Master's thesis Ørsted as a gravity mission satellite

FACULTY OF SCIENCE
UNIVERSITY OF COPENHAGEN
Master’s thesis
Ørsted as a gravity mission
satellite
Sofie Louise Sandberg Sørensen
Niels Bohr Institute
Academic advisor: Carl Christian Tscherning
Submitted: 16/01/06

Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 The Ørsted Satellite 3
2.1 Instruments on board Ørsted . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Satellite-to-satellite Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Ørsted Data 6
3.1 RINEX Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 JPL Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 ECI Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Basic Theory 8
4.1 Spherical Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 The Gravitational Potential of the Earth . . . . . . . . . . . . . . . . . . . 10
4.3 The Normal Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.4 Chauvenet’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.5 Rotation between CTS and CIS . . . . . . . . . . . . . . . . . . . . . . . . 13
4.6 Simpson’s Rule of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Selecting the Data 16
5.1 Solar Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Evaluation of the GPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 Ground Track of the Satellite . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Gravity Field Determination 20
6.1 Dedicated Gravity Mission Satellites . . . . . . . . . . . . . . . . . . . . . 21
6.1.1 CHAMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1.2 GRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1.3 GOCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 Ørsted as a Gravity Mission Satellite. . . . . . . . . . . . . . . . . . . . . . 23
7 Methods for Gravity Field Determination from State Vectors Only 25
7.1 The Acceleration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.2 The Energy Conservation Theorem . . . . . . . . . . . . . . . . . . . . . . 26
7.2.1 Tidal Gravitational Disturbances Caused by the Sun and Moon . . 26
7.2.2 Rotational Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.2.3 Non-Gravitational Disturbing Potential . . . . . . . . . . . . . . . . 28

8 Deriving The Gravitational Potential 29
8.1 Calculation of Vkin, Vrot, Vsun and Vmoon . . . . . . . . . . . . . . . . . . . . 29
8.2 Determination of E0 and F . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.3 Calculation of the Gravitational Potential Vearth . . . . . . . . . . . . . . . 39
9 Theory of Least Squares Collocation 41
9.1 The Covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
10 Methods of Calculating the Spherical Harmonic Coefficients 45
10.1 Method 1: Using Numerical Integration . . . . . . . . . . . . . . . . . . . . 45
10.2 Method 2: Using Orthogonality Properties of the Spherical Harmonics . . . 46
10.3 The Kepler Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 47
11 Collo.f90 50
11.1 Program outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11.2 Programming Problems and Solutions . . . . . . . . . . . . . . . . . . . . . 55
11.2.1 The multi-grid function . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.2.2 Iterative Improvement . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.2.3 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
11.2.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12 Deriving the Ørsted Gravitational Potential Field Model ORSTED05 60
12.1 Calculation of GM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
12.2 Comparing Results from Method 1 and Method 2 . . . . . . . . . . . . . . 60
12.3 Collo.f90 error test on EGM96 test data . . . . . . . . . . . . . . . . . . . 62
12.4 The Model ORSTED05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
13 Final Considerations and Discussion 69
14 Conclusion 74
A Data Files 77
A.1 ECI file example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2 JPL file example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.3 RINEX file example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B EGM96 coefficients 78
C Fortran programs 79

Abstract
The Ørsted satellite was launched in 1999, with the main goal of modeling the Earth’s
magnetic field.
In this thesis Ørsted has been evaluated as a gravity mission satellite, and the goal has
been to derive a gravity field model, based only on Ørsted data.
By using satellites like Ørsted as gravity mission satellites, it will in theory be possible to
gain new information on the variations in time of the gravity field, because these satellites
have been orbiting the Earth for a longer period of time that the dedicated gravity mission
satellites. In this thesis, only the static gravity field has been modeled.
The Ørsted data, used in this project, are GPS measurements giving the state vector
of the satellite. Since the kinetic energy can be calculated directly from the state vector,
the energy conservation method has been used to derive the gravitational potential from
the state vectors.
When using the energy conservation method, information on the non-gravitational accelerations
are needed. These are not measured by the Ørsted satellite, and therefore an
approximations of these has been found and applied.
Least squares collocation has been used to derive a global gravity field model from
the calculated gravitational potential. Together with the least squares collocation, two
different methods of calculating the spherical harmonic coefficients of the gravity model
has been used and compared; a method using numerical integration and a method using
the orthogonality properties of the Legendre polynomials.
It was shown that the two methods were equally accurate, if the grid used in the numerical
integration method, was chosen sufficiently dense. The method using the orthogonality
properties was chosen for the full modeling.
An Ørsted gravitational potential model, ORSTED05, was found which contains spherical
harmonic coefficients up to degree and order 6. The upper limit of obtainable coefficients
was determined by comparison with EGM96 coefficients and evaluation of the modeling
errors from collo.f90, the program which has been developed for the model calculations.
The ORSTED05 potential field model, with the term ¯ C20 subtracted, is shown in the figure
below.

Sammendrag
Den danske satellit Ørsted blev sendt i kredsløb om Jorden i 1999. Hovedfom˚alet med
Ørsted var at kortlægge Jordens magnetfelt.
I dette speciale er mulighederne for, at benytte Ørsted satellitten som tyngde missionssatellit
blevet evalueret, og form˚alet med specialet har været, at bestemme en model af
Jordens tyngdefelt kun baseret p˚a data fra Ørsted.
Ved at benytte satellitter som Ørsted i tyngdefelts-modelleringer er det i princippet
muligt at forbedre billedet af de tidslige variationer af tyngdefeltet, da denne slags satellitter
har været i kredsløb omkring Jorden i længere tid end de egentlige tyngdemissionssatellitter.
I dette projekt er kun det statiske tyngdefelt blevet modelleret.
De Ørsted data, som er blevet benyttet i dette speciale, er GPS-data der indeholder
information om satellittens position og hastighed (state vector). Da den kinetiske energi af
satellitten kan beregnes direkte ud fra state vector’en, er energibevarelses metoden blevet
valgt til bestemmelse af Jorden’s tyngdepotential. N˚ar man benytter energibevarelses metoden,
er det nødvendigt at have kendskab til satellittens ikke-gravitationelle accelerationer,
for at kunne isolere og bestemme de accelerationer, der skyldes Jordens tyngdepotential.
De ikke-gravitationelle accelerationer m˚ales ikke af Ørsted, og derfor har det, i dette
projekt, været nødvendigt at finde en m˚ade at approksimere disse p˚a.
Mindste kvadraters kollokation er blevet benyttet til at bestemme en global model
af Jorden’s tyngdepotential. I dette speciale er to forskellige metoder blevet benyttet
sammen med kollokation, til at bestemme kuglefunktions-koefficienterne i denne model.
Ved den ene metode benyttes numerisk integration, og ved den anden metode udnyttes
ortogonalitetsegenskaberne af Legendre funktionerne til bestemmelsen af koefficienterne.
Den globale model, ORSTED05, der er fundet i dette speciale, er en kuglefunktionsudvikling
af Jorden’s tyngdepotential der indeholder led op til grad og orden 6. Den
øvre grænse af led der kunne bestemmes ud fra Ørsted data, og som indg˚ar i ORSTED05,
er blevet bestemt p˚a baggrund af sammenligning med EGM96 samt overvejelser af de
modellerings-fejl der genereres af collo.f90, der er det program der er blevet udviklet
til at udføre model beregningerne. ORSTED05 potentialfelts-modellen, med ¯ C20 leddet
fratrukket, er vist i figuren herunder.

I would like to thank
Acknowledgments
- C. C. Tscherning for being a committed and helpful academic advisor.
- Martin B. Sørensen for valuable help on working with the Ørsted data.
- Anders Lynge Esbensen for great support,
both personal and programming, throughout the project.
- Henriette Hjorth for sharing office and opinions with me.
- Eva Howe for always wanting to help.

1 Introduction
1. Introduction
The main purpose of this thesis, will be to explore the possibilities of deriving a model
of the Earths gravity field, based only on data from the danish micro satellite Ørsted. It
has to my knowledge not been done before, due to the fact that Ørsted is not designed for
gravity field measurement but for magnetic measurements.
The modeling will be based on the energy conservation method and least squares collocation,
and the aim is to determine which level of accuracy can be achieved for such an
Ørsted gravity field model.
When using the energy conservation method, information of the non-gravitational potential,
such as the friction acting on the satellite, is needed. But since there is no accelerometer
placed on board Ørsted, there are no direct measurements of this. Therefore,
one of the major challenges of this thesis will be to find an acceptable method to estimate
this non-gravitational potential.
Before getting started, one must recognize that it will not be possible to obtain better
- or even as good - models derived from Ørsted data, as those derived from data
from the dedicated gravity mission satellites like for example CHAMP [Howe et al., 2003],
[Gerlach et al., 2003a]. The reasons being the lack of non-gravitational acceleration data,
and the fact that Ørsted is orbiting the earth at a much higher altitude than the dedicated
gravity mission satellites.
The Ørsted satellite and mission is explained in section 2, and the different available data
files from Ørsted are described in section 3.
In section 4 the basic theory, needed in this project, is explained. This section includes
for instance an introduction to spherical harmonics, a description of the use of Chauvenet’s
criterion and the theory of rotation between the Conventional Terrestrial System (CTS)
and the Conventional Inertial System (CIS).
The selection of usable data for this project is described in section 5.
Gravity field determination in general is described in section 6, in which three different
dedicated gravity missions are also described. In this section Ørsted is evaluated as a
gravity mission satellite, based on considerations of the instruments on board the satellite
and the satellites orbital parameters.
The theory behind the energy conservation method is described in section 7 and the
calculations in practice in section 8.
In order to derive a model from the calculated gravitational potential, least squares
collocation is used. The theory behind this method is explained in section 9. In section 10
three different methods of calculating the spherical harmonic coefficients of the model are
explained, and two of these methods are evaluated in section 12. The gravity modeling in
practice is executed by the program collo.f90, which has been developed during this project,
and this program is described in section 11. In section 11.2 some of the programming
problems and solutions related to collo.f90 are described.
1

1. Introduction
The result of this thesis is the gravitational potential field model ORSTED05, and in
section 12 the ORSTED05 model is derived and described.
In section 13 the main tasks of the thesis are summarized and the choices, made throughout
the project, are discussed.
In section 14 the final conclusions on the thesis is made.
1.1 Motivation
I believe that it will be of great importance, if it can be shown to be possible, to derive
useful information on the Earth’s gravity field from Ørsted data.
This will mean that the number of satellites, which can be used for the purpose of
gravity field modeling, will increase. The reason being that other satellites similar to
Ørsted also could be shown to be useful, in spite of the fact that they are not designed
specifically for gravity field measurements.
Gravity field derivations from satellites like Ørsted will, for example, give us new information
on the variations of the gravity field in time. Hence these satellites have been
orbiting the earth for a longer period of time, than the specific gravity mission satellites.
Furthermore, these satellites can contribute with additional information to the ongoing
static gravity field modeling, since the coverage of the Earths surface will become better
when the number of useful satellites increases. The result will be a more accurate and
detailed model of the earth’s gravity field.
Gravity field models are of great importance in a wide variety of sciences. The solidearth
geophysicists can derive information on the structure of the earth, and phenomena
like sea level changes and land uplift can also be observed, when studying the gravity field
variations. Furthermore glaciologists and oceanographers use gravity field variations to
investigate the movement of ice sheets and ocean circulation.
Satellites provide important data for the ongoing gravity field modeling, because their
measurements cover the Earth much better than it would ever be possible to do by measurements
on earth only. But since satellite data is not nearly as accurate as earth measurements,
a combination of both data is often used in modern gravity field modeling.
2

2 The Ørsted Satellite
2. The Ørsted Satellite
The gravity field modeling which is the goal of this thesis, will be based only on data from
Ørsted. Therefore, the satellite and the Ørsted mission is shortly described in this section.
The Ørsted satellite is the first and only danish satellite, and it is the result of a collaboration
between institutions of research and private companies in Denmark. The satellite
is seen in figure 1.
The satellite was launched on the 23rd of February 1999, and it is still functional today.
It transmits data to the Earth several times per day. Some facts about the Ørsted satellite
are listed in the box below [E. F. Christensen, 1994].
Facts on the Ørsted satellite.
-Inclination : 96.1 degrees.
-Orbital period : ∼ 100 minutes.
-Apogee altitude : ∼ 850 km.
-Perigee altitude : ∼ 450 km
-Weight : 60.7 kg.
-Total cost: 126.1 mill dkr.
-Eccentricity 0.015
The Ørsted satellite is designed for measurements of the Earths magnetic field and it’s
variations in time. The first magnetic measurements made by Ørsted were, at the time, the
most precise ever made from space. It resulted in the Ørsted Initial Field Model (OIFM).
Besides mapping the Earth’s magnetic field, other purposes of the Ørsted mission has
been to investigate the electric current systems in the upper atmosphere and magnetosphere,
and to increase our knowledge of the connection between the solar wind and the
Earths magnetic field. A GPS receiver, placed on board the Ørsted satellite, has made it
possible also to derive information on some meteorological conditions in the atmosphere,
such as temperature and air humidity, for further information see
http://esa-spaceweather.net/sda/gpsvalidation/.
2.1 Instruments on board Ørsted
An Overhaus magnetometer is placed on board Ørsted, which measures the strength of
the magnetic field. It is located at the end of a long boom, in order to prevent it from
measuring the magnetic disturbances from the satellite’s electrical instruments. A vector
magnetometer, also placed on the boom, measures the components of the field.
3

2. The Ørsted Satellite
Figure 1: The Ørsted satellite.
There are two GPS receivers on board Ørsted; a TANS 2 receiver produced by Trimble
and a TurboRogue receiver produced by Jet Propulsion laboratories (JPL). These are used
for determination of the exact position of Ørsted.
The majority of the time, the TANS 2 receiver is used for the positioning of the satellite,
though it is not nearly as accurate at the TurboRogue receiver. This is due to the fact
that the TurboRogue uses a lot of electric power compared to the TANS 2.
The GPS measurements do not give information on the attitude of the satellite in space.
For this purpose there is a star camera on board the satellite. It continuously takes images
of the visible stars and compare these with star catalogues. This gives a very precise
determination of attitude.
For the purpose of determining the density of charged particles (for example protons,
electrons and helium nuclea) in space, there is also a particle detector on board.
The power for all these instruments is provided by solar panels. They also charge
batteries, which provide the power for the instruments, whenever the satellite is in the
shadow of the Earth.
The location of the individual instruments on the satellite is shown in figure 1
[E. F. Christensen, 1994].
2.2 Satellite-to-satellite Tracking
The Ørsted satellite is part of a configuration called satellite-to-satellite tracking in highlow
mode (SST-HL) [Seeber, 2003]. The low orbiting Ørsted satellite, with an average
4

