distribution for these waveform data. For
POSEIDON, the 50% retracking points comprise 15%, 50% and 59% in bins 30-32
for bands of 0-2 km, 6-8 km and 8-10 km, respectively. For ERS-2, they are 20%,
45% and 45% in bins 31-33 for bands of 0-2 km, 6-8 km and 8-10 km, respectively.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 95
POSEIDON has a higher percentage than that of ERS-2 near the tracking gate with
increasing distance. It was found from the calculation that 94% of POSEIDON
waveforms represent their mid-point of leading edge very close to the tracking gate
in bins 30-32 when beyond 10-20 km of the shoreline, while 84% of ERS-2 data
show in bins 31-33 for the same distance.
% of observations
Poseidon
60
40
0−2 km N=2814
20
0
0
60
10 20 30 40 50 60
40
2−4 km N=3349
20
0
0
60
10 20 30 40 50 60
4−6 km N=4207
40
20
0
0
60
10 20 30 40 50 60
40
6−8 km N=4895
20
0
0
60
10 20 30 40 50 60
40
8−10 km N=6187
20
0
0 10 20 30 40 50 60
Bins
% of observations
ERS−2
60
40
0−2 km N=1075
20
0
0
60
10 20 30 40 50 60
40
2−4 km N=1457
20
0
0
60
10 20 30 40 50 60
40
4−6 km N=1830
20
0
0
60
10 20 30 40 50 60
6−8 km N=2316
40
20
0
0
60
10 20 30 40 50 60
40
8−10 km N=2504
20
0
0 10 20 30 40 50 60
Bins
Figure 4.13 Fifty percent retracking point frequency distributions for 5 cycles of
Poseidon waveform data (left) and 1 cycle of ERS-2 waveform data (right). They are
shown as the percentage of observations in each bin for five 2 km-wide bands around
the Australian coastline.
Tables 4.2 and 4.3 show the statistics of the differences between 50% retracking
point and the tracking gate for POSEIDON and ERS-2, respectively. The 50%
retracking points are obviously offset from the tracking gate by 2.5±5.9 bins for
POSEIDON and by 2.2±6.3 bins for ERS-2 in the band of 0-2 km. These offsets lead
to a mean range difference of about 1m for both altimeters. POSEIDON’s mean

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 96
range difference drops below 10 cm in the 6-8 km band, whereas ERS-2 does not
decrease to below 10 cm until 10-20 km from the shoreline.
Table 4.2 Differences between 50% retracking points and the tracking gate (bin 31.5)
for 5 cycles (January 1998 to January 1999) of Poseidon waveforms (each bin has
the range distance of ~0.469 m).
Band
(km)
Mean
(bins)
Range difference
(cm)
RMS
(bins)
STD
(bins)
Number of
20 Hz Points
0 – 2 2.46 115 ±5.94 ±5.40 2814
2 – 4 1.14 53 ±3.72 ±3.54 3349
4 – 6 -0.38 -18 ±2.49 ±2.46 4207
6 – 8 -0.15 7 ±1.67 ±1.66 4895
8 - 10 0.10 5 ±0.96 ±0.96 6187
10 - 20 0.14 6 ±0.72 ±0.74 23820
Table 4.3 Differences between 50% retracking points and the tracker point (bin 32.5)
for 1 cycle (March to April 1999) of ERS-2 waveforms (each bin with the range
width of ~0.454m).
Band
(km)
Mean
(bins)
Range difference
(cm)
RMS
(bins)
STD
(bins)
Number of
20 Hz Points
0 – 2 2.20 100 ±6.32 ±5.93 1075
2 – 4 0.27 12 ±5.82 ±5.81 1457
4 – 6 -0.47 -21 ±4.74 ±4.72 1830
6 – 8 -0.29 -13 ±3.27 ±3.25 2316
8 - 10 -0.23 -10 ±2.56 ±2.55 2504
10 - 20 -0.14 -6 ±1.94 ±1.95 14224
From Figure 4.13 and the values in Tables 4.2 and 4.3, this 50% retracking analysis
indicates that a distance band of ~8 km and ~10 km are evidently the contaminated
area for POSEIDON and ERS-2, respectively. The effect even extends to ~10-20 km
for some ERS-2 data. Thus, there is a clear difference between POSEIDON and
ERS-2 results from this analysis. It seems that contaminated distance for POSEIDON
is shorter than that for ERS-2.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 97
Finally, from reviewing the waveform products, it was found that ERS-2 provides
20 Hz ground locations for 20 Hz ground waveforms, while POSEIDON provides
1 Hz locations for 20 Hz waveforms. Thus, if a location for POSEIDON is given in a
band, the waveforms 3 km either side of the band also contribute to the calculation in
the band. This means that the results in Table 4.2 are actually computed for a larger
band than 2 km, which is a plausible explanation for the different contaminated
distances, as well as the different sized footprint and altimetric characteristics (also
see Section 4.2.1).
4. 6 Waveform Contamination Analysis
The waveform contamination from near the coastline differs among the altimeter
ground tracks, depending on whether the groundtrack is crossing the shoreline from
water to land or from land to water (e.g., Brooks et al., 1997). In this Section,
examples of ERS-2 altimeter ground tracks at the Australian coast (two water-to-land
and two land-to-water) are analysed to understand the effects of the coastal
topography on waveforms. Both SSH profiles (without retracking) and relevant
waveforms are used for this analysis. Unlike the previous Section, distances in this
Section are defined as the along-track distance to the coastline. The analysis in this
Section focuses on the waveform itself. A comparison of the altimeter-derived SSH
with the geoid heights will be given in Sections 1.2.1 and 5.8.2 to support the results
obtained in this Section.
4.6.1 Waveform Contamination for Water-to-Land Ground Tracks
(a) Tasman Sea (Eastern Australia)
As an ERS-2 altimeter groundtrack from cycle 43 (ascending pass 21792)
approaches land northwestward (Figure 4.14) in the Tasman Sea, the 20 Hz SSH
profile initially shows small but smooth undulation, and then rises greatly in the five
late SSH samples near the shoreline. These relevant 20 successive waveforms are
plotted via the bins (or gates) in Figure 4.15, which correspond to a ~7 km alongtrack
distance. The locations of waveforms in Figure 4.15 correspond to the last 20
SSH measurements numbered in Figure 4.14.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 98
Land - 20
SSH (m)
30
Tasman Sea
- 15
- 10
- 5
-30˚ 18'
153˚ 06' 153˚ 09' 153˚ 12'
- 1
-30˚ 21'
-30˚ 24'
Figure 4.14 The 20 Hz SSH profile of pass 21792 (ascending track, cycle 43 of
ERS-2), approaching the eastern Australian shoreline from the Tasman Sea. The
number beside the SSH samples corresponds to the waveform in Figure 4.15.
1000
1
1000
2
1000
3
1000
4
Counts
500
500
500
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
5
1000
500
6
1500
1000
500
7
1000
500
8
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
9
1000
500
10
1000
500
11
1000
500
12
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
13
2000
1000
14
1500
1000
500
15
2000
1000
16
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
2000
1000
17
2000
1000
18
4000
2000
19
8000
6000
4000
2000
20
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.15 20 Hz waveforms of pass 21792 (ascending track, cycle 43 of ERS-2),
approaching Australian eastern shoreline in the Tasman Sea. The number in the top
left-hand corner is related to that in Figure 4.14.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 99
Waveform one in Figure 4.15 occurs when the groundtrack is ~5.4 km from land,
and the land reflection does not yet appear. The land reflections appear clearly first in
the gate (or bin) near gate 45 in the waveform 14 (Figure 4.15), then moving
gradually closer to the tracking gate 32.5 in the successive waveforms 15-20. In the
16th waveform (Figure 4.15), the ocean return is discernible near the tracking gate,
while the land returns begin near the tracking gate. As the groundtrack approaches
the land, the ocean return remains invisible on the leading edge and appears in the
late gates, while the land returns are around the tracking gate in the range window.
Waveforms 1 to 13 (Figure 4.15) basically show the ocean waveform characteristics
with a clear single ramp. From waveforms 14 to 18, a single peak can be observed,
but the shape is different from the open-ocean waveform. The last two (19 and 20)
have a narrow width and significant peaks of the power, showing sharply rising and
descending shape. Beginning with the waveform 16, the altimeter tracker appears to
be responding more to the off-nadir higher and brighter land return than the at-nadir
water return. As a result, the measured range is too short, and calculated at-nadir sea
surface heights corresponding to waveforms 16-20 will be biased too high (Figure
4.14).
(b) Indian Ocean (Western Australia)
Another example of an ERS-2 altimeter groundtrack (cycle 43 and ascending pass
21450) approaching the land is shown in Figure 4.16, where the coast is dominated
by sand dunes (Thom, 1984). In the area where the groundtrack intersects the
shoreline, the land elevation in this area is between ~0-65 m above mean sea level
from the Australian 9" DEM (version 2, Hutchinson, 2001, described in Section
2.5.4). The SSH profile (related to a ~80 km along-track distance) shows no apparent
change in Figure 4.16 when the groundtrack is offshore further than ~3.2 km (before
the SSH sample 10). As the groundtrack gets closer to the shoreline after the tenth
SSH samples, the SSH begins to rise.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 100
SSH (m)
-25
-26˚ 24'
20 -
Land
10 -
-26˚ 27'
1 -
-26˚ 30'
-26˚ 33'
-26˚ 36'
Indian
Ocean
-26˚ 39'
113˚ 18' 113˚ 21'
Figure 4.16 The 20 Hz SSH profile of pass 21450 (ascending track, cycle 43 of
ERS-2), approaching the western Australian shoreline from the Indian Ocean. The
number beside the SSH samples corresponds to the waveform in Figure 4.17.
Counts
10000
5000
1
1000
500
2
1000
500
3
1000
500
4
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1500
1000
500
5
1000
500
6
1000
500
7
1000
500
8
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1500
1000
500
9
1500 10
1000
500
1000
500
11
1500
1000
500
12
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
13
1500
1000
500
14
1000 15
500
1000
500
16
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
17
2000
1000
18
4000
2000
19
4000
2000
20
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.17 The 20 Hz waveforms of pass 21450 (ascending track, cycle 43 of ERS-
2), approaching Australian western shoreline in the Indian Ocean. The number is
related to that in Figure 4.16.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 101
Figure 4.17 shows 1-second (20 Hz) waveforms related to the last 20 SSHs in
Figure 4.16. The distance to the coastline from the location of the first waveform is
~3.2 km. The first waveform shows a specular type with a high peak of ~10,000
counts, implying that it is reflected from calm or standing water. As a result, the
relative SSH (the first one in Figure 4.17) is measured too short, because the range is
too long. The following waveforms (two to nine) show ocean-like shapes, but do not
contain obvious land returns until the tenth waveform. The higher coastal land does
not show significant effects on the waveforms. This is because the range window of
64 gates is only related to a difference between surface elevations no larger than
29 m for ERS-1/2. Therefore, the land topography more than ~30 m with respect to
mean sea level around the groundtrack at this area does not appear as obvious land
returns in the waveform range window.
However, from waveforms 10 to 17 (Figure 4.18), the land returns appear to have
affected the altimeter tracker response, and the waveforms have moved slightly to the
left with respect to the tracking gate in the range window. In the last three waveforms
(18-20) in Figure 4.17, the ocean returns have moved to the later gates and the land
returns moved close to the tracking gate. This means that the shorter range
measurement is obtained from the land returns reflected from higher surfaces. As a
result, the calculated SSH will be too high.
4.6.2 Waveform Contamination for Land-to-Water Ground Tracks
(c) Indian Ocean (Northwestern Australia)
Figure 4.18 shows an ERS-2 20 Hz SSH profile from an ascending pass 21636 (cycle
43), which is going from land to water northwestward. This is a tidal plain coast area
(Thom, 1984). The land-to-water crossover occurs at the 16th SSH in Figure 4.18. At
this time, the terrain is higher than the ocean with respect to the radar pulse front, and
the land reflections appear dominantly in the waveforms so that the SSH results are,
probably, calculated from the land returns. As the satellite goes further from the
shoreline, because of the off-nadir angle, the altimeter still measures the closest slant
range to the higher land. As a result, in this case the effects of land on the calculated
SSHs last nearly three seconds (~21 km along-track distance).

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 102
The three-second successive waveforms, which correspond to the SSHs from 1-60 in
Figure 4.18, have been plotted via the bins or gates in Figures 4.20, 4.21 and 4.22.
Most waveforms seem to be specular due to still water in Figure 4.20. From the first
to fifth waveforms, still water produces waveforms with higher peaks and narrow
width. Lower power signals, which begin around the 40th gate from the sixth to 19th
waveforms, are ocean returns as the groundtrack crosses the shoreline from the land
to water.
Beginning with waveform 20 in Figure 4.19, the initial radar returns are from the sea
surface and the land returns move towards the late gates. Between waveforms 28 to
33 in Figure 4.20, the off-nadir terrain and at-nadir sea surfaces are approximately
equidistant from the altimeter, producing waveforms with double ramps. By
waveform 34 when the groundtrack is 3.6 km to the shoreline (Figure 4.20), the land
returns have disappeared.
Although the sea surface returns are dominant in waveforms 34 through 40 in Figure
4.20 and from 41 to 54 in Figure 4.21, these waveforms are misaligned with respect
to the pre-designed tracking gate (32.5) such that the measured ranges are too long,
and the calculated SSHs are too low (see Figure 4.18). In the rest of waveforms from
55 to 60 in Figure 4.21, the half power point shifts slightly to the left with respect to
the tracking gate. After three seconds (about ~21 km along the track or ~6 km from
shore), waveforms become centred in the range window (i.e., at the tracking gate).
(d) Southern Ocean (Southern Australia)
The 20 Hz SSH profile from cycle 43 descending pass 21886 traverses
southwestward across the shoreline from land to the Southern Ocean is shown in
Figure 4.22. There is low land elevation along the coastline (Hutchinson et al., 2001).
The SSH profile drops fast after the groundtrack enters the water, and then rises
gradually. It changes to the normal ocean waveform and thus normal SSH after
~2.5 seconds (~17 km from the shoreline) in this example.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 103
SSH (m)
-20
Indian Ocean
-22˚ 03'
-22˚ 06'
-22˚ 09'
-22˚ 12'
60 -
-22˚ 15'
50 -
-22˚ 18'
40 -
30 -
20 -
10 -
1 -
Land
114˚ 24' 114˚ 27' 114˚ 30'
Figure 4.18 The 20 Hz SSH profile of pass 21636 (ascending track, cycle 43 of
ERS-2), receding from the northwestern Australian shoreline to the Indian Ocean.
x 10 4
Counts
15000
10000
5000
1
2
1
2
2000
1000
3
15000
10000
5000
4
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
3000
2000
1000
5
1000
500
6
1500 7
1000
500
1500
1000
500
8
Counts
2000
1000
0
0 16 32 48 64
9
2000
1000
0
0 16 32 48 64
10
8000
6000
4000
2000
0
0 16 32 48 64
11
6000
4000
2000
0
0 16 32 48 64
12
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
6000
4000
2000
13
3000 14
2000
1000
2000
1000
15
1000
500
16
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
17
1000
500
18
1500
1000
500
19
1000
500
20
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.19 20 Hz waveforms of pass 21636 at the 1st second (~0-7 km along-track
distance) from the shoreline (ascending track, cycle 43 of ERS-2), receding from the
northwestern Australian shoreline to the Indian Ocean.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 104
Counts
1000
500
21
1500
1000
500
22
1500 23
1000
500
1500 24
1000
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
2000
1000
25
2000 26
1000
1500 27
1000
500
1500 28
1000
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1500
1000
500
29
1500
1000
500
30
1000 31
500
1500
1000
500
32
Counts
1500
1000
0
0 16 32 48 64
500
33
1000
0
0 16 32 48 64
500
34
0
0 16 32 48 64
1000 35
500
1000
0
0 16 32 48 64
500
36
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
800
600
400
200
37
0
0 16 32 48 64
Bins
1000
500
38
0
0 16 32 48 64
Bins
800
600
400
200
39
0
0 16 32 48 64
Bins
1000
500
40
0
0 16 32 48 64
Bins
Figure 4.20 20 Hz waveforms of pass 21636 at the 2nd second (~7-14 km alongtrack
distance) from the shoreline (ascending track, cycle 43 of ERS-2), receding
from northwestern Australian shoreline to the Indian Ocean.
Counts
800 41
600
400
200
0
0 16 32 48 64
1000 42
500
0
0 16 32 48 64
1000 43
500
0
0 16 32 48 64
1500
1000
500
44
0
0 16 32 48 64
1000
45
1000 46
1000
47
1000 48
Counts
500
500
500
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
49
1000 50
500
1000 51
500
1000
500
52
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
53
1000
500
54
1000
500
55
1000
500
56
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
57
1000
500
58
1000
500
59
1000
500
60
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.21 20 Hz waveforms of pass 21636 at the 3rd second (~14-21 km alongtrack
distance) from the shoreline (ascending track, cycle 43 of ERS-2), receding
from the northwestern Australian shoreline to the Indian Ocean.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 105
When the groundtrack exits the land, the first three seconds of 60 waveforms related
to the SSH profile have been plotted in Figures 4.24, 4.25 and 4.26. The first five
waveforms in Figure 4.23 show mostly the response to the land reflections. After
waveform six, which is at a distance of ~2.1 km from the shoreline, the leading edges
of the waveforms shift to the earlier gates. This leftward shift continues to the
waveform 25 in Figure 4.24. After that, the waveforms move back gradually towards
the alignment of the ocean waveform through the rest of waveforms in Figure 4.24
and all waveforms in Figure 4.25. Waveforms after 54 in Figure 4.25, when the
ground track is more than ~17.9 km from shore, become stably aligned with respect
to the tracking gate, appearing to be normal open-ocean returns.
Waveforms in this example do not show obvious high power and the land returns
except for the first six waveforms in Figure 4.23, because there is a beach area here
without the steep slope and sharp changes of the topography (Thom, 1984). However,
apparent waveform contamination still lasts about three seconds (~21 km along-track
distance). This will be discussed in Section 4.6.4
Land
-15
- 20
- 10
- 1
-32˚ 18'
SSH (m)
- 60
- 50
- 40
- 30
-32˚ 21'
-32˚ 24'
Southern Ocean
126˚ 45' 126˚ 48'
-32˚ 27'
-32˚ 30'
-32˚ 33'
Figure 4.22 The 20 Hz SSH profile of pass 21886 (descending track, cycle 43 of
ERS-2), receding from the southern Australian shoreline to the Southern Ocean.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 106
4000
1
4000 2
4000
3
3000 4
Counts
2000
2000
2000
2000
1000
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
4000
5
3000 6
2000 7
2000 8
Counts
2000
2000
1000
1000
1000
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1500
1000
500
9
1000
500
10
1000 11
500
1000
500
12
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
800
600
400
200
13
0
0 16 32 48 64
1000
500
14
0
0 16 32 48 64
600 15
400
200
0
0 16 32 48 64
800
600
400
200
16
0
0 16 32 48 64
Counts
800
600
400
200
17
600
400
200
18
600
400
200
19
600
400
200
20
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.23 20 Hz waveforms of pass 21886 at the 1st second (0-7 km along-track
distance) from the shoreline (descending track, cycle 43 of ERS-2), receding from
the southern Australian shoreline to the Southern Ocean.
Counts
800
600
400
200
21
800
600
400
200
22
600
400
200
23
800
600
400
200
24
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
800
600
400
200
25
800 26
600
400
200
800
600
400
200
27
800
600
400
200
28
Counts
0
0 16 32 48 64
800
600
400
200
29
0
0 16 32 48 64
1000
0
0 16 32 48 64
500
30
0
0 16 32 48 64
1000
0
0 16 32 48 64
500
31
0
0 16 32 48 64
1000
0
0 16 32 48 64
500
32
0
0 16 32 48 64
Counts
800
600
400
200
33
0
0 16 32 48 64
1000
500
34
0
0 16 32 48 64
1000 35
500
0
0 16 32 48 64
1000 36
500
0
0 16 32 48 64
1000
37
1000
38
1000
39
1000
40
Counts
500
500
500
500
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.24 20 Hz waveforms of pass 21886 at the 2nd second from the shoreline
(ascending track, cycle 43 of ERS-2), receding Australian southern shoreline to the
Southern Ocean.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 107
1000
41
1000 42
1000 43
1000 44
Counts
500
500
500
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
45
1000 46
500
1000
500
47
1000
500
48
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
1000
49
1000 50
1000
51
1000
52
Counts
500
500
500
500
Counts
1000
0
0 16 32 48 64
500
53
1500
1000
0
0 16 32 48 64
500
54
1000
0
0 16 32 48 64
500
55
0
0 16 32 48 64
1000 56
500
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
0
0 16 32 48 64
Counts
1000
500
57
1000
500
58
1000
500
59
1000
500
60
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
0
0 16 32 48 64
Bins
Figure 4.25 20 Hz waveforms of pass 21886 at the 3rd second from the shoreline
(ascending track, cycle 43 of ERS-2), receding Australian southern shoreline to the
Southern Ocean.
4.6.3 Contamination Caused by the Altimeter’s Operation
As can be seen from Figures 4.19 to 4.26, after obvious returns from the land or
inland coastal still water disappear from the range window, waveforms continue to
take a second or more of time to align to the tracking gate. The corresponding alongtrack
distance can be longer to ~21 km from the shoreline, which cannot be
explained by the land effects within the altimeter footprint. The reasons for this are
not clear, but it may be linked to the operation of the altimeter. The altimeter works
often in the normal track mode over sea surfaces, while this is not the case over nonocean
surfaces. After a water-to-land or land-to-water transition, the altimeter usually
takes a second or more of time to reacquire the signal over the changed surface
(Brooks et al., 1997; Strawbridge and Laxon, 1994, cited in Scott et al., 1994).
In addition, the ocean waveform should show a linear trailing edge (see Section 5.4.3)
based upon the theoretical waveform model. This means that the ocean waveform
samples can be shifted rightwards in the range window without a change of the
relevant position of the retracking gate. To demonstrate this, 20 waveforms at a

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 108
second time span along descending groundtrack 21886 related to waveforms 21-40 in
Figure 4.24 are chosen. This group of waveforms is characterised to have retracking
gates from 23.3-27.3, which all depart from the pre-designed tracking gate (32.5).
These waveforms are shifted rightwards with respect to the centre of the range
window by 0, 5, 10, 15, 20, or 25 gates (corresponding to the range of 0 m, 2.27 m,
4.54 m, 6.81 m, 9.08 m, or 11.36 m, respectively). Using the least squares fitting
algorithm (Chapter 5), an ocean model (see Section 5.4.1) is fitted to each waveform
to estimate the retracking gate. Differences between the retracking estimates before
and after shift are computed. Descriptive statistics of the differences are then
calculated and listed in Table 4.4.
Table 4.4 Descriptive statistics of the differences of retracking gate estimates before
and after shifting rightward waveform samples in the range window (20 Hz
waveforms from pass 21886 at the 2nd second from the coastline, 1 gate≈0.4542 m)
Shift
(gates)
Number of
waveforms
Min
(gates)
Max
(gates)
Mean
(gates)
STD
(gates)
5 20 -0.10 0.03 -0.02 0.03
10 20 -0.13 0.07 -0.02 0.05
15 19 -0.22 0.09 -0.01 0.08
20 19 -0.20 0.24 0.01 0.09
25 19 -0.26 0.23 -0.01 0.13
As can be seen from Table 4.4, when shifting waveforms 5 and 10 gates rightward in
the range window, the retracking gate estimates have a good agreement with those
before shifting. When shifting 15, 20, or 25 gates, one waveform fails to obtain the
retracking gate estimate, while others still agree well with the estimate before
shifting. The STD increases with increasing the shifting gates. Because of the fading
noise, the maximum range error caused by this shifting is ~6 cm. The fitting
algorithm keeps working until shifting the position of the leading edge rightward in
the range window to gate 57. This testing result indicates that the retracking gate can
be successfully estimated by the fitting procedure.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 109
Therefore, these test results support that the original tracking mode still continues for
some time, when the returns reflected from the surface changed present in the range
window. The results also demonstrate that the altimeter’s operating feature is one of
waveform contamination sources in coasts, and this operating feature influences
mainly waveforms along the ground tracks that leave land to water (cf. Figures 4.19
and 4.23). It has been noted from previous discussions (Sections 4.6.1 and 4.6.2) that
this problem does not appear to happen to the data from water to land ground tracks
where the altimeter probably remains in the normal ocean mode till getting very
close to coast. These data are, thus, less contaminated.
This result also implies a tremendous influence on the use of untracked SSH data in
coastal regions. Only the data from water to land ground tracks could be used in the
area vicinity (e.g., ~5 km) of the coastline without waveform retracking
4.6.4 Summary of Waveform Contamination in Coasts
Considering the groundtrack approaching land, as it gets closer to the coasts, the land
returns within the altimeter footprint begin to appear in the waveform range window.
As long as the range to the nadir ocean surface is shorter than the range to the offnadir
land surfaces, the ocean returns will appear earlier in the range window.
However, as the land returns move into the AGC gate, the altimeter tracker tends to
track the off-nadir terrain, leading the calculated at-nadir SSH too high. The affected
distance is about 7 km along the groundtrack from the coastline for ERS-1/2.
On the other hand, for ground tracks leaving land to water, the altimeter tracker
continues to track the off-nadir land surfaces and responds more to the off-nadir
shorter range at the time when the ground track is just receding from land to water.
When the altimeter ground track goes further into the ocean, land returns move back
to the late gates. Waveforms are, meanwhile, shifted left first and then right with
respect to the centre of the range window (e.g., Figures 4.21 and 4.22). This shift can
last three seconds or more of time, indicating a longer along-track contaminated
distance of ~21 km (cf. Section 4.6.2).
If considering only the shape variations of the waveform in the proximity of the
coasts, the following three typical characteristics can be summarised:

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 110
(1) The land returns appear in the range window together with the ocean surface
scattering. Because of the different surface reflectivity and elevations, the
power superposition of the ocean returns and land returns changes both the
waveform shape and magnitude of the amplitude with increasing the
importance of the land return in the range window. This, in turn, causes the
gate location of the mean-surface retracking point to vary from 50% of the
amplitude (i.e., the half-power point) as the land return signal begins to
increasingly dominate the waveform shape. The ocean returns cannot be
recovered when the land returns predominate in the waveform.
(2) Waveforms shift towards the left or right as a function of along-track distance
with respect to the tracking gate in the range window, but do not significantly
change the magnitude of the waveform’s amplitude. Land returns are not
obviously visible, and the contamination cannot be explained by the land
effects, because the distance to the shoreline may be larger than the radius of
the footprint. In this case, waveform contamination results from the altimeteroperating
feature.
(3) Inland standing water causes very high peaks in the waveform. This type of
specular waveform appears with the large magnitude of the amplitude and the
narrow width of power. These should not be retracked unless the information
on the inland water is required.
4. 7 An Experiment of the Land Effects on Waveforms Using the DEM Data
A basic feature of the waveform is that it can be considered as the cumulative energy
returned from each illuminated surface scatterer, sampled by a series of gates in the
range window (Femenias et al., 1993; Martin et al., 1983). Although most coastal
waveforms have an ocean-like shape with a single ramp, there are a percentage of
waveforms (~20%) that show non-ocean-like shapes, particularly over areas closer to
land (Section 6.2). These waveforms consist of ocean and land returns, hence leading
to a mixed waveform shape. There are two possibilities for these waveforms. The
first is that the ocean return could be recovered from these observed waveforms
using waveform retracking. The alternative is that it might be impossible to recover
the ocean return if it is obscured by the land return when the ground track is very

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 111
close to the coastline, say, ~0-2.5 km, and range to the ocean surface is no longer
recoverable.
In most cases of waveform retracking, the relation between the derived surface and
the actual surface are unknown, as independent measures of the ground truth are not
generally available. A digital elevation model (DEM), however, is a representation of
the terrain using the mean elevation information. Moreover, it includes the coastline
information including the main outline of the land, bays, the outer edge of mangrove,
swamps, closed-off narrow inlets, and watercourses at or near their mouths
(Hutchinson et al., 2001). Therefore, a high-resolution DEM can provide the land
elevation for dealing with mixed waveforms, though the disadvantage is that it is
generated by the land-averaged elevation at each cell. Other alternatives are the
optical satellite remote sensing, topographic maps or air photography, but these are
time consuming. Thus, a high-resolution DEM is still a useful external data source.
Because the well-developed ocean surface scattering model (e.g., Brown, 1977) is
deduced under the assumptions of the surface statistical homogeneity (see Section
3.2), these assumptions are not valid over non-ocean surfaces. Thus, to use the
independent DEM data, some other methods must be developed. The idea is that if
the land returns contained in the waveform can be determined separately using an
independent data source, they might be removed from the shape-mixed waveform to
leave the ocean return alone. Thus, the ocean surface can be recovered. To achieve it,
however, detailed information about the topography along the coastline is necessary.
The topography should include the land and inland water (e.g., lakes and rivers). Part
of the effort, which has been performed and designed in this study, will be presented
in this Section.
4.7.1 Concept of Land Effects Using the DEM Model
As stated in Chapter 2, the altimeter’s beam-limited footprint is the area within
which the beam attenuation is 3 dB or less. It has a larger footprint size than that of
the pulse-limited footprint (see Section 4.2.1). It is in this larger footprint size that
the land returns have higher probability of appearing in the range window. Therefore,
when discussing the land effects on the waveform, it is necessary to account for the
beam-limited footprint rather than the pulse-limited footprint.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 112
Consider two different reflected surfaces of land and water within the beam-limited
footprint from which the power backscatters to the altimeter. The land elevation
above local instantaneous sea level is H . The coordinate of this location can be
defined by the radial distance from the altimeter to the point on the sea surface, the
off-nadir angle ξ , the azimuth angle χ about the axis defined by the line between
the altimeter and the nadir point. Because of the approximate spherical geometry of
the Earth, it is convenient to replace the angle ξ with the colatitude angle φ
subtended by rays from the centre of the Earth to the nadir point and the off-nadir
'
point H ( χ , ξ,
t)
= H ( χ,
φ,
t)
(see Figure 4.26). Define Rφ ( χ,
t)
to be the radial
distance from the altimeter to H ( χ , φ,
t)
. The reflected power is received from this
specular reflector at the two-way travel time t if H ( χ , φ,
t)
falls within the footprint.
To the lowest order for the small off-nadir angles ξ relevant to the altimeter, the
'
distance Rφ ( χ,
t)
is related to the distance R from the altimeter to the mean sea level
at the angle φ by
R ( χ,
φ,
t)
≈ R − H ( χ,
φ,
t)
(4.3)
'
φ
A more precise relation is easily obtained from the geometry in Figure 4.26, but this
'
approximate solution is adequate for the present purpose. In addition, Rφ ( χ,
t)
can
be also related to the satellite altitude alt(t) and the geoid height N( χ , φ , t)
above a
reference ellipsoid as
R ( χ,
φ,
t)
≈ alt(
t)
− H ( χ,
φ,
t)
− N(
χ,
φ,
t)
(4.4)
'
φ
Thus, on the one hand, the range to the land surface (
'
R
φ
) can be estimated by
Equation (4.4), On the other hand, it can be obtained from the altimeter-measured
range ( χ , φ , t)
to the surface and the retracking correction dr 2 (Equation 5.7, see
R s
Section 5.4 for details) estimated by the nine-parameter function (Equation 3.41) as
R ( χ,
φ,
t)
= R
'
φ
s
( χ , φ , t)
+ dr2
= R ( χ , φ , t)
+ G
s
2M
( β − g )
32
0
(4.5)

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 113
alt
R
ξ
R'φ
Rφ
H
Re
N
mean sea
level
reference
ellipsoid
φ
Figure 4.26 A schematic geometrical description of the radar return from a point on
the land surface at off-nadir angle ξ and associated colatitude angle φ where the
land elevation above the mean sea level is H . The distance from the altimeter to the
'
land surface is R
φ
.
where β
32
is the retracking gate estimate in units of waveform gates (related to the
second ramp),
G 2 M
= t p
⋅ c / 2 is the conversion from numbered gates to metres, t p
is
the pulse width, and g
0
is the location in gates of the altimeter tracking gate. From
Equations (4.4) and (4.5), β
32
can be related to the land elevation H ( χ , φ,
t)
by
β = a + bH ( χ , φ , )
(4.6)
32
t
where
a
alt( t)
− N(
χ , φ , t)
− R ( χ , φ , t)
s
= g
0
+
(4.7)
G2M

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 114
and
= 1/
(4.8)
b G2M
Equation (4.6) describes the relationship between the retracking gate estimate and the
land elevation. At a certain time t , β
32
is only linearly related to the land elevation
H ( χ , φ,
t)
at the reflector’s location. Therefore, if the location of β 32
at the range
window can be determined by the independent DEM data source, it can be used as a
constrained condition or initial value of the nine-parameter function. This is because
an appropriate initial value or constraint will make the fitting algorithm converge
rapidly and (probably) accurately. However, it is also clear from Equation (4.6) that
several surface reflectors at the time t within the satellite footprint can have the same
slant range to the satellite. Thus, β
32
is not a unique function of the land elevation.
This can be also presented from the simple test shown below
4.7.2 A Preliminary Test of Land Effects Using the DEM Model
When testing this method, the following steps have been taken for each observing
location ( λ
0
,ϕ
0
) along satellite ground tracks near land.
(1) The algorithm estimates the shortest spherical distance s to the coastline and
extracts the location ( λ , ϕ ) at the point using a shoreline model, at which the
c
c
ground track intercepts the coastline.
(2) A search radius centred at the location ( λ
c
, ϕ
c
) is then determined according to
the value of s and the size of the satellite footprint, which can range from
2-8 km. The land elevations and geoid heights within the search area are
extracted at the point of each elevation cell using the Australian 9" DEM
(Hutchinson, 2001) and AUSGeoid98 (Featherstone et al., 2001) geoid grid
(2´×2´), respectively.
(3) Retracking gate estimates β
32
( χ i
, φ i
, t)
are obtained using Equation (4.6),
where i = 1, 2, ···, n is the number of the 9" cells within the search radius.
Because of the width of the range window, the maximum difference of the

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 115
elevations must be less than 28 m. Thus, elevations higher than 28 m were not
used in the computation.
Figure 4.27 shows two plots of β
32
( χ i
, φ i
, t)
via the land elevation H ( χ
i
, φi
, t)
. The
discontinuous retracking gate estimates in Figure 4.27 are caused by the non-uniform
elevation distribution within the search radius. As can be seen, all land elevations
within the search radius contribute to the return power, thus causing the problem of
non-uniqueness. This is a typical problem for non-ocean surfaces, because several
targets with the same slant range to the altimeter exist within the footprint.
Retracking gate eatimates
(bins)
60
55
50
45
40
35
30
0 1 2 3 4 5 6 7 8
Land elevations (m)
Retracking gate estimates
(bins)
60
55
50
45
40
35
30
0 1 2 3 4 5 6 7 8
Land elevations (m)
Figure 4.27 Retracking gate estimates corresponding to the land elevation. The
search radius is 2.7 km (top) and 3.5 km (bottom).
Thus, although Equation (4.6) is simple, it is difficult to effectively implement.
Theoretically, the altimeter starts to track off-nadir higher and brighter land when the

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 116
distance between the satellite and the land surface is shorter than that to the sea
surface. In practice however, of these land reflectors illuminated by the satellite, it is
hard to determine, based on only the DEM, which one importantly contributes to the
relevant mid-point on the second ramp of the waveform. Also, information about the
surface reflectivity of land and water is important, because the radar returns are
directly related to the nature of the scattering surface illuminated by the transmitted
electromagnetic single. However, the DEM cannot provide the water information on
land.
To overcome these problems, a more detailed function based directly on the radar
equation is necessary. Wingham’s method (1995) may be a way to solve the
problem. However, the DEM contains only land elevation and no water information
over land. This is another difficulty when trying to create the function for the land
return, because the water reflector, even a layer of very shallow water (e.g., after
rain), can make the return power much higher (Berry, 2002). In addition, the DEM
represents the average elevation of a cell (cf. Hutchinson, 2001), while returns reflect
from the facets with different elevations. Another condition that needs to be thought
about is the land slope. Brooks (2002) has found that the land slope affects the
returns reflected from the land near the coasts as well. Some higher land may scatter
forward the radar pulse so that it does not show land return on the waveform. Since
20 Hz waveform data require a spatial resolution of ~350 m, the high-resolution
DEM and detailed topography information are necessary to estimate land returns.
Because of the time limitation of this study, this can only be a topic of future
research.
4. 8 Summary
Five cycles of POSEIDON and one cycle of ERS-2 20 Hz waveform data have been
used to quantify a broad, contaminated boundary around the Australian coast. This
contamination is assumed to have been caused by backscatter from the land and the
more variable coastal sea surface states, coupled with the footprint size of the radars
and the incorrect positioning of the groundtrack.
Since bins 31-33 for the ERS-2 altimeter and bins 30-32 for the POSEIDON
altimeter are the expected bins for the tracking gate over oceans, three major

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 117
conclusions can be drawn from this study. First, the ERS-2 altimetry shows
differences in contamination as a function of the sub-area along the Australian
shoreline. The maximum contaminated distances vary from ~8 km to ~22 km,
depending upon locations and features of the shoreline. Second, both altimeters show
differences in contamination as a function of the distance from the shoreline. Finally,
the contaminated distance for POSEIDON is less than that for ERS-2. A mean
contaminated distance of ~8 km for POSEIDON and ~10 km for ERS-2 can be
observed using the 50% of threshold retracking points in the whole Australian
coastal region.
The mean waveforms and standard deviations of the mean in different distance bands
provide an indication of the contamination influencing the shape and variability of
the returned waveforms in coastal regions. They can provide a reasonable
explanation for the observed variations of the waveform in the proximity of coasts. A
typical contaminated boundary can be ascertained from Australian shoreline to
~8 km and ~10 km for POSEIDON and ERS-2, respectively. Beyond 8 or 10 km (to
350 km), the mean waveform shapes for both altimeters match the mean waveform
shape observed over ocean surfaces, though they do not agree exactly with the
theoretical waveform shape (Section 4.5).
The different reflector characteristics lead to different types of waveforms. In
addition, the contamination distance depends on how the satellite ground track
crosses the shoreline (i.e., approaching or leaving the land). In general, the
waveforms along the altimeter groundtrack leaving land to water suffer longer
contaminated distances than those along the track approaching land from water. In
both cases, the ocean surface sometimes can no longer be recovered from
contaminated waveforms obscured significantly by the land return, when the ground
track is much closer to the coastline, say, ~0-2.5 km, because the land returns
become predominant in the range window.
A preliminary test has been performed to calculate the retracking gate estimate
related to the land returns in the range window using an external independent data
source of the DEM and AUSGeoid98 models in coastal regions. Analysing results
show that non-uniqueness of the estimate over land near coasts is still a main
problem that may need to be solved in the future.

Chapter 4. WAVEFORM CONTAMINATION CLOSE TO AUSTRALIAN COASTS 118
Contamination leads to errors in the on-board calculation of the range measurements.
It is recommended from the results in this Chapter that such contaminated
measurements must be detected prior to their being included in geodetic and
oceanographic solutions in coasts. Until then, altimeter SSH or range data should be
interpreted with some caution for distances less than, say, ~22 km from a coastline,
and discarded altogether for distances less than 4 km.
This Chapter utilises different methods (e.g., statistic and threshold analysis, and
using additional DEM data source) to show that the land topography does influence
the waveform shape and thus the data. This can also be achieved using some other
methods, such as to plot the distance from the coastline against near shore land
height (within 0.5 km of coast) to see if there is a correlation between them.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 119
5. COASTAL WAVEFORM RETRACKING SYSTEM: DESIGN AND
IMPLEMENTATION
5. 1 Introduction
The techniques used to retrack ocean and non-ocean waveforms have been discussed
in Chapter 3. They can also be applied to the coastal waveform (e.g., Anzenhofer et
al., 2000; Deng et al., 2001). During waveform data processing and analysis, it has
been found in this study that fitting the waveform to a well-developed analytical
function (i.e., the Brown (1977) ocean model) is capable of retracking most of the
individual altimeter waveforms (~99% of data, see Table 6.1 in Section 6.2.2) over
open oceans. Similarly, using a 50% threshold level can give reasonable estimates of
the absolute range to the at-nadir sea surface. Using only a single retracker in coasts,
however, limits the precision of the recovered SSH due mainly to the disagreement
of the contaminated waveform with the theoretical/ideal waveform.
Therefore, it has become clear that no single retracking algorithm can always deal
with the diverse waveforms in coastal regions. A coastal waveform retracking system
which consists of different retrackers (e.g., the parametric fitting and the threshold
algorithms), thus, is necessary when attempting to extract the precise SSH from
retracked range measurements.
This Chapter presents the implementation of both fitting and threshold retracking
algorithms, and develops a coastal retracking system that is based upon a detailed
pre-analysis of the coastal waveforms. Issues involved in the algorithm development
are fitting functions, the iterative least squares procedure, linearisation of the
parameters, determination of the initial estimates of parameters, and the weight
scheme. A new method will be introduced that uses iterative weights to detect the
outliers in the waveforms ensuring the effective convergence of the fitting procedure.
For the threshold retracking algorithm, a method will be described that selects an
appropriate threshold level for coastal waveforms. Finally, a coastal retracking
system will be developed, which makes it possible to improve the SSH precision by
reprocessing the altimeter waveforms, correcting the range measurements, and
extracting precise SSH data from these corrected ranges.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 120
5. 2 Coastal Waveform Retracking System Design
A comprehensive analysis of waveform shapes at the Australian coast (Chapter 4)
led to the need to develop a coastal waveform retracking system. The primary idea of
the retracking system is that retrackers must cover most of the coastal waveforms,
and they can complement each other. Since the SSH data can only be recovered from
ocean returns, it is impossible for the system to reprocess the waveforms dominated
by the land return near land to extract SSHs. This means that it is not essential to
have as many retrackers as used over land, where much more diverse waveforms are
found (cf. Berry, 2000). However, it is significant that the system can correctly
categorise the waveforms, so that they can be retracked using an adequate retracker.
This concern generated an additional waveform classification.
Ideally, a physically realistic model should be used to retrack coastal waveforms,
because the waveform measured by a radar altimeter links not only to the geometric
surface of the ocean, but also to oceanic geophysical features. An appropriate model
will provide a correct understanding of the relationship between the geophysical
parameters of interest and the radar return from coastal regions.
With this consideration, the ocean model without non-linear wave parameters
(presented in Section 5.5.1) is first used to retrack ocean-like waveforms. To deal
with narrow, high-power waveforms reflected from calm coastal or inland water
surfaces, the threshold retracking technique with 50% threshold level is used as well.
It was found from an analysis of retracked results that both retrackers could not deal
with some waveforms in close proximity to the land. Fitting an ocean model to nonocean-like
waveforms causes failure of the fitting algorithm. This led to the use of
other retrackers of the five- and nine-β-parameter fitting functions because of their
ability to retrack non-ocean waveforms (presented in Section 5.5.3). The 50%
threshold level cannot always give improved results, particularly in the area closer to
the coastline. To validate the threshold level of the retracking algorithm, a statistical
analysis has been performed to suggest the quality of the retracked SSH data (Section
5.7.2). The evaluation turned out a detailed selection of the threshold levels.
When retracking individual waveforms in coasts using all possible retrackers, results
show improved SSHs, but biases exist among different algorithms. Thus, an

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 121
Figure 5.1 Block diagram of the coastal retracking system

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 122
algorithm assessment has been implemented to analyse and estimate these biases. All
these investigations motivate the development of the coastal retracking system.
Finally, a coastal retracking system has been developed to retrack altimeter
waveforms in Australian coasts, which involves several external input data,
waveform data editing, waveform classification, fitting algorithms and threshold
techniques. The flow diagram in Figure 5.1 illustrates how altimeter waveforms are
reprocessed to give the corrected SSH data. The procedure follows the main steps as:
(1) Data editing - edit out the waveforms that do not show the leading edge in the
range window, retain waveforms that show a leading edge regardless of
complexity of the whole waveform;
(2) Waveform classification - sort waveforms based on different shapes, and
indicate which retracker will be used for each;
(3) Fitting algorithms - make initial estimates of the parameters, determine
observational weights, and run iteration procedure;
(4) Threshold retracking - retrack waveforms with the narrow width and high peak,
as well as the waveforms that failed with the fitting algorithms; and
(5) Corrections - include the biases between each method, range correction, and
the geophysical and environmental corrections.
The description of each step will be presented in this Chapter and the ancillary input
data sets needed by the system are described in Section 2.5. An important factor in
the system is to accurately classify each return waveform so that it can be performed
by an appropriate retracker. Retracking results and analysis at the Australian coast
will be shown in Chapter 6.
5. 3 Extraction of the Geodetic Parameter: the Sea Surface Height
After retracking, the retracked range is then
Retracked Range = Observed Range + Retracking Range Correction
Since the retracking system consists of fitting algorithms and threshold retracker, the
biases between them (discussed later in Section 5. 8) must be applied to the range
measurements. Biases can be calculated by retracking the same cycle of ocean

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 123
waveform data. The corrected range in this study is based upon the retracked range
from the fitting algorithm so that the bias is applied to the range which is retracked
using the threshold retracker by
Corrected Range = Retracked Range + Bias
In addition, calculating an accurate SSH from the altimetric measurement involves
accounting for not only the waveform range corrections, which are discussed in this
study, but also correcting the range for atmospheric propagation delay and applying
appropriate ocean-tide correction to remove the time-varying tidal effect. After these
corrections are applied to the range, the SSH can then be calculated from the satellite
altitude and corrected range as
Corrected SSH = Orbit - Corrected Altimeter Range - Corrections.
Up to now, the corrected SSH data are used at 20 Hz. They can produce 1 Hz or 2 Hz
data products for geodetic applications.
5. 4 Analysis of Waveform Shapes
5.4.1 Determination of the Peaks in the Waveform
The peaks in the waveform (cf. Figure 5.2) can be estimated from an analysis of the
second derivative of the waveform. The first derivative, W ′(t)
, is approximated by
the first forward differences with respect to the gate number
W ( tk
+ 1)
−W
( tk
)
W '( t)
=
∆t
(5.1)
where W t ) is the waveform sample at the time t , ∆ t is the time interval between
( k
waveforms t
k + 1
and t
k
, and k = 1L64
(depending on the altimeter) is the gate
number of the waveform.
Using results of contiguous first forward differences, the second derivative of the
waveform, W ′′(t
) , can be approximated by second forward differences as

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 124
W ′(
t W ′
k + 1)
− ( tk
)
W ′ ( t)
=
∆t
(5.2)
If there are n p peaks (n p = 6 in this study) in the waveform, the gate locations of all
possible peaks L max (i) (i = 1, 2, ···, n), then, can be determined, where the second
derivatives cross zero from negative to positive, and vice versa for all minimum
locations L min (i).
Finally, the peak amplitudes, W amp (i), can be found from the waveform
corresponding to the gates of L max (i). All peaks are checked to determine whether
they are larger than an input criterion, which is obtained prior to the retracking
procedure by analysing ocean waveform amplitude features. Peaks less than this
criterion are not significant. In addition, if the interval between two peaks is too
small, only the larger one is selected as the peak. After editing, the remaining peak(s)
will be used to classify the waveforms. The maximum number of waveform peaks
(ERS-1/2) in Australian coastal regions is found to be three. Results of a detailed
analysis will be given later in Chapter 6.
5.4.2 Waveform Classification
The shape of the radar altimeter waveform depends strongly on the reflecting surface
properties. Small amplitude roughness and slopes of the surface will cause a
broadening of the leading edge of the return. Multiple reflecting surfaces within the
altimeter footprint will cause the waveform to be the superposition of the different
returns from each surface elevation, thus forming the multi-peaks. Over these
multiple reflecting surfaces near the land, the waveform shape cannot follow the
Brown (1977) model.
Waveform classification is based upon a wide ranging analysis of the waveform
shapes at the Australian coast. After observing a number of ERS-2 coastal
waveforms, they can be sorted into nine categories (Figure 5.2) depending on the
waveform shape features, such as amplitude, width, and measure of peaks. Of these
waveform features, the peaks can be ascertained using the method described in
Section 5.4.1, and waveform amplitude and width can be determined using the

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 125
OCOG method (Wingham et al., 1986). A detailed waveform analysis by
geographical area will be given in Chapter 6.
The shape characteristics of the waveform revealed in Figure 5.2 are generally
associated with a particular type of surface. Of these waveforms in coastal regions,
most of them (~80%) present a shape of ocean-like single ramp without the
significant peak in the range window (Figure 5.2a), though the waveform profiles
may shift left or right with respect to the predesigned tracking gate. This type of
return has been found in this study over ocean surfaces at any distance up to 300 km
from the coastline. Waveforms shown in Figure 5.2b to Figure 5.2g are usually found
when the altimeter simultaneously illuminates both water and land surfaces, where
the elevation difference between both surfaces within the pulse-limited footprint is
less than ~28 m (due to the scale of the range window). Because of two or more
elevation surfaces in the footprint, such waveforms occur ~0-10 km from the
coastline (Chapter 3). The narrow-peaked waveform (Figure 5.2h) occurs over the
standing water near coastline, such as bays, gulfs, estuaries, and harbours, as well as
inland lakes. A distinct feature of this type of waveform is its extremely high
amplitude (typically ~8400 counts in average from Section 6.2.2).
Therefore, waveform categories in coastal regions indicate the specific reasons why a
retracking system is appropriate in coasts. The primary factor is that many waveform
shapes observed in coastal regions are not encountered in data sets over open oceans.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 126
(a) No Significant Peak
(b) Single Pre-peak
Power (counts)
2000
1500
1000
500
0
1
6
11
16
21
26
31
36
41
46
51
56
61
B i n s
Power (counts)
3000
2500
2000
1500
1000
500
0
1
6
11
16
21
26
31
36
41
46
51
56
61
B i n s
Power (counts)
3000
2500
2000
1500
1000
500
0
(c) Single Post-peak
Power (counts)
2500
2000
1500
1000
500
0
(d) Single Middle-peak
1
6
11
16
21
26
31
36
41
46
51
56
61
1
6
11
16
21
26
31
36
41
46
51
56
61
B i n s
B i n s
Power (counts)
4000
3000
2000
1000
0
(e) Double Pre-peaks
Power (counts)
2000
1500
1000
500
0
(f) Double Post-peaks
1
6
11
16
21
26
31
36
41
46
51
56
61
1
6
11
16
21
26
31
36
41
46
51
56
61
B i n s
B i n s
(g) Multi Peaks (h) Sharp Peak
Power (counts)
2000
1500
1000
500
0
Power (counts)
20000
15000
10000
5000
0
1
6
11
16
21
26
31
36
41
46
51
56
61
1
6
11
16
21
26
31
36
41
46
51
56
61
B i n s
B i n s
Figure 5.2 Typical ERS-2 waveform categories in Australian coasts

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 127
5. 5 Fitting Functions
To use fitting algorithms, an analytic expression of coastal waveforms is essential,
which describes the interaction of the radar pulse with the scattering surface. As
stated, estimating non-linear wave parameters is not the aim of this study. In fact, it
is impossible to accurately estimate non-linear wave parameters using contaminated
waveforms in coasts. Therefore, the fitting functions used in this study aim at the
determination of the range correction through the revised estimate of the tracking
gate. In the proposed system, an ocean model and the β-parameter models will be
used as fitting functions, described next.
5.5.1 The Ocean Model
In Chapter 3, a modified expression (Equation 3.31) for the waveform over non-
Gaussian ocean surface was presented. The sea surface skewness λ cs
in Equation
(3.33) usually takes a small magnitude. Rodriguez and Martin (1994) estimate the
skewness using TOPEX Ku-band waveforms (cycles 3-27) as between 0 - 0.11 for
both ascending and descending passes, which is obtained by averaging estimates
over all cycles. Therefore, it can be safely neglected in the waveform fitting function
without the loss of the retracking accuracy. For λ = 0 (i.e., neglecting skewness),
Equation (3.31) is the linear surface case and the result can be reduced to a form
similar to the Brown (1977) model as
cs
1 ⎡ ⎛ τ ⎞ ⎤ ⎡ ⎛ d ⎞⎤
P( t)
= PN + A0
Aξ ⎢erf
⎜ ⎟ + 1⎥
exp⎢−
d⎜τ
+ ⎟
2
⎥
(5.3)
⎣ ⎝ 2 ⎠ ⎦ ⎣ ⎝ 2 ⎠⎦
in which five fitting parameters are:
(1) The thermal noise level P
N
,
(2) The amplitude scaling term A 0
,
(3) The tracking-gate-related time t 0
(i.e., the time corresponding to the mean sea
surface),
(4) The risetime σ , and
(5) The off-nadir angle ξ (represented by Aξ
Equation 3.7).