2. The Ørsted Satellite
Figure 2: Illustration of satellite-to-satellite tracking in high-low mode.
height of approximately 700km, is tracked by the high orbiting GPS satellites, with a
height of approximately 26.000km, and its position is thereby determined relative to an
Earth-fixed coordinate system, CTS. The constellation of SST-HL can be seen in figure 2.
The position of the low orbiting satellite is found by continuously measuring the ranges
and range changes between this and several GPS satellites (at least 4), which positions are
known with high accuracy. For the purpose of gravity field measurements, the low orbiting
satellite (in this case Ørsted) is used as a probe in the Earths gravity field. Information
on the gravity field can be derived by studying the motion of the low orbiting satellite.
5

3. Ørsted Data
3 Ørsted Data
The Ørsted data, which will be used in this thesis, consists purely of GPS-data. This
GPS-data is available in three different types of files; RINEX files, JPL files and ECI files.
Only GPS-data from the TurboRogue receiver will be used, since this is the most accurate
of the two GPS receivers on board Ørsted, see section 2. The Ørsted GPS-data have been
collected and processed by DMI.
In the following, the three types of GPS data files are described, and in appendix A
examples of the files are given.
3.1 RINEX Files
The RINEX files are the post processed GPS data files, and both the JPL and ECI files
are derived from the RINEX files. RINEX is short for Receiver Independent Exchange
Format, and it is a file format which has been developed, in order to minimize the potential
problems of international collaboration. It is an internationally accepted data format,
by both users and manufacturers and being receiver independent, the RINEX data from
different receiver types can easily be processed together. The RINEX files, used in this
thesis, contain information on which GPS satellites, the TurboRogue GPS receiver has
been in contact with, for every single measurement. Besides this information the time,
range (pseudo-range) and phase of each signal are given in the RINEX files. The time
given in the RINEX files, is the arrival time of the received signals, and it is given in GPS
time.
The pseudo-range P R is defined as:
P R = d + c ∗ (dtreceiver − dtsatellite + B), (1)
where d is the distance between the receiver antenna and the satellite antenna, c is
the velocity of light, dtreceiver is the receiver clock offset and dtsatellite is the satellite clock
offsets. B are the sum of other biases, as for example atmospheric delays. The pseudorange
is given in meters.
The phases given in the RINEX files are the carrier phases of both L1 and L2.
For further information on the RINEX file format see for example
http://igscb.jpl.nasa.gov/igscb/data/format/rinex2.txt.
3.2 JPL Files
The JPL (Jet Propulsion Laboratories) files are processed GPS data files. The JPL files
contain information on the state vector ¯ VCT S of the satellite, calculated from the information
in the RINEX files. The state vector is given every 10th second in Cartesian
coordinates in an earth-fixed coordinate system CTS.
¯VCT S = (¯x, ¯v)CT S = (x, y, z, vx, vy, vz)CT S, (2)
6

3. Ørsted Data
where ¯x is the position and ¯v is the velocity of Ørsted. The units of the state vector are
[km] and [ km]
of the position and the velocity respectively. The time is, as in the RINEX
s
files, given in GPS time.
The ¯ VCT S state vector is calculated directly from the RINEX file, since the GPS data
is given relative to a system of reference points on the Earth, meaning relative to the CTS
system.
3.3 ECI Files
The ECI (Earth Centered Inertial) files contain information on the state vector, ¯ VCIS, of
Ørsted’s motion relative to the Earth-centered inertial system CIS.
¯VCIS = (¯x, ¯v)CIS = (x, y, z, vx, vy, vz)CIS, (3)
The data in the ECI files are calculated from the data in the JPL files by rotating the
vectors from the Earth-fixed system (CTS) to the inertial system (CIS):
rCIS = P −1 N −1 S −1 rCT S
where P is the precession matrix, N is the nutation matrix and S is the matrix, which
rotates from the space-fixed system to the Earth-fixed system. The rotation from CTS to
CIS is fully explained in section 4.5.
The time in the ECI files is given in UTC time. Throughout the period of time relevant
in this thesis, the difference between GPS time and UTC time has been 13 seconds.
GPS time − UTC1999 = GPS time − UTC2000 = +13seconds (5)
The latter is an important information when comparison between the state vectors in
the JPL files and the ECI files must be made.
7
(4)

4. Basic Theory
4 Basic Theory
In this section some of the basic theory needed in this project, is explained. For example,
some potential field theory, a method for numerical integration and rotation between coordinate
systems is described.
Firstly the spherical harmonics are introduced, since these functions are of primary importance
in the gravity field modeling.
4.1 Spherical Harmonics.
The functions, Y c
nm and Y s
nm, are called the Laplace surface spherical harmonics
[Torge, 1999].
Y c
nm = Pnm(cosθ)cosmλ (6)
Y s
nm = Pnm(cosθ)sinmλ
They are an orthogonal basis for the Hilbert space on the unit sphere. Any continuous
function on a sphere, as for example the Earth’s gravity field, can be represented by a
linear combination of these functions. The functions Pnm(cosθ) are the associated Legendre
functions of degree n and order m.
The orthogonality of the basis functions, are due to:
Y i
nmY k
n ′ m ′dσ = 0, (7)
form:
σ
for n = n ′ , m = m ′ or i = k. For n = n ′ , m = m ′ and i = k the latter equation has the
σ
Y i
nmY i
nm =
4π
2n+1
2π (n+m)!
2n+1
for m = 0
for m = 0
(n−m)!
This means that the fully normalized spherical harmonics can be written as follows
[Torge, 1999] :
⎧
⎨ 2n+1
¯Ynm
4π
=
⎩
Ynm for m = 0
2n+1 (n−m)!
2π (n+m)! Ynm
(9)
for m = 0
The Laplace surface spherical harmonics of order zero (m = 0) are called the zonal
harmonics. They have no dependence on the longitude λ since:
Y c
n0 = Pn0(cosθ)cos0λ = Pn0(cosθ) (10)
Y s
n0 = Pn0(cosθ)sin0λ = 0
8
(8)

(a) (b) (c)
4. Basic Theory
Figure 3: Spherical harmonics. (a) Zonal harmonics with n = 3 and m = 0. (b) Tesseral
harmonics with n = 5 and m = 2. (c) Sectorial harmonics with n = 3 and m = 3.
The zonal harmonics have n nodes (zeros) on the interval (0 ≤ θ ≤ π), separating zones
of alternating signs on the unit sphere.
The spherical harmonics with m = 0 are longitude dependent and are called the tesseral
harmonics. These contain 2m nodes in the interval 0 ≤ λ ≤ π and (n − m) nodes in the
interval 0 ≤ θ ≤ π.
The last kind of harmonics are called the sectorial harmonics, and are defined by same
degree and order (n = m). These have no latitude dependency. Examples of the different
kinds of spherical harmonics are shown in figure 3.
9

4. Basic Theory
4.2 The Gravitational Potential of the Earth
The gravitational field of the Earth is invariant to rotation, meaning that:
curl ¯g = 0, (11)
with ¯g being the gravitational acceleration vector [Torge, 1999], [Seeber, 2003]. This
characteristic means that the following is valid:
¯g = grad V, (12)
where V is the Earth’s gravitational potential. The gravitational potential has the
form:
1
V (r) = G
dm , lim V = 0, (13)
earth r − r ′ r→∞
where r − r ′ is the distance from the mass volume dm(r ′ , θ ′ , λ ′ ) to the point P (r, θ, λ).
Hence the gravitational potential in point P is the work needed, in order to move a unit
mass from r = ∞ to the point P .
The Earth’s gravitational potential, V , is a continuous function with continuous derivatives
of all orders, and it furthermore fulfills Laplace’s differential equation of second order
in places with no mass:
∆V = 0 (14)
Therefore, the Earth’s gravitational potential field is a harmonic function, and can be
represented by a spherical harmonic expansion, as mentioned in section 4.1. The spherical
harmonic expansion transforms the volume integral in eq. 13 into an infinite series by
rewriting the reciprocal distance, 1
r−r ′ , into a series ([Torge, 1999],eq. 3.79):
1 1
=
r − r ′ r
∞ ′ r
n Pn(cos Ψ) (15)
r
n=0
where Ψ is the angle distance between dm and P. Combining equation 13 and 15 the
following expression can be derived, see also section 4.1:
V = GM
(1 +
r
∞
n
n=2 m=0
a
n Pnm(cos θ)(Cnm cos mλ + Snm sin mλ)) (16)
r
where a is the equatorial radius of earth, a = 6378136.6m. The term with n = 1 is zero
since the center of the coordinate system is chosen to be at the Earth’s center of mass.
These equations are valid if all the mass of the Earth is assumed enclosed.
The latter equation can equivalently be expressed in terms of the fully normalized
Legendre functions ¯ Pnm and coefficients ¯ Cnm and ¯ Snm:
10

V = GM
(1 +
r
∞
n
n=2 m=0
4. Basic Theory
a
n ¯Pnm(cos θ)(
r
¯ Cnm cos mλ + ¯ Snm sin mλ)) (17)
In geodesy, equation 16 is often written with another notation, namely:
V = GM
(1 −
r
∞
n
n=2 m=0
a
n Pnm(cos θ)(Jnm cos mλ + Knm sin mλ)) (18)
r
The spherical harmonic expansion is conventionally divided into two parts:
V = U + T, (19)
where T is called the disturbing potential and U is the normal potential defined subsequently.
The spherical harmonic expansion of the Earth’s gravitational potential, given i eq. 16,
simply represents a spectral decomposition of the gravitational field. The amplitude of
each term in the expansion are determined by the harmonic coefficients Cnm and Snm (or
Jnm and Knm), which are mass integrals describing the mass distribution in the Earth.
The zero order term , GM also called the Keplerian term, represents the potential of a
r
homogeneous or radially layered Earth. The coefficient C20 = −J20 is called the dynamical
form factor and it represents the flattening of the Earth.
4.3 The Normal Potential
The normal potential, U, is the potential of the normal gravity, γ:
γ = grad U (20)
The normal gravity field is the external gravity field of an equi-potential ellipsoid. The
normal potential consists of a gravitational, Vellipsoid, and a centrifugal, Zellipsoid, term.
U = Vellipsoid + Zellipsoid, (21)
The spherical harmonic expansion of the normal potential has the form:
U = GM
(1 −
r
∞
n=1
( a
r )2n J2nP2n(cosθ)) + ω2
2 r2 sin 2 (θ), (22)
where a is the semi major axis of the ellipsoid, J2n are the even harmonic coefficients
and P2n are the even Legendre functions.
The odd terms of Jn are zero due to the symmetry of the ellipsoid with respect to the
equator of the earth. Due to the symmetry of the ellipsoid with respect to the rotational
axis of the earth, U contains no tesseral terms.
In this thesis, the normal potential will be used as a reference field.
11

4. Basic Theory
4.4 Chauvenet’s Criterion
When evaluation of noisy data is needed, Chauvenet’s criterion can be used. In other
words, Chauvenet’s criterion can be used to filter data sets.
In this thesis, Chauvenet’s criterion will be used to evaluate the TurboRogue GPS
measurements, and in the following the criterion is described, [Taylor, 1997].
If we suppose having a series of N measurements, xi, of a quantity x, which probability
distribution is governed by a normal distribution. The mean, ¯x, and standard deviation,
σx, of the measurements can be estimated using the following equations.
¯x = 1
N
xi
(23)
N
i=1
σx = 1
N
(xi − ¯x)
N − 1
2 (24)
Each measurement can be evaluated by calculating the value tsus of the measurement.
tsus is the difference between the value of the measurement and the mean value of the
distribution, given in units of the standard deviations.
tsus =
i=1
|x − ¯x|
Then the probability, p, that a legitimate measurement would differ from ¯x by more
than tsus standard deviations is determined.
σx
p = P rob(outside(tsusσx))
(25)
= 1 − P rob(inside(tsusσx)) (26)
= 1 − 1
tsus
√ e
2π
−z2 /2
dz
The latter equation must be calculated on a computer, since it is not possible to evaluate
it analytically. p is converted into number of measurements, n, by multiplying the
probability p by the number of measurements N. n is statistically the number of points
which differ at least tsus standard deviations from ¯x, and therefore the size of n determines
whether a measurement must be rejected or accepted.
−tsus
n = pN (27)
if (n ≥ 0.5) then accept data
if (n < 0.5) then reject data
If indeed a measurement is rejected, and thereby filtered out, the mean and standard
deviation of the remaining measurements must be recalculated, in order to evaluate the
next measurement.
12

4.5 Rotation between CTS and CIS
4. Basic Theory
The Ørsted JPL state vectors are given in the CTS system, which is centered in the Earth’s
center of mass and is rotating with Earth. In this thesis, the energy conservation equation
will be derived in the CIS system, which is also centered in the Earth’s center of mass,
but is fixed to some fundamental stars. Therefore a rotation between the two systems is
needed when transforming the JPL state vectors into the ECI state vectors, described in
section 3.
The theory of this section is based primarily on [Seeber, 2003]. A rotation of a vector
r from the Earth-fixed system CTS to the space-fixed system CIS, must take into account
both precession, nutation and Earth rotation:
rCIS = P −1 N −1 S −1 rCT S, (28)
where P is the precession matrix, N is the nutation matrix and S is the matrix which
transforms the instantaneously space-fixed system to the Earth-fixed system.
Because of the equatorial bulge on the Earth, the gravitational attraction from the sun
and the moon induces a torque on the rotating Earth. Due to this torque, the Earth’s
rotation axis moves in space with respect to an inertial system. It sweeps out a cone,
centered on the ecliptic pole.
This motion is not smooth, and it can be divided into parts of different periods. The
motion of the longest period (∼26000 years) is called the precession, and it is shown
schematically in figure 4.
Figure 4: Figure showing precession and nutation.
The precession matrix, P, is given by:
13

4. Basic Theory
P = R3(−z)R2(θ)R3(ζ) (29)
The matrices R1, R2 and R3 are the rotation matrices, which rotate around the x-axis,
y-axis and z-axis respectively.
The angles z, θ and ζ are functions of time, T since 01.01.2000 12:00 counted in Julian
centuries:
z = 0. ◦ 6406161T + 0. ◦ 0003041T 2 + 0. ◦ 0000051T 3
θ = 0. ◦ 5567530T − 0. ◦ 0001185T 2 − 0. ◦ 0000116T 3
ζ = 0. ◦ 6406161T + 0. ◦ 0000839T 2 + 0. ◦ 0000050T 3
Besides the precission, there are additional periodic motions of the Earth’s rotation
axis. These are called the nutation, and are also shown in figure 4. The main periods of
the nutation are 13.66 days, half-year, one-year, 9.3 years and 18.6 years. The nutational
motion is caused by the variations in the relative position of the moon, sun and Earth.
The nutation matrix, N, is given by:
with the angles ε, ∆ψ and ∆ε given by:
(30)
N = R1(−ε − ∆ε)R3(−∆ψ)R1(ε), (31)
ε = 23 ◦ 26 ′ 21 ′′ .448 − 46 ′′ .815T − 0 ′′ .00059T 2 − 0 ′′ .001813T 3
∆ψ = −17 ′′ .1996 sin Ω − 1 ′′ .3187 sin(2λM − 2D) − 0 ′′ .2274 sin(2λM) (32)
∆ε = 9 ′′ .2025 cos Ω − 0 ′′ 5736 cos(2λM − 2D) + 0 ′′ .0977 cos(2λM),
where Ω is the mean ecliptic longitude of the lunar ascending node, D is the mean
elongation of the moon to the sun and λM is the mean ecliptic longitude of the moon.
The transformation matrix account for the instantaneous polar coordinates xp, yp and
the Greenwich apparent sidereal time GAST :
S = R2(−xp)R1(−yp)R3(GAST ) (33)
14

4.6 Simpson’s Rule of Integration
4. Basic Theory
Simpson’s rule of integration is a method used for estimating integrals numerically
[Kreyszig, 1999].
The function f(x) will be piecewise quadratic approximated, in order to calculate the
numerical approximation, ˜ J, of the integral.
b
a
f(x)dx ≈ ˜ J (34)
For the use of Simpson’s rule, the integration interval x ∈ [a, b] must be divided into
an even number of subintervals, n, of equal length h = (b − a)/n. The points are thereby
given by xr = a + r ∗ h.
Using Simpson’s rule, two sub-intervals are used at the same time. This is why n must
be an even number.
A parabola is drawn through the three points (xr−1, f(xr−1)), (xr, f(xr)), (xr+1, f(xr+1))
defining the two sub-intervals, see figure 5
Figure 5: Simpson’s rule of integration
The area, A, under the parabola, which is highlighted in figure 5, is given by:
A = h
3 [f(xr−1) + 4f(xr) + f(xr+1)]
By adding all of the areas, Simpson’s rule of integration of the whole function, has the
form:
˜J = h
3 (f0 + 4f1 + 2f2 + 4f3 + · · · + 2fn−2 + 4fn−1 + fn), (35)
where fr = f(xr).
15

5. Selecting the Data
5 Selecting the Data
The first task in this project has been to select the Ørsted GPS-data, which subsequently
will be used for the modeling of the Earth’s gravity field. Since the TurboRogue GPS
receiver is the most accurate of the two GPS receivers on board Ørsted, only data from
this receiver will be used in the modeling. The disadvantage of this choice is, that this
receiver only has been turned on for limited periods of time.
5.1 Solar Activity
There are some things that have to be taken into consideration, when selecting the data.
Firstly, one has an interest in eliminating some of the disturbing effects which can influence
the later calculations, as for example the solar activity. Variations in the solar
activity induce changes in the atmospheric density, which again result in changes in the
deacceleration of the satellite due to the increased friction.
Hence, by choosing to work with data from a period of time, in which the solar activity
does not vary significantly in magnitude, the estimate of the non-gravitational accelerations
acting on Ørsted, will be simplified.
For the purpose of selecting a period of data, where the solar activity has been somewhat
stable, the radio flux has been plotted as a function of time since Ørsted was launched in
1999, see figure 6.
Solar flares generate electron beams, which again generate electromagnetic waves which
can be observed on Earth. These electromagnetic waves are called the radio flux, and it
is a good indicator of the magnitude of the solar activity [Kivelson and Russell, 1997]. In
figure 6, the red line shows the variations of the radio flux, and the green and blue marks
indicate which days, the TurboRogue GPS-receiver on board Ørsted, has been turned on.
The period of data, which runs from 1999-09-02 to 2000-02-22, has been chosen for the
further calculations, because it is a period of time in which the solar activity was rather
stable, and at the same time it contains many TurboRogue GPS-measurements not too far
spaced in time. In figure 6 the period of data, that has been selected, is marked with blue.
5.2 Evaluation of the GPS Data
Originally the chosen period of time contained more than 400.000 TurboRogue GPSmeasurements.
It can be assumed that they are not all equally accurate. Unfortunately
there are no direct uncertainty estimates, and another parameter therefore must be found
to evaluate the data.
For this purpose, it would be preferable to know the PDOP values of every single GPS
positioning, since these could be used for weighting the individual measurements in the
modeling. The PDOP value is an indicator of the geometry of the GPS-satellite constellation
used to determine the position, and it is given by:
16

Figure 6: Solar activity indicated by the radio flux.
17
5. Selecting the Data

5. Selecting the Data
a b
Figure 7: (a) The position of the satellite in the X-Y plane before the filtering. (b) After
the filtering
P DOP = Vmax
, (36)
V
where V is the volume formed by the satellite constellation and Vmax is the largest
volume possible expanded by the satellite constellation ([Seeber, 2003], p.301).
Unfortunately the PDOP values for each measurement are not given, and they would
be both difficult and time consuming to calculate. Therefore another parameter must be
used in the determination of whether a measurement is good (accurate enough) or not.
The accuracy of a GPS-measurement depends strongly on the number of satellites used
in the positioning, and therefore this parameter will be used in the selection of data. The
choice is made, that all measurements based on signals from 6 or more GPS satellites will
be considered as good measurements.
The program rinex pass2.f90, see appendix C, has been developed for the purpose
of filtering the GPS-data. The output of this program is a file containing only data
from the GPS-measurements based on 6 or more satellites. Figure 7 shows the effect
of rinex pass2.f90, by showing the data points of the satellite for one day in the X-Y plane
both before and after the filtering. It is seen that the filtering leaves ”holes” in the orbit
of the satellite.
The chosen period of time contains 254463 good GPS-measurements. The distribution
of good measurements in the chosen period, can be seen in figure 8.
5.3 Ground Track of the Satellite
It must be ensured that the ground track of the satellite, in the chosen period of time, covers
the whole surface of the Earth. This must be fulfilled, in order to derive a satisfactory
18

Figure 8: Distribution of good data in the chosen period.
5. Selecting the Data
model, which describes the complete gravity potential field of the Earth, without any
”holes”. The ground track of the Ørsted satellite, in the selected period, is seen in figure 9.
It is seen that the ground track covers the Earth satisfactory except for the polar regions.
The polar gaps are a result of the inclination of the satellite.
Now the data, which will be used in the modeling of the Earth’s gravity field, has been
selected, based on considerations of the solar activity, accuracy of the positioning and the
coverage of the ground track of the satellite.
Figure 9: The ground track of Ørsted in the chosen period of time.
19

6. Gravity Field Determination
6 Gravity Field Determination
There are many different ways of determining the Earth’s gravity field by satellite. In this
section three different gravity mission satellites, using different measuring techniques will
be described. This is done to clarify the limitations and possibilities of Ørsted as a gravity
mission satellite.
The result of a gravity field determination can be represented in many different ways.
The aim is to produce a result which also can be used as a tool in other projects. Therefore
the goal of this thesis is not only to calculate the values of the gravity potential from
the Ørsted data, but rather to be able to determine a global model representing the results.
The global gravity model determined in this project, will be expressed by a spherical
harmonic expansion of the gravitational potential field. This type of global model has
been chosen because of the advantages of spherical harmonics, when describing a function
on a sphere, see also section 4.1. Furthermore the models widely used today, such as
EGM96 [Lemoine et al., 1998], are spherical harmonic series representing the gravitational
potential field. By choosing this kind of representation, the results found in this thesis can
and will be directly compared to EGM96.
Figure shows the EGM96 model to degree and order 360, represented by the geoid
heights.
Figure 10: The geoid model EGM96.
Many different parameters can be used to represent the Earth’s gravity field. For example
both the gravitational potential, the geoid height and the gravitational acceleration
represents the field, and these parameters can be derived from one another. The choice of
representation depends on the purpose of the modelling. In this thesis, the gravity field
will be represented by the gravitational potential field.
Contrary to Ørsted, the satellites CHAMP, GRACE and GOCE are dedicated gravity
mission satellites. They have different characteristics, but they are designed to complement
each other in the goal of refining the knowledge of the Earths gravity field. This is
20

6. Gravity Field Determination
seen in figure 11, which shows the expected geoid errors of the harmonic coefficients of the
gravity model.
Figure 11: Geoid error estimates on the spherical harmonic coefficients. It is seen that
the three missions complement each other. ([Seeber, 2003] fig. 10.3)
6.1 Dedicated Gravity Mission Satellites
6.1.1 CHAMP
The CHAMP (Challenging Mini-Satellite Payload) satellite is, like the Ørsted satellite,
part of a SST-HL constellation, see section 2.2. [Seeber, 2003]
CHAMP was launched in July 2000 into a near polar orbit of height 450 km. The
satellite will drop to a height of approximately 300 km during a period of about 5 years
due to external forces, such as friction. This decrease in height makes the satellite able to
measure signal of varying wavelength throughout it’s lifetime.
CHAMP acts like a probe in the Earth’s gravitational field, which orbit perturbations
are registered by the GPS satellites. On board the satellite are accelerometers measuring
the non-gravitational accelerations of the satellite, in order to extract these disturbances
from the signal. The satellite is seen in figure 12. For further information on the CHAMP
satellite, see http://www.gfz-potsdam.de/pb1/op/champ/index CHAMP.html
6.1.2 GRACE
The two GRACE (Gravity Recovery and Climate Experiment) satellites are identical and
are orbiting the Earth at an altitude of about 500 km in a near polar orbit [Seeber, 2003].
The satellites are seen in figure 13.
The distance between the two satellites is approximately 220 km. The GRACE mission
is using the concept of satellite-to-satellite tracking in low-low mode (SST-LL). Each
21

6. Gravity Field Determination
Figure 12: The CHAMP satellite - part of a SST-HL mode constellation.
satellite transmits signals, which are received by the other satellite, in order to obtain a
very precise determination of the relative motion of the satellites.
The goal of SST in general is to setup equations connecting the coefficients of the
terrestrial gravity field, Cnm and Snm, with the observed radial velocity range rate, ˙ρ, and
the range rate change, ¨ρ. The parameters ˙ρ and ¨ρ are not actually measured, but are the
result of the data processing.
The relative motion is caused by the different accelerations of the two satellites, caused
by the different gravitational attraction from the Earth, and from non-gravitational accelerations.
As with the CHAMP satellite, the non-gravitational accelerations of the GRACE
satellites are measured by 3D accelerometers placed in the centre of mass of both satellites.
These measurements are done in order to separate the gravitational and non-gravitational
disturbances and thereby determine the gravity field of the earth. For more information
on the GRACE mission satellites, see for example: http://www.csr.utexas.edu/grace/.
Figure 13: The GRACE satellite - part of a SST-LL mode constellation.
22

6.1.3 GOCE
6. Gravity Field Determination
The GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite is planned
for launch in 2006. The satellite is seen in figure 14. The principles of this gravity mission
is GPS SST-HL and satellite gravity gradiometry (SGG). The satellite will be send into
a near polar orbit of a height of approximately 250 km. A gravity gradiometer on board
GOCE provides two measurements of the accelerations in every direction with a distance
of about 50 cm, from which the difference in acceleration can be derived.
The advantage of the SGG method compared to SST-HL and SST-LL is the fact that
the non-gravitational accelerations are the same for all the accelerometers since they are
placed in the same spacecraft. Therefore they cancel out when finding the acceleration
differences. With SGG, the second derivatives of the gravitational potential, or gravity
gradients, are determined.
The satellite is designed to be drag free, meaning that the cross section area in the
direction of maximum drag is minimised. There is an attitude and drag controller on board
the satellite, and the satellites attitude will be corrected continuously using thrusters. The
correction demands fuel, and therefore the lifetime of the satellite will be relatively short,
namely about 2 years. As seen in figure 11, GOCE is expected to refine the higher order
coefficients of the gravitational potential field.
The GOCE mission is described in detail at:
http://www.esa.int/esaLP/ESA1MK1VMOC LPgoce 0.html.
Figure 14: The GOCE satellite.
6.2 Ørsted as a Gravity Mission Satellite.
When comparing Ørsted with GRACE, CHAMP and GOCE, it is clear that it is most
comparable with CHAMP, mainly because they are both part of a SST-HL constellation.
There are several differences between CHAMP and Ørsted though, which make the gravity
derivations based on Ørsted data more complicated than the CHAMP derivations. There
are three important criteria which a satellite must fulfil, in order to qualify as a good
23