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 128
Equation (5.3) will be the model used in this study for the purpose of waveform
retracking. For simplicity of presentation, it will be called the revised ocean model
hereafter.
After waveform retracking, the retracking range correction, dr , is given as
dr = G ⋅ t / t − )
(5.4)
2M
(
0 p
g
0
where
G 2 M
= t p
⋅ c / 2 is the conversion from numbered gates to metres, t p
is the
pulse width, and
g
o
is the location in gates of the altimeter tracking gate.
The revised ocean model (Equation 5.3) describes the waveform reflected from the
linear Gaussian ocean surface. Compared to the Brown model (Equation 3.12), the
differences between them are that the ocean model includes the effects of the Earth’s
curvature and the modified Bessel function in Equation (3.2) has been replaced by an
approximated expression.
It is recalled briefly here about the series expansion of the Bessel function (Equation
3.17). In practice, it has been expanded to different orders (Parsons, 1979; cf. Davis,
1993; Hayne, 1980; Rodriguez, 1988). When retracking waveforms to determine the
range correction using the Brown (1977) model, it is generally expanded to zero
order (Parsons, 1979; e.g., Davis, 1993). This means that I ( β t ) 1 , and the
0
=
Bessel function does not change its value with varying pulse return times and
corresponds to a zero off-nadir angle.
The approximation used in Equation (5.3) comes from Rodriguez (1988), who
2
expands the Bessel function to first order and keeps terms of order β t and lower,
then replaces series by exp( β
2 t / 4)
. The error introduced from this approximation is
less than 1% (Rodriguez, 1988). This approximation is more reasonable for actual
waveforms, because it takes account of the non-zero off-nadir angle and different
pulse return times.
In Equation (5.3), the sea surface roughness (through the composite risetime) and the
off-nadir angle are two main factors that affect the shape of the waveform. Mean
ocean-return waveforms in Figure 5.3 show the family of the curves, which result

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 129
from varying the RMS wave height (SWH from 0 - 20 m with a 2 m interval). As can
be seen from Figure 5.3, varying SWH changes the rise time (or the slope) of the
leading edge of the waveform. The rougher the sea state, the longer the rise time (i.e.,
lower gradient) of the waveform’s leading edge.
550
500
450
SWH = 0 m
400
Power (counts)
350
300
250
200
SWH = 20 m
150
100
50
0
0 10 20 30 40 50 60
Bins
Figure 5.3 Variation of the leading edge’s slope with varying SWH (0 - 20 m),
modelling waveforms using the ocean model ( ξ = 0 °).
In addition to the surface roughness, another factor that can affect the waveform rise
time is the surface slope or altimeter antenna off-nadir pointing angle. Figure 5.4
shows the effects of the off-nadir angle on the waveform, for which SWH = 4 m is a
typical situation of the sea surface and the numbers give the values of the off-nadir
angles. As can be seen from Figure 5.4, there are three effects:
(1) The off-nadir angle ξ changes not only the slope of the trailing edge, but also
the slope of the leading edge. Both slopes of the leading and trailing edges
decrease with increasing ξ . However, the location (or range to the mean sea
surface) of the mid-point on the leading edge of the waveform does not change
with varying ξ .

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 130
500.0
Power (counts)
400.0
300.0
200.0
ξ=0.0
ξ=0.2
ξ=0.4
ξ=0.6
100.0
ξ=0.8
ξ=1.0
0.0
0 8 16 24 32 40 48 56 64 72 80
Bins
Figure 5.4 Effect of the off-nadir angle (in degrees) on the amplitude, leading edge,
and especially the slope of the trailing edge, modelling waveforms using the ocean
model (SWH = 4 m).
(2) The magnitude of the amplitude of the waveform is affected as well. Since an
off-nadir angle changes the decay of the trailing edge, it will affect the value of
the AGC gate and hence the AGC scaling of the waveform.
(3) The slope of the trailing edge changes approximately linearly with off-nadir
angle (Barrick and Lipa, 1985). The trailing edge’s slope changes its direction
at ξ ≈ 0.6° (Figure 5.4). The reason is that as the surface slope increases from
zero, the pulse-limited footprint moves from the centre of the beam and the
leading edge part of the waveform (both leading edge and amplitude) is
attenuated by the antenna gain function. At steeper angles ( ξ >1°), the
attenuation becomes so great that the AGC loop can no longer compensate the
loss in signal strength (Martin et al, 1983; Brrick and Lipa, 1985). As a result,
loss of tracking will occur.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 131
5.5.2 Five- and Nine-β-Parameter Models
The β-parameter models (Equations 3.40 and 3.43) are also employed in the
retracking system because of their capability of dealing with complex waveforms
reflected from single or multiple scattering surfaces. They are developed based upon
a modified version of the Brown (1977) ocean return function (Parsons, 1979), but
add an additional slope parameter after the ramp (discussed later in Section 5.5.3). A
description of the five- and nine-β-parameter models can be found in Section 3.5.1.
The retracking corrections dr can be computed by the following formulas.
(a) The Five-β-Parameter Model
dr = G 2 M
⋅ ( β − g
0
)
(5.5)
where β
30
is the retracking gate estimate in units of waveform gates.
30
(b) The Nine-β-Parameter Model
dr = G 2 M
⋅ ( β − g )
(5.6)
1
30 0
dr = G 2 M
⋅ ( β − g )
(5.7)
2
60 0
where β
60
is the retracking gate estimate in units of waveform gates (related to the
second ramp), dr 1 is the correction related to the range to at-nadir mean sea level,
and dr 2 is the correction related to the range to the off-nadir (assumed) land surface.
5.5.3 Comparison between the Ocean and Five-β-Parameter Models
Both ocean and five-β-parameter models are fundamentally based upon the Brown
(1977) model. Comparison between them will give qualitative guidance to selecting
an adequate fitting function for the retracking procedure. Therefore, an assessment of
these functions will be discussed in this Section. In order to compare them directly, it
is necessary to simplify the ocean model (Equation 5.3). By assuming a zero offnadir
angle (i.e., A = 0 ), Equation (5.3) can be rewritten as
ξ

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 132
2
⎡
⎤
2
1 ⎛ t − t ⎞ ⎡ ⎛
⎞⎤
0
− ασ
ασ
P(
t)
= P + ⎢ ⎜
⎟
+ 1⎥
exp⎢−
⎜ − −
⎟
N
PA
erf
α t t0
⎥
(5.8)
2 ⎢⎣
⎝ 2σ
⎠ ⎥⎦
⎣ ⎝ 2 ⎠⎦
2
in which P A
= A0
is the amplitude scaling term. The term ασ in Equation (5.8) can
be computed by substituting the altimeter parameters from Table 2.1. The computed
2
values of ασ as a function of SWH are listed in Table 5.1. It can be seen, assuming
2
that a typical situation of the ocean surface (i.e., SWH = 4 m), the values of ασ are
approximately 0.16 ns for both ERS-1/2 and TOPEX. Thus, this term may be safely
neglected for the purpose of the waveform retracking, and Equation (5.8) can be
reduced further to
Table 5.1 ασ² Values for ERS-1/2 (ocean mode) and TOPEX
SWH (m) ERS-1/2 (ns) TOPEX (ns)
0.5 0.01 0.01
2.0 0.05 0.05
4.0 0.16 0.16
6.0 0.35 0.35
8.0 0.62 0.61
10.0 0.96 0.95
12.0 1.38 1.36
14.0 1.88 1.85
16.0 2.45 2.41
18.0 3.10 3.05
20.0 3.82 3.76
P
1
2
⎡
⎢
⎣
⎛ t − t
[ − ( t − )]
0
( t)
= PN
+ PA
erf ⎜ ⎟ + 1 exp α t0
⎝
⎛ t − t
⎝ σ
⎞
2σ
⎠
⎤
⎥
⎦
0
= PN
+ PA
P⎜
⎟exp
α
⎞
⎠
[ − ( t − t )]
0
(5.9)
where P (x)
is the normal probability distribution (Equation 3.42).

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 133
Comparing Equation (5.9) with Equations (3.40) and (3.43) [set i = 1 ], the only
difference between the two functions is in the trailing edge of the waveform. In the
region of the trailing edge, the normal probability distribution P (x)
is equal to unity
for SWHs less than 10 m. Thus, in this region, Equations (5.9), (3.40), and (3.43) can
be written as P t
(t)
, y t
(t)
and y te
(t)
as
[ − ( t − )]
P ( t)
= P + P exp α t
(5.10)
t
y t
N
A
1
+ β
21
1
( β )
51
1
0
( t)
= β + Q
(5.11)
and
y t
1
+ β
21
exp
( − β )
( t)
= β
Q
(5.12)
51
1
From the linear ocean surface (Equation 5.11), β
51
is the slope of the trailing edge.
In addition, Equations (5.10) and (5.12) can be rewritten more simply as
( P ( t)
P ) = [ ln( P ) +α t ] −α
t
ln (5.13)
t
−
N
A 0
and
ln
( y t
( t)
β1 ) = ln( β
21
) − β
51Q1
− (5.14)
where α and β
51
are also the slope of the trailing edge of the ocean model and the
five-β-parameter model with a exponential decay trailing edge, respectively.
The slope of the trailing edge in the ocean model is related to the physical parameters
of the radar antenna pattern, the Earth’s radius, and the off-nadir angle, according to
4c
1
α =
cos(2ξ
)
γ h (1 + h / R)
(5.15)
However, it can be seen from Equations (5.11) and (5.14) that the slope of the
trailing edge depends on only a non-physical (or empirical) parameter, β 51
. In this
sense, the five-β-parameter model is derived partly from the ocean model (Equation
5.3) or the Brown (1977) model (Equation 3.12) in the case of a linear ocean surface.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 134
The slope of the waveform trailing is presented using an empirical parameter rather
than the antenna off-nadir angle in the waveform. Therefore, the five-β-parameter
model may be defined as a semi-empirical surface-scattering model.
Variations of the slope of the waveform’s trailing edge modelled using the five-βparameter
function (with linear trailing edge) for SWH = 4 m, β 21
= 100 (counts),
and − 0.03
≤ β
51
≤ 0. 03 (0.01 interval) are shown in Figure 5.5. From this, two
effects are evident:
1000.0
β5=0.03
800.0
β5=0.02
Power (counts)
600.0
400.0
β5=0.01
β5=0.0
β5=−0.01
200.0
β5=−0.02
0.0
β5=−0.03
0 8 16 24 32 40 48 56 64 72 80
Bins
Figure 5.5 Variations of the trailing edge’s slope with varying parameter β 5 ,
modelling waveforms using the five-β-parameter parameter function (linear trailing
edge, SWH = 4 m, β
21
= 500 counts, β 5 = β
51).
(1) As the value of β 51
increases, the power of the waveform increases, i.e., the
values of β 51
affect the amplitude. Similarly as for the ocean model, the
position of the mid-point on the leading edge of the waveform is not influenced
by variations of the slope in the trailing edge.
(2) The slope of the trailing edge increases with increasing values of β 51
. The
descending trailing edge of the waveform changes to a rising trailing edge after
β
51= 0.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 135
The results in Figure 5.5 show a common feature with Figure 5.4, in which the
trailing edge region is nearly linear. However, β
51
can vary to a greater extent than
the slope of the ocean model. This makes it possible that the five-β-parameter can fit
more complex waveform shapes over non-ocean surfaces.
Figure 5.6 shows modelling ERS-2 waveforms using the five-β-parameter function
(Equation 3.43, with an exponential decayed trailing edge) for SWH = 4 m, β 21
=
100 counts, and − 0.03
≤ β
51
≤ 0. 03 (0.01 interval). From Figure 5.6:
(1) Although β
51
mainly affects the slope of the trailing edge and the amplitude of
the waveform, it slightly changes the slope of the leading edge. However, this
change does not vary the mid-point location on the leading edge.
(2) Unlike the effects in the case of the linear trailing edge, the exponential decay
trailing edge decreases both the slope and the amplitude with increasing β 51
.
Again, the slope of the trailing edge changes its direction after β
51= 0.
1500.0
β5=−0.03
Power (counts)
1000.0
500.0
β5=−0.02
β5=−0.01
β5=0.0
β5=0.01
β5=0.02
β5=0.03
0.0
0 8 16 24 32 40 48 56 64 72 80
Bins
Figure 5.6 Variations of the trailing edge’s slope with varying parameter β 5 ,
modelling waveforms using the five-β-parameter parameter function (exponential
decayed trailing edge, SWH = 4 m, β
21
= 500 counts, β 5 = β
51).

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 136
In addition, if the above analysis is performed on Equation (5.3), one can obtain the
trailing edge part as
2
⎡ ⎛ ασ ⎞⎤
P(
t)
= P + exp⎢−
⎜ − −
⎟
N
A α t t0
⎥
(5.16)
⎣ ⎝ 2 ⎠⎦
and
ln
[ P(
t)
− P ]
N
⎡ ⎛
= ln( A)
+ ⎢−α
⎜t
− t
⎣ ⎝
⎡ ⎛
= ⎢ln(
A)
+ α
⎜t
⎣ ⎝
0
0
2
ασ ⎞⎤
−
⎟⎥
2 ⎠⎦
2
ασ ⎞⎤
+
⎟⎥
−αt
2 ⎠⎦
(5.17)
in which
A
0
= A A , and α is the slope of the trailing edge of the ocean waveform.
ξ
Since only ξ is a variable in the function α (ξ ) related to an altimeter
(Equation 5.15), the trailing edge’s slope varies with variation of ξ . This is the same
result as that discussed in Section 5.5.1, in which ξ has been found to be a dominant
factor of the trailing edge’s slope.
By comparison, it is evident that different models have different slopes of the trailing
edge and variations in the slope of the trailing edge do not change the position of the
mid-point on the leading edge of the waveform. In coastal regions, however,
waveforms show diverse shapes with different slopes (see Figure 5.2) of the trailing
edge. An inappropriate slope description may degrade the parameter estimates of the
mid-point (i.e., retracking gate) on the leading edge or cause a failure of the fitting
procedure. The above analysis is significant for understanding the differences among
the fitting functions. More importantly, it is the different slope characters of the
trailing edges that make these functions complement each other for retracking
different waveform shapes in coastal regions.
5. 6 Iterative Nonlinear Fitting Procedure
No matter whether or not the convolution functions contain non-linear wave
parameters, the solution of the fitting functions is characterised by a non-linear
model. Thus, there is no straightforward method to optimally fit the model to the

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 137
return waveforms and extract the model parameters. This is further complicated due
to the noisy and shape-mixed features of measured waveforms in coastal regions. All
fitting models describe the mean (or idealised) power, whereas real waveforms
measured at the altimeter fluctuate about the mean because of the fading noise,
variations in the surface geometry, and inhomogeneities near the land. Therefore, the
determination of model parameters is based on an iterative least squares approach
and an appropriate weighting scheme.
5.6.1 Solutions of the Waveform Retracking Fitting Function
In general, non-linear models in the standard statistical techniques are linearised by
using approximate values via a first-order Taylor series expansion of the unknown
parameters. The corrections to the approximate values become the unknown
parameters and solutions for the corrections are iterated in order to improve the
approximate values. Taking y i
as a function of the estimated parameter
k = 1,
2, K,m
, it can be written as:
x
k
,
y
=
( x )
i
F i
m×1
(5.18)
where
i = 1,
2, L,
n is the bin-number of the waveform samples in range window
(e.g., n = 64 for ERS-1/2). Assuming x = x 0
+ δ x , y
i
is approximated at x 0
in the
parameter-space by the Taylor series expansion to the first order as
∧
y ≈ y = F x ) + F′
( x ) δ x
i
i
i
(
0 i 0
(5.19)
where F x ) is an approximation computed by the fitting function using the
i
( 0
approximated values of the unknowns, and
F′
( x
i
0
)
⎛
⎜ ∂y
=
⎜ ∂x1
⎝
∂y
∂x
L
∂y
∂x
x 2
0 x
m
0
x0
⎞
⎟
⎟
⎠
(5.20)
is a matrix populated by the partial derivatives of the functional model with respect
to the unknowns. From Equations (5.19) and (5.20), the function is formed as

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 138
V = Aδ x − L
(5.21)
where A = F′
x ) is the design matrix, L = l − F x ) is the vector of the n
( 0
observations determined by the measured waveforms and the first-order
approximations at x
0
. Equation (5.21) can be solved by the least squares approach:
( 0
T −1
T
δ x = ( A PA)
( A PL)
(5.22)
where P is the weight matrix of the observations. The standard deviations of the
estimated parameters are given as
2 T −1
Q x
= σ
0
( A PA)
,
(5.23)
σ
2
0
=
T
V PV
n − m
.
(5.24)
The initial approximation of x
0
is corrected to produce revised estimates
x = x 0
+ δ x . Then, the x replaces the original x 0
as new a priori values and the
iterative procedure is repeated N times until a defined relative error criterion is
satisfied, or a maximum number of iterations is reached.
5.6.2 Linearisation of the Waveform Retracking Fitting Function
Having expressed the return power as analytic functions, the linearisation of these
functions can be explicitly implemented by computing the partial derivatives of each
parameter. Taking the revised ocean model (Equation 5.3) as an example, the five
partial derivatives of the modelled waveform with respect to the parameters are given
by
∂P
∂
P N
= 1
(5.25)
∂P =
∂A
P
0
A 0
(5.26)

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 139
∂P
∂t
0
=
A0
⎧ 2
2 ∂u
∂v
Aξ
exp( −v)
⎨ exp( −u
) − [ erf ( u)
+ 1]
2 ⎩ π ∂t0
∂t0
⎫
⎬
⎭
(5.27)
∂P
=
∂σ
A
2
0
⎧
Aξ
exp( −v)
⎨
⎩
2
exp( −u
π
2
∂u
) −
∂σ
[ erf ( u)
+ 1]
∂v
⎫
⎬
∂σ
⎭
(5.28)
∂P
2
∂ξ
=
A0
2
+
⎪⎧
1
Aξ
exp( −v)
⎨
⎪⎩ A
2
exp( −u
π
2
ξ
∂u
)
2
∂ξ
[ erf ( u)
+ 1]
−
∂Aξ
2
∂ξ
[ erf ( u)
+ 1]
∂v
2
∂ξ
⎫
⎬
⎭
(5.29)
where
u =
τ
2
(5.30)
⎛ d ⎞
v = d⎜τ
+ ⎟
⎝ 2 ⎠
(5.31)
It should be noted that the derivative with respect to ξ 2 has been computed rather
than the derivative with respect to ξ in Equation (5.29). The reason for this is that
the derivative with respect to ξ could be zero if ξ → 0
, and hence the off-nadir
angle cannot be estimated. Thus, assuming that the two sinusoidal terms in Equations
(3.14) and (3.15) can be approximated to the first order for small angles, ξ can be
2
replaced by ξ to avoid the problem. An additional problem with the introduction of
2
ξ is that the estimate of ξ 2 could contain negative values due to the presence of
noise. However, this effect can be neglected in the study, as the aim is to extract the
retracking gate estimate rather than the off-nadir angle. For research that is interested
only in the off-nadir estimate, other approaches can be used (cf. Barrick and Lipa,
1985).
5.6.3 Initial Estimates of the Unknown Model Parameters
High-quality initial estimates of the parameters are important to reduce the number
of iterations and guarantee a convergence in the least squares estimation (and hence

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 140
computational time) that the non-linear least squares procedure takes to converge. If
the initial parameters cannot be estimated precisely, the parameter corrections after
one iteration might be very large, causing the algorithm to diverge rapidly or to
converge on the incorrect result. However, there is no predefined method that
determines the optimum a priori values for the altimeter waveforms. Therefore, to
generate initial parameter estimates, different schemes were tested and evaluated
based on the speed of convergence, the mean-squared error, and ability to fit a wide
variety of waveforms. In accordance with the testing results of these three factors, the
OCOG method can also provide additional information when making initial
estimates of the parameters (see below).
(1) The Ocean Model
Before ascertaining the initial parameter estimates, the initial estimate of the thermal
noise (or noise baseline),
0
P
N
, is determined by calculating the average amplitude of
the waveform samples over five of earlier gates (e.g., gates 6-10 for POSEIDON)
P
1
10
0
N
= ∑W i
5 i=
6
(5.32)
where
W
i
is the i -th measured waveform sample. This value is then subtracted from
each waveform sample and the OCOG method is applied to the waveforms to
calculate relative parameters of the rectangle (described in Section 3.5.2). The results
from the OCOG method are used to determine the initial approximations of the
amplitude term (
0
A
0
) and the time corresponding to the location of the mid-point at
the range bins ( t 0 0
). The waveform amplitude ( A ) is set to be the initial estimate of
0
A
0
, and t 0 0
can be obtained from the gate number of the leading edge ( LEG ) from
the OCOG by
t
0
0
= LEG ⋅T
(5.33)
where T is the duration of the radar pulse ( T = 3. 03ns for ERS-1/2 and T = 3. 125ns
for TOPEX/POSEIDON).

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 141
The initial approximation of the RMS surface roughness (i.e., the composite
0
risetime), σ
cr
, can be estimated by substituting SWH = 2cσ
s
into Equation 3.13 to
give
σ
0
cr
⎡
= ⎢
⎢⎣
( f t
T )
2
2
1/ 2
⎛ SWH ⎞
2 ⎥ ⎥ ⎤
+ ⎜ ⎟
⎝ c ⎠ ⎦
(5.34)
where factor f t
is an approximation (0.425 for T/P and 0.526 for ERS-1/2) of the
compressed pulse shape by a Gaussian function (Marth et al., 1993; Batoula et al.,
1999), SWH = 4
m, which is the common case over ocean surfaces. The initial
2
approximation of the (small) off-nadir angle squared, ξ , can be set to zero.
(2) The Five-β-parameter Model
The initial approximations for the five- β-parameter model are determined using the
method from Anzenhofer et al. (2001), but modified slightly to account for the
thermal noise. The initial estimate of the parameter β
1
is set using Equation (5.32).
The maximum amplitude of the waveform after removing the thermal noise (item 2)
is defined as the initial estimate of the parameter β
2
. The gate number of the leading
edge from the OCOG is taken for the initial approximation of the parameter β
3
. The
initial estimate of the waveform risetime β
4
is set empirically to 1.3 gates, and the
slope of the trailing edge β
5
can be set to zero.
(3) The Nine-β-parameter Model
Measured waveform shapes related to this function usually show double ramps
(Figure 3.6). Of these parameters, the initial estimate of the thermal noise can be set
using Equation (5.32). The initial estimate for the location of the first ramp (usually
related to the mean sea surface) is determined by locating the first sample gate,
whose slope exceeds a threshold value (e.g., 50 counts/gate for ERS-1/2). Once this
threshold level is exceeded, the next five gates are checked to find the maximum
slope. The gate number corresponding to this maximum slope is used as the initial
approximation for the location of the first ramp. The same method can be used to
determine the initial estimate for the second ramp location. Other initial estimates

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 142
can be defined using the similar method described above for the five-β-parameter
model.
5.6.4 Data Weight and Outlier Detection Scheme
Before fitting, a priori weights are determined using the method from Anzenhofer et
al. (2000). It is acknowledged that the heavier weight emphasis should be put on
fitting to the ramp locations which define the retracking correction (e.g., Zwally,
1996; Brenner et al., 1993; Anzenhofer et al., 2000). Using the parameters estimated
from the OCOG retracker, the location of the leading edge, tp , can be estimated.
Then a priori weights of the ERS-1/2 waveform can be determined empirically for
the ocean model and the five-parameter model as
0
p i
= 200 for t ≤ tp -1
(5.35)
= 100 for tp-1
< t ≤ tp + 2
0
p i
(5.36)
0
p i
= 50 for tp + 2 < t ≤ 64
(5.37)
This is a reasonable weight scheme because it accounts for the characteristics and
distribution of the useful information in the waveform. The heaviest weights are put
on the waveform samples in the first plateau (i.e., thermal noise region) before the
leading edge so that they will be thought to be relatively accurate measurements in
the adjustment. The second heaviest weights are applied to the samples of the leading
edge itself. The lightest weights are given to the waveform samples in the trailing
edge, making them contribute less to the solutions (cf. Figure 3.5).
In general, the data in the trailing edge show obvious undulations caused by the
‘fading’ noise, which results in the incoherent superposition of signals from different
reflecting facets (Partington, 1991; Marth et al., 1993; Quartly and Srokosz, 2001).
The noise in the trailing edge greatly influences the iterative fitting procedure and
even accurate results cannot be estimated (Tokmakian et al., 1994). To solve the
problem, two approaches have been used. The first is to average waveforms in the
time span of several seconds, and then to fit a model to this averaged waveform (e.g.,

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 143
Tokmakian et al., 1994; Hayne and Hancock III, 1990). The second is to lightly
weight the waveform samples in the trailing edge (Anzenhofer et al., 2000; Zwally,
1996).
However, the averaging procedure is not always appropriate in coastal regions,
because ocean waveforms will be distorted by contaminated waveforms within the
averaging period. Therefore, the second method is used in the study. According to
Anzenhofer’s (2000) weight scheme, the data after the ramp have been
downweighted in the solution (see Equation 5.37). However, it is found in this study
that some unexpected undulations in the trailing edge significantly affect the solution.
To overcome this problem, an outlier detection approach from an iterative weight
scheme is used in this study.
In this iterative procedure, the initial least squares adjustment is conducted using the
0
a priori values ( β
k 0
) and a priori weights ( p
i
). After the first adjustment, a weight
function
w
i
is given as (Li, 1988)
⎧ 1
⎪ for vi
> 0.7σ
0
vi
w
i
= ⎨
(5.38)
⎪
1
for vi
≤ 0.7σ
0
⎪⎩
0.7σ
0
2
where v
i
and σ
0
, which is the estimation of the unit weight variance, are computed
by Equations (5.21) and (5.24), respectively. The next step then is to determine the
weight (
j+1
p
i
) for the next adjustment using wi
as follows:
p =
j+1
i
p
j
i
w
i
(5.39)
where j means the j-th iteration in the adjustment. In the meantime, estimates of the
β parameters take the place of the originals and repeat the entire cycle to produce
improved estimates of the β parameters. In this way, the observations considered to
contain the outlier will be downweighted but still be kept in the dataset, while nonoutlier
observations will maintain full weight (though downweighted with respect to
the other data in the range window) after the iterative cycle.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 144
5.6.5 Iterative Procedure
The procedure taken here is similar to that used by Anzenhofer et al. (2000), Zwally
(1996) and Davis (1993). The parameter estimates are obtained by an iterative
procedure as follows:
(1) Estimate initial values of the model parameters and the weights of the
measured waveforms.
(2) Compute the new values for the least squares approach using the first-order
Taylor expansion and measured waveforms (Equation 5.21).
(3) Compute a correction to the current estimates of the unknown parameters
(Equation 5.22), and the standard deviations of the parameters (Equation 5.23).
(4) Update the parameter estimates for the new initial estimates.
(5) Repeat steps 2-4 until some exit criterion in Table 5.2 is satisfied.
From the flowchart of the fitting algorithm (Figure 5.1), after corrected parameter
estimates are obtained from Equation (5.22), several exit criteria are applied to
determine whether the fitting iteration continues.
(1) The first exit criterion is based upon an absolute difference specified for each
unknown parameter between ( i +1)
and i iterations. Table 5.2 lists the
absolute difference values for the ocean model parameters. If the absolute
difference of all parameter estimates from two successive iterations is less than
these criteria, the fitting iteration terminates.
(2) The second exit criterion is based upon the relative change in the standard
2
deviation of the unit weight, σ
0
. If the relative change of the σ
0
is less than
2% from one iteration to the next, the fitting loop is ended.
(3) The final exit criterion is the maximum number of the iterations. The
maximum number in this study is set to 20 to avoid infinite looping situation in
the case of divergence. In practice, a flag which records the status of the fitting
loop will be returned with the parameter estimates. The parameter estimates are
retained if the algorithm exits before 20 iterations and the flag does not show
iteration failure, while they are discarded if the other criteria fail in 20
iterations or the loop ends at the maximum iterative number.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 145
In addition, for each iteration step, some constraints are applied, such as amplitude
parameter estimate must be larger than zero.
Table 5.2 Criteria of the absolute differences between two iterations (the ocean
model for ERS-1/2)
Parameters Absolute Differences Units
P
N

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 146
Equation (5.32) using five unaliased waveform samples before doing retracking.
Once the estimate of P
N
is obtained, the threshold level, T
l
, is then set to
T = P + T
(5.40)
l
N
LO
where T
LO
is the threshold level determined using the selection scheme (d) in Section
5.7.2. The retracking location on the leading edge of the waveform (or at the range
window),
g
r
, is linearly interpolated between the bins adjacent to T
l
using
=
T
−1+
−W
l k −1
g
r
g
k
(5.41)
Wk
−Wk
−1
where
g
k
is the location of the first gate exceeding T l
. The g r
is set to ( g
k
−1)
in
the case of W k
W
1
in Equation (5.41). Then, the range correction can be obtained
= k −
from a bin-to-metre conversion factor timing the gate difference between this
estimate and the designed tracking gate as
dr = G ⋅ ( g −
0
)
(5.42)
2M
r
g
where g
0
is the expected tracking gate in units of waveform gates.
5.7.2 Selection of the Threshold Level
As stated, Brooks et al. (1997) select the threshold level for mixed waveform shapes
in coastal regions based upon a visual examination method. These are two concerns
related to this selection. The first is that threshold values are higher than both the
noise and the minor waveform leakage in the early gates. The second is that the
values are sufficiently lower than the sought-after ocean returns. Brooks et al. (ibid.)
determine different threshold values for 18 TOPEX ground tracks near land using
both concerns. Then, they are used to retrack the coastal waveform along the same
pass. This method is good for selecting the threshold level because it takes account
for the strength of the power backscattered from different reflecting surfaces.
However, because an automatic selection of the threshold level is not available, this
restrains the technique from retracking a wide range of the waveforms in coastal
regions.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 147
Considering the shape variations of the waveform in the proximity of the coasts
analysed in Section 4.6, the selection of the threshold level used in the coastal
retracking system differs from these previously published methods used over ice
sheets (Zwally, 1996; Davis, 1995; Davis, 1997) and coastal regions (Brooks et al.,
1997). Four threshold schemes are investigated before arriving at the final threshold
level.
(a) Fixed Counts
The threshold level of the fixed count is defined here as the 50% of the meanwaveform
amplitude in counts. For each satellite ground pass, the mean waveform is
formed from one or two seconds of purely ocean waveforms prior to the coastal
waveforms along it. The power in counts corresponding to the 50% of the meanwaveform
amplitude is the fixed count for the pass. This fixed count is, then, used as
the threshold level to retrack the waveforms along the same pass into coastal regions.
The gate location corresponding to this value is linearly interpolated between the
bins adjacent to the crossing of the fixed counts on the leading edge of the coastal
waveform, being the estimate of the retracking gate. This scheme, in fact, determines
the retracking gate estimates using the 50% of the ocean-waveform amplitude, but it
assumes that the amplitude of the ocean returns does not change with respect to the
reflecting surfaces.
(b) 50% of the Waveform Amplitude
Over open oceans where the waveform is completely dominated by surface scattering,
the half-power point (i.e., the 50% threshold retracking point) best represents the
expected tracking-gate estimate which corresponds to the range to the at-nadir
surface within the altimeter pulse-limited footprint. Therefore, the second selecting
scheme is to choose the 50% threshold point as the threshold level.
(c) 30% and 50% of the Waveform Amplitude
Factors that affect the waveform’s shape (or amplitude) are primarily the off-nadir
angle or the surface slope and the surface geometry. Compared with brighter and
higher land, where returns usually show higher power, ocean returns that appear in
the range window tend to be lower power. Therefore, in the case of the mixed
waveform of land and water, an appropriate low threshold level is suitable such that
retracking can locate the half-power point of the leading edge of ocean returns.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 148
Mean Power and STD (counts)
2500
2000
1500
1000
500
0
Mean waveform
STD
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
Bins
Mean Power and STD (counts)
1200
1000
800
600
400
200
0
Mean waveform
STD
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
Bins
Figure 5.7 Mean waveforms and the standard deviation in two 5 km-wide bands of
0-5 km (top) and 5-10 km (bottom) from the Australian coastline.
Therefore, in this third scheme, two threshold levels are selected depending on the
distance between the observing point along the satellite groundtrack and the coast.
The 30% threshold level is only selected for the waveforms within 5 km distance
from the coast, while a threshold level of the 50% of the waveform amplitude is
chosen after 5 km from the coastline. The distance of 5 km is determined based upon
an analysis of the mean waveforms (Section 4.5.2). To determine the proper
threshold level within different distance bands from the coast, the average waveform
is formed using one cycle of ERS-2 20 Hz waveforms from the coastal regions of

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 149
interest within six 5 km-wide bands from the Australian coastline. Figure 5.7 shows
the mean waveforms and STD in two 5 km-wide bands of 0-5 km and 5-10 km from
the coastline. Beyond 10 km from the coastline, the mean waveforms show no
significant change to the 5-10 km band.
The position of the threshold crossing corresponds to the mid-point of the leading
edge. The 30% of the waveform amplitude (or 20% of the amplitude in some areas,
discussed below) is selected to be the threshold level for retracking waveforms
within 0-5 km from the coasts. When the distance from the coast is larger than 5 km,
the ocean scattering becomes predominant in the waveform returns so that the 50%
of the waveform amplitude can be used as an estimate of the threshold level.
(d) Varying Threshold Level
Although the 30% threshold level should effectively be a low-power strength
retracker in coastal regions for the mixed waveforms of land and water, it might
cause two problems. The first one is that waveforms without contamination will be
retracked inaccurately by this lower-level retracker. The second is that changing the
threshold level at 5 km distance from the coast might cause a step jump to the
retracked range measurements. To avoid both problems, a scheme of varying
threshold level is developed.
Different threshold levels are set depending on whether or not the waveform is
contaminated. If the waveform classification indicates a distorted waveform, the
threshold level is set to 30% (or 20%) of the waveform amplitude, otherwise the 50%
of waveform amplitude is set. Compared with the waveform shape, the distance from
the coast is not significant in this criterion. It links the threshold level to the
contaminated waveform, avoiding the step change that might occur in corrected
range measurements from the threshold level (c).
5.7.3 Results of Selecting the Threshold Levels
In order to ascertain the appropriate threshold level, one cycle of 20 Hz ERS-2
waveform data (March to April 1999) and ten (15º×15º) areas around the Australian
coast are used (Figure 5.8). The AUSGeoid98 (2'×2') geoid grid is used as an
external ‘ground truth’ for the result comparisons (see also Section 2.5.1). Because

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 150
the waveform contamination strongly depends on both the distance from the coast
and land surface topography along the coastline, computations are conducted in six 5
km-wide bands around the coasts. Waveforms have been threshold retracked using
the above schemes, respectively. The range correction calculated from waveform
retracking and geophysical corrections from the data products are then applied to the
measured range. The retracked SSH data are then obtained from the satellite altitudes
and corrected range measurements.
105˚
0˚
120˚
135˚
150˚
165˚
0˚
1 2
-15˚
3
4 5
6
-15˚
-30˚
-30˚
7
8 9
10
-45˚
105˚
120˚
135˚
150˚
-45˚
165˚
Figure 5.8 Ten 15º×15º coastal areas around Australia, showing the area number
(Lambert projection).
The comparison between the instantaneous SSH and the geoid height is an
appropriate tool to investigate the overall quality of the threshold-level selecting
schemes. For each along-track altimeter SSH measurement after retracking, the
gridded AUSGeoid98 geoid height is bispline-interpolated to the longitude and
latitude of the point. The difference between the mean SSH and geoid height can be
obtained by subtracting the interpolated geoid height from the corresponding SSH.
Then, the descriptive statistics are computed from this difference. The instantaneous
SSH, of course, is not equivalent to the geoid (see Section 2.5.2). The difference

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 151
between them includes the time variation (e.g., tides) and dynamic SST. However,
averaging can remove the time variation and so the remaining difference is between
the mean SSH and the geoid height. Since the precisely regional geoid model
AUSGeoid98 is used, the mean difference can be a good approximation of the
dynamic SST, which is of the order of ~1-2 m (e.g., Chelton et al., 2001). Therefore,
the verification of any improvement in the SSH will only be seen in the standard
deviations of the mean difference, with a smaller value indicating an improvement in
the determination of the threshold level.
Figures 5.9 and 5.10 show mean differences of the SSH and geoid heights and
standard deviations in a distance out to 5 km around Australia’s coasts, respectively.
Before retracking, the magnitudes of mean differences in ten areas are less than 2 m,
but the standard deviations (STD) are large in most of the areas. In area one, where
the land topography involves rocks and small barrier reefs in the north Australian
coasts, the STD is even larger (±19.93 m). The STDs are small (±0.47 m and ±0.49
m) in south-western and south-eastern areas seven and ten, where the coastal relief
comprises rocks, cliffs and large barrier reefs (cf. Thom, 1984).
After threshold retracking using scheme (b), the STD decreases in six areas.
However, it increases in the other four areas compared with the STD before
retracking. This suggests that the 50% threshold level is not an appropriate selection
for the waveforms within 5 km from the coast. The reason for this is that the mostly
varying land topography and standing water within this distance can heavily
contaminate the waveforms, so that many waveform shapes do not follow the ocean
model. After retracking using schemes (c) and (d), the STD shows improvements in
seven areas. Of them, the results from the scheme (d) show the most significant
improvement than any others. In contrast, scheme (a) gives the least improvement,
and even worse results in most areas after retracking compared with the results
before retracking. For areas 5, 8 and 10, an appropriate threshold level of 20% of the
waveform amplitude has been found from the tests.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 152
2.5
2
unretracked
50% threshold level
30% threshold level
varying threshold level
fixed counts
1.5
Mean Differences (m)
1
0.5
0
−0.5
−1
1 2 3 4 5 6 7 8 9 10
Areas
Figure 5.9 Mean differences between the SSH data after retracking and AUSGeoid98
geoid heights in ten areas 5 km from the Australian coastline.
25
20
unretracked
50% threshold level
30% threshold level
varying threshold level
fixed counts
15
STD (m)
10
5
0
1 2 3 4 5 6 7 8 9 10
Areas
Figure 5.10 Standard deviations of the mean differences in ten areas 5 km from the
Australian coastline.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 153
Figures 5.11 and 5.12 show mean differences of the SSH and geoid heights and
standard deviations in a distance band of 5-10 km from Australia’s coasts,
respectively. As can be seen from the ordinate scale in Figures 5.11 and 5.12, both
mean differences and the STDs decrease with increasing the distance from the coasts.
It is evident that the SSH is improved after retracking no matter which threshold
level is used. The schemes (b) and (d) show similar results in most areas. This
implies that less waveform contamination occurs in these areas, which gives the
same result as that shown in Chapter 4. The results from scheme (a) present
improvements in this distance band as well. However, the results from Figures 5.11
and 5.12 suggest that scheme (d) is the best threshold level.
The mean differences and the STD in four other 5 km-wide distance bands (10-30
km from the coast) were computed and analysed. When the distance between the
ground track and coastline is far more than 10 km, schemes (b) and (d) generate a
very good agreement for both the mean and STD from the results after retracking.
Results suggest that less waveform contamination occurs in these areas and the
threshold level can be set to 50% of the waveform amplitude.
During testing, it became evident that the varying threshold level can be used to
effectively threshold retrack most waveforms in coastal regions, since it takes into
account the coastal waveform characteristics. The count-fixed level (scheme a)
cannot follow the variations of the scattering surfaces. It assumes that the retracking
amplitude of the ocean waveform does not change with varying the surface
roughness or topography. However, this is not the actual case of the waveform even
over ocean surfaces. The 50% threshold retracking point is set too high to catch the
ocean returns for the contaminated waveforms in coasts. Setting the 30% threshold
level within 5 km from the coast and the 50% threshold after 5 km (scheme c)
presents an improved result. However, changing the threshold level suddenly at a
distance of 5 km causes a step change in the range measurements after retracking. In
contrast, the varying threshold level cannot only generate improved SSH data, but
also avoids this step change.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 154
1.4
1.2
unretracked
50% threshold level
varying threshold level
fixed counts
1
Mean Differences (m)
0.8
0.6
0.4
0.2
0
−0.2
1 2 3 4 5 6 7 8 9 10
Areas
Figure 5.11 Mean differences between the SSH data after retracking and
AUSGeoid98 geoid heights in ten areas 5-10 km from the Australian coastline.
12
10
unretracked
50% threshold level
varying threshold level
fixed counts
8
STD (m)
6
4
2
0
1 2 3 4 5 6 7 8 9 10
Areas
Figure 5.12 Standard deviations of the mean differences in ten areas 5-10 km from
the Australian coastline.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 155
It should be pointed out that the majority of coastal altimeter waveforms exhibit an
ocean-like shape with a single ramp (see Chapter 6). Only a small percentage of them
show a significant peak or a combination of ocean and land returns. Consequently,
the use of the 30% and 50% threshold levels over large areas of the coastal regions is
more reasonable from these results. In the test, it has also been found that only 30%
of threshold level sometimes might not be sufficiently low to intercept the soughtafter
ocean returns from the waveforms with double ramps. The 20% retracking point
has been found in this investigation to be the best threshold level for areas five, eight,
and ten (cf. Figure 5.8).
5. 8 An Assessment of Biases among Waveform Retracking Algorithms
In this study, waveform retracking is carried out in coastal regions. Thus, the first
concern in this Section is whether or not the retracked data in coasts matches the
unretracked data in open oceans, and if not, to find the difference (or bias) between
them. In fact, biases have been found in this investigation and in a previous study
(Anzenhofer et al., 2000). In the results of Anzenhofer et al. (2000), a bias
(~66.4±10.7 cm) has shown between the ERS-1 GDR unretracked range and the
retracked range using the five-β-parameter function.
In addition, there is no ‘universal’ retracking algorithm that can deal with all types of
the waveform data (Berry et al., 1998; Martin et al., 1983; e.g., Berry, 2000).
Therefore, a bias question is: if different retrackers are used to retrack individual
waveform data, is there any bias exiting between among retrackers? The second
concern is to estimate if there is a bias among different retracking algorithms, and to
quantify and correct for it.
5.8.1 Analysis Methods and Data
Both the iterative least squares method (Section 5.6.5) and threshold retracking
methods (Section 5.7.1) are used to assess the performance of different retracking
algorithms in coastal regions through waveform retracking. These retrackers include
the 50% and 30% threshold levels, the ocean model and five-β-parameter model. The
five- β -parameter function with an exponential trailing edge is not used, because the

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 156
results of Anzenhofer et al. (2000) indicate that the bias between the five- β -
parameter functions with the linear and exponential trailing edges is insignificant
(~1.3±10.8 cm). For convenience of presentation, hereafter the five- β -parameter
model is called the five-parameter model.
The data used are cycles 42 and 43 of ERS-2 20 Hz waveforms over an ocean area
around Australia out to 350 km from the coastline. Contaminated waveforms in
coasts (obtained in Chapter 4) are retracked first to investigate how retracking
improves the range and thus the SSH, and to find biases. Then, waveforms in an
ocean area of 50-350 km off the coastline are retracked to estimate biases between
both algorithms and SSHs before and after retracking. Waveforms are retracked
using each retracker to estimate the range correction and then the retracked SSH.
Retracked SSHs are used to compute the SSH differences between different
retrackers at each geographical location. The bias is finally estimated by fitting a
cubic polynomial function to the SSH differences.
Both before and after waveform retracking, it is likely that there is still a small
percentage (~4% in this study) of SSH data that may present as outlier measurements.
Therefore, the Z-score is employed as an outlier detection method when computing
the bias
where
∆SSH i
− ∆SSH
Z
i
=
(5.1)
STD
∆ SSH is the difference of the SSH data at the same geographical position
(assumed being normal distribution), STD is the standard deviation of the mean
∆ SSH , Z
i
is the Z-score and
the variable
Z
i
< 3 is selected. The value of
Z
i
> 3 indicates that
SSH
i
is not within 3 STD of the sample mean, and may be considered
as an outlier and deleted.
The above analysis method is similar to the collinear analysis technique, which is
usually applied to repeated altimeter data to assess the mesoscale variability of the
sea surface (e.g., Cheney and Marsh, 1983; Sandwell and McAdoo, 1990), average
repeated SSHs to estimate a mean sea level and its variations (Nerem, 1995; Leben et
al., 1990; e.g., Deng et al., 1997), or compute gravity anomalies (e.g., McAdoo,

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 157
1990). Since the inter-comparison here is conducted between different retracking
algorithms using the same data set, the analysis is also different from the collinear
technique, which involves the time variation and the small separation of the same
ground track (< 1 km) occurring between different repeat orbits. This makes it more
effective to estimate the biases among the different retracking methods.
5.8.2 Retracking Results and Analysis
All contaminated ERS-2 waveform (20 Hz, cycles 42 and 43, from March to May
1999) estimated in Chapter 4 at the Australian coast have been retracked using the
fitting and threshold algorithms. Figure 5.13 shows two typical examples of the
retracking results, plotting the SSH (20 Hz) profile versus along-track distances. In
order to evaluate how the waveform is contaminated near the coast and improved by
the retracking techniques, the geoid height has been bispline interpolated at the each
location along the track using the AUSGeoid98 grid. The geoid height profiles are
also plotted versus along track distances in Figure 5.13. The profiles at the top of
Figure 5.13 represent a ground track 21085 approaching land from water, while those
at the bottom of the Figure is another ground track 21364 leaving land to water.
As stated in Sections 2.5.2 and 5.7.3, it is acknowledged that the geoid height does
not equal the SSH measured by the altimeter. There is a dynamic SST between them,
which can be of the order of ~1-2 m. This difference is evidenced in Figure 5.13.
However, the absolute SST cannot be determined at this time due to the uncertainty
in the geoid undulation (cf. Chelton et al., 2001). In this case, applying a SST
correction may cause an additional error to the SSH, which might be larger than the
errors of other corrections (Hwang, 2003). Therefore, the SST is not applied to the
SSH data in Australian coastal regions (Chapters 5 and 6) when an absolute height
comparison is conducted. The difference between the SSH and geoid height is not
taken as an indicator for the purpose of comparison or analysis. Instead, the profile’s
trends of the SSH and geoid or the along-track slopes will be considered as a more
appropriate indicator to examine the waveform contamination and the improvement
of the retracking procedure.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 158
−18.00
−20.00
−22.00
−24.00
SSH (m)
−26.00
−28.00
−30.00
−32.00
Geoid height
Unretracked
Ocean model
Five−parameter model
50% threshold level
−34.00
0.0 5.0 10.0 15.0 20.0 25.0
Along−track distance from the coastline (km)
−8.00
−10.00
SSH (m)
−12.00
−14.00
−16.00
Geoid height
Unretracked
Ocean model
Five−parameter model
50% threshold level
−18.00
0.0 5.0 10.0 15.0 20.0 25.0
Along−track distance from the coastline (km)
Figure 5.13 Geoid height and SSH profiles along ground tracks 21085 (top) and
21364 (bottom) (ERS-2, cycle 42) before and after retracking using the ocean model,
the five-parameter model and the 50% threshold level in coastal regions.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 159
As can be seen from Figure 5.13, the typical near-land effects on unretracked SSH
profiles begins at the along-track distance of ~20 km and ~17.5 km from the
coastline for the track 21085 (top picture) and track 21364 (bottom picture),
respectively. When approaching the coastline, unretracked SSHs decrease and then
increase gradually (see also Section 4.6). The difference of the minimum SSH height
and the normal SSH height can be large, up to ~ -5 m within ~10 km for both tracks.
In contrast to the change of unretracked SSH profile, the geoid heights show a linear
trend within the whole distance.
By comparison of the geoid heights with unretracked SSHs, it is evident that
unretracked SSHs are affected by waveform contamination in coastal regions.
In addition, according to the shape of the unretracked SSH profiles, the along-track
SSH and geoid gradients are computed and listed in Tables 5.3 and 5.4. The results
of the track 21085 (Table 5.3) show that along-track geoid gradients computed from
the AUSGeoid98 grid are small, from 9.2 ppm to 10.9 ppm when the track is
0-7.5 km and 7.6-19.5 km from the coastline. Corresponding to the same distances,
SSH gradients computed from unretracked SSHs are 975.6 ppm and -509.2 ppm,
respectively. They are much larger than the maximum geoid gradient value of ~150
ppm in Australia (Johnston and Featherstone, 1998; Friedlieb et al., 1997; cf.
Featherstone et al., 2001). After 19.5 km from the coastline, geoid gradients
computed using AUSGeoid98 geoid grid and SSH gradients computed using
unretracked SSHs show small values of 12.5 ppm and -6.4 ppm, respectively. These
results, thus, suggest as well that unretracked SSHs are in error within an along-track
distance of 0-19.5 km from the coastline.
Table 5.3 Along-track geoid and SSH gradients (track 21085)
Gradients
Names
(ppm)
ds = 0.0 - 7.5 km ds = 7.6 - 19.5 km ds = 19.8 - 26.0 km
Geoid height 9.2 10.9 12.5
Unretracked SSH 975.6 -509.2 -6.4
Analysing the along-track SSH and geoid gradients listed in Table 5.4 for the track
21364 can also find that unretracked SSHs are affected within a distance of 0-20 km