6. Gravity Field Determination
gravity mission satellite, and these are stated below. ([Seeber, 2003] p. 471)
-The satellite must have a relatively low orbital height (200-500 km).
-The GPS-tracking of the satellite must be uninterrupted.
-There must be instruments on board, which make it possible to separate the gravitational
and non-gravitational accelerations.
Ørsted does not fully satisfy the first criteria, since the apogee altitude of Ørsted is
about 850 km and the perigee altitude is approximately 450 km. The criteria stems from
the fact that the gravitational acceleration of a satellite caused by a mass anomaly in the
Earth, decreases with the square of the increasing orbital height, r. Furthermore the term
A = ( ae
r )n , describes the increasing attenuation of the gravitational field with increasing
orbital height. As a result of this, high orbiting satellites will only be able to measure
the components of the field, which have long wavelengths. Because of this, the Ørsted
data is expected only to reveal the low degree harmonic coefficients, if any at all, of the
gravitational field.
The TurboRogue GPS-tracking cannot fully fulfil the second criteria, since it is only
turned on for limited periods of time as it is seen in section 5.2. Though this is compensated
for by the fact that data from a long period of time will be used in further calculations. In
this way a satisfactory coverage can be obtained, in spite of the interrupted GPS-tracking.
The last criteria is not fulfilled by Ørsted either, since there is no accelerometer on
board, to measure the non-gravitational accelerations. Therefore a method, in which these
accelerations can be modelled and separated from the gravitational accelerations, must
be used in this thesis, in order to be able to convert the state vectors into gravitational
potential.
One advantage of the Ørsted satellite, when evaluating it as a gravity mission satellite,
is that it is orbiting the Earth in a near polar orbit, see section 2. This results in a very
good ground track coverage of the Earth, which is also clearly seen in figure 9 where it is
also seen that the polar gaps are of a limited size.
All in all, it must be concluded that Ørsted has many limitations as a gravity mission
satellite, and it will be interesting to see the level of accuracy which can be achieved.
24

7. Methods for Gravity Field Determination from State Vectors Only
7 Methods for Gravity Field Determination from State
Vectors Only
The only available data from Ørsted are the state vectors, and this is the reason why
the energy conservation method has been chosen for the calculations of the gravitational
potential in this project. Hence the kinetic energy can be directly calculated from the
state vector. In the following, the theory of the energy conservation theorem will be fully
explained.
Another way of calculating the gravitational potential field only from state vectors is
the acceleration method. This method is also described shortly subsequently, although it
has not been used in this thesis.
7.1 The Acceleration Method
When using the acceleration method, the accelerations, Ā, of the satellite are calculated
from the state vectors, by deriving the first order derivatives of the velocity, ¯v, or the
second order derivatives of the position, ¯x.
Ā = d¯v
dt = d2¯x dt2 (37)
In order to calculate the first and second order derivatives from the state vectors, these
must be changed from a point-wise function into a continuous function. This can for
example be done by the use of Newton interpolation [Abt, 2004], [Reubelt et al., 2003],
[Ditmar and van Eck van der Sluijs, 2004].
When the interpolation is executed, the accelerations of the satellite can be calculated
directly from the continuous state vector function. The acceleration found is the total
acceleration of the satellite, and from this the tidal accelerations and non gravitational
acceleration must be subtracted in order to isolate the accelerations only caused by the
Earth’s gravitational pull.
Agravitational = A − Anon−gravitational − Atidal
The acceleration method can for example be used in combination with Least squares
collocation, to derive a global model of the gravity field.
The main reason why the acceleration method has not been used in this thesis, is that
the method was not fully developed or tested, when this project was initiated.
Furthermore the acceleration method is more sensitive to noise in the state vector, than
the energy conservation method is [Abt, 2004].
25
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7. Methods for Gravity Field Determination from State Vectors Only
7.2 The Energy Conservation Theorem
The energy conservation method has been chosen for the calculation of the Earths gravitational
potential field in this thesis.
In this section, the basic ideas and considerations behind the energy conservation
method will be explained. As mentioned earlier, the energy conservation method has
been chosen for this project, because the kinetic energy can be directly calculated from the
state vectors [Visser et al., 2003], [Gerlach et al., 2003b].
One of the most basic theorems of physics is the energy conservation theorem
[Knudsen and Hjorth, 1996].
The mechanical energy of a particle is
conserved for motion in a conservative
force field.
This theorem does also apply on the Ørsted satellite in the Earth centered inertial coordinate
system CIS. If it is assumed that the only force acting on Ørsted is the gravitational
force from the Earth, the energy conservation theorem will have the simple form:
Ekin + Epot = constant, (39)
where Ekin is the kinetic energy of Ørsted and Epot is the potential energy. Since the
potential energy is caused only by the Earth’s gravitational pull, it will subsequently be
referred to as Eearth.
Equivalently, the latter equation may be expressed in terms of potentials instead of
energy, namely by dividing each term with the mass, m, of the satellite.
Ekin
m
− Eearth
m = Vkin − Vearth = E0, (40)
where E0 is a constant, Vkin is the kinetic potential and Vearth is the Earth’s gravitational
potential.
The satellite and the Earth can not be considered an isolated system though, since
there are other forces than the gravitational force from the Earth acting on Ørsted, and
these must also be present in the energy conservation equation.
7.2.1 Tidal Gravitational Disturbances Caused by the Sun and Moon
The sun and the moon induce tidal gravitational disturbances, Vsun and Vmoon, on the
Earth. These disturbances generate a movement of masses, which again influence the
satellite, and this effect must also be present in the energy equation.
In principle the tidal effects caused by the other planets in the solar system, must also
be considered. Though the gravitational tidal influences generated by these planets are of
such a small size, that they are assumed negligible. When including the disturbing solid
26

7. Methods for Gravity Field Determination from State Vectors Only
Figure 15: Figure showing angles and distances used in equation 42 and 43.
Earth tides in the energy conservation equation, and expressing it in terms of potentials,
it has the form:
Vkin + Vearth + Vsun + Vmoon = E0
The size of the solid Earth tidal potential caused by the sun and the moon are given
by equation 19 and equation 20 respectively [Seeber, 2003]:
Vsun = − k2
2
Vmoon = − k2
2
Gmsun
r 3 sun
Gmmoon
r 3 moon
(41)
a 5 e
r 3 ((3 − 15 cos(θsun) 2 ) + 6 cos(θsun)) (42)
a 5 e
r 3 ((3 − 15 cos(θmoon) 2 ) + 6 cos(θmoon)), (43)
where k2 is the Love number which describes the elasticity of the Earth, ae is the
equatorial radius of the Earth, msun is the mass of the sun and mmoon is the mass of the
moon. The distances and angles, used in the equations, are shown in figure 15.
7.2.2 Rotational Potential
In this thesis, the calculations considering the energy conservation will be executed in
the conventional inertial system, CIS. In this system, the gravitational force field is not a
conservative force field due to the rotation of the Earth.
Since the energy conservation theorem only applies in a conservative force field, a rotational
potential, Vrot, must be included in the energy conservation equation, to compensate
for this.
Vkin + Vearth + Vsun + Vmoon + Vrot = E0
27
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7. Methods for Gravity Field Determination from State Vectors Only
The rotational potential is derived in [Jekeli, 1999], and has the form:
Vrot = −ω(xvy − yvx), (45)
−5 rad
where ω = 7.292115 · 10 is the rotational velocity of the Earth.
s
7.2.3 Non-Gravitational Disturbing Potential
There are also non-gravitational forces, acting on the satellite in its orbit. One of these
forces is the friction force, caused by particles in the atmosphere de-accelerating the satellite.
The friction force is dependent on many different parameters such as solar activity,
the composition of the atmosphere, the velocity, altitude, geometry and attitude of the
satellite.
Including the non-gravitational disturbing potential, F , in the energy conservation
theorem, it now has the form:
Vkin + Vearth + Vsun + Vmoon + Vrot + F = E0
Theoretically, the non-gravitational potential can be calculated by [Howe et al., 2003]:
F = (v · a)dt, (47)
where v is the velocity vector and a is the non-gravitational acceleration vector of the
satellite. But since the non-gravitational acceleration of Ørsted is not known, another
approach to the determination of F must be found. This approach will be explained in
section 8.2.
If the latter equation is assumed to be the complete energy conservation equation, it
is possible to isolate and calculate the Earths gravitational potential, Vearth, if the other
terms in the equation can be calculated.
−Vearth = Vkin + Vsun + Vmoon + Vrot + F − E0
This equation shows the basic idea behind calculating the Earths gravitational potential,
using the energy conservation theorem.
28
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8 Deriving The Gravitational Potential
8. Deriving The Gravitational Potential
One of the most important tasks in this project is, to derive the gravitational potential
from the state vectors, in every single GPS measuring point. It has proved to be a process,
which demands many different considerations. Given are the state vectors in both CIS and
CTS coordinates, and from these the Earth’s gravity potential must be derived.
For this purpose the program stat2pot2 has been developed. It is based on the program
stat2pot programmed by C.C.Tscherning (personal communication) for the work
with different dedicated gravity mission satellites. The most significant difference between
stat2pot and stat2pot2 is the fact that the non-gravitational accelerations of Ørsted are
not known, and therefore must be calculated in stat2pot2, whereas they are used as input
in stat2pot.
The gravitational potential, Vearth, is calculated using the conservation of energy theorem,
which is explained in section 7.2. The energy conservation method has also been used
by others for gravity field modeling [Visser et al., 2003], [Gerlach et al., 2003b], [Gerlach et al., 2003a],
[Howe, 2005]. In this section, the computation of each term in the theorem is explained
and all of the considerations, related to these computations, are likewise described. All
of the calculations, described in this section, are executed by stat2pot2. A flow chart of
stat2pot2 is shown in figure 16 and the program can be seen in appendix C.
As mentioned in section 7.2, the coordinate system chosen for the calculations is the
CIS, and in this inertial system the conservation of energy theorem has the form:
Vearth = Vkin − Vrot − Vsun − Vmoon − E0 + F (49)
In the latter equation, the signs of every term has been incorporated. Hence the difference
between this equation and equation 48. This has been done to be able to work with
Vearth as a positive term.
The first thing that is done in stat2pot2, is to open and read the ECI and JPL files.
Therefore this is described as activity 1 in the flowchart of stat2pot2.f90 in figure 16.
8.1 Calculation of Vkin, Vrot, Vsun and Vmoon
The kinetic energy per mass, Vkin, of the satellite in the CIS coordinate system is directly
calculated from the CIS state vector by:
Vkin = 1
2 v2 CIS = 1
2 (v2 x + v 2 y + v 2 z)CIS
This calculation is executed as activity 2 in the flowchart of stat2pot2.f90. The kinetic
potential is plotted as a function of time in figure 17. Figure 17(a) shows the whole period
of time, and figure 17(b) shows a time period corresponding approximately to one orbit
(∼100 minutes).
29
(50)

8. Deriving The Gravitational Potential
[irec=irec+1]
Start program stat2pot2.f90
[Start counting : irec=1]
1:Open and read line "irec" of the inputfiles
2: Calculate kinetic energy per mass
3: Calculate rotational potential
4: Define referencefield: normalpotential
5: Calculate tidal potential from Sun and Moon
6: Calculate E0-F
7: Write position and the terms calculated in 3-7 to file
[if not end of file]
[if end of file]
8: Finds linear regression on (E0-F)(dt) and subtract the line found
9: Use Chauvenet's criterion to filter the data from 8
10: The line found in 8 is added to (E0-F)(dt) again
11: Linear regression is done on the filtered (E0-F)(dt) to determine E0 and F
12: Calculate gravitational potential from every data point
Diagram: stat2pot2 Page 1
13: Write position and gravitational potential to file
End program
Input files are the
JPL and ECI files
Filename :
Vjord.out
Filename:
collodata.inp
Figure 16: Flowchart of the program stat2pot2.f90
30

8. Deriving The Gravitational Potential
(a) (b)
Figure 17: The kinetic energy per mass as a function of time, Vkin(t). (a) The entire
period of time. (b) Approximately one orbit.
The rotational potential, Vrot, is also calculated from each GPS-measurement directly
from the CIS state vector by [Jekeli, 1999]:
Vrot = ω(xvy − yvx)CIS
This is done as activity number 3 in stat2pot2.f90, In figure 18, the rotational potential
as a function of time is shown.
The disturbing potential of the solid Earth tidal effects caused by the Sun, Vsun, and
the Moon, Vmoon, are functions of both distance, latitude and longitude of the satellite in
each measuring point [Seeber, 2003]. This potential is calculated in activity 5, figure 16.
Vsun/moon = k2
2
Gmsun/moon
r 3 sun/moon
(51)
a 5 e
r 3 ((3 − 15 cos(θsun/moon) 2 ) + 6 cos(θsun/moon)), (52)
Vsun and Vmoon are plotted as functions of time in figure 19 and 20 respectively.
The minimum, maximum and mean values of the four terms ,Vkin, Vrot, Vsun and Vmoon
are shown in table 1. The values in table 1 give an impression of the order of magnitude
of the individual terms.
It is seen that the kinetic energy per mass and the rotational potential are the most
dominant by far. The small size of the tidal disturbances caused by the sun and the moon,
indicate that it in fact was legitimate, to discard the tidal potential disturbances caused
by other planets, as stated in section 7.2.1, since these terms would be even smaller and
insignificant.
31

8. Deriving The Gravitational Potential
(a) (b)
Figure 18: The rotational potential as a function of time, Vrot(t) for (a) the whole period
of time and (b) approximately one orbit.
(a) (b)
Figure 19: The tidal potential caused by the Sun, as a function of time, Vsun(t). (a) For
the whole period of time and (b) for approximately one orbit.
32

8. Deriving The Gravitational Potential
(a) (b)
Figure 20: The tidal potential caused by the Moon, as a function of time, Vmoon(t). (a)
The entire time period and (b) approximately one orbit.
Term min. value max. value mean value
Vkin 26989069.76 28837625.66 27899061.48
Vrot -441914.88 -434234.46 -438169.90
Vsun -3.88 7.74 -0.57
Vmoon -7.02 14.48 -1.64
Table 1: Max.-, min.- and mean values of Vkin, Vrot, Vsun and Vmoon. Units are [ m2
s s ].
33

8. Deriving The Gravitational Potential
Left is now to determine the constant E0 and the non-gravitational potential, F , in
order to finally be able to calculate the Earth’s gravity potential Vearth from every GPS
data point. The equation of energy conservation, eq. 49, is rewritten with the known and
the unknown parameters separated:
8.2 Determination of E0 and F
E0 − F + Vearth = Vkin − Vrot − Vsun − Vmoon
As stated in the preface, one of the major challenges of this thesis is to find a valid way to
calculate the non-gravitational potential, F , since the non-nn-gravitational accelerations of
the satellite are not known. The fact that the friction force acting on the satellite depends
on many different parameters, as stated in section 7.2.3, makes it difficult to set up a model
describing the force.
There are several different methods of estimating the non-gravitational potential without
directly modeling it. The way that E0 and F are determined in this thesis, will be
explained in this section.
In order to estimate and examine the non-gravitational potential of the satellite as
a function of time, the gravity potential of the earth, Vearth, must be known according
to equation 53. Since this is obviously not the case, it is approximated with an already
known reference potential field. The simpler the reference field is, the better, since one of
the goals of this thesis is to use as little a priori knowledge as possible, in the computation
of the Earths gravity field. But since it is not possible to estimate F and E0 without any
knowledge of Vearth, some a priori gravity potential field, Vearth,ref., is necessary.
E0 − F = Vkin − Vrot − Vsun − Vmoon − Vearth,ref.
With the aim of keeping the a priori information at a minimum, the gravity potential
field is, as a start, approximated by:
Vearth,ref1 = GM
r =
GM
(x 2 + y 2 + z 2 )
When this reference field is calculated, and subtracted, in every measuring point, E0−F
can be examined, see figure 21.
Here E0 − F is shown as a function of longitude and latitude, and the large bulge near
equator indicates that the first approximation, Vearth,ref1., of the gravity potential was too
simple.
Intuitively the values of E0 − F should be almost constant, except from a small decrease in
time, due to the non-gravitational forces acting on the satellite. The bulge is a consequence
of the fact that the ellipsoidal form of the earth was not taken into account in the chosen
reference potential field Vearth,ref1.. Based on this observation it is clear that a new reference
field, containing more information about the earth’s gravity field must be used. The first
34
(53)
(54)
(55)

8. Deriving The Gravitational Potential
Figure 21: (E0 − F ) calculated, using Vearth,ref1. = GM
r
4 terms of the normal potential, without the centrifugal term, is used as the reference
potential instead. See section 4.3, [Torge, 1999].
Vearth,ref2. = GM
(1 +
r
3
n=1
( ae
r )2n J2nP2n(cosθ)), (56)
where ae = 6378136.4m is the Earth’s mean equatorial radius. This reference field is
defined in activity 4 in the stat2pot2 flowchart.
When the latter reference field is calculated and subtracted in every measuring point,
E0 − F can be examined again, see figure 22.
In figure 22 it is seen that E0 − F calculated from Vearth,ref2. is almost constant all over
the earth, which indicates that the chosen reference field contains sufficient information.
Based on these considerations, Vearth,ref2. will be used as the reference field in the further
calculations. E0 − F is calculated as activity 6 in stat2pot2, and it is plotted as a function
of time in figure 23.
When looking at figure 23, the overall trend is clear. Even though there are of course
daily variations in the friction potential due to changes in the atmosphere’s density etc, the
non-gravitational potential, F , is decreasing at an almost constant rate in time when looking
at the dominating trend during the chosen period of time. Based on this observation,
the friction is assumed linear in time, meaning that it can be written as:
F = A · t, (57)
35

8. Deriving The Gravitational Potential
Figure 22: (E0 − F ) calculated, using the normal potential as reference field, Vearth,ref2..
Figure 23: (E0 − F )(dt)
36

8. Deriving The Gravitational Potential
where A is a constant and t is the time. This is clearly an approximation, and the
approach is chosen mainly because of its apparent simplicity. The later result must tell
whether or not it is a valid one. If it is shown not to be valid, an other approach will be
used.
When (E0 − F )(dt) is found, linear regression is used to find the best fit of the function
to a straight line, and according to the following statement, E0 is found as the crossing of
the straight line with the y-axis:
dt = t − t0
(E0 − F )(t0) = E0(t0) − F (t0) = E0(t0) − 0 = E0, (58)
due to the fact that F is assumed linear in time and zero at t0
Before fitting (E0 − F )(dt) by linear regression, some considerations must be made.
It is clearly seen in figure 21 - 23 that some of the (E0 − F ) values are so extreme, that
they most likely are a result of an error in the GPS positioning. Therefore, a criterion
to exclude extreme error-based data must be stated, since the deviant data points will
wrongly influence the linear regression and thereby also the determination of E0 and F .
Chauvenets criterion is used to filter out very deviant or error-based data. Chauvenets
criterion is described in section 4.4. In order to use Chauvenets criterion on a data set, the
data set must have mean 0. In order to fulfil this requirement, linear regression is used to
fit the data shown i figure 23 by a straight line, a ∗ t + b, and the value of the found line is
then subtracted in every data point.
C(t) = (E0 − F )(t) − (a · t + b) (59)
Another requirement is that the data must be governed by a normal distribution. To
check if this is actually the case a histogram, showing the distribution of the data, is seen
in figure 24. Only the central part of the distribution is shown in figure 24, since the tails of
the distribution are very long due to the very extreme measurements. Though it is clearly
seen that the data are in fact governed by a normal distribution. Therefore Chauvenets
criterion, will be used to filter this data set, C(t), with mean zero.
Finally, the line (a ∗ t + b) is added to the filtered data again, and the result is the data
set, which is seen marked with red, in figure 25. These steps are executed in activity 8-10
in stat2pot2.
The number of data points before the filtering was 203443 and the number of points
after the filtering is 202125, which corresponds to a 0.647% decrease in data points due to
the filtering. As stated in eq. 58, E0 can now be derived by finding the crossing between
a line found by linear regression and the y-axis, and the rate of the decrease in F is found
as the slope of the line. The straight line, found by linear regression as activity 11, figure
16 is described by the function:
−F + E0 = At + B = −3.699402494578402 · 10 −4 · t − 27501928.8889737 (60)
37

8. Deriving The Gravitational Potential
Figure 24: Histogram of the data set C with mean 0.
Figure 25: The filtered data and straight line found by linear regression.
38

The line is shown in figure 25 marked with green.
The result is:
E0 = −27501928.89
F = 3.699402494578402 ∗ 10 −4 · t
8. Deriving The Gravitational Potential
8.3 Calculation of the Gravitational Potential Vearth
The gravitational potential in all the measuring points can now be calculated, hence all of
the necessary terms have been calculated, see eq. 49.
In figure 26 the calculated gravity potential in satellite altitude, can be seen as a
function of longitude and latitude. The gravitational potential is calculated as activity 12
in stat2pot2.f90.
Figure 26: Earths gravity potential [ m2
s 2 ] in every measuring point.
An important part of the thesis has now been completed, since the gravity potential of
the Earth has been derived from the state vectors in every measuring point.
The minimum-, maximum- and mean value of the calculated gravitational potential are
listed in table 2.
The result shown in figure 26 is the full gravitational potential derived from every data
point. In order to show this result in a more intuitive way, the first two terms of the spherical
harmonic series of the gravitational potential GM GM R and ( r r r )2C20P20(cos(θ)) cos(mλ)
have been subtracted. This has been done, since these terms are completely dominating,
hence by subtracting them in every point, the variations of the field can be seen. This
39

8. Deriving The Gravitational Potential
min. value max. value mean value
54932086.69 56781344.53 55843136.9841374
Table 2: Maximum-, minimum- and mean value of the gravitational potential in satellite
altitude.
anomalous gravitational potential for a selected period of the data can be seen in figure
27.
Figure 27: Earths anomalous gravitational potential [ m2
s 2 ] in every data point.
The next major task will be to derive a model which describes the gravitational potential
all over the Earth, from the gravitational potential just calculated. The model will be
expressed in terms of spherical harmonics.
40

9 Theory of Least Squares Collocation
9. Theory of Least Squares Collocation
The theory behind the least squares collocation method is described in this section, and it
is based on
[Moritz, 1980], [Heiskanen and Moritz, 1967], [Moritz, 1978], [Goovaerts, 1997].
Least squares collocation is a mathematical statistical method, which can be used to
determine the anomalous gravitational potential field, T , through a combination of various
geodetic observations, ¯ l. The measurements and the gravitational potential are assumed
statistically correlated.
Least squares collocation is widely used as a tool for interpolation and for transforming
between different types, distributions and qualities of measurements. One of the great
advantages is that the correlation of one field, such as the the gravitational potential, can
be transformed into the correlation of another potential field, such as the gravity anomaly
field. This is possible because these gravity field parameters are connected through linear
functionals.
In the following, T represents the anomalous gravitational potential:
T = V − Vr, (61)
with V being the Earth’s external gravitational potential, and Vr a reference potential.
Two examples of the linear functionals relating T and other gravity parameters by ¯ l = ¯ L ¯ T ,
are listed below ([Torge, 1999] eq. 6.101b, 6.97b):
N = T
1
= T (62)
γ0 γ0
∆g = − ∂T 2T
−
∂r r =
− ∂
2
− T,
∂r r
where N is the geoid height and ∆g is the gravity anomaly. γ0 is the normal potential
at the ellipsoid, see section 4.3.
The basis functions:
GM
a n Pnm(cos θ) cos mλ (63)
r r
GM
a n Pnm(cos θ) sin mλ
r r
expand the Hilbert space, which is a space containing the functions defined in a spherical
coordinate system. In least squares collocation, these basis functions are weighted such
that the overall estimation error, mP , is minimized [Moritz, 1980]
mP 2 = M(ɛP 2 ) (64)
ɛP = Testimated(P ) − Tmeasured(P ),
41

9. Theory of Least Squares Collocation
where M is the averaging or mean function and ɛP is the local estimation error in point
P.
Using this definition, least squares collocation finds the ”best” estimate of the potential
field, based on the measured data.
The estimated value T ′ (A) of the field in point A is given by the following equation,
assuming that the data is noise-free:
T ′ (A) = ¯Cst ¯C −1¯l (65)
T ′ ⎡
⎤
K1,1 K1,2 . . . K1,i
⎢ K2,1 K2,2 . . . K2,i
⎥
(A) = [ KA,1 KA,2 . . . KA,i ] ⎢
⎥
⎣ . . . ⎦
Ki,1 Ki,2 . . . Ki,i
−1 ⎡
where ¯Cst is a vector containing the covariances, K, between all of the i data points
and the point A, in which the estimation is wanted. The estimation (or interpolation)
points can for example be defined by a grid, covering the Earth.
The matrix ¯C, is the auto covariance matrix, containing the covariances between all of
the data points. It is a symmetric matrix, since K1,2 = K2,1. The covariance function K
is explained subsequently.
The vector ¯ l is the observation vector, and it contains the measurements in all the i
data points.
The inverse of the matrix ¯C exists because it has full rank, due to the fact that covariance
matrices are positive definite, and that no observations are used more than once,
meaning that they are all linear independent.
As mentioned, equation 65 is only valid if the observations are noise free which is never
the case for real measurements. Therefore the noise must be accounted for in equation 65,
and this is done by adding a matrix ¯N, containing the noise, to the auto covariance matrix
¯C. In this way, the noise may also be minimized.
T ′ (A) = ¯Cst[¯C + ¯N] −1¯l (66)
If it can be assumed that all of the observations have the same variance, and that the
noise on the data are uncorrelated, the noise matrix is simplified and becomes a diagonal
matrix containing the variance, n 2 .
¯N =
⎡
⎢
⎣
n 2 0
...
0 n 2
42
⎤
⎢
⎣
l1
l2
.
li
⎤
⎥
⎦
⎥
⎦ (67)

9.1 The Covariance Function
9. Theory of Least Squares Collocation
A covariance function exists which describes the correlation between all the measurements,
hence it describes the form of the field, and this information is needed when doing an estimation
or interpolation. [Moritz, 1980], [Moritz, 1978], [Torge, 1999], [Heiskanen and Moritz, 1967]
The spacial covariance function, K(P, Q), between two points P = (θ, λ) and Q =
(θ ′ , λ ′ ) on the sphere r=R, separated by the angle ψ is given by ([Moritz, 1980] eq. 10-9):
K(P, Q) = M(T (P ), T (Q)) (68)
= 1
8π2 2π π 2π
T (θ, λ) · T (θ ′ , λ ′ ) sin θdθdλdα.
λ=0
θ=0
α=0
M is an averaging or mean operator and α is the azimuth [Moritz, 1978].
When the disturbing potential is written as a spherical harmonic series ([Torge, 1999]
eq.6.4):
T (r, θ, λ) = GM
r
∞
n=2
a
n n
r
m=0
or equivalently with fully normalized coefficients:
T (r, θ, λ) = GM
r
n=2
m=0
[Cnm cos(mλ) + Snm sin(mλ)]Pnm(cos(θ)), (69)
∞
a
n n
[
r
¯ Cnm cos(mλ) + ¯ Snm sin(mλ)] ¯ Pnm(cos(θ)), (70)
and the above mentioned covariance function is extended into space outside r=R, it
can also be written as a spherical harmonic expansion ([Torge, 1999] eq.6.191a):
K(P, Q) =
∞
n=2
σn 2
2 n+1
R
Pn(cos ψ). (71)
(rP rQ)
The angle ψ is calculated from the relationship ([Moritz, 1980] eq. 10-3):
cos ψ = cos θ cos θ ′ + sin θ sin θ ′ cos(λ ′ − λ) (72)
The coefficients σn, called the degree variances, are determined by the relationship
([Torge, 1999] eq.6.192 and 6.141 ):
σn 2 =
GM
R
2 n
m=0
( ¯ C 2 nm + ¯ S 2 nm) (73)
Two parameters characterize the covariance function, namely the variance, K0, the
correlation length, ξ ([Moritz, 1978] eq.35):
43

9. Theory of Least Squares Collocation
Figure 28: Characteristic parameters of the covariance function.
K0 = K(ψ = 0) = var(T )
K(ξ) = 1
2 K0
These parameters are also shown in figure 28.
If the covariance between two points is zero, the points in question are statistically
uncorrelated. As the distance between the two points increase, the covariance decreases
since the points become more and more uncorrelated. The covariance does not converge at
zero, but tends to oscillates between small values around zero. This oscillation is caused by
the fact that the gravitational potential in a point is not only affected by the local density
variations, but also from large scale density variations. Positive covariances indicates that
the two points have potentials of same size and sign, and negative covariances indicates
potentials of same size but opposite signs.
44
(74)
(75)