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 160
from the coastline. After 20 km, geoid gradients computed from AUSGeoid98 geoid
grid and SSH gradients computed from unretracked SSHs take small values of
8.5 ppm and -21.4 ppm, respectively.
Table 5.4 Along-track geoid and SSH gradients (track 21364)
Names
Gradients (ppm)
ds = 0.0 - 4.7 km ds = 4.8 - 17.5 km ds = 17.5 - 23.5 km
Geoid height 14.2 16.4 8.5
Unretracked SSH 803.3 -363.8 -21.4
It is also apparent from Figure 5.13 that the retracking techniques, irrespective of the
method used, have extended the SSH profile several kilometres shoreward. For the
track 21085 (top picture in Figure 5.13), retracking improves the SSHs over a
distance of ~5-20 km from the coastline. A similar improvement of the SSH can be
found from the track 21364 (bottom figure in Figure 5.13), where the SSH profile
has been extended shoreward ~14 km along track. Although the analysis is only for
two tracks, they are good representative of the ground tracks closer to the coastline.
5.8.3 Biases among Retracking Algorithms
Apart from the improvement in SSHs after retracking, it is evident that there is the
bias among different retracking algorithms, in particular between the fitting
algorithm and the threshold retracking (50% threshold level) algorithm (Figure 5.13).
In order to investigate the biases among different algorithms and even between the
SSH data before and after retracking, waveform data over a open ocean area
(50-350 km from the coastline, -30°S

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 161
A typical example of the SSH profiles before and after retracking is plotted as a
function of the latitude in Figure 5.14. It is evident that the differences of the SSH
are caused by the bias. The histograms of the differences ( ∆ SSH ) between the SSH
data retracked by different methods are shown in Figures 5.15 and 5.16. From
Figure 5.15, the differences of the ocean model and the five-parameter model are
small, between -5 cm and +5 cm (see also Table 5.5 and 5.6) and nearly unidentified,
though the differences do not exactly follow a normal distribution. The results from
the differences between 50% threshold retracking and two fitting functions
(Figure 5.16) show, however, an obvious bias.
Figure 5.14 Retracked and unretracked SSH profiles over open oceans, showing the
biases among them.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 162
Figure 5.15 Histogram of the differences between the ocean model and fiveparameter
model.
Figure 5.16 Histograms of the differences between the 50% threshold retracking
technique and the ocean model as well as the five-parameter model.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 163
Using cycles 42 and 43 of ERS-2 waveform data, the biases have been estimated and
the results are listed in Tables 5.5 and 5.6, respectively. The bias between two fitting
models is small and almost statistically insignificant (Table 5.5). This is because they
are both derived from the Brown model. However, it is significant to find the bias
values of ~57 cm and ~56 cm between the 50% threshold algorithm and the fiveparameter
model and the ocean model, respectively (Table 5.6). These values
indicate that the range retracked by the fitting model is ~57 cm or ~56 cm longer
than that retracked by 50% threshold retracking algorithm.
Table 5.5 Biases between different retracking techniques from fitting a quadratic
function to the ERS-2 SSH (cycle 42) differences over an ocean area (-30°S

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 164
account of noise in the waveform (especially the trailing edge) and how the noise is
affecting the leading edge from one waveform to the next, while the fitting algorithm
is still affected by the noise in the trailing edge. The limited 3-4 sample bins of the
leading edge are also restricted the precision of the estimates for both algorithms.
Thus, the bias problem results from the noise effect in the fitting algorithm and too
simplistic assumptions in the threshold retracking. The biases will be studied further
in the next section.
5.8.4 Biases between the SSH Data before and after Retracking
From Figure 5.13, it is also apparent that there is a difference between the SSH data
before and after retracking. Figures 5.17 and 5.18 show histograms of differences of
the SSH data before and after retracking using different algorithms. It is evident that
the differences of the SSHs before and after retracking appear to follow a normal
distribution, but their means obviously are not equal to zero and indicate the biases.
Figure 5.17 Histograms of the differences between unretracked SSH data and
retracked SSH data using the ocean model and the five-parameter model.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 165
Figure 5.18 Histogram of the differences between untracked SSH data and retracked
SSH dada using the 50% threshold retracking technique.
The biases have also been estimated and the results from cycles 42 and 43 are listed
in Table 5.7 and Table 5.8, respectively. The differences computed from all
retracking algorithms indicate that there are biases between retracked and
unretracked the SSH data.
From Table 5.7 and Table 5.8, a bias of ~30 cm exists between retracked SSH data
(by 50% threshold retracking) and the unretracked SSH data, indicating the measured
range is too long. The bias of ~27 cm between the SSH data retracked by fitting
approaches and the unretracked SSH data suggests the measured range is too short.
Theoretically, it is not necessary to retrack waveforms over open oceans as the onboard
tracking algorithm is based on the Brown model and the assumptions of the
homogeneous statistics of the model are generally satisfied over ocean surfaces.
However, a bias still found between the SSH data retracked by the ocean model and
the unretracked SSH data in this study. Therefore, the subsequent analysis here aims
at understanding the reasons that cause the biases.

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 166
Table 5.7 Biases between the ERS-2 (cycle 42) SSHs before and after retracking
over an ocean area (-30°S

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 167
samples in the range window. Since a sample or gate corresponds to a distance of
~0.4542 m (ERS-1/2), it should be possible that the error is caused by the retracking
procedure itself.
From the above analysis, if one is interested in using data closer to the coastline and
retracking the coastal waveform data, the bias between the SSH data before and after
retracking should be removed when combining the retracked and unretracked data.
Three approaches can be used:
• Accept the already-applied instrument corrections and use the range provided
by the GDR products over open oceans. The corrected range in coastal regions
is obtained by applying the range correction estimated from retracking
procedure to the measured range from the waveform products. Then, these two
types of range measurements can be used together without the bias between
them.
• Use the range measurements from waveform products, retrack all waveforms
(both over open ocean and coastal regions), and then apply the range
corrections obtained from the retracking procedure to all measurements.
• Retrack only the waveform data in coastal regions, but estimate an absolute
bias between the altimeter-derived SSH (without retracking) and independent
sea surface measured at the tide gauge. The bias, then, is used to correct the
SSH over open oceans.
5. 9 Summary
A coastal retracking system has been developed. Central to the system is the use of
two retracking techniques to derive an improved geodetic parameter, namely SSH,
which include both the iterative least squares fitting algorithm and the threshold
retracking algorithm. Retrackers adopted in the system are the ocean fitting model,
five-parameter fitting models, nine-parameter fitting models, and the varying
threshold levels, in which five- and nine-parameter models are used for the
waveforms with both the linear and exponential trailing edges.
By quantitative comparison of the ocean model and the five-parameter model, it has
been found that the slope of the trailing edge modelled by the five-parameter model
has a larger span of variation than the slope modelled by the ocean model. This

Chapter 5. COASTAL WAVEFORM RETRACKING SYSTEM 168
advantage, in turn, gives the five-parameter model greater capability of fitting nonocean-like
waveforms in coasts than the ocean model.
A waveform classification procedure is given in this Chapter, which enables the
waveform to be sorted and retracked by an appropriate retracker. To overcome the
‘noise’ in waveforms caused by fading noise and sources, a weight iterative scheme
has been developed.
Threshold retracking can be an improved method in coastal regions, if an appropriate
threshold level is selected for the retracking. Comparison of retracked SSH data with
available ground truth of the AUSGeoid98 geoid heights confirms that the 50%
threshold level is the best level for open-ocean waveform, but not an adequate level
for the contaminated ocean waveforms in coastal regions. Instead, a varying
threshold level, which is the 30% threshold level for contaminated waveforms and
50% threshold level for uncontaminated waveforms, can make a good agreement
between the retracked SSH data and the geoid heights.
Biases exist both between retracked SSH data from different retracking algorithms
and SSH data before and after retracking. Biases have been analysed and estimated
by retracking ocean waveforms in this Chapter using two cycles of ERS-2 waveform
data readily available at the time of this study. This finding suggests that biases must
be estimated before retracking and applied to the data to obtain consistent results
after retracking.

Chapter 6. WAVEFORM RETRACKING APPLICATION 169
6. WAVEFORM RETRACKING APPLICATION TO THE AUSTRALIAN
COAST
6. 1 Introduction
The principal objective of this Chapter is to determine to what extent the SSH data
can be improved by the retracking system designed and developed in Chapter 5 in
coastal regions. The quantitative distance estimation and detailed analysis of
waveform contamination for the Australian coastal regions has been addressed in
Chapter 4. As part of the development and implementation of the retracking system,
it is important to apply the system to the coastal regions and to assess the quality of
the retracked altimeter-derived SSH data. Therefore, the coastal retracking system
will be tested over a small part of the Australian coast (0-30 km from the coastline,
Figure 4.3) using two cycles (42 and 43, March to May 1999) of ERS-2 20 Hz
waveform data. To obtain a general idea of the impact of retracking on the altimeterderived
SSH, the quality checks of the retracking results consist of the comparisons
with external data source and validation of internal results. The external independent
reference used is the AUSGeoid98 geoid model. The results of retracking and
comparisons with unretracked and external control data will be discussed in this
Chapter.
Following the steps of the retracking block diagram in Figure 5.1, results of
waveform classification and waveform retracking will be presented and discussed in
Sections 6.2 and 6.3, respectively. The internal validation of the quality of the
retracked altimeter data will be conducted by validation of the single track SSH data
(Section 6.4), comparisons between SSH data before and after retracking (Section
6.5) and a collinear analysis (Section 6.6). SSH data before and after retracking will
be compared with the geoid heights interpolated from the AUSGeoid98 model along
ground tracks for the external comparison (Sections 6.7).
In addition, validating the retracked data will involve using ~2.5 years of ERS-1
20 Hz waveform data from different phases in the Taiwan Strait. Retracked SSH data
will be used to estimate the gravity anomalies. The results will then be compared

Chapter 6. WAVEFORM RETRACKING APPLICATION 170
with the external ground control of the ship-track gravity anomalies. The results will
be given in Chapter 7.
6. 2 Waveform Classification in Australian Coastal Regions
As shown in Chapter 4, most of contaminated waveforms in coastal regions present
an ocean-like single ramp. However, the returns also present some other non-oceanlike
shapes, such as a superposition of the energy reflected from each scattering
surface within the satellite footprint, which represent each distinct elevation when
approaching the shoreline. Therefore, sorting data based on waveform shapes is
essential before retracking them. In this Section, the contaminated waveform data
estimated in Chapter 4 will be classified by using the methods presented in Section
5.3.
6.2.1 Typical Waveform Shapes
Contaminated waveforms shown in Chapter 4 around the Australian coast have been
categorised. A detailed example of waveform categorisation given here is from two
ascending (21636 and 21679) and one descending (21815) ground tracks in a
15º×15º area (105°≤λ

Chapter 6. WAVEFORM RETRACKING APPLICATION 171
108˚
110˚
112˚
114˚
116˚
118˚
120˚
-16˚ -16˚
-18˚ 21815
-18˚
21679
-20˚ -20˚
-22˚ -22˚
Indian
Ocean
21636
-24˚ -24˚
-26˚ -26˚
-28˚ -28˚
-30˚ -30˚
108˚
110˚
112˚
114˚
116˚
118˚
120˚
Figure 6.1 Part of ERS-2 ground tracks off the northwest Australian coast, showing
the orbit number of each ground track. The highlighted diamonds along tracks will
be shown in detail waveform shapes later in this Section.
Figure 6.3 also shows waveforms from the pass 21815 but crossing an island
(cf. Figure 6.1). The blank area (~ -20.9°

Chapter 6. WAVEFORM RETRACKING APPLICATION 172
reflected from both land and standing water, and finally change to the typical ocean
shapes at around -22.1° latitude. The waveform shifting in the range window is
obviously observed in Figure 6.4, which is probably due to the operation manner of
the altimeter discussed in Section 4.6.3, and thus causing a longer contaminated
distance for this track.
Figure 6.5 shows waveforms from an ascending ground track 21679 (cf. Figure 6.1)
passing a coastal area of Shark Bay (113°≤λ

Chapter 6. WAVEFORM RETRACKING APPLICATION 173
1000 2000
Counts
-18.60 -18.55 -18.50 -18.45 -18.40 -18.35 -18.30 -18.25
Latitude (degrees)
16 32 48 64
Bins
Figure 6.2 Typical ocean waveforms from pass 21815 (cf. Figure 6.1).
Counts
2500 5000
-21.0 -20.9 -20.8 -20.7 -20.6 -20.5 -20.4 -20.3 -20.2
Latitude (degrees)
16 32 48 64
Bins
Figure 6.3 Waveforms recorded when the ground track 21815 is approaching an
island at latitude ~ -20.5º and leaving the island to water at ~ -20.9º. The data gap is
due to the island (cf. Figure 6.1).

Chapter 6. WAVEFORM RETRACKING APPLICATION 174
Counts
2000 4000 6000 8000 10000
-22.35 -22.30 -22.25 -22.20 -22.15 -22.10
Latitude (degrees)
16 32 48 64
Bins
Figure 6.4 Waveforms leaving land to water (pass 21636 in Figure 6.1). The highpeaked
specular responses are probably due to off-nadir brighter calm water. The
waveform shifting (~ -22.25°

Chapter 6. WAVEFORM RETRACKING APPLICATION 175
Counts
5000 10000 15000
64
48
32
Bins
16
-26.0 -25.9 -25.8 -25.7 -25.6 -25.5 -25.4 -25.3 -25.2 -25.1
Latitude (degrees)
Figure 6.5 Typical ERS-2 radar altimeter contaminated waveforms (pass 21679)
across water, land, and calm/still sea surfaces, showing high-peaked waveform
shapes reflected from the calm water surfaces, mixed shapes of land and water, and
ocean-like shapes.

Chapter 6. WAVEFORM RETRACKING APPLICATION 176
6.2.2 Results of Waveform Categories
The detailed categories of contaminated waveforms in ten Australian coastal regions
(see Figure 5.8) have been conducted and the general statistics of the waveform
categories for these ten coastal areas are listed in Table 6.1. The following
conclusions can be drawn from Table 6.1.
Table 6.1 Statistics of ERS-2 (cycle 42) contaminated waveform shapes in ten
Australian coastal areas (cf. Figure 5.8)
Waveform
categories
% of
waveforms
Mean distance
from coastline
(km)
Mean width of
waveforms
(bins)
Mean peak of
waveforms
(counts)
Ocean * 99.97 168.04 25.8 1304.1
Ocean-like 80.19 10.50 24.9 1380.3
Single pre-peaked 4.64 2.31 10.7 2785.1
Single mid-peaked 0.54 2.54 9.8 2615.0
Single post-peaked 1.35 3.12 11.8 3027.0
Double pre-peaked 0.55 3.08 14.8 2280.9
Double post-peaked 0.73 2.15 16.1 2271.9
Multi-peaked 0.18 2.30 13.0 2117.5
Sharp/High-peaked 4.35 1.73 3.3 8439.5
Unusable 7.51 6.03 34.7 1234.6
* Ocean waveforms over open oceans 50-350 km from the coastline, with 50%
threshold retracking point estimates between bins/gates 31-33.
1. Most contaminated waveforms (80.19%) have ocean-like shapes. The mean
width is similar to that of ocean waveforms, whereas the mean peak value is
slightly larger than ocean waveforms.
2. Of other shapes, the highest peak of the power is the high-peaked waveform
and the next is the single post-peaked waveform, but mean peak values of all
other shapes are larger than the mean peak of the ocean-like waveform.
3. The high-peaked waveform has the narrowest mean waveform width (3.3 bins).
The second narrowest width (9.8 bins) belongs to the single mid-peaked
waveform. Other non-ocean-like waveform widths are still much smaller than
the ocean-like waveform.

Chapter 6. WAVEFORM RETRACKING APPLICATION 177
4. Although ~80% of contaminated waveforms in coastal regions take an oceanlike
shape, other shapes of waveforms are found much closer to the coastline
from this data set (1.73 - 3.12 km in average), in particular the high-peaked
waveform (1.73 km from the coastline in average). This fact indicates further
that using only the ocean model or five-parameter model cannot re-process all
waveforms in coastal regions. Other retrackers are more significant in such
areas to remove errors caused by waveform contamination.
In addition, from a waveform categorisation in this study, high-peaked waveforms
have been found to take a larger percentage of contaminated waveforms (from 5.4%
to 6.9%) in north-west Australian coastal areas 1 and 2 because of the rugged
coastline, complex sea states and inland/standing water.
6. 3 Waveform Retracking Results and Discussion
Given the above observations, 20 Hz waveforms from cycles 42 and 43 offshore
Australia are re-processed using the coastal retracking system developed in Chapter 5
out to a distance of 30 km. Choosing 0-30 km is based on the observation in Chapter
4, where the maximum contaminated distance can occur out to ~22 km from the
coastline.
6.3.1 Data Editing
The first step of retracking is data editing to check whether or not the return presents
a leading edge (i.e., ramp) in the range window. This is conducted through the simple
heuristic assessment method used by Scott et al. (1994). The maximum 8.2% of noramp
(i.e., unusable) waveforms are filtered out from this step (see Table 6.2 and
Table 6.3). The second step is to classify each return waveform so that it can be
retracked by an appropriate retracker.
6.3.2 Waveform Retracking Using Different Algorithms
Next, a specific retracker based upon the classification is applied to each waveform,
which is either a fitting or threshold retracking algorithm. When using the iterative
procedure, an adequate fitting function must be selected and the procedure is

Chapter 6. WAVEFORM RETRACKING APPLICATION 178
repeated until a pre-defined existing criterion is satisfied (see Section 5.5.5). Since a
linear solution to a non-linear problem is used, sometimes the fitting algorithm does
not work (Zwally, 1996; Davis, 1995). Therefore, the threshold-retracking algorithm
is used to replace the iterative least squares fit to compute the retracking correction if
and when the fitting procedure fails. The threshold retracking technique thus works
for both the high-peaked waveforms and those fail to the fitting algorithm. The
system will select automatically the threshold level, which is 30% or 50% of the
waveform amplitude, for each waveform depending on its shape. As stated in Section
5.6.3, the 50% threshold level is used to retrack the ocean-like waveforms, while
other shapes of waveforms will be retracked by the 30% threshold level.
The percentages of the waveforms retracked by fitting and threshold algorithms are
listed in Tables 6.2 and 6.3. It can be seen that 77.7% - 94.9% (cycle 42) and 83.5% -
96.2% (cycle 43) of waveforms in this area can be retracked by the iterative least
squares fitting procedure. The rest of the waveforms (the last column) still require
threshold retracking. They are 4.8% - 18.1% of waveforms for cycle 42 and 3.8% -
15.6% of waveforms for cycle 43, respectively. The percentages of using different
retrackers vary from area to area depending on the reflecting surface topography.
Table 6.2 Waveform data status and percentage of the waveforms retracked using
fitting and threshold retracking algorithms in Australian coastal regions (cycle 42)
Area
No. of
Unusual
Fit
Threshold
number
Areas
data
data
%
%
%
1 120°≤λ

Chapter 6. WAVEFORM RETRACKING APPLICATION 179
Table 6.3 Waveform data status and percentage of the waveforms retracked using
fitting and threshold retracking algorithms in Australian coastal regions (cycle 43)
Area
No. of
Unusual
Fit
Threshold
number
Areas
data
data
%
%
%
1 120°≤λ

Chapter 6. WAVEFORM RETRACKING APPLICATION 180
Table 6.4 Percentages of ERS-2 waveforms fitted using different functions in
Australian coastal regions (cycle 42).
Area
No. of fit
Ocean
E-five-
Nine-
number
Areas
data
model
parameter a
parameter b
%
%
%
1 120°≤λ

Chapter 6. WAVEFORM RETRACKING APPLICATION 181
Figure 6.6 Typical coastal waveforms fitted using the following models:
(a): the five-parameter model with an exponential trailing edge,
(b) and (c): the ocean model,
(d) and (f): the nine-parameter model with linear trailing edge,
(e): the nine-parameter model with an exponential trailing edge.
waveforms are retracked by the fitting algorithm, while only 2% of waveforms need
to be threshold retracked. The percentage of fitting algorithm increases with
increasing distance from the coastline, confirming that the waveform shapes agree
well with the typical ocean waveform after 5 km offshore. This was also seen in
Chapter 4.
Figure 6.6 shows six typical coastal waveforms and the functional fits to them. It also
shows that different fitting functions work well to deal with different shapes of
waveforms in coastal regions. As stated, most of waveforms are single peaked (or
ramp) pulses. The ocean model fits these waveforms very well as shown in Figure
6.6(b) and (c). Some waveforms near land show higher power and double ramps,
which need to be fitted by the five-parameter function with an exponential trailing
edge as shown in Figure 6.6(a), or the nine-parameter function with either a linear

Chapter 6. WAVEFORM RETRACKING APPLICATION 182
trailing edge (cf. Section 3.5) in Figure 6.6(d) and (e) or an exponential trailing edge
(Figure 6.6(e), cf. Section 3.5).
6. 4 Analysis of the Single-Track Retracked Results
After retracking, two types of corrections from atmospheric propagation delays and
geophysical models supplied with the waveform data by NRSC (1995) are applied to
both retracked and unretracked range measurements to obtain the SSH. Since the EM
bias correction caused by ocean surface waves is not supplied with the waveform
data products used in this study, it is not applied to unretracked range measurements.
However, the retracking procedure removes the need for this correction (Brenner et
al., 1993; Hayne, 2002). The geophysical and environmental corrections applied are
dry and wet tropospheric range corrections, ionospheric correction, and tide
correction. The tide correction is the sum of the ocean tide from the Schwiderski
model and the ocean loading tide from the Ray-Sanchez model (NRSC, 1995). The
Doppler range correction is not applied to the range measurement, because most
values were found to exceed the error criteria of ±55 cm given by NRSC (1995) as
stated in Section 2.4.2.
As mentioned in Section 5.8.3, the relative biases among different retracking
algorithms are determined by retracking the waveforms over open oceans. They are
then applied to the range measurements retracked by the threshold retracking
algorithm. Finally, the altimeter-derived SSH is based on a reference of the ocean
model. This aims to remove the biases caused by reprocessing the waveform data
using different algorithms. As the determined biases depend strongly on the
algorithm, they can also be determined by other methods. For example, the method
used by Dong et al. (2002) may be employed for this purpose, which validates the
absolute bias between the altimeter-derived sea level and that derived by the tide
gauge data using the local geoid model, SST model, and data measured at the tide
gauge equipped with Global Positioning System (GPS) receivers. By using a similar
method, it is possible to estimate the absolute biases between different retracking
algorithms, but this remains an open issue for future work because of data
availability limitations at the time of this study.

Chapter 6. WAVEFORM RETRACKING APPLICATION 183
To compare the results from the retracking system with those retracked by a single
retracker, two ground tracks (21085 and 21364) are chosen, which are the same
tracks as these presented in Figure 5.13 (Section 5.8.2). The SSH (20 Hz) and geoid
height profiles (AUSGeoid98) are also plotted via the along-track distance from the
coastline in Figure 6.7. As stated in Sections 2.5.2 and 5.8.2, this aims to compare
the profile trends of the SSH and geoid height for investigating the contamination
caused by coastal sea states.
To compare results with those in Figure 5.13 (Section 5.8.2), the pictures have the
same X- and Y-scales as those in Figure 5.13. It is evident when comparing Figure
6.7 with Figure 5.13 that erroneous SSHs caused by the fitting algorithm when
approaching the coastline (0-5 km) in Figure 5.13 have been removed in Figure 6.7.
The bias between fitting and threshold retracking algorithms is also removed after
applying the bias correction determined in Section 5.8 (cf. Figures 6.7 and 5.13).
Because of use of the retracking system, more SSH data can be derived from
corrected range measurements than the SSH data from a single fitting retracker, thus
keeping the useful high-frequency singles in coastal regions. The numbers of SSH
data obtained from the system and a single retracker (the ocean model) are N system
retracking = 78 and N single retracker = 59 for track 21085 and N system retracking = 73 and N single
retracker = 63 for track 21364.
As can be seen from Figures 5.13 and 6.7, both single retracker and the retracking
system can improve the SSH profile shoreward several kilometres. More importantly,
the retracking system comprising several retrackers can improve further the SSHs
shoreward ~2.5 km from ~5 km to ~2.5 km for both tracks. However, it is important
to note that the SSH profiles cannot be closed completely to the coastline because of
land-dominated returns discussed in Chapter 4. In the case of tracks 21085 and
21363, there are still an along-track distance of ~2.5 km that cannot be recorded by
waveform retracking.

Chapter 6. WAVEFORM RETRACKING APPLICATION 184
−18.00
−20.00
−22.00
−24.00
SSH (m)
−26.00
−28.00
−30.00
−32.00
Geoid height
Unretracked
Retracked
−34.00
0.0 5.0 10.0 15.0 20.0 25.0
Along−track distance from the coastline (km)
−8.00
−10.00
SSH (m)
−12.00
−14.00
−16.00
Geoid height
Unretracked
Retracked
−18.00
0.0 5.0 10.0 15.0 20.0 25.0
Along−track distance from the coastline (km)
Figure 6.7 Geoid height and SSH profiles (ERS-2, cycle 42) along ground track
21085 (top) and 21364 (bottom) before and after retracking using the coastal
retracking system, showing the same profiles as those in Figure 5.13.