10. Methods of Calculating the Spherical Harmonic Coefficients
10 Methods of Calculating the Spherical Harmonic
Coefficients
If the anomalous potential field T is known everywhere on a sphere with surface area σ, the
fully normalized spherical harmonic coefficients, ¯ Cnm and ¯ Snm, can be calculated directly
using the following equation ([Torge, 1999], eq. 6.138a):
¯Cnm =
¯Snm =
1
4πGM
1
4πGM
σ
σ
r
n r T (θ, λ)
a
¯ Pnm(cos(θ)) cos(mλ)dσ (76)
r
n r T (θ, λ)
a
¯ Pnm(cos(θ)) sin(mλ)dσ,
but since this is clearly not the case, other methods must be used in order to calculate
the coefficients. The two methods used in this project, are explained in the following
sections. The first method is the numerical integration method, which subsequently will be
called method 1, and the other method using the orthogonality properties of the spherical
harmonics, will be called method 2. These two methods are used together with least
squares collocation to derive an Ørsted gravity field model.
The Kepler element method is also described in this section, though this method has
not been used here.
10.1 Method 1: Using Numerical Integration
Since the anomalous potential T is not known everywhere, but only in the data points,
an interpolation of the field can be found using least squares collocation, such that it is
possible to execute a numerical calculation of the integrations in eq. 76, and determine the
spherical harmonic coefficients.
Equation 66 is used to determine the anomalous potential T ′ in a number of points
defined by a grid covering the Earth, by interpolating between the data points. When this
interpolation of the field is executed, numerical integration can be used to calculate the
coefficients by [Moritz, 1980]:
¯Cnm =
¯Snm =
1
4πGM
1
4πGM
σ
σ
r
n r T
a
′ (θ, λ) ¯ Pnm(cos(θ)) cos(mλ)dσ
r
n r T
a
′ (θ, λ) ¯ Pnm(cos(θ)) sin(mλ)dσ, (77)
Here Simpson’s rule of numerical integration has been chosen for the calculations, and
this method is explained in section 4.6. When using Simpson’s numerical integration,
the grid defining the interpolation points must be equi-angular and it must have an even
number of subintervals in both latitude and longitude.
An example of such a grid is shown in figure 29, and here the number of grid points is
9 × 9 = 81.
45

10. Methods of Calculating the Spherical Harmonic Coefficients
Figure 29: Example of grid.
10.2 Method 2: Using Orthogonality Properties of the Spherical
Harmonics
The method, which is based on the orthogonality properties of the spherical harmonics,
is another method of calculating the spherical harmonic coefficients ¯ Cnm and ¯ Snm,
[Tscherning, 2001]. This method will be described in this section.
The estimated anomalous potential in the point P , T ′ (P ), can be calculated by equation
66, using least squares collocation. This equation can be rewritten in the following
way:
T ′ (P ) = ¯C T st(P )[¯C + ¯N] −1¯ l
= ¯C T st(P ) ¯ k (78)
N
= K(P, Qi)ki
i=1
Where N is the number of observations, K is the covariance and the elements ki can
be calculated directly by ki = ([ ¯C + ¯N] −1¯l)i.
The fully normalized coefficients are given by equation 76, and by substituting the
expression for the anomalous potential just derived and equation 71 into this equation of
¯Cnm, the following is obtained:
¯Cnm =
1
N
rP
n
rP
4πGM σ R
i=1
¯Pnm(cos θ) cos(mλ)dσ.
46
∞
j=2
σj 2
2 j+1
R
Pj(cos ψ)ki · (79)
rP rQi

10. Methods of Calculating the Spherical Harmonic Coefficients
Here rP is the altitude of point P from equation 78 and φ is the angle between the point
P and the measurepoint Qi.
In the following, only the ¯ Cnm coefficient will be derived. Equivalent derivations of ¯ Snm
are easily found.
Rearranging the latter equation,
and normalizing the Legendre polynomial Pj(cos ψ)
2j+1
by the normalization factor , the following expression is obtained:
2
¯Cnm =
σ
1 r
4πGM
n+1
P
Rn N ∞
σj
i=1 j=2
2
2 j+1
R
ki · (80)
rP rQi
2j + 1
¯Pj(cos ψ)ki
2
¯ Pnm(cos θ) cos(mλ)dσ
Here it is seen that the point P is cancled out from the equation. By writing the
equation with spherical harmonics instead of Legendre polynomials, using the relation:
¯Pj(cos ψ) =
j
k=−j
¯Yjk(θP , λP ) · ¯ Yjk(θQi , λQi ),
and also using the orthogonality properties of the spherical harmonics:
equation 79 has the form:
σ
¯Cnm = R
4πGM
¯Yjk · ¯ Ynmdσ = 0, if j = n or k = m
N
i=1
σn 2
2 n+1
R 2n + 1
ki
¯Ynm(Qi) (81)
rP rQi
2
From the latter equation, the harmonic coefficients can be directly calculated, if the
σn’s are known.
10.3 The Kepler Element Method
As mentioned, the Kepler element method is another way of finding the coefficients of the
spherical harmonic expansion of the gravitational potential of the earth, see section 4.2.
This method is an alternative to the two methods, which are used in this project and are
described in section 10.1 and 10.2.
When using the Kepler element method [Kaula, 2000][Seeber, 2003], the motion of the
satellite in question must be expressed in terms of the Kepler elements (or Keplerian orbital
parameters):
47

10. Methods of Calculating the Spherical Harmonic Coefficients
a semi-major axis of the satellite orbit
e numerical eccentricity
i orbit inclination
Ω right ascension of ascending node
ω argument of perigee
ν true anomaly
The Kepler elements are also shown in figure 30, in which Pe is the pericenter and S is
the position of the satellite.
Figure 30: Figure showing the Keplerian orbital parameters ([Seeber, 2003], fig.3.4).
Equations as the folowing can be derived. This equation gives the variation of the
Kepler element, Ω in time is given by ([Kaula, 2000], p.39):
dΩ
dt = 3nC20ae 2 cos i
2(1 − e 2 ) 2 , (82)
with ae being the equatorial radius of the earth and n = √ GMa −3/2
From this equation, the coefficient C20 can be directly calculated.
From similar expressions, the rest of the coefficients can be derived ([Kaula, 2000],p.40).
Equation 82 is valid when assuming that:
da de
= 0 and = 0. (83)
dt dt
And this is clearly not the case due to the non-gravitational forces which are indeed
acting on the satellite and disturbing orbit. If the Kepler element method should be used,
48

10. Methods of Calculating the Spherical Harmonic Coefficients
one would have to estimate an average of the semi-major axis of the satellite orbit, which
would induce an error, and this is the reason why this method has not been used in this
thesis.
49

11. Collo.f90
11 Collo.f90
The program collo.f90 has been developed during this project, with the purpose of calculating
the spherical harmonic coefficients of the Ørsted gravitational potential model. The
program can be seen in appendix C. The calculations are based on least-squares collocation,
which is described in section 9 and the two methods of calculating the coefficients
described in section 10.1 and 10.2.
11.1 Program outline
In this section, the program outline of the main program collo.f90 and of subroutine ’collocation’
will be described.
Main Program ’collo.f90’
A flowchart of the main program collo.f90 is seen in figures 31.
Figure 31: Flowchart of collo.f90
50

11. Collo.f90
The first activity in collo.f90 is to read the arguments from the command line which
are the input file name and the parameter nte. The input file for collo.f90 is the file
collodata.inp which is generated by stat2pot2.f90, see section 8 figure 16.
The parameter nte indirectly defines the number of input data which will be used in
the following calculations. For example by choosing nte = 2, every second input data will
be used and by choosing nte = 3, every third input data will be used etc. This is what is
shown in activity 2 and 3 in the flowchart of collo.f90.
The theory of least squares collocation which is explained in section 9 is valid when wanting
to determine the anomalous potential, T . Often, the anomalous potential is defined
as T = V − U, see equation 19. This is not a good definition here since the purpose of
collo.f90 is also to determine the coefficients of the normal potential U, see eq. 22, and it
therefore should not be subtracted from the data. Therefore the anomalous potential will
be defined as T = V − GM in the following.
r
Hence, in order to determine the anomalous potential, the first term of the spherical
harmonic expansion of the gravitational potential, namely GM must be calculated and
r
subtracted from the data. The latter is executed in activity 4. The calculation of GM
r will
be described in section 12.1.
As described in activity 5 in figure 31, the subroutine ’collocation’ is called to execute
the calculations on the selected data. The subroutine ’collocation’ contains the essential
parts of the collo.f90 program, and it is described in the following.
Subroutine ’collocation’
A flowchart of ’collocation’ is shown in figure 32. One of the basic ideas behind this
program is to estimate the individual coefficients in an iterative process, as it is outlined
the flowchart.
The first activity shown in the ’collocation’ flowchart is the definition of a grid. This
grid will be used when performing the numerical integration used in method 1, see section
10.1. An example of such an equi-angular grid is shown in figure 29.
In the program, the grid points are defined by two vectors containing the intervals in
longitude λ and latitude θ:
(λ1, λ2, λ3, . . . , λj) = [0; 2π
; 2π] (84)
j
(θ1, θ2, θ3, . . . , θj) = [0; π
; π],
j
such that j × j is the number of grid points. The chosen grid height is mean satellite
height.
In activity 2, an initial guess of the coefficient ¯ C20 is used as input (it is known that
¯S20 = 0), and based on this guess the degree variance σ2, see eq. 73, is calculated as
51

11. Collo.f90
Figure 32: Flowchart of subroutine collocation
activity number 3. The final result of the coefficient estimation is independent on this
initial guess, since an iterative process is used. The number of iterations needed to obtain
a satisfactory result is dependent on the initial guess, though.
The next task in the program is to calculate the elements of the auto covariance matrix
¯ C, based on the degree variances found in activity 3, and this is done in activity 4.
Hence in the iterative process, the covariance is calculated from the current guess of the
coefficients. This auto covariance matrix is used in activity 5 to solve the matrix equation
¯C ¯ k = ¯ lobs for ¯ k. ¯ lobs is the vector containing all of the chosen input data.
In the program, an ordering of the different matrices used must be determined, such that
the multiplication of the matrices is done correctly. The auto covariance matrix is ordered
as follows:
52

⎛
K(l1, l1)
⎜
¯C
⎜ K(l2, l1)
= ⎜
⎝ .
K(l1, l2)
K(l2, l2)
.
. . .
. . .
...
⎞
K(l1, li)
K(l2, li) ⎟
. ⎠
K(li, l1) K(li, l2) . . . K(li, li)
11. Collo.f90
Since the auto covariance matrix is symmetric, such that K(l1, l2) = K(l2, l1), it is
reduced to:
⎛
K(l1, l1)
⎜
¯C
⎜ 0
= ⎜
⎝ .
K(l1, l2)
K(l2, l2)
.
. . .
. . .
.. .
⎞
K(l1, li)
K(l2, li) ⎟
. ⎠
0 0 . . . K(li, li)
At this point, two different methods are used for the calculation of d(nm), which is the
estimate of the (nm) coefficient. In activity 9, method 2 described in section 10.2 is used
to estimate the intermediate coefficient d(nm) which can be either ¯ Cnm or ¯ Snm. This is
done using only the vector ¯ k.
In activity 6-8 method 1, which is described in section 10.1, is executed. In activity 6
the cross covariance matrix ¯Cst, see eq. 65, is calculated such that the vector ¯t = ¯Cst ¯ k can
be calculated in activity 7. ¯Cst is written as a matrix containing the covariances between
all of the grid points and all the data points instead of being a vector. This is done in
order to execute all of the calculations at once, in stead of calculating the interpolation
in the grid points one at the time. This choice is time saving, which is a very important
element when dealing with many data. The ordering of the cross covariance is as follows:
⎛
K((θ1, λ1), l1)
⎜
K((θ2, λ1), l1)
⎜ .
¯Cst
⎜
= ⎜ K((θj, λ1), l1)
⎜ K((θ1, λ2), l1)
⎜
⎝ .
K((θ1, λ1), l2)
K((θ2, λ1), l2)
.
K((θj, λ1), l2)
K((θ1, λ2), l2)
.
. . .
. . .
...
. . .
. . .
...
⎞
K((θ1, λ1), li)
K((θ2, λ1), li) ⎟
. ⎟
K((θj, λ1), li) ⎟
K((θ1, λ1), li) ⎟
. ⎠
K((θj, λj), l1) K((θj, λj), l2) . . . K((θj, λj), li)
The ¯t vector contains all of the interpolation values, T ′ , in all of the grid points, and it
has the form:
53
(85)
(86)

11. Collo.f90
⎛
⎜
¯t
⎜
= ⎜
⎝
T ′ (θ1, λ1)
T ′ (θ2, λ1)
.
T ′ (θ1, λ2)
T ′ (θ2, λ2)
.
T ′ (θ?, λ?)
Finally the estimate of the coefficient d(nm) i calculated using numerical integration in
activity 8.
Independently of the method used, the d(nm) coefficient is added to the sum of calculated
d(nm) coefficients (nm) ′ in activity 10. In the first iteration (nm) ′ = 0 since the
calculated d(nm) is the first contribution to the coefficient.
In activity 11, the d(nm) potential is subtracted from the data, in order to be able to
perform further calculations on the residual potential, such that the spherical harmonic
coefficient estimation is done in an iterative way.
The d(nm) estimate is now being evaluated. If the found d(nm) is less than 1·10 −16 , it is
assumed that the full coefficient has been found and no further iteration on this coefficient
will be executed. As a result, the order or degree is increased, such that the next coefficient
can be calculated. This is what is shown in activity 12b. This can be done because the
¯Cnm and ¯ Snm coefficients are all orthogonal.
Assuming that the full coefficient has been found and is equal to (nm) ′ this is printed
to the screen, and (nm) ′ is set to zero for the calcuation of the next coefficient, see activity
13. When the next coefficient must be found, another first guess must be made, and this
is determined based on the EGM96 coefficients as it is stated in activity 14. The guess is
based on the EGM96 model coefficients to decrease the number of iterations, and thereby
also the computational time needed to obtain a satisfactory result.
If, on the other hand, the coefficient d(nm) is larger that 1 · 10 −16 , it is assumed that
the full coefficient is still not found, and the guess for the new iteration is set to d(nm)/2
and hereafter another iteration of the program is initiated. The covariance function of this
new iteration has the same form as in the first iteration, see eq. 71, only with the degree
variances calculated from the new guess d(nm)/2.
54
⎞
⎟
⎠
(87)