Chapter 6. WAVEFORM RETRACKING APPLICATION 185
Comparing the SSH profiles before and after retracking in Figure 6.7, the SSH
profiles after retracking show a good agreement with the geoid-height profile from
~2.5 km to ~25 km to the coastline. However, it is also apparent that the retracking
has added some measurement noise. Compared with unretracked SSHs, the retracked
SSH profile becomes nosier. The noise in the retracked SSH profile might be caused
by, firstly, the shorter wavelength signal, which suggests that on-board tracking
algorithm cannot follow well temporal variations of the scattering surface; secondly,
unmodelled non-linear parameters in this study (e.g., the skewness). Thus, a low-pass
filtering procedure is still essential when using 20 Hz SSH data after retracking for
the geophysical application in coastal regions. The search radius of the filter depends
on the applications, say 5-7 km for the recovery of gravity anomalies.
6. 5 Analysis of the SSH Data Before and After Retracking
The comparison is performed by calculating the difference dh between the retracked
SSH h r and unretracked SSH h g at the same geographic location along each ground
track. This difference (dh=h r − h g ) represents the sum of the bias of SSH data before
and after retracking, and tracking error that can be removed by waveform retracking.
The mean difference and standard deviation (STD) are calculated at six
5 km wide distance bands. Choosing different distance bands allows one to see how
close to the coastline the altimeter-observed ranges are improved by the retracking
procedure. Results are listed in Tables 6.6 and 6.7 for cycles 42 and 43, respectively.
It should be noted that the bias between SSH before and after retracking (described
in Section 5.8.4) is not applied to the individual SSH data. Instead, the biases can be
applied to the mean difference between the SSHs before and after retracking in
Tables 6.6 and 6.7 to analyse the error removed by the retracking procedure.
Therefore, results in Tables 6.6 and 6.7 contain the biases, which are -27.8±12.4 cm
(cycle 42) and -27.5±10.8 cm (cycle 43) (ibid.). It is recalled here that the bias could
be estimated and removed by using the methods mentioned in Section 5.8.
Tables 6.6 and 6.7 show that STD of the SSH differences decreases with increasing
the offshore distance. The STD decreases to less than 1 m for distances greater than
15 km from the coastline. Because the comparison is conducted at the same along-

Chapter 6. WAVEFORM RETRACKING APPLICATION 186
track point and same time, the satellite orbit error and any other random errors can be
removed by taking the difference of the retracked and unretracked SSH. Therefore,
the large values of the STD indicate, on one hand, that the significant error occurs
due to waveform contamination near the coastline. On the other hand, the retracking
algorithm itself introduces the additional error in the SSH data near land, because the
ocean returns are difficult to recover from land-dominated waveforms.
Table 6.6 Descriptive statistics of ERS-2 20 Hz SSH differences before and after
retracking (dh = h r – h g , cycle 42).
Distance
to
coastline
(km)
Number
of
SSH
Mean
difference
(m)
Max
difference
of SSH
(m)
Min.
difference
of SSH
(m)
STD
±(m)
0-5 5074 -0.79 9.25 -9.99 2.35
5-10 6517 -0.20 9.43 -9.82 1.31
10-15 7416 -0.24 6.62 -8.15 0.99
15-20 7313 -0.25 7.43 -6.89 0.71
20-25 7728 -0.30 7.71 -8.38 0.57
25-30 7496 -0.28 5.69 -9.13 0.57
Table 6.7 Descriptive statistics of ERS-2 20 Hz SSH differences before and after
retracking (dh = h r – h g , cycle 43).
Distance
to
coastline
(km)
Number
of
SSH
Mean
difference
(m)
Max
difference
of SSH
(m)
Min.
difference
of SSH
(m)
STD
±(m)
0-5 6218 -0.79 8.41 -9.99 2.43
5-10 8924 -0.14 9.89 -9.82 1.39
10-15 9820 -0.31 6.63 -9.47 1.02
15-20 9850 -0.30 8.11 -9.91 0.84
20-25 10140 -0.31 4.71 -5.70 0.65
25-30 9660 -0.32 7.05 -6.31 0.56
If considering only the amount of the bias, the important amount of error removed by
waveform retracking is -51.2 cm and 7.8 cm for cycle 42 in distance bands of 0-5 km
and 5-10 km, respectively, while the results do not show a significant amount of

Chapter 6. WAVEFORM RETRACKING APPLICATION 187
error in the rest of the four distance bands. Similar results can be found for cycle 43,
for which the amount of error in distance bands of 0-5 km and 5-10 km is -51.5 cm
and 13.5 cm, respectively. Analysis of the results suggests that waveform retracking
can effectively remove the altimeter tracking error in the vicinity of coastal regions.
When the satellite ground track goes further to the open ocean, the tracking error
becomes less significant. In the case of this study, the amount of tracking error is
~-3 cm. This agrees well with the result (-2.37 cm) of Brenner et al. (1993).
6. 6 Collinear Analysis
From the analysis in Sections 6.4 and 6.5, it is evident that retracking can remove
errors caused by altimeter on-board tracking algorithm due to coastal sea states and
land topography. In this Section, the collinear differences, which are computed using
altimeter-derived SSH profiles obtained along identical ground tracks, are employed
to estimate how the retracking procedure improves the altimeter data in coastal
regions. The differences are also used to qualify altimeter data before and after
retracking.
6.6.1 Methodology
Over open oceans, ERS-2 SSH profiles from different cycles contain primarily geoid
and SST information, as well as the random measurement error, which includes
errors introduced by satellite orbit error, altimeter range measurement error, and
temporal variability of the ocean surface, such as tides (Cheney et al., 1983). In
coastal regions, however, waveform contamination causes an additional error. The
relative collinear offset between the profiles is
dH
= H 2
− H1
+ ε , where the SSH
measurements at the same point, H
1
and H
2
, occur at different times of cycles t 1
and t 2
, and ε is a random measurement error.
The collinear differences remove the long-wavelength geoid, the quasi-timeindependent
SST and variations of the ocean surface, representing thus the amount of
orbit error, altimeter range measurement error, and the temporal variations of the
surface over open oceans. As stated, the random error includes the tracking error
caused by waveform contamination for unretracked data in coastal regions. After

Chapter 6. WAVEFORM RETRACKING APPLICATION 188
removing the satellite orbit error, SSH data from repeat cycles can also be used to
determine the mean sea surface (e.g., Deng et al., 1997; Nerem, 1995; Wang and
Rapp, 1991). Thus, the STD of the difference dH here is used as a measure of the
quality of a retracking algorithm’s range estimate. If errors caused by waveform
contamination can be removed by retracking algorithms, this will result in smaller
STDs. By comparing the STD of the collinear difference dH before and after
retracking, it can be determined whether or not the retracked SSHs are improved by
the retracking system.
The requirement of the collinear analysis is that data from different cycles must be
reduced to a reference orbit, so that the comparison can be conducted on the same
basis (Cheney et al., 1983; Wang and Rapp, 1991). In reality, the satellite ground
tracks do not repeat exactly due to orbital variations caused by gravitational
influences and drag effects. The cross-track deviation is ~1 km for most satellites (cf.
Brenner et al., 1990; Wang et al., 1991). Therefore, 1-second normal points are
usually generated (ibid.), and the geoid gradient correction is applied to data (Wang
and Rapp, 1991) for the reduction of the observed data to the reference orbit to
obtain exactly ‘collinear’ data. The cross-track geoid gradient correction is applied to
correct the effects of the geoid slope caused by non-repeated tracks (ibid.).
6.6.2 Discussion of Results
In this study, 20 Hz SSH data are retained to see possible high-frequency signals in
coastal regions. Since cycle 43 contains more orbits and thus more complete data
coverage, it is chosen as a reference. SSH data from cycle 42 are then reduced to
relevant points along ground tracks of cycle 43 using the method developed to
compute the collinear difference between SSHs from different cycles by Deng et al.
(1996). The cross-track geoid gradient correction is not applied to the data here,
because data came from two adjacent cycles and the effects of the geoid slope can be
neglected.
The descriptive statistics of the collinear SSH differences (20 Hz) computed from
cycles 42 and 43 have been calculated both before and after retracking. Results have
been plotted via six 5-km-wide distance bands in Figures 6.8 and 6.9. There are high
values of STD for dH using both unretracked and retracked data. The reasons may

Chapter 6. WAVEFORM RETRACKING APPLICATION 189
be the satellite orbit error and temporal variability of the ocean surface. As can be
seen from Figure 6.9, the STDs of dH after retracking are smaller than STDs before
retracking in all distance bands, in particular 0-15 km from the coastline. It is clear
from this comparison that retracking improves the precision of altimeter range
estimates in all distance bands to a different extent, in particular in two distance
bands of 0-5 km and 10-15 km. After 15 km from the coastline and in 5-10 km
distance, the STDs after retracking are still slightly smaller than STD values before
retracking, indicating improvement from waveform retracking as well.
Before Retracking
After retracking
Mean differences (m)
0.2
0.1
0
-0.1
-0.2
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.8 Mean differences of the collinear SSH data of cycles 42 and 43 before and
after retracking in six 5 km wide distance bands.
Before retracking
After retracking
STD (m)
2.5
2
1.5
1
0.5
0
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.9 STD (positive values) of the collinear 20 Hz SSH differences of cycles 42
and 43 before and after retracking in six 5 km wide distance bands, showing
improvement from SSH data after retracking.

Chapter 6. WAVEFORM RETRACKING APPLICATION 190
6. 7 Evaluation of SSH Data Using Geoid Heights
6.7.1 Comparison with the AUSGeoid98 Geoid Model
The collinear analysis of SSH data along repeating tracks can determine the internal
consistency (i.e., precision) of altimeter measurements. In order to assess the
accuracy of the retracked data and to determine how the altimeter retracked SSH
profiles represent the true sea surface, an independently surveyed reference surface,
referenced to the same ellipsoid, is required. The surface used here is the
AUSGeoid98 2′×2′ geoid grid (Featherstone et al., 2001). This geoid height model,
actually, is not an absolutely independent ground truth, because it uses altimeterderived
free-air gravity anomalies offshore (ibid.). However, as stated in Section
2.5.1, altimeter-derived gravity and ship track gravity data were used for
computation of the geoid in a ‘draping’ method such that the dense ship-track gravity
contribute significantly to the grid signal near the Australian coast, especially
0-30 km from the coastline, while the altimeter-derived gravity data contribute
mainly to the area which is far away from the Australian coast and without the shiptrack
data (cf. Kirby and Forsberg, 1998; Featherstone et al., 2003, Figure 1).
Therefore, the AUSGeoid98 geoid model can be thought of as a quasi-independent
reference.
To first order, the altimeter-observed mean sea surface represents the geoid
(Mantripp, 1996). This means that comparison of the geoid height and SSH can
provide assessment of the quality of retracked SSH data. It is also recalled that the
SST is not removed from the SSH data around the Australian coast due to the reasons
stated in Section 2.5.2. As stated in Section 5.7.3, since the quasi-independent
reference AUSGeoid98 is used, again, the verification of any improvement in the
SSH will only be seen in the standard deviations of the mean difference, with a
smaller value indicating an improvement in the determination of the altimeterderived
SSH.
For each along-track altimeter-derived SSH both before and after retracking, the
gridded geoid height is bispline interpolated with longitude and latitude of the
location. Then it is subtracted from the corresponding SSH to give the difference
between the SSH and geoid height. The statistics of differences is used to quantify

Chapter 6. WAVEFORM RETRACKING APPLICATION 191
the quality of the retracking system. Again, the STD is used to measure how the
waveform retracking system works rather than the mean difference between the
collinear differences.
6.7.2 Results of the Comparison between SSH Data and Geoid Heights
The descriptive statistics of differences between SSH data (20 Hz) before and after
retracking and gridded geoid height has been computed. For convenience of
comparison, mean differences and the STD of the differences have been plotted via
six 5-km-wide distance bands in Figures 6.10 and 6.11 for cycle 42, and Figures 6.12
and 6.13 for cycle 43, in which the STD is presented only using the positive value.
Again, the STDs show large values for both SSH data before and after retracking.
The reasons may be the temporal variations of the ocean surface and other incorrect
corrections, such as ocean tides and wet tropospheric range corrections, but retracked
SSH data show a higher precision than unretracked data.
Before retracking
After retracking
Mean difference (m)
1
0.8
0.6
0.4
0.2
0
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.10 Mean differences of geoid heights (AUSGeoid98) and SSH data before
and after retracking in six 5 km wide distance bands (cycle 42).
For cycle 42, Figures 6.10 and 6.11 indicate that both mean difference and STD
decrease when using retracked SSH data. The values of STD drop down to ~1 m and
the significant improvement occur in distances 0-15 km to the coastline. After 15 km
from the coastline, there is no significant decrease in the value of the STD of
differences. From Figures 6.12 and 6.13 for cycle 43, similar results can be obtained.
These results suggest that more precise SSH profile can be obtained from retracked
SSH data, especially in distance of 0-15 km from the coastline.

Chapter 6. WAVEFORM RETRACKING APPLICATION 192
Before retracking
After retracking
STD (m)
3
2.5
2
1.5
1
0.5
0
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.11 STD (positive values) of the mean difference before and after retracking
in six 5 km wide distance bands, showing improvement on SSH data after retracking
(cycle 42).
Before retracking
After retracking
Mean differences (m)
1
0.8
0.6
0.4
0.2
0
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.12 Mean differences between geoid heights (AUSGeoid98) and SSH data
before and after retracking in six 5 km wide distance bands (cycle 43).
Before retracking
After retracking
STD (m)
3
2.5
2
1.5
1
0.5
0
0-5 5-10 10-15 15-20 20-25 25-30
Distances from the coastline (km)
Figure 6.13 STD (positive values) of the mean difference before and after retracking
in six 5 km wide distance bands, showing improvement on SSH data after retracking
(cycle 43).

Chapter 6. WAVEFORM RETRACKING APPLICATION 193
Figures 6.11 and 6.13 show that the STD of differences after retracking drops down
to ±1 m when the distance to the coastline is larger than 10 km for cycle 42 and 5 km
for cycle 43, respectively. However, the STD before retracking decreases to less than
±1 m when the distance to the coastline is more than 20 km for both cycles. This
indicates clearly that the retracked SSH can be determined accurately to the same
level as SSHs over open oceans after offshore 10 km or even 5 km (e.g., cycle 43),
while unretracked SSH data can have the same accuracy after only 15 km from the
coastline. This means that these improved SSHs after retracking in coastal regions
can be used with sufficient accuracy to compute coastal geoidal undulations, coastal
ocean circulations, gravity anomalies and ocean tides. The results from Figures 6.11
and 6.13 also show a better accuracy at retracked SSH data in Australian coastal
regions than unretracked SSH data. This demonstrates that the coastal waveform
retracking system developed in this study has improved and expended altimeterobserved
measurements ~10 km shoreward in generally in Australian coastal regions
comparing with unretracked SSH data when using AUSGeoid98 as a quasiindependent
reference.
6. 8 Summary
Regional investigations around the Australian coast have demonstrated that the
waveform retracking system developed in Chapter 5 can effectively reprocess 20 Hz
waveform data and remove most errors caused by waveform contamination in these
areas. Two cycles (42 and 43, March to May 1999) of ERS-2 20 Hz waveform data
sets offshore 30 km around the Australia’s coast have been used to test the coastal
retracking system developed in this study. The whole coastal region is divided into
10 15°×15° sub areas. SSH data both before and after retracking are derived from
uncorrected and corrected range measurements for the purpose of comparison and
validation. Altimeter-derived SSH data used are 20 Hz without applying any filtering
procedure to avoid any loss of high frequency signals. Using these SSH data and the
AUSGeoid98 model, several analyses have been conducted in this Chapter.
The results of waveform classification show that most coastal waveforms (80.19%)
have ocean-like shapes with a single ramp but without significant peaks. Of the
remaining ~12.34% of waveforms, predominant waveforms have single pre-peaked

Chapter 6. WAVEFORM RETRACKING APPLICATION 194
(4.65%) and high-peaked (4.35%) waveform shapes. Others, such as single midpeaked
and post-peaked, double pre-peaked and post-peaked, and multi-peaked
waveforms, are from 0.73% to 1.35%.
The categorised waveform is then retracked by an appropriate retracker depending on
its shape. Retracking results at the Australian coast show that 77.7% - 96.2% of
waveforms can be retracked using least squares iterative fitting algorithm, while
3.8% - 18.1% of waveforms are needed to be retracked by the threshold retracking
algorithm. The percentages of using different retrackers vary from area to area. Of
these waveforms retracked by fitting algorithm, over 98.3% of waveforms are fitted
using the ocean model and the rest of waveforms are fitted by the five-parameter
model with an exponential trailing edge and the nine-parameter model with either a
linear trailing edge or an exponential trailing edge. It has been found that waveforms
retracked by the threshold retracking algorithm decrease from 35% in a distance of
0-5 km from the coastline to 4% in a distance of 5-10 km from the coastline. After
10 km offshore, only 2% of waveforms need to be threshold retracked.
By comparison of along-track altimeter-derived SSH data before and after retracking,
the largest errors caused by waveform contamination are found in a distance band of
0-5 km, which is ~-51.2 cm (cycle 42) and ~-51.5 cm (cycle 43). Errors of 7.8 cm
and 13.5 cm are found in the distance band of 5-10 km from the coastline. Beyond 10
km offshore, the error has been found to be less significant (~-3 cm).
Results from single track analysis indicate that the coastal retracking system can deal
with more waveforms to derive thus more SSH data than a single retracker (e.g., the
ocean model). More importantly, the system can make further the SSH profile closer
to the coastline ~2.5-5 km. This is an improvement of the single retracker, at which
the SSH profile presents without showing effects of waveform contamination after
5 km from the coastline.
In general, by comparing altimeter-derived along-track SSH data before and after
retracking with AUSGeoid98 geoid heights, results show that the altimeter-derived
SSH profiles can be extended ~5-10 km shoreward by the use of a coastal retracking
system and range corrections. These improved SSHs show a same accuracy level as
those over open oceans, thus indicating a great potential to apply them with sufficient

Chapter 6. WAVEFORM RETRACKING APPLICATION 195
accuracy to altimetric geodetic applications in coastal regions. However, SSH
profiles at a distance of 0-5 km to the coastline cannot be recovered by the waveform
retracking procedure at the same precision level as SSH data after 5 km from the
coastline.
The problem found from this research is that, in general, the retracked SSHs become
nosier compared to the unretracked data. Therefore, a low-pass filtering process is
necessary before applying these retracked data.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 196
7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES NEAR
TAIWAN FROM WAVEFORM RETRACKING
7. 1 Introduction
Satellite altimetry has provided the most valuable of data sets for the recovery of the
marine gravity field. Numerous local and global marine gravity anomalies have been
created using a variety of successful techniques. Papers reporting current results
include those of Hwang and Parsons (1995), Sandwell and Smith (1997), Andersen
and Knudsen (1998), Tziavos et al. (1998), Hwang (1998), Andersen et al. (2001),
and Hwang et al. (2002b). The precision and spatial resolution of marine gravity
anomalies derived from satellite altimetry have also been significantly improved in
open oceans over the past 25 years (Tapley and Kim, 2001; Rapp and Yi, 1997).
Using the high-density data collected from multi-satellite missions, the precision of
the derived global gravity fields is reported to have a range of ~±3 mgal to
~±14 mgal from the comparisons of ship-track gravity and altimeter-derived gravity
measurements (e.g., Tapley and Kim, 2001; Hwang et al., 2001).
However, the precision of altimeter-derived gravity anomalies in coastal regions is
not at the same level as those in open oceans. The differences between predicted
anomalies from altimeter data and ship-track gravity data tend to become larger in
coastal zones (e.g., Hwang et al., 1998b; Andersen and Knudersen, 2000;
Featherstone, 2002; Hwang et al., 2002b). Hwang et al. (2002b) estimate a global
gravity anomaly grid using Seasat, GEOSAT, ERS1-2, TOPEX/POSEIDON
altimeter data and report a precision of ~±13 mgal in the Fiji Plateau and
Mediterranean Sea, where the SSH data are likely affected by land and islands.
Featherstone (2002) investigates the differences around the Australian coastal
regions between ship-track gravity data and the most recent altimeter-derived gravity
grids of GMGA97 (Hwang et al., 1998b), KMS01 (Andersen et al., 2001), and
Sandwell’s version 9.2 (Sandwell and Smith, 1997). The result shows that larger
differences (>100 mgal in magnitude) occur in areas closer to the coastal regions,
which are consistent for all altimeter grids (see Figures 11-14 of Featherstone, 2002).

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 197
The reasons for these larger differences observed near the coastal regions are
probably as follows.
• Land topography and rugged sea states contaminate coastal altimeter waveform
measurements, causing SSHs to be in error. For example, TOPEX/POSEIDON
altimeter waveform data are affected by land to ~0-34.8 km from the coastline
in some coastal regions (Brooks et al., 1997); ERS-2 altimeter waveform data
can be contaminated to a distance range of ~0-22 km off the Australian
coastline (cf. Chapter 4).
• Altimeter measurements cannot be properly corrected due to inaccurate
geophysical corrections such as shallow-sea tides and the atmospheric delay.
The recent analyses in accuracy of these corrections are given, e.g., by Shum et
al. (1998), Andersen and Knudsen (2000), and Hwang et al. (1998b). In
addition, waveform contamination introduces errors to the SWH for the SSB
correction, which will be discussed later.
• There are still large biases in ship-track gravity data due to improper or
impossible data cleaning (i.e., crossover analysis, poor navigation in position,
and Eötvös corrections) so that they cannot always offer definitive control on
the accuracy of the altimeter-derived gravity anomalies.
• Lack of altimeter data over the land near the coastline presents a problem that
the gravity computation over coastal waters will be theoretically inaccurate.
This is because the gravity recovery procedure requires uniform data around
the computation point (see Eq. 15 of Hwang et al., 1998b), but this may not be
the case when the point is close to the coastline.
• The FFT technique requires the data to be on a regular grid. Incomplete data
with gaps in coastal regions will cause the edge effect (i.e., Gibbs phenomenon)
in recovery of gravity anomalies.
In considering the first reason above, the error caused by altimeter waveform
contamination in coastal regions can be mostly removed using waveform-retracking
(e.g., Brooks et al., 1997; Deng et al., 2001). This Chapter focuses on improving the
accuracy of altimeter-derived gravity anomalies through retracking ERS-1 altimeter
waveform data in the Taiwan Strait over an area bound by 119°E ≤ λ ≤ 122°E and
23°N ≤ ϕ ≤ 26°N. For the third cause above, there are some well-adjusted ship-track

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 198
gravity data in the study area (see Section 2.5.3) which can be used to conduct an
analysis between them and altimeter-derived gravity anomalies (not withstanding the
errors in these data; mentioned above).
The Taiwan Strait is characterised by shallow water (ocean depth less than 200 m),
small islands, and rugged coastlines. Thus, it is assumed reasonably representative of
the altimeter waveform features that may be encountered in most coastal regions.
Nearly 2.5 years of ERS-1 altimeter waveform data over the area include the
measurements from the ice, multi-disciplinary and geodetic phases, producing denser
data coverage for the recovery of the gravity anomalies. These are the reasons that
this area is chosen.
The data and area signatures, waveform retracking methods and SSH extraction will
be described in Sections 7.2, 7.3 and 7.4, respectively. In order to assess how the
retracking procedure improves the accuracy of the altimeter-derived gravity
anomalies, two SSH sets (i.e., before and after retracking) are extracted from
waveform data using the same method except for the retracking procedure.
Retracking is carried out over a larger area including the China Seas (i.e., the East,
South, Yellow Seas and Bohai). The description in Section 7.5 will relate to the
recovery of gravity anomalies from altimeter-derived data both before and after
waveform retracking from the inverse Vening Meinesz formula (Hwang, 1998a)
using the 1-D FFT technique. Comparisons of recovered anomalies from estimates
by these two data sets to ship-track gravity data will be described. It should be
acknowledged here that although these ship-track gravity data may contain errors,
they are the only external data that can be used to indirectly verify the tracking
algorithm’s differences.
7. 2 Data and Area
As stated in Section 2.4.1, ~2.5 years of 20 Hz ERS-1 waveform data from AWF
products are used in a coastal region close to China and Taiwan bounded by
110°E ≤ λ ≤ 129°E and 20°N ≤ ϕ ≤ 42°N. Figure 7.1 shows the geographical
distribution of the observations, where the study area for the gravity anomalies in this
Chapter is represented by a small rectangle (118°E ≤ λ ≤ 123°E and

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 199
22°N ≤ ϕ ≤ 27°N). Among the ERS-1 repeat cycles, the geodetic mission (GM)
enables the acquisition of a high density of altimeter measurements and thus
improves the spatial resolution of the altimeter-derived SSH and hence the
determination of marine gravity anomalies.
Figure 7.1 Distribution of ERS-1 ground tracks
As the original purpose of the data used in this area was to investigate the shallow
water tides, observations with the ocean depth deeper than 200 m were omitted,
making no data in the eastern area of Taiwan (Figure 7.1). This presents a problem
for the recovery of the gravity anomaly in this part of the study area, where the final
marine gravity field will be dominated by the reference gravity field and cannot show
the high frequency components (i.e., ~20 km wavelength) because of the lack of data.
The study area is a narrow strait (roughly ~110 km wide) between Taiwan and China
linking the East China Sea to the South China Sea (named the Taiwan Strait). The
computation area is 1° larger than the final gridded gravity field (119°E ≤ λ ≤ 122°E

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 200
and 23°N ≤ ϕ ≤ 26°N) at the edges to avoid the Gibbs effect of the FFT technique.
Table 7.1 shows a summary of altimeter observations in the area.
Table 7.1 Summary of ERS-1 observation cycles bounded by 118°E ≤ λ ≤ 123°E and
22°N ≤ ϕ ≤ 27°N
Repeat cycle
Time period
Number
Ascending
Descending
(days)
of cycles
pass
pass
3 23/12/1993 - 10/04/1994 36 0 1
35 13/07/1993 - 23/12/1993
21/03/1995 - 02/06/1996
5
13
8 6
168 04/10/1994 - 24/03/1994 1 35 24
Although only ERS-1 altimeter data are used, an adequate density (4×4 minutes) of
measurements is the main characteristic of the data set readily available at the time of
this research. From Table 7.1, the 3-day repeat cycle shows only one descending
ground track, but the 168-day repeat cycle presents 35 ascending and 24 descending
tracks. Together with eight ascending and six descending tracks from the 35-day
repeat cycle, satellite ground tracks from three types of repeat cycles create a denser
coverage of data than that using only single repeat cycle of data over this area,
increasing greatly the spatial resolution (Figure 7.1). This will contribute most to the
high-frequency parts of the coastal marine gravity field.
To assess the accuracy of the gravity anomaly from satellite altimeter, the ship-track
gravity data at 397 points described in Section 2.5.3 around Taiwan are used in this
study (Figure 2.5).
7. 3 Waveform Retracking
To retrack the return waveforms in this area, the iterative least squares procedure
(Section 5.5) is used to fit an analytic function representing the expected return to the

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 201
waveform. The forms of the analytic function used in this study are the fiveparameter
model (Equation 3.40 and n =1) and the ocean model (Equation 5.3).
7.3.1 Retracking Scheme
Waveform contamination occurs mostly in the vicinity of land and islands (see
Chapter 4). Over ocean surfaces, the contamination occurs only over regions where
the rugged sea state or the off-nadir satellite attitude excursions affect the shape of
the return (Brenner et al., 1993). However, research conducted by Fairhead et al.
(2001) and Sandwell (2003) indicates that waveform retracking is required over all
oceans to improve the resolution and quality of the marine gravity field. Therefore,
all 2.5 years of ERS-1 20 Hz waveforms shown in Figure 7.1 are retracked. Before
retracking, an edit criterion (Section 5.2) was applied to remove waveforms in which
there is no leading edge in the waveform sample window, which removed ~2% of the
data.
When fitting, a priori values and weights are determined using the method presented
in Section 5.5.4. Retracking results are obtained by retracking the waveform over
ocean surfaces and in coastal regions using the fitting functions of either the ocean
model or five-parameter model. Each waveform is first classified into ocean-like and
non-ocean shapes so that it can be retracked by the appropriate functional fit. The
iterative procedure is repeated until the unknown parameter estimate ( t 0
or β
3
)
converges to within 0.1% or
−4
σ
0
−σ
0
< 10 . It converges in typically 2-4
i
i −1
iterations. In addition, the five-parameter function replaces the ocean model to
compute the retracking correction when the ocean model fails.
The retracking procedure is conducted by using either the five-parameter model or
the ocean model. Retracking was first performed using both models. A collinear
analysis was applied to one cycle of 35-day repeat of 1 Hz SSH data (from July to
August 1993) to investigate whether there is a serious bias between both methods.
The difference ( dh ) between the retracked SSH data calculated from the ocean
model and five-parameter model was computed and analysed. Figure 7.2 shows a
histogram of dh in which the mean SSH difference is -0.7±13.5 cm (3070
observations). Of the residuals, 95% are less than 10 cm and 83% are less than 5 cm,

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 202
indicating a good agreement between both methods and a similar result as ERS-2
(see Section 5.8). This result also means that the combination of both ocean and fiveparameter
models can maintain a consistent measurement noise level across the
entirety of the sea surface profile. Since the ocean model physically represents the
returns over oceans, this is the reason that both models are finally chosen to be the
main retracking functions in this study.
Frequency of data
800
600
400
200
0
-1.45
-1.27
-1.08
-0.90
-0.71
-0.52
-0.33
-0.14
0.05
dh (m)
0.23
0.42
0.61
0.80
0.99
1.18
1.37
Figure 7.2 The histogram of dh (the SSH difference between the five-parameter
model and the ocean model (mean -0.7 cm; STD 13.5 cm; 3070 observations).
7.3.2 Examples of Waveform Retracking Results
Figure 7.3 shows one ascending (north-westwards) and one descending ground track
(south-westwards) in the study area. The ascending track (A1) goes from water to
land. The descending track (D1) recedes from land to water. It is parallel to the
coastline at ~37°N latitude, and then intercepts the land at ~32.4°N latitude. The
track then re-commences as D2 and goes to the Taiwan Strait. The 20 Hz waveforms
have been retracked using the methods mentioned above and compressed to 1 Hz
SSH data; the before/after retracking 1 Hz SSH results, plotted versus latitude, are
shown in Figure 7.4 for these three tracks. The profiles from top to bottom of
Figure 7.4 show the SSH from tracks A1, D1 and D2. The SSH derived from the
original (i.e., not retracked) and retracked (using the five-parameter and ocean
models) data are shown. It is apparent that the retracking has removed some
measurement noise near the coastal regions caused by land topography and
rugged/rough sea states.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 203
Figure 7.3 Three-ground tracks in the study area
As for the SSH derived from waveforms before retracking, Figure 7.4a does not
show the obvious land effect on the unretracked SSH data when the ascending track
(A1) approaches land at ~23.5°N latitude, while the contamination which may be
caused by rough sea states occurs from ~21.8°N to ~22.4°N latitudes. Descending
track (D1, Figure 7.4b) shows obvious sharp changes in unretracked SSH results
when the track gets close to the land. Similar sharp changes can also be found in
unretracked SSH data along the descending track (D2, Figure 7.4c). It is evident in
Figure 7.4 that the retracking, no matter which method, has removed the ‘error’
(unexpected sharp changes) in the SSH and the highly variable measured sea surface
is replaced by the smoothed profile. The SSH data from retracked waveforms using
the five-parameter model or ocean model show similar results (Figure 7.4).

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 204
Sea Surface Height (metre)
14.0
12.0
10.0
8.0
Five−parameter
Ocean + five−parameter
without retracking
6.0
21.7 22.2 22.7 23.2
Latitude (degree)
18.0
Sea Surface Height (metre)
16.0
14.0
12.0
10.0
8.0
Five−parameter
Ocean + five−parameter
without retracking
6.0
32.0 34.0 36.0 38.0 40.0
Latitude (degree)
sea Surface Height (metre)
21.0
20.0
19.0
18.0
17.0
16.0
15.0
Five−parameter
Ocean + five−parameter
without retracking
14.0
22.0 23.0 24.0 25.0 26.0
Latitude (degree)
Figure 7.4 SSHs from ground track A1 (a), D1 (b), and D2 (c) in Figure 7.3 before
and after retracking using the retrackers of the five-parameter and ocean models.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 205
The results in Figure 7.4 show that it is necessary to retrack all waveforms in this
area due to its characteristics of shallow water, rugged coastlines and rough sea states.
It also can be seen from Figure 7.4 that the retracking methods used can extend the
reliability of the measurements shoreward. From this analysis, there is a small bias
between the SSH before and after retracking (cf. Figure 7.4). A similar problem has
also been found when using ERS-2 waveform data around Australian coastal regions
(Section 5.8.4), where a bias of ~27 cm is estimated from two cycles of waveform
data. Since the along-track geoid slope will be used in this study to calculate the
gravity anomaly, the differentiation operation can effectively remove this bias. Thus,
its effect on the SSH can be reduced. However, it cannot be neglected when
analysing the absolute temporal variation in SSHs or mixing the original data (before
retracking) and retracked data.
7. 4 The SSH Extracted from Waveforms
To assess the results from the above retracking, the SSH is extracted using two
approaches. The first is to create the 1 Hz SSH data set from the 20 Hz data source
without waveform retracking, while the other is to retrack the 20 Hz waveform and
then to ‘compress’ the retracked SSH to the 1 Hz SSH set. No matter which method
is used, a quadratic function is fitted to 20 Hz SSHs and Pope’s (1976) tau-test is
applied to detect the erroneous SSH data. Final 1 Hz SSHs are calculated at the midpoint
of the fitting function after removing the outlying observations.
7.4.1 Corrections
To obtain the data set of the SSH, geophysical and environmental corrections are
applied to the range measurement. The corrections for atmospheric propagation
delays are obtained directly from the AWF waveform product. The AWF product did
not supply the inverse barometer correction, which is related to the hydrostatic
response of the sea surface with increasing or decreasing atmospheric pressure.
Instead, it was computed using the surface atmospheric pressure which is available
indirectly via the dry tropospheric correction obtained from the AWF. The tidal
correction applied is the superposition of the elastic ocean tides (i.e., the ocean tide
and loading tide) derived from CRS4.0 global ocean tide model (Eanes, 1999), and

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 206
the solid Earth tide supplied by the AWF. The vertical (radial) velocity of the
satellite causes a frequency Doppler shift which affects the time delay measurement
and thus the range. For each elementary measurement, the Doppler correction is
added to the altimeter range. Table 7.2 lists the variability of the corrections applied
in the area and their maximum and minimum magnitude. From Table 7.2, the change
of the SSB was found to cause an error in the SSH data in the study area.
Table 7.2 The variability of the range corrections to deliver SSH in the study area
(cf. Figure 7.1)
Correction Min (m) Max (m)
Dry tropospheric delay 2.27 2.30
Wet tropospheric delay 0.12 0.40
Ionospheric delay 0.05 0.09
Inverted barometer 0.05 0.17
SSB -1.23 -0.02
Elastic Ocean tides -2.15 0.22
Solid Earth tides -0.10 0.03
Doppler shift -0.01 0.01
As stated in Section 3.3.3, the SSB is the difference between the apparent sea level
measured by an altimeter and the true mean sea level. Theoretical understanding of
the SSB, particularly the EMB, remains limited, so the SSB correction is calculated
from the SWH and the wind speed using an empirical model derived from analysis of
altimeter data itself (AVISO/Altimeter, 1998; Chelton et al., 2001). The SWH, as
computed by the on-satellite processor (MacArthur, 1980, cited in Martin et al.,
1983), is a measure of the ocean wave amplitude. Over the coastal regions near the
land, the SWH computed on-satellite is not useful due to the effects of land
topography and generally more rough sea states. Therefore, the SSB may no longer
be correct in these areas.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 207
The SSB’s effect on the SSH is analysed through SSH profiles along the three
ground tracks in Figure 7.3. SSH profiles along tracks are plotted based upon
whether they are retracked or unretracked, applied by all geophysical corrections
including SSB or not. Besides the SSB, parts of corrections of dry tropospheric delay
and elastic ocean tides are chosen to plot along each track (from Figures 7.5 to 7.7),
because they also show the larger variety in magnitude (Table 7.2). As can be seen,
the SSB shows an obvious step change in this coastal region, but it is not expected to
change significantly and suddenly over a few kilometres unless a rougher sea state is
present.
Compared to the SSB, other corrections of dry tropospheric delay and ocean tides
show a relatively smooth change. As a result, the SSH profiles are shifted upward a
value of the sum of corrections after they are applied to the range measurements.
However, since the SSB should be subtracted from the SSH, it doubles its effect on
the SSH when the step change occurs. Therefore, the SSB is not applied to the range
measurements in this study. This may leave an error for both unretracked and
retracked SSH data. However, as the retracking procedure partly removes the ocean
wave influence on the waveform, the need of the range measurement for the SSB has
been reduced. This is the reason that the retracked SSH shows the smoothing profiles
in Figures 7.5, 7.6 and 7.7. In addition to the SSB, there are some sharp changes to
be found in SSH profiles without retracking even in ocean areas far away from the
coastline (see, Figures 7.5 and 7.6), while waveform retracking, again, obviously
removes these unexpected changes. This implies that the SSH cannot be precisely
computed from the on-satellite tracking algorithm in this coastal and semi-enclosed
sea.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 208
14.0
Sea Surface Height (metre)
10.0
6.0
2.0
−2.0
21.7 22.2 22.7 23.2
Latitude (degree)
Figure 7.5 SSH profiles and parts of corrections along the track A1.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 209
18.0
Sea Surface Height (metre)
14.0
10.0
6.0
2.0
−2.0
32.0 34.0 36.0 38.0 40.0
Latitude (degree)
Figure 7.6 SSH profiles and parts of corrections along the track D1.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 210
22.0
Sea Surface Height (metre)
18.0
14.0
10.0
6.0
2.0
−2.0
22.0 23.0 24.0 25.0
Latitude (degree)
Figure 7.7 SSH profiles and parts of corrections along the track D2.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 211
7.4.2 The Weight Function
After waveform retracking, both retracked and unretracked 20 Hz SSH data derived
from waveforms along ground tracks are created to 1 Hz SSH sets. Before the
calculation of the gravity anomaly using these SSH data, a procedure of data filtering
is necessary to delete possible outliers. The 1 Hz data are then separately smoothed
and edited by a low-pass filter to remove the noise in the measurements and ensure
the accuracy of altimeter-derived gravity anomalies. Because of the advantage of the
data filter, a Gaussian gridding kernel is chosen as the weight function for removing
outliers in SSH data and computing the average gradients along ascending and
descending tracks (stated below). For a point ( ϕ , λ)
from the observation ds
kilometres, the formula is
2
⎡ 1 ⎛ ds ⎞ ⎤
w(
ϕ , λ)
= exp⎢−
⎜ ⎟ ⎥
ds ≤ R s
. (7.1)
⎢⎣
2 ⎝ σ ⎠ ⎥⎦
where R s
= D / 2 is the search radius of the weight function, D is the search
window size, and σ = D / 6 . The value of D is important for this function due to the
degree of smoothness increasing with it. From Equation (7.1), an influenced point
from the observation will gain a small weight near the edge of the search window no
matter how large the size is. Thus, a large value of D will over smooth the filtered
results, while a small value of D will still leave unreasonable results. In this study,
D =70 km is chosen for the along track SSH smoothing. This value ensures that there
are about 10 seconds [time] of SSH data to be included in each filtering procedure
for obtaining a robust result. For calculating along-track average gradients on a 2'×2'
grid, the value of D is chosen to be the diagonal of the cell (~5 km) so that all SSH
data can be searched (see Section 7.5.2).
7.4.3 Data Filtering and Editing
Data filtering will smooth out any high-frequency signal that is larger than the
proposed resolution to avoid data aliasing. SSH data filtering is a moving average
procedure, which uses a weight function (Equation 7.1, D = 70 km) and slides along
the track by 1 second of data (i.e., 20 SSHs) for each calculation. This procedure is

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 212
carried out at all observation points, resulting in a filtered SSH set. The final SSH at
( ϕ , λ)
is given as:
SSH ( ϕ , λ)
= ∑(
SSH )
i
wi
∑ wi
.
(7.2)
i
i
where
w represents the value of the weight at the point ( ϕ , λ)
for the particular i th
i
observation within the distance D .
Data that disagree by more than three times of STD are flagged as outliers and
replaced by the filtered values. In addition, when the number of SSH data within D
are less than 4, the smoothing procedure is not conducted, and all data are deleted.
Approximately 1% of the unretracked and 2% of retracked initial data sets are
eliminated using this filter, implying that retracked data show more short-wavelength
signal than the unretracked data. Figures 7.8 and 7.9 show filtered (dark curves) and
unfiltered (light curves) SSH results before and after retracking along ground tracks
in Figure 7.3 through the above Gaussan smooth procedure, respectively. As can be
seen, the Gaussian weight function removes the measurement error in observations.
7. 5 Marine Gravity Anomalies from Satellite Altimetry
There have been several approaches developed to compute gravity anomalies from
satellite altimeter data over the past two decades. The four main types of the
philosophies and computational procedures have been summarised by Featherstone
(2002). Each of them has advantages and disadvantages. Featherstone (ibid.) shows
that there are some significant differences among marine gravity anomalies
computed using different methods from satellite altimetry around Australia, and
these tend to become larger in coastal areas. Therefore, it is still an open question to
say which one is better than others.
One of them is the approach described by Hwang (1998a), which is used in this study
due to its numerically efficient computation and no requirement of crossover
adjustment (e.g., Sandwell and Smith, 1997; Hwang, 1998a; Andersen and Knudsen,
1998). In this approach, the gravity anomaly is calculated from the along-track
deflections of the vertical (or along-track SSH slopes) derived from satellite altimeter

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 213
Figure 7.8 Untracked SSH profiles, showing the filtered SSH data (dark curves) and
unfiltered SSH data (light curves) along the tracks A1 (a), D1 (b) and D2 (c) in
Figure 7.3.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 214
Figure 7.9 Retracked SSH profiles, showing filtered SSH data (dark curves) and
unfiltered SSH data (light curves) along the tracks A1 (a), D1 (b) and D2 (c) in
Figure 7.3.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 215
data using the inverse Vening Meinesz formula (Hwang, 1998a). The use of alongtrack
geoid slope data has been proved to be one of effective techniques used in the
calculation of gravity anomalies from satellite altimeter data, as the differentiation
operation effectively reduces the effect of the long wavelength radial orbit error (e.g.,
Sandwell, 1984; Hwang and Parsons, 1995; Sandwell, 1997).
Before calculating along-track deflections of the vertical, the quasi time-independent
estimates of dynamic ocean topography must be removed from the altimeter-derived
SSHs. Such removal is conceptually required because the ocean surface is not an
equipotential surface, but the geoid is. The remove-compute-restore technique is
used, in which the deflections of the vertical (i.e., ξ north-south and η east-west)
implied by EGM96 (Lemoine et al., 1998) global geopotential model to spherical
harmonic degree 360 is used as a reference field, and it is removed from the raw
deflections of the vertical to yield a residual deflections of the vertical set. These
north and east component of deflections of the vertical are then interpolated onto a
regular grid using least squares collocation (Moritz, 1980) after a procedure of
determining outliers in the data (cf. Hwang et al., 2001). The inverse Vening
Meinesz formula (Hwang, 1998a) is employed to convert these gridded north-south
and west-east components to gravity anomalies via the 1-D FFT technique. The final
gravity anomalies are obtained by adding back the eliminated 360 degree EGM96
reference field.
The approach has been successfully applied to recover marine gravity anomalies
from multi-satellite altimeter data (e.g., SEASAT, GEOSAT, ERS-1 and
TOPEX/POSEIDON) both at global and regional scales (Hwang, 1998a; Hwang et
al., 2002b). The advantage of this computational procedure is that data are carefully
prepared using the gridding and filtering techniques so that the data can be extracted
efficiently. It is noted that ERS-1 altimeter data close to the coastline (~20 km from
the coastline line) had not been used by Hwang (2001) to avoid data contamination
in coastal regions. In this study, however, data used are only from the ERS-1
altimeter in a coastal region. This presents a necessity for this study that some slight
modifications (see Section 7.5.1) should be conducted in the computing procedure to
use all potential retracked SSHs in coastal regions when recovering the gravity
anomalies through the inverse Vening Meinesz formula.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 216
7.5.1 Along-track Deflections of the Vertical
Before computing the along-track deflections of the vertical, in order to obtain
geoidal height from SSHs, the value of the quasi-time independent SST interpolated
from the model of Levitus et al. (1997) is subtracted from the SSH. In addition, the
reference gravity field of the EGM96 to harmonic degree 360 is also removed from
the geoidal height, meaning the along-track deflections of the vertical (generated
from below Equation 7.3) are residual deflections of the vertical or residual geoidgradients.
Taking the first horizontal derivatives of the altimeter-derived SSHs along track
generates the along-track deflections of the vertical (DOV), which is the negative
gradient of the geoid (Sandwell, 1992; Olgiati et al., 1995; and Hwang et al., 1998b).
The along-track gradient, ε , is calculated at the midpoint between two successive
observations by the formula
ε
k
=
N
k +1
− N
∆s
k
(7.3)
where
N
k
is the geoidal height at the observation k , and ∆ s is the distance between
observations k + 1 and k . In addition, the following Equation is less noisy for the
along-track DOV:
ε
N
1
− N
2∆s
k + k −1
k
=
(7.4)
7.5.2 Averaging Gradients
Usually, before computing the along-track geoid slope, the SSHs from repeat
missions are reduced to a reference orbit (see Section 6.6) and then averaged them to
remove the time-variant ocean signals and measurement noise (e.g., Hwang and
Parsons, 1995; Hwang et al., 2002a; Rapp and Yi, 1997; Sandwell and Smith, 1997).
However, lack of a reference orbit makes it difficult to directly average geoid heights
for ERS-1/35 repeat data. Therefore, a new method has been developed to average
along-track geoid gradients from repeat cycles in this coastal region. This procedure
averages the geoid slope rather than the geoid height, thus avoiding the introduction

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 217
of offsets at data gaps (cf. McAdoo, 1990; Kirby, 1996) and reducing the timevariant
signals.
The data used contain 3-day and 35-day repeat missions, and geodetic mission. Of
these missions, the repeat mission repeats ground tracks to within 1 km, while the
ground track of ERS-1/GM is unpredictable (with respect to the 35-day repeat track).
For the repeat missions, the same pass (ascending or descending ground track) from
different cycles does not repeat exactly due to some causes, such as orbital variations
caused by gravitational influence and drag effects. Orbital manoeuvres have been
used in the satellite to maintain a ground track within some specified linear tolerance
of a nominal or reference track (Wang and Rapp, 1991). For most satellite altimeters
such as GEOSAT and T/P, the aim is to achieve cross-track deviations on the order
of 1 km (Wang and Rapp, 1991). This is the same case as the ERS satellite. Taking
this data characteristic in mind, ideally the averaging method should retain the alongtrack
resolution, but also average the data into the spaces between tracks from
different repeat cycles so that a complete along-track gradient set can be generated.
In this new averaging procedure, residual geoid heights at any location are first
sorted and found in a 2′×2′ cell. A weighted mean geoid gradient, ε , is then
computed at this cell using a Gaussian weight function (7.1). The search radius
(~2.5 km) is chosen to be half the diagonal of the cell so that all SSHs in a cell can be
searched. In addition, the standard deviation of the mean geoid gradient is calculated
in the cell by the formula
∑
( ε − ε )
i
2
i
σ
k
=
(7.5)
n −1
where n is the number of the sea surface slopes in the cell. When n =
1, it is
impossible to calculate σ
k
from (7.5). In this case, σ
k
is given a value of 0.21 µrad
which is based upon an empirical noise level of ERS-1 (cf. Hwang et al., 2001). This
usually happens to the mean slopes formed from the geodetic phase.
After averaging gradients and before forming the regular grids for the use of 1-D
FFT algorithm, a similar outlier-removing procedure to Hwang et al. (2002b) is
applied to the mean gradients to remove residual erroneous data. Unlike Hwang’s

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 218
scheme, where a 4′×4′ cell was used due to a high density of the ground coverage
from multi-satellite altimeter data, the size of the cell is increased to 8′×8′. The
reasons for this are as follows. First, the data used comes only from the ERS-1
altimetry, thus a small cell cannot provide enough data points (some 4′×4′ cells may
contain only one data point from the GM phase) and produce larger degrees of
freedom to make the result more reliable, leading to improper detection of outliers.
Second, since the study area is a coastal region, a small cell will cause all data to be
eliminated when close to the coastline. However, it should be acknowledged that a
larger cell might oversmooth the data.
Averaging is carried out separately on the ascending and descending groundtracks.
Table 7.3 lists the descriptive statistics results from averaging gradients in this way
and outlier detection. Before detecting outliers, the mean value of the averaged
slopes (before retracking, both ascending and descending) is 0.13 µrad and their
standard deviation is ±3.01 µrad, while the mean of the slopes (after retracking, both
ascending and descending) is 0.25 µrad and their standard deviation is ±2.61 µrad.
After detecting outliers, the standard deviations reduce to ±2.34 µrad and ±1.82 µrad
before and after retracking, respectively. There are 15% and 13% of outliers which
are detected from geoid gradient data sets before and after waveform retracking,
respectively. From Table 7.3, the number of final residual geoid gradients before
retracking is more than that after retracking. Again, this shows that the retracking
procedure is effective.
Table 7.3 Descriptive statistics of the averaged residual geoid gradients before and
after the outlier detection on a 8′×8′ cell.
Before detecting outliers After detecting outliers
Methods Mean
(µrad)
STD
(µrad)
Number
of slopes
Mean
(µrad)
STD
(µrad)
Number
of slopes
Before retracking 0.13 ±3.01 5949 -0.01 ±2.34 5177
After retracking 0.25 ±2.61 5806 0.14 ±1.82 4953

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 219
7.5.3 Gravity Anomaly Computation
After removing outliers, the along-track geoid slopes are converted to regular grids
of east (η) and north (ξ) vertical deflection using least-squares collocation. The
covariance functions used here are taken from Hwang and Parsons (1995). Then, the
calculations for the determination of residual gravity anomalies are carried out using
the inverse Vening Meinesz formula Equation (23) of Hwang (1998a), and the
innermost zone effects is estimated by Equation (47) of Hwang (ibid.) using 1-D FFT
and the remove-restore technique. As stated, a 1° border is used to avoid Gibbs edge
effects. The flow diagram in Figure 7.10 illustrates how altimeter waveform
observations are processed and converted to gravity anomalies. The procedure
follows the steps described in Section 7.4 for extracting the SSHs and estimating the
gravity anomalies in this Section. Two gravity-anomaly grids (2′×2′) are generated
by this procedure: a grid without waveform retracking; and a grid with waveform
retracking. Figures 7.11 and 7.12 show the equivalent free air anomalies relative to
GRS80 normal gravity estimated by unretracked and retracked ERS-1 SSH data.
7.5.4 Comparisons between Marine and Ship Gravity Anomalies
The accuracy of the gridded free air anomaly in Figures 7.11 and 7.12 can be
assessed by comparing the ship-track gravity data with anomaly fields generated
using altimeter data. The comparison is made along ship tracks in the Taiwan Strait
in the vicinity of Taiwan (Section 2.4.3), where the ship data available is shown in
Figure 2.5 and the altimeter ground tracks are shown in Figure 7.1. Each of the
gridded anomaly data sets derived from altimeter SSH data before and after
retracking is interpolated using a bispline function to the position of the ship-derived
gravity anomaly. The number of the ship-track gravity data used in the comparisons
in this area is 397 points. The descriptive statistics of the differences are shown in
Table 7.4. Considering the retracked results in the comparison, it can be seen that the
solution is better than that which uses unretracked SSHs in the anomaly estimation.
The standard deviation of the difference (altimeter-derived anomalies minus ship
anomalies) of ±13.93 mgal before retracking is larger than that of ±9.92 mgal after
retracking. Therefore, the precision of the gridded altimeter-derived gravity anomaly
has been improved by ~4 mgal.