11.2 Programming Problems and Solutions
11. Collo.f90
During the development of the program collo.f90 and especially the subroutine collocation.f90,
many problems arose and solutions had to be found. In this section, these problems
and solutions are described and discussed. For the process of programming problem
solving, [Press et al., 1992] has been a great help. In order to be able to find and eliminate
the programming problems, the program has continuously been tested on test data.
The testdata consists of potentials directly calculated from the coefficients of the EGM96
model, see [Lemoine et al., 1998] and appendix B.
The advantage of testing the program on the EGM96 test data, is that it is easy to
conclude whether the result is satisfactory or not. This is not easy with results found
from the Ørsted data, since it is not yet known how much gravity information these data
actually contain.
11.2.1 The multi-grid function
Any arithmetic operation using floating numbers, are introducing roundoff error. These
roundoff errors accumulate, when the number of calculations increase, and may consequently
swamp the true solution to the problem. For linear equation systems, this happens
if the number of unknowns (N) is too large [Press et al., 1992].
In this project, several thousand data points must be evaluated, and hence a system of
equations with dimension equal to the number of data equations, must be solved. To test
if roundoff errors are a problem, the program was tested on the EGM96 test data, and it
was found that the solution, found by collo.f90 ,was indeed swamped in roundoff errors.
In order to minimize the roundoff errors, a multi-grid function was implemented in collocation.f90.
This multi-grid function divides the Surface of the Earth into boxes containing
only a fraction of the data points. The calculations given in activity 5 in the ’collocation’
flowchart (figure 32), are executed within each box, meaning that the size of the equation
systems is decreased dramatically. The decrease in number of equations, should decrease
the roundoff errors and thereby improve the solution.
It is important though, that the number of points in each box is neither too large such
that roundoff errors dominate the solutions of some boxes, nor that it is too small such
that possible measuring errors dominate the results instead of being averaged out. Also, it
is important that the box size is sufficiently large, such that the boxes in the polar regions
are not empty due to the polar gaps of the Ørsted satellite coverage. To continuously check
this condition, the boxes and the number of data are plotted in plots like the one shown
in figure 33.
Another advantage of dividing the problems into smaller linear systems is a great improvement
in the computational speed, because the calculations of the boxes can be executed
paralleled.
One disadvantage of using the multi-grid method, is that the points in one box are treated
as if they are completely uncorrelated with points from all other boxes. This is not a
55

11. Collo.f90
180˚
180˚
225˚
225˚
270˚
270˚
315˚
315˚
0˚
80˚ 80˚
60˚ 60˚
40˚ 40˚
20˚ 20˚
0˚ 0˚
-20˚ -20˚
-40˚ -40˚
-60˚ -60˚
-80˚ -80˚
0˚
Figure 33: Boxes used in the multi-grid function and the number of data, N, in every
box.
fair assumption, since points lying close to a box edge will indeed be correlated with the
points in the neighboring boxes. Therefore the concept of overlap was also implemented in
collo.f90 such that the data of all of the neighboring boxes are also used when solving for
each box. In this way solutions of each box is found 9 times, and the final result is then
given by the average value of these 9 solutions.
The choice of using the multi-grid function did reduce the roundoff errors and the results
of the computations were improved considerably. But unfortunately the errors were not
eliminated, and therefore additional programming considerations were necessary. These
will be described subsequently.
11.2.2 Iterative Improvement
A problem that can occur, when working with large sets of equations, is that it is not
always possible to obtain a level of precision, which is comparable with the limit of the
computer. Sometimes even one or two figures can be lost, for matrices which were far
from singular. An effective way of restoring the full computer precision is to use ”iterative
improvement” of the solution [Press et al., 1992]. This method is shown schematically in
figure 34.
In order to obtain the wanted level of precision, it was chosen to implement the method
of iterative improvement in the subroutine collocation.f90. In the following, the activities
mentioned refer to the flowchart of collocation.f90, see figure 32.
The iterative improvement was implemented by calculating a first estimate of the coefficient,
d(nm) in activity 9 and adding this to the the (nm) ′ coefficient in activity 10.
Before the first iteration (nm) ′ = 0 and after the last iteration (nm) ′ will be the the sum
of all of the calculated d(nm) coefficients and the result is the full estimated coefficient
either ¯ Cnm or ¯ Snm. In activity 11 the d(nm) coefficient is subtracted from the data and in
45˚
45˚
56
90˚
90˚
135˚
135˚
180˚
180˚
N
95
90
85
80
75
70
65
60
55

11. Collo.f90
Figure 34: The principle of iterative improvement of the solution to Ā · ¯x = ¯ b. The
first guess ¯x + δ¯x is multiplied with Ā to give ¯ b + δ ¯ b. ¯ b is subtracted to give δ ¯ b which is
inverted to give δ¯x. This is subtracted from the first guess to givean improved solution ¯x.
([Press et al., 1992] figure 2.5.1)
activity 9, a new coefficient d(nm) is calculated on the residual data set.
11.2.3 Decomposition
When wanting to optimize a program which deals with large matrices, matrix decomposition
can be introduced with great computational benefit.
Assuming that the matrix Ā can be written as a product of two matrices, such that
Ā = ¯ L · Ū. Here Ū is an upper triangular matrix containing elements in and above
the diagonal while ¯ L is a lower triangular matrix containing elements in and below the
diagonal.
Such a LU decomposition can be used to solve the following linear equation; Ā · ¯x =
( ¯ L· Ū)· ¯x = ¯ L·( Ū· ¯x) = ¯ b. This is done by first solving for the vector ¯y such that ¯ L· ¯y = ¯ b
and then Ū · ¯x = ¯y [Press et al., 1992].
The great computational advantage of this kind of decomposition, is that the solution
of a triangular set of equations is quite trivial.
In the case of Ā being symmetric and positive definite, it has a more efficient triangular
decomposition called Cholesky decomposition. A positive definite matrix Ā is defined
by
¯v · Ā · ¯v > 0 for all ¯v
Cholesky decomposition is extremely stable numerically without any pivoting at all,
and it is used by the LAPACK subroutines used in collocation.f90 for the symmetric and
positive definite auto covariance matrix C in activity 5 in the flowchart, figure 32.
57

11. Collo.f90
LAPACK is a collection of standard subroutine packages, which has been developed at
Argonne National Laboratories and is published, well documented and available for free
use, see http://www.netlib.org/lapack/.
Several of these subroutines has been used in collocation.f90 since these are developed
to solve problems involving large systems of equations, and it was indeed found that the
use of the LAPACK subroutines did give better results.
11.2.4 Regularization
A problem is said to be ill-posed if the following criteria are not fulfilled (see e.g.
http://en.wikipedia.org/wiki/Ill-posed problem):
1. A solution exists
2. The solution is unique
3. The solution depends continuously on the initial data.
If a problem is ill-posed, it must be reformulated in order to be solved numerically.
This reformulation is done by regularization, where new assumptions on the solution are
added, as for example smoothness. Therefore regularization is often a trade off between
smoothness and data fitting.
The condition number, κ, is a parameter which defines whether a problem of the form
Ā¯x = ¯ b in numerical analysis is well-posed or not.
κ( Ā) = Ā−1 · Ā (88)
A well-conditioned (well-posed) problem has a low condition number, while a problem
with a high condition number is said to be ill-conditioned (ill-posed).
In collocation.f90 the regularization is performed by adding a number α to the diagonal
of the auto covariance matrix, C ′ = C + αI, in activity 4, figure 32. The number α has
been chosen by trial and error, in the search for a good trade off between smoothness and
data fitting.
As stated in [Metzler and Pail, 2005] the presence of polar gaps in the distribution of
data points is partly responsible for the ill-posed problem of calculating the gravitational
potential from GOCE data. The problem becomes ill-posed because the basis functions (the
spherical harmonics) are globally defined, and therefore dependent on globally distributed
data. Therefore a Spherical Cap Regularization Approach (SCRA) has been developed for
GOCE data. Since the inclination of Ørsted also creates polar gaps, similar considerations
has been made in this project, and regularization has been used to stabilize the set of linear
equations.
To determine whether the problem is ill-posed because of to the polar gaps, synthetic
data generated from the EGM96 model has been used to cover the polar gaps. This
synthetic data has been added to the real Ørsted data and collo.f90 has been used to
find a solution. It was shown that this Ørsted-synthetic solution did only vary little (a
few percent) from the Ørsted-only solution, and it therefore must be concluded that the
58

11. Collo.f90
presence of polar gaps is not a significant problem in this project. This may be due to
the use of the multi-grid function which divides the Earth into boxes which also cover the
polar regions and do all contain a certain amount of data, see 11.2.1
59

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
12 Deriving the Ørsted Gravitational Potential Field
Model ORSTED05
In this section the Ørsted gravitational potential model will be derived, expressed as a
spherical harmonic series. The program collo.f90 has been developed during the project
for this purpose, and it was described in section 11.
12.1 Calculation of GM
Since the term GM is by far the most dominating term, it is assumed that it can be
r
determined by a simple method described in the following. Since GM is the dominant term
r
of the gravitational potential, we disregard all other terms.
Hence all of the calculated values of the potential multiplied with the distance Vearth · r
are added up, and divided by the number of measurements, N. This means that GM is
simply approximated by the mean of the Vearth · r. This can only be done due to the great
and the following terms in the gravitational potential.
difference in size of GM
r
GM ≈
N i=1 (Vearth(i) · r(i))
N
= (GM)approx
Using the latter approximation, the following result is achieved:
(GM)approx = 398514652297117m 3 /s 2
The difference between the approximated and the ’true’ value is evaluated in percent:
(GM)true − (GM)approx
(GM)true
= (3.986009 ∗ 1014 m 3 /s 2 − 398514652297117m 3 /s 2 )
3.986009 ∗ 10 14 m 3 /s 2
= −2.1638 ∗ 10 −4 ≈ −0.0216%
The latter equation shows that it is an acceptable approximative method to find GM,
since the difference is of such a small size.
Hence, the anomalous potential, T , can now be used in the further calculations. This
is done in activity number 4 in the flowchart of collo.f90, which is shown in figure 31.
12.2 Comparing Results from Method 1 and Method 2
Both method 1 and 2 have been tested on the EGM96 test data to determine the spherical
harmonic coefficients of degree 2 to 4. This has been done in order to find which of the
two methods is the fastest and most accurate, when used for the full model determination.
The results of the two methods are compared to each other and to the equivalent
EGM96 coefficients. In table 3, the coefficients found by the two methods are listed.
In this test, 2520 test data points generated from EGM96 were used, and the grid used
in method 1 has 20 × 20 points. The overlap multi-grid method has been used, and the
60
(89)

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
largest system of equations to be solved is of size 64 while the smallest is of size 42. The
box size is approximately 12 ◦ × 12 ◦ and the number of multi-grid boxes is 15 × 30.
n m ¯ Cnm
Method 2 Method 1
¯Snm
¯Cnm
¯Snm
2 0 −0.48445497 · 10−3 0 −0.48447268 · 10−3 0
2 1 −0.99037859 · 10−10 −0.82713757 · 10−10 −0.10038595 · 10−9 −0.93629297 · 10 10
2 2 0.20923471 · 10−5 -0.10190048 · 10−5 0.20809231 · 10−5 −0.10426476 · 10−5 3 0 0.92019799 · 10−6 0 0.96282684 · 10−6 0
3 1 0.16576926 · 10−5 0.20233258 · 10−6 0.16588057 · 10−5 0.24617314 · 10−6 3 2 0.62848981 · 10−6 −0.42572507 · 10−6 0.62489276 · 10−6 −0.44617471 · 10−6 3 3 0.35437913 · 10−6 0.72325547 · 10−6 0.41794515 · 10−6 0.80057489 · 10−6 4 0 0.12895032 · 10−5 0 0.13100918 · 10−5 0
4 1 −0.29858347 · 10−6 −0.25282049 · 10−6 −0.28729297 · 10−6 −0.28710544 · 10−6 4 2 0.23121831 · 10−6 0.26341738 · 10−6 0.24554158 · 10−6 0.29924076 · 10−6 4 3 0.42740805 · 10−6 −0.76283535 · 10−7 0.42692479 · 10−6 −0.59335934 · 10−7 4 4 −0.56833794 · 10−7 0.86072184 · 10−7 −0.50901670 · 10−7 0.87327298 · 10−7 Table 3: Spherical harmonic coefficients found by two different methods.
From table 3 it is seen that the two methods give similar although not identical results.
For the purpose of comparison, the lowest order coefficients of the EGM96 model are listed
in appendix 10. When comparing the results given in table 3 to the EGM96 coefficients it
is seen that that the results are of the same order of magnitude as the EGM96 coefficients,
but that they are not identical.
This difference must be due to errors in the calculations done by collo.f90, and will be
discussed later.
To determine which of the two methods should be used for the further calculations, a
more detailed examination of the determination of ¯ C20 has been executed.
For this examination, the number of data points has been held fixed, while the number
of integration grid points used in method 1, has been increased.
In table 4, the coefficient ¯ C20 found by method 2 and the computation time tcomp are
seen, and in table 5 the values of the ¯ C20 coefficient found by method 1 are listed together
with the chosen number of grid points, Ngrid, and the computation time, tcomp. The number
of boxes used in the multi-grid function is the same as in the first test leading to the results
in table 3.
When comparing the results given in table 5 and 4, it is seen that they are almost
identical. This means that both methods are usable, but there are major differences when
looking at the computation time.
It is intuitively clear that the numerical integration depends on the chosen number of
grid points, and it is seen in the tables that at constant number of data points, an increase
61