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 220
Figure 7.10 Flow diagram for the SSH extraction from waveforms and recovery of
gravity anomalies

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 221
119˚
26˚
120˚
121˚
122˚
26˚
183.79
89.96
25˚
25˚
66.44
49.49
34.99
m
gal
21.45
7.90
24˚
24˚
-6.59
-23.55
-47.07
-147.56
23˚
119˚
120˚
121˚
23˚
122˚
Figure 7.11 Gravity anomalies from altimeter data before retracking (units in mgal)
119˚
26˚
120˚
121˚
122˚
26˚
187.70
91.47
25˚
25˚
67.90
50.91
36.39
m
gal
22.82
9.25
24˚
24˚
-5.27
-22.27
-45.83
-143.78
23˚
119˚
120˚
121˚
23˚
122˚
Figure 7.12 Gravity anomalies from altimeter data after retracking (units in mgal)

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 222
It is acknowledged that there is a low accuracy from Table 7.4 for both altimeterderived
gravity anomalies before and after retracking with respect to other studies,
e.g., Rapp and Yi (1997). As stated in Section 2.5.3, however, the accuracy of the
ship-track gravity data after the crossover adjustment is ±11.2 mgal, although these
data has been subjected to a data cleaning processes. This means that there is still the
error existing in the ship-track gravity data. Therefore, the error of the ship-track
gravity data should be a possible reason that causes larger STDs in Table 7.4.
Figures 7.13 and 7.14 show the different results along ship tracks using unretracked
and retracked altimeter data, respectively. As can be seen from Figure 7.13,
numerous larger differences with magnitudes greater than +15 mgal occur in any
ship track not being limited to near the coast. As stated, the ERS-1 waveform data
are contaminated in this area not only in coastal regions but also in the water area far
from the shoreline (cf. Figures 7.5 and 7.6). The larger differences here are more
likely to have been subjected to some form of waveform contamination. This
becomes more evident when comparing Figure 7.13 with Figure 7.14 (after
retracking), where larger differences (>15 mgal in magnitude) appear along the ship
tracks that are very much closer to Taiwan’s coast. These larger differences near the
coastline may be due either to the limited distance that waveform retracking can
improve or the Gibbs edge effect at the coastline.
Table 7.4 Descriptive statistics of the differences between altimeter-derived grids and
the 397 ship-track free-air gravity anomalies (units in mgal)
Grid Max Min Mean STD RMS
Before retracking 32.71 -41.62 -4.86 ±13.93 ±14.74
After retracking 19.82 -41.08 -6.09 ±9.92 ±11.63

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 223
119˚
26˚
120˚
121˚
122˚
26˚
25˚
25˚
35
25
15
5
-5
m
gal
24˚
24˚
-15
-25
-35
-45
23˚
119˚
120˚
121˚
23˚
122˚
Figure 7.13 Differences between altimeter-derived (before retracking) and ship-track
gravity anomalies
119˚
26˚
120˚
121˚
122˚
26˚
25˚
25˚
35
25
15
5
-5
m
gal
24˚
24˚
-15
-25
-35
-45
23˚
119˚
120˚
121˚
23˚
122˚
Figure 7.14 Differences between altimeter-derived (after retracking) and ship-track
gravity anomalies

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 224
In interpreting the anomaly difference here, one should recall that different altimeter
SSH data sets are extracted from the same waveform source except for the retracking
procedure. All other calculations are carried out using the same procedure. Therefore,
the results shown in Table 7.4 reflect the improvements only from waveform
retracking in coastal regions.
7. 6 Summary
Gravity anomaly predictions were made from ~2.5 years of ERS-1 SSH data sets
before and after retracking using the inverse Vening Meinesz formula and the results
were compared with ship-track gravity data in the Taiwan Strait. The comparison of
results demonstrated that waveform retracking has improved the quality of the SSH
in coastal regions near the China Sea and Taiwan Strait, thus improving the accuracy
of the altimeter-derived gravity field. The agreements between the predicted and
ship-track gravity anomalies are ±13.9 mgal before retracking and ±9.9 mgal after
retracking.
The analysis of extracting the SSHs from altimeter waveform data suggests that the
SSB correction is inaccurate when it is computed using the empirical model in the
coastal region, due mainly to the incorrect SWH estimated from the on-satellite
processor. Waveform retracking can remove the need for the ocean wave influence
and thus reduce the SSB error, while without or using inaccurate SSB corrections
causes the error to the unretracked SSH data. In addition, the method of averaging
residual geoid gradients developed for the AWF data product in this study has been
demonstrated as being effective. When applied this averaging method to the
unretracked SSHs, an accuracy of ~ ±14 mgal can be obtained for the altimeterderived
gravity anomalies, which is a similar level to that in Hwang et al. (2001).
Results from this Chapter show a great potential of providing accurate gravity field
information in coastal regions by using satellite altimeter SSH data after waveform
retracking. For further research, to overcome the problem of altimeter data
discontinuity very near to the coastline, combining the uniform coverage of the
altimeter data with ship-borne gravity and land gravity data will provide an accurate
high-resolution gravity field in coastal regions. It is also necessary to improve the un-

Chapter 7. IMPROVED COASTAL MARINE GRAVITY ANOMALIES 225
modelled or poorly modelled geophysical corrections (e.g., SSB, wet tropospheric
delay, sallow-water tides). However, this was not a specific topic of this research.

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 226
8. CONCLUSIONS AND RECOMMENDATIONS
8. 1 Summary of Dissertation
Satellite radar altimetry is able to provide globally homogeneous and repeated sea
surface information with very high accuracy and resolution. However, the accuracy
of the satellite altimeter-based SSH degrades in coastal regions due to altimeter
waveform contamination, inaccurate geophysical corrections, and enhanced ocean
variability. Because of lower precision of the data in coastal areas, improvement of
the quality of altimeter data is an important issue before using them for geodetic and
oceanographic applications. When considering only the improvement of the altimeter
data themselves, the approach used is to retrack waveforms to estimate the range
correction, and then to derive the SSH in coastal regions. This approach has shown
the altimeter’s potential in coastal regions. However, previous research on retracking
procedures reprocess the coastal waveform using either merely an existing retracker,
which is used to retrack data over non-ocean surfaces, or a non-automatic method
that is hard to retrack a wide range of altimeter data near land. Compared with
previous work, this is the first time that this research has carried out the waveform
contamination in such a detailed way in a wide range of coastal regions.
In an effort to precisely recover the altimeter-derived SSH data to the greatest extent
close to the coastline, a dedicated coastal waveform retracking system has been
proposed in this thesis. This system is developed based upon a systematic analysis of
altimeter radar waveforms in Australian coastal regions. It has the advantage of
containing several retrackers required to reprocess the waveform data automatically
depending on the shapes of the waveform, while offering the precise SSH data sets
derived from corrected range measurements that are estimated through adequately
modelling the altimeter waveform at each along-track location in coastal regions.
A coastal retracking system has been designed and developed (Chapter 5). Central to
the system is the use of two retracking techniques to derive accurate geophysical
parameters, which include both the iterative least squares fitting algorithm and the
threshold retracking algorithm. Retrackers adopted in the system are the ocean fitting
model, five-parameter fitting models, nine-parameter fitting models, and the varying

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 227
threshold levels, in which five- and nine-parameter models are applied to the
waveforms with both linear and exponential trailing edges. In order to obtain
optimum and consistent estimates of the range correction, detailed investigations
have been performed to test and select appropriate retrackers. A waveform
classification procedure is also developed, which enables the waveform to be sorted
and retracked by an appropriate retracker. To dear with the fading ‘noise’ in the
waveforms, a weight iterative scheme has been developed.
Data used are 20 Hz waveforms from two cycles of ERS-2 altimeter mission, five
cycles of POSEIDON mission, and nearly 2.5 years of ERS-1 multi-missions. The
AUSGeoid98 geoid model and ship-track gravity anomalies in the Taiwan Strait
have been used as (partly) independent ground truth. In conjunction with the
shoreline model, an Australian bathymetric grid, and a water/land grid, these data
sets have been used in this research for the purpose of developing and implementing
this system.
The validation of the retracking algorithms has been tested using these data sources
in both the Australian coast and the Taiwan Strait. Results from a number of
comparisons and tests will be specifically concluded in Section 8.2. Generally
speaking, regional investigations around the Australian coast have demonstrated that
a waveform retracking system can effectively remove errors caused by waveform
contamination in the area due to the presence of land, coastal sea states (e.g. inland
or still water), and even the altimeter-operating feature. This study indicates that the
altimeter-derived SSH data along a ground track, as supplied in GDR products, can
be extended 5-15 km kilometres shoreward by the use of a coastal retracking system
developed in this study.
To investigate how retracked SSH data can improve the accuracy of altimeterderived
coastal marine gravity fields in coastal regions, an application has been
conducted using the ERS-1 20 Hz waveform data (

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 228
averaging geoid gradients has been developed for the data used due to lack of the
reference orbit information. Two altimeter-based gravity anomaly grids are created
using the altimeter data before and after retracking. Results are then compared with
the ship-track gravity data. Compared with the result of ±13.9 mgal before retracking,
an improved accuracy of ±9.9 mgal is obtained after retracking. In addition, it has
been found that the SSB correction is inaccurate in coastal regions because of the
coastal waveform contamination.
8. 2 Specific Conclusions
This research has investigated the use of a coastal retracking procedure to reprocess
satellite radar altimeter waveform data for extracting improved SSH data sets in
coastal regions. The investigations conducted involve a quantitative estimation of the
waveform-contaminated distance, the development and calibration of the coastal
waveform retracking system, a comprehensive analysis of waveform data, the
implementation of the retracking system, and applications of the system in Australian
coastal regions and the Taiwan Strait. The key conclusions of the research are made
below.
8.2.1 The Quantitative Estimation of the Contaminated Distance
Five cycles of POSEIDON and one cycle of ERS-2 20 Hz waveform data offshore
0-350 km have been used to quantify a broad, contaminated boundary around the
Australian coast (Chapter 4). This contamination is assumed to have been caused by
backscatter from the land and the more variable coastal sea surface states, coupled
with the footprint size of the radar altimeters, the direction that the satellite ground
track crosses the coastline, and the operating feature of the radar altimeter. In
addition, a preliminary experiment of land effects on the waveform has been
performed using waveforms, the Australian 9"×9" DEM and the AUSGeoid98 2'×2'
geoid model.
Using the mean waveforms and standard deviations of the mean in different distance
bands (Section 4.5.2), a typical contaminated boundary can be ascertained from the
Australian shoreline to ~8 km and ~10 km for POSEIDON and ERS-2, respectively.
Beyond 8 km or 10 km (to 350 km), the mean waveform shapes for both altimeters

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 229
match the mean waveform shape observed over ocean surfaces, though they do not
agree exactly with the theoretical waveform due to the fading noise and the wraparound
error.
Using the 50% threshold retracking point as the estimate of the pre-design tracking
gate (Sections 4.5.2 and 4.5.3), three main conclusions can be drawn. First, the
ERS-2 altimetry shows differences in contamination as a function of the sub-area
along the Australian shoreline. The maximum contaminated distances vary from
~8 km to ~22 km, depending upon locations and features of the shoreline. Second,
both POSEIDON and ERS-2 altimeters show differences in contamination as a
function of the distance from the shoreline, while ocean depths do not show an
important correlation with the contaminated distance (Section 4.5.1). Finally, the
contaminated distance for POSEIDON is less than that for ERS-2. A mean
contaminated distance of ~8 km for POSEIDON and ~10 km for ERS-2 can also be
observed using the 50% of threshold retracking points in the whole Australian
coastal region.
The ERS-2 altimeter also shows distance differences (Section 4.6.4) in waveform
contamination depending on different manners of the ground track crossing the
shoreline (i.e., approaching or leaving the land). In general, the waveforms along the
altimeter ground track leaving land to water suffer longer contaminated distance than
those along the track approaching land from water. The contaminated distance for a
land-approaching track is ~7 km along the ground track for ERS-2. On the other
hand for a satellite ground track leaving land to water, the along-track contaminated
distance can be as long as ~21 km from the coastline. In both cases, the ocean
surface sometimes can be no longer recoverable from contaminated waveforms
obscured significantly by the land return, when the ground track is very closer to the
coastline, say ~0-2.5 km.
The reasons that cause the distance differences of waveform contamination can be
the specific reflected topography (both land and sea) in coastal regions. From a
ERS-2 waveform shifting test (Section 4.6.3), the result shows that altimeter’s
operating feature should be one of reasons that cause the longer distance of distorted
waveforms along-track leaving land to water. This type of distorted waveform

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 230
usually shows usually the shift forward and backward with respect to the expected
tracking gate in the range window, but do not significantly change the magnitude of
the waveform’s amplitude. Land returns are not obviously visible in these
contaminated waveforms, because the distance between the observing location and
the land is usually beyond the satellite footprint. Using TOPEX waveform data,
Brooks et al. (1997) find also the same problem that longer contaminated distances
of the waveform occur upon the descending tracks. Thus, the result of the shifting
test in this research can apply to not only the ERS-2 altimeter, but also TOPEX.
Contamination leads to errors in the on-board calculation of the range measurements
so that the accuracy of altimeter range measurements in such areas is lower and thus
presents the problem for extending the use of altimeter data to the shoreline. It is
recommended from the quantitative contaminated distance in this study that the SSH
data within the distance provided by the GDR products are not reliable and must be
detected prior to their being included in geodetic and oceanographic solutions in
coastal regions. It is also suggested that waveform data reprocessing is essential to
improve altimeter-derived SSH data in these areas. For most GDRs users without
waveform products, it is recommended that altimeter SSH or range data should be
interpreted with some caution for distances less than say ~22 km from a coastline,
and discarded altogether for distances less than 5 km.
8.2.2 Practical Implementation of the Coastal Waveform Retracking System
Regionally comprehensive investigations of waveform characteristics and validation
of the coastal retracking system around the Australian coast have demonstrated that
diverse waveform shapes exist, and the retracking of 20 Hz waveform data can
effectively remove most errors caused by waveform contamination in these areas.
Two cycles of ERS-2 20 Hz waveform data sets offshore 30 km around Australian
coastal regions have been used for implementation of the coastal retracking system
developed in this study. The AUSGeoid98 model was again used as a ground truth
for the purpose of comparison. In an effort to improve the altimeter-derived SSH
results, the following conclusions can be made:
(1) Results of waveform classification show that most coastal waveforms (80.19%)
have ocean-like shapes with a single ramp but without significant peaks. Of the

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 231
remaining ~20% of waveforms, predominant waveforms have single prepeaked
(4.65%) and high-peaked (4.35%) waveform shapes. Others, such as
single mid-peaked and post-peaked, double pre-peaked and post-peaked, and
multi-peaked waveforms, take the percentages of 0.73% - 1.35%. Although
there is only a small percentage of non-ocean-like waveforms, the statistical
results show that these waveforms are much closer to the coastline (1.73-3.12
km in average), indicating that a retracking system containing several
retrackers is necessary.
(2) Retracking results in Australian coastal regions show that 77.7% - 96.2% of
waveforms can be retracked using a least squares iterative fitting algorithm,
while 3.8% - 18.1% of waveforms have to be retracked by the threshold
retracking algorithm. The waveforms show differences in percentages of using
different retrackers as a function of the sub-area along ground tracks. Of these
waveforms retracked by the fitting algorithm, over 98.3% of waveforms are
fitted using the ocean model and the rest of waveforms are fitted by the fiveparameter
model with an exponential trailing edge and the nine-parameter
model with either a linear trailing edge or an exponential trailing edge. It has
been found that the use of the threshold retracker decreases with increasing
offshore distance from 35% in an offshore distance of 0-5 km to 4% in a
offshore distance of 5-10 km. After 10 km offshore, only 2% of waveforms are
threshold retracked, which agrees well with the estimation of the contaminated
distance in Chapter 4 and indicates that the method developed to categorise
coastal waveforms is effective.
(3) Results from single track analysis indicate that the coastal retracking system
has a capability of dealing with diverse waveforms in coastal regions. This
advantage means that the system derives more accurate SSH data than a single
retracker. By comparing altimeter-derived along-track SSH data before and
after retracking with AUSGeoid98 geoid heights, results show that the
altimeter-derived SSH profiles can be extended ~5-10 km shoreward by the use
of coastal retracking system and range corrections. However, SSH profiles at a
distance of 0-5 km to the coastline cannot be recovered by waveform
retracking procedure at the same precision level as SSH data after 5 km from

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 232
the coastline. This is most probably due to predominant land returns in the
waveform range window and the incorrect geophysical corrections in these
near land areas.
In summary, the effect of a coastal waveform retracking system, which contains a set
of methods, computing programs and retrackers for reprocessing waveform data, has
been demonstrated by the improvement observed in Australian coastal regions. When
using two cycles of ERS-2 20 Hz waveform data and the AUSGeoid98 geoid grid as
an independent ground reference, results after retracking provide a typical
improvement of ~5 km on the offshore distance. Compared with the contaminated
distance of ~10 km before retracking, the use of the retracking system is able to
reduce this contaminated distance to ~5 km.
8.2.3 Development and Validation of a Coastal Retracking System
The development and validation of the coastal retracking system includes design and
investigation of algorithms, selection of the threshold levels, estimation of biases
between different retrackers and the SSH data before and after retracking, as well as
the tests on the effectiveness of the system.
(d) Retracker Investigation
An ocean return model has been deduced and is a modified convolutional solution of
the radar returns based upon the Brown (1977) model. This ocean model without the
non-linear ocean wave parameter has been used as a main retracker in this research
due to its clearly physical description of the ocean surface. By quantitative
comparison to the five-parameter model, it has found that the slope of the trailing
edge modelled by the five-parameter model has a larger span of variation than the
slope modelled by the ocean model. This advantage makes, in turn, the fiveparameter
model hold a greater capability of fitting non-ocean-like or irregular
waveforms in coastal regions than the ocean model. Thus, the five-parameter model
replaces the ocean model for these irregular shapes of waveforms in the
implementation of the system.
Selection of the threshold level has been performed by comparing an available
ground truth of the AUSGeoid98 geoid heights with the SSH data before and after

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 233
threshold retracking, in ten 15°×15° sub-areas around the Australian coastal regions
and six 5 km wide distance bands. Results confirm that the 50% threshold level is the
best level for open-ocean waveform, but not an adequate level for the contaminated
ocean waveforms in coastal regions. Instead, a varying threshold level, which is the
30% threshold level for contaminated waveforms and 50% threshold level for
uncontaminated waveforms, can give a good agreement between the retracked SSH
data and the geoid heights. This result of the threshold level broadly agrees with that
used by Brooks et al. (1997).
(b) Bias Evaluation
Biases exist both between retracked SSH data from different retracking algorithms
and SSH data before and after retracking. Biases have been analysed and estimated
by retracking ocean waveforms in Section 5.8 using two cycles of ERS-2 waveforms
and can be concluded as follows, which suggest that biases must be estimated before
retracking and applied to the data to obtain consistent results after retracking.
(1) Bias between the SSH data retracked by the ocean and five-parameter fitting
algorithms is −0.85±1.35 cm, which is not statistically significant.
(2) Bias between the SSH data retracked by the 50% threshold retracker and the
fitting algorithms (both the ocean and five-parameter models) is
+56.35±5.84 cm.
(3) Biases between altimeter-derived SSH data before and after retracking are
+27.58±8.78 cm for fitting algorithms and −30±3.58 cm for the 50% threshold
retracking algorithm.
It should be pointed out that the differences between the standard deviations are
partly due to the different numbers of the waveform data used for the comparison.
This means that it is not reliable to determine which retracker is better though a
direct comparison of the STDs. Thus, an independently external ocean surface
reference is necessary to estimate absolute biases precisely. However, as any
gravimetric determination of the geoid is deficient in the zero- and first-degree terms
(Heiskanen and Moritz, 1967; Featherstone et al., 1996), the geoid heights
interpolated from AUSGeoid98 are not taken as a reference to ascertain the absolute
bias between the SSH before and after retracking.

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 234
For the biases existing between the SSH data before and after retracking, however,
there is alternative approach that can be adopted to remove them. Because the bias is
caused by the instrument and the on-board tracking algorithm (Hayne and Hancock
III, 1990; Chelton, 2001; Hayne, 2002), it can be corrected from the instrument
correction. The correction can be computed by a ground data processing system or
by waveform retracking. The recent research of Dong et al. (2002) show that the
absolute bias can also be determined using altimeter data and GPS levelling at tide
gauges.
(c) Algorithm Validation
To test how effective the retracker is, retracking using only a single retracker has
been conducted in two distinct coastal areas of Australia and the Taiwan Strait. The
AUSGeoid98 geoid heights and the ship track gravity anomalies are used as external
input data sets of the ground truth.
When comparing the geoid heights with the SSH data in Australian coastal regions
(Section 5.8.2), the emphasis is on the shapes of the height profiles and the changes
of the along-track slope. Results from single track comparison show that the SSH
gradients in the vicinity of the land before retracking are large varying from
~363.8-975.6 mm/km over offshore along-track distances of 0-19.5 km, being much
larger than the maximum geoid gradient value of ~150 mm/km in Australia (cf.,
Johnston and Featherstone, 1998; Friedlieb et al., 1997; Featherstone et al., 2001).
After retracking, the SSH gradients decrease by ~16.0 mm/km over offshore alongtrack
distance of 4.7 km − 19.5 km, agreeing well with the geoid gradient
(~16.4 mm/km) at the same distance. Results indicate strongly that retracking, no
matter which retracker is used, can improve the altimeter-derived SSH in coastal
regions.
Using ~2.5 years of ERS-1 SSH data sets before and after retracking, gravity
anomaly predictions in the Taiwan Strait are made from the inverse Vening Meinesz
formula (Hwang, 1998). Retrackers used are the fitting algorithm including the ocean
and five-parameter models. Results are then compared with limited ship-track gravity
data. The comparison of results demonstrated that waveform retracking has improved
the quality of the SSH in coastal regions near the China Sea and Taiwan Strait, thus

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 235
improving the accuracy of the altimeter-derived gravity field. Improved residual
geoid gradients and later on gridded gravity anomaly estimation are obtained in this
coastal region when the SSHs after retracking are used instead of unretracked SSH
data. The STD of agreements between the predicted and ship-track gravity anomalies
are ±13.9 mgal before retracking and ±9.9 mgal after retracking. This shows a great
potential of providing accurate gravity field information in coastal regions by using
satellite altimeter SSH data after waveform retracking.
8.2.4 Other Conclusions
A preliminary test has been performed to calculate the retracking gate estimate
related to the land returns in the range window using external independent data
source of the DEM model in coastal regions. Analysing results show that nonuniqueness
of the estimate over land near coastal regions is still a main problem that
should be solved in the future.
The analysis on extracting the SSHs from altimeter waveform data suggests that the
SSB correction is inaccurate when it is computed using the empirical model in the
coastal region due mainly to the incorrect SWH estimated from the on-satellite
processor. Waveform retracking can remove the need for the ocean wave influence
and thus reduce the SSB error, while an error is caused to the unretracked SSH data
by the need of this correction no matter whether or not the SSB is applied. In
addition, the method of averaging residual geoid gradients developed for the AWF
data product in this study has been demonstrated being effective, because a similar
accuracy level (~±14 mgal) as Hwang et al. (2001) is obtained using unretracked
altimeter data.
8. 3 Recommendations for Future Work
The improvement of the altimeter SSH data in Australian waters and the Taiwan
Strait has demonstrated that the coastal waveform retracking system developed in
this research has the potential of being applied to other coastal regions, as well as
other satellite altimeter waveform data sets. With higher precision of the altimeter
measurements over open oceans, demand for improving the precision of the data in
coastal regions will become increasingly desirable to realise the global coverage of

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 236
the altimetric missions. Therefore, recommendations and further work will be given
in this Section.
8.3.1 Extraction of Ocean Returns from the Mixed Waveform Shape
Current retracking algorithms including those in the coastal system cannot recover
the ocean returns that appear in the late gates behind the land returns in the range
window when the ground track is very much closer to the coastline (Brooks et al.,
1997). However, if the land return in the waveform can be estimated and removed, it
should be possible to recover the ocean return. As stated, this can be achieved only
when other external and high-resolution reflected surface topography are available.
With new satellite altimetry missions, improved DEM created from satelliteobserved
data, and detailed topography information, developing new waveform
retrackers for this type of waveform will become an area requiring further research
effort.
8.3.2 Improvement of the Correction Algorithms near Land
This research has demonstrated that waveform contamination can be removed in a
certain offshore distance using waveform retracking techniques. However, waveform
retracking can improve only the range measurements. The incorrect geophysical and
atmospheric corrections still affect the precision of the altimeter-derived SSH data
sets, which are the direct measurements in geodetic applications. Improvement of the
corrections in coastal regions has received attention from several researchers, such as
Shum et al. (1998) and Anzenhofer et al. (2000). To fulfil the satellite altimetric
potential in coastal regions, developing geophysical and environmental correction
algorithms will be another future research field.
As concluded in this research, the SSB correction causes a large error in the SSH
data near the coastline. In general, the SSH data before used for geodetic applications
must apply an instrument correction to remove the effects of the SSB. However, how
can the SSH be improved by such incorrect SSB correction in coastal regions?
Fortunately, retracking procedures also removes the need for the SSB correction.
Therefore, retracking in a wide range of coastal regions will also be one for future
research.

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 237
8.3.3 Improvement of the Altimeter Operation
From this research, the operation of the altimeter affects the waveform, in particular
waveforms along ground tracks leaving land to water. This result suggests that it is
essential to improve the on-board tracking algorithm based on this operating feature
or the design of the altimeter operation.
8.3.4 Geodetic Application
For further research of determining gravity anomalies in coastal regions, and to
overcome the problem of altimeter data discontinuity very near to the coastline,
combining the uniform coverage of the altimeter data with shipborne gravity and
land gravity data will provide a very accurate high-resolution gravity field in coastal
regions.

238
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