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
Ndata
Method 2
¯C20(orth) tcomp[sec]
2520 −0.48445497 · 10−3 4.020252
Table 4: Results of the determination of ¯ C20 using the orthogonality of Legendre
polynomials-method. Ndata is the number of data points used as input and tcomp is the
computation time.
Method 1
Ndata 2520 2520 2520 2520
Ngrid 9 × 9 15 × 15 25 × 25 45 × 45
¯C20(int)[10 −3 ] -0.48453368 -0.48457688 -0.48446463 -0.48445504
tcomp[sec] 7.156447 8.528533 10.64067 15.74098
diff. 0.01624 0.02516 0.0019939 1.444 · 10 −5
Table 5: Results of the determination of ¯ C20 using the numerical integration method.
Ndata is the number of data points used as input, Ngrid is the number of grid points, diff is
is the relative deviation between the result found by method 1 and 2 in percent and tcomp is
the computation time.
in grid points causes the result from the method 1, to become more and more similar to
the result from method 2.
From this fact it must be concluded that in order to obtain identical results by the two
methods, the grid must be very dense, and therefore method 2 is preferable, due to the
low computation time. Method 2 will therefore be used in the following.
12.3 Collo.f90 error test on EGM96 test data
As described in the latter section, method 2 has been chosen for the full spherical harmonic
coefficient determination, leading to the Ørsted potential model ORSTED05. Many
considerations on eliminating programming errors were made while developing collo.f90,
and these were described in section 11.2. To examine the collo.f90 program accuracy, the
program has been tested on EGM96 generated data. Since this data is noise-free it can be
assumed that errors determined from this test run entirely is a result of roundoff errors,
instability and other problems caused by collo.f90. These errors are the so-called model
errors.
The spherical harmonic coefficients up to degree and order 6, which are the result of
this test run on EGM96 generated data, are shown in table 6.
It is important to know how well the program collo.f90 can determine the coefficients of
62

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
Coefficients found from EGM96 test data
n m Cnm
¯ ¯Snm Erel(Cnm) Erel(Snm)
2 0 −0.48453712 · 10−3 0 0.08
2 1 −0.18703043 · 10−9 −0.15624559 · 10−9 0.02 -113.07
2 2 0.24100089 · 10−5 −0.12058656 · 10−5 -1.19 -13.88
3 0 0.10425540 · 10−5 0 8.91 0.00
3 1 0.19925720 · 10−5 0.22108229 · 10−6 -1.84 -11.04
3 2 0.78616977 · 10−6 −0.56062048 · 10−6 -13.09 -9.44
3 3 0.49183355 · 10−6 0.10355649 · 10−5 -31.79 -26.78
4 0 0.17563786 · 10−5 0 225.33 0.00
4 1 −0.38399811 · 10−6 −0.32358149 · 10−6 -28.40 -31.65
4 2 0.83840095 · 10−7 0.47463012 · 10−6 -76.09 -28.38
4 3 0.64279324 · 10−6 −0.11453986 · 10−6 -35.12 -42.99
4 4 −0.71160763 · 10−7 0.17085108 · 10−6 -62.26 -44.68
5 0 0.30338033 · 10−7 0 -55.73 0.00
5 1 0.71377222 · 10−7 −0.18314154 · 10−7 -214.94 -80.60
5 2 0.42677613 · 10−6 −0.17286081 · 10−6 -34.59 -46.54
5 3 −0.22963628 · 10−6 −0.16159178 · 10−6 -49.19 -24.79
5 4 −0.13503873 · 10−6 0.12584736 · 10−7 -54.27 -74.66
5 5 0.68520757 · 10−7 −0.31734344 · 10−6 -60.84 -52.59
6 0 −0.53044678 · 10−6 0 253.73 0
6 1 −0.37407780 · 10−7 −0.13960414 · 10−7 -50.84 -153.10
6 2 0.16893802 · 10−7 −0.13834618 · 10−6 -64.93 -62.98
6 3 0.30180637 · 10−7 −0.74285183 · 10−8 -47.21 -182.29
6 4 −0.40652870 · 10−7 −0.19502480 · 10−6 -52.85 -58.63
6 5 −0.12069107 · 10−6 −0.22178500 · 10−6 -54.82 -58.66
6 6 −0.67023069 · 10−10 −0.71904223 · 10−7 -100.69 -69.69
Table 6: Spherical harmonic coefficients and their relative errors, determined from
EGM96 test data.
a known noise-free field, when evaluating the coefficients found from Ørsted data. In figure
35 it is shown how much the results in table 6 deviates from the true EGM96 coefficients
as a function of degree and order, given by the relative error. This relative error Erel is
given by:
(nm)estimated − (nm)EGM96
Erel =
· 100%, (90)
(nm)EGM96
, where (nm) is the coefficient in question.
It is seen that the relative error from coefficient to coefficient varies a lot, the smallest
being 0.02% on coefficient ¯ C21 and the largest being 253.73% on coefficient ¯ C60 . It is
difficult to determine from where exactly these errors stem. It has not been possible,
63

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
¯Cnm
Figure 35: Relative error on the coefficients from table 6 given in %.
during this project, to increase the accuracy of collo.f90 further than these results show.
Neither has it been possible to explain the great variation in the relative error on the
coefficients.
When examining the relative error estimates in figure 35, it is seen that for the majority
of the coefficients, the relative error is negative. This means that the coefficients
estimated by collo.f90 are often too small. Unfortunately it has not been possible to find
an explanation of this either.
In order to convert the relative errors shown in table 35 into errors given in potentials,
the potential field of the coefficients in table 6 has been calculated and subtracted from
the potential field calculated from the true EGM96 coefficients up to order and degree 6.
The potential fields are calculated in ellipsoid height 400km. The result is seen in figure
36.
When examining the result in figure 36, a pattern in the relative errors is seen. The
general pattern shows that the relative errors changes sign in three ”belts” across the earth.
This is a result of the coefficient C40 having a large relative error. It also can be noted
that the error in general follows the amplitude of the field. This is result of the fact that
collo.f90 in general underestimates the coefficients.
12.4 The Model ORSTED05
In table 7 the spherical harmonic coefficients up to degree and order 6, calculated from the
Ørsted potential data by collo.f90, are listed. These calculations are based on 30319 Ørsted
data, and the multi-grid box size is 4.5 ◦ × 4.5 ◦ . The box with the least data contains 0 and
the one with the most contains 26 data. Hence the maximum number of equations solved
64
¯Snm

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
Figure 36: Difference between the collo.f90 potential field calculated from EGM96 test
data, and the true EGM96 potential field [ m2
s 2 ].
simultaneously by collo.f90 is 9 × 26 = 234.
The relative error on the ORSTED05 coefficients are shown in figure 37. The relative
errors are found by comparison with the true EGM96 coefficients.
¯Cnm
Figure 37: Relative error on the coefficients from table 7 given in %.
It is seen that the relative errors are generally higher that the ones shown in figure 35,
which was off course expected, since the Ørsted data contain noise.
It was experienced that the result of the Ørsted modeling depended to some extend on
the input parameter which defines the number of data points, and on the box size chosen
for the multi-grid function. Therefore some effort was made to find the best box size for
the calculation of ORSTED05.
65
¯Snm

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
Coefficients found from Ørsted data
n m Cnm
¯ ¯Snm Erel(Cnm) Erel(Snm)
2 0 −0.48430152 · 10−3 0 0.03 0
2 1 0.20918906 · 10−10 0.11526763 · 10−9 -111.19 -90.36
2 2 0.54956953 · 10−7 0.37681631 · 10−7 -97.75 -102.69
3 0 0.49093894 · 10−6 0 -48.71 0
3 1 0.65031951 · 10−7 0.15956609 · 10−6 -96.80 -35.79
3 2 −0.25690223 · 10−8 0.25657993 · 10−7 -100.28 -104.14
3 3 −0.20055761 · 10−7 −0.45693563 · 10−7 -102.78 -103.23
4 0 0.31344998 · 10−6 0 -41.94 0
4 1 0.26589597 · 10−8 0.13587627 · 10−7 -100.50 -102.87
4 2 0.10974940 · 10−6 −0.10235117 · 10−6 -68.71 -115.45
4 3 0.54036757 · 10−7 0.26473021 · 10−8 -94.55 -101.32
4 4 −0.68176511 · 10−7 0.29326264 · 10−7 -63.84 -90.50
5 0 −0.82332096 · 10−8 0 -112.01 0
5 1 0.17745786 · 10−8 0.19571046 · 10−7 -102.86 -120.73
5 2 −0.29136486 · 10−7 0.38796906 · 10−7 -104.47 -112.00
5 3 0.21662527 · 10−7 −0.18331542 · 10−7 -104.79 -91.47
5 4 0.10677581 · 10−7 0.14816158 · 10−7 -103.62 -70.17
5 5 0.26782299 · 10−7 0.35216889 · 10−7 -84.69 -105.26
6 0 0.27876531 · 10−7 0 -118.59 0
6 1 0.22034075 · 10−8 0.18914277 · 10−7 -102.90 -28.05
6 2 0.69187286 · 10−7 −0.79483553 · 10−7 43.62 -78.73
6 3 0.62885721 · 10−8 0.55044245 · 10−8 -89.00 -39.02
6 4 0.19604171 · 10−7 0.10455823 · 10−7 -122.74 -102.22
6 5 0.16016148 · 10−7 0.17809609 · 10−7 -106.00 -103.32
6 6 0.26971528 · 10−7 0.32539687 · 10−7 178.74 -113.72
Table 7: Spherical harmonic coefficients and the relative errors, determined from Ørsted
data.
66

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
The reason why only the coefficients up to degree and order 6 has been listed in this
section and the latter, is that it was found that the relative error on the coefficients, found
from the Ørsted data, increased significantly at the higher degree coefficients. For example
the relative error on coefficient ¯ C70 is Erel = −725.32%. This relative error could not be
explained by model errors, since the relative error on the ¯ C70 coefficient found from EGM96
generated data was only Erel = −52.14%.
Based on this, it was determined that the potential, derived from Ørsted data does at
least contain information on the Earth’s gravity field, which can be described by the first
coefficients up to degree and order 6. Therefore the model ORSTED05 will only consists
of these 25 coefficients and the GM
r found in section 12.1.
For the purpose of evaluating the ORSTED05 model, the potential of the EGM96
coefficients from degree 2 to 6 is shown in figure 38. As expected it is seen that the ¯ C20 is
dominating the image, and in order to get a more detailed potential image, this coefficient
has been subtracted, and the resulting field is shown in figure 39.
Figure 38: The potential field from the true EGM96 coefficients up to degree and order
6 [ m2
s 2 ].
In figure 40, the ORSTED05 potential field calculated from the coefficients in table 7,
is shown and in figure 41 this field is shown with the ¯ C20 term subtracted.
Comparing figure 41 and 39 it is seen that the model calculated from Orsted data
indeed looks similar to EGM96. Obviously there are some derivations, but main features
such as the large minimum over India and the maximums over Europe and Indonesia are
in fact observed.
From this it is seen that it has been possible to derive an all-Ørsted model, which indeed
contains some information on the Earth’s gravity field.
67

12. Deriving the Ørsted Gravitational Potential Field Model ORSTED05
Figure 39: The potential field from the true EGM96 coefficients up to degree and order
6 with ¯ C20 subtracted [ m2
s 2 ].
Figure 40: The potential field from the true ORSTED05 coefficients [ m2
s 2 ].
Figure 41: The potential field from the ORSTED05 coefficients with ¯ C20 subtracted [ m2
s 2 ].
68

13 Final Considerations and Discussion
13. Final Considerations and Discussion
In this section, I will summarize and comment on the choices made and results found in
this thesis. This is done in acknowledgment of the fact that many of the choices made, has
merely been one of many possible.
In section 6.2 the Ørsted satellite was evaluated as a gravity mission satellite. Three
criteria were stated which must be fulfilled by a good gravity mission satellite, namely a
low orbital height, uninterrupted GPS tracking and finally it must have instruments on
board which measure the non-gravitational accelerations.
It was concluded though, that Ørsted does not fulfill any of these criteria completely,
and therefore the expectations for the result of the gravity field modeling were not high.
It was concluded that it would probably only be possible to derive the coefficients of low
degree of the spherical harmonic expansion of the gravitational potential.
A gravity potential field model, ORSTED05, was derived from the Ørsted data. This
model contains coefficients up to degree and order 6, which to some extent are comparable
to the known EGM96 coefficients.
Subsequently, some comments on the different parts of this thesis are made. Firstly the
data selection is discussed.
Data Selection.
The task of selecting the Ørsted GPS data, which should be used in the gravity field
modeling, is described in section 5. The period of data running from 1999-09-02 to 2000-
02-22 was chosen.
This period was chosen based on the considerations that the period should be stable in
respect to the solar activity, and the period should also contain a certain amount of useful
data.
It would have been preferable to work with data from a later period. The reason being
that the height of the satellite decreases with time, and thereby the number of spherical
harmonic coefficients which can be calculated will increase. This is due to the attenuation
of the gravitational field with height, which was mentioned in section 6.2.
During the data selection, a choice was made of accepting all measurements based on
tracking by 6 or more GPS satellites. It was estimated that 6 satellites were a minimum,
since no DOP values of the measurements were available, but that it is very likely to have
a low DOP value when having at least 6 satellites.
If a higher number of satellites needed for accepting a measurement had been chosen,
the uncertainties of the measurements would decrease, since the uncertainty of a GPS
69

13. Final Considerations and Discussion
a b
Figure 42: The position of the satellite in the X-Y plane (a) after filtering with 7 satellites
(b) after filtering with 8 satellites.
measurement depends on the number of satellites available for the measurement. A decrease
in measurement uncertainty would be an advantage in the modeling, but the choice
of a higher number of satellites would be in conflict with the criteria for uninterrupted
measurements, which is stated in section 6.2.
The direct result of changing the number of satellites from 6 to 7 or 8 is shown in figure
42, which shows that when the number of satellites needed for a good measurement was
increased to 7 or 8, the GPS measurement can indeed not be considered uninterrupted,
and this was why the choice of 6 satellites was used when selecting the data.
Figure 42 should be compared with figure 7, and it is seen that the increase in satellites
does create many more ”holes” in the measurements, and it therefore is concluded that
the choice of 6 satellites was preferable.
The Energy Conservation Method.
The energy conservation method was chosen for gravitational potential derivations in this
project. This choice was made primarily due to the fact that the kinetic energy can be
calculated directly from the state vectors. Furthermore the method had earlier been tested
by e.g Eva Howe on CHAMP data with success [Howe et al., 2003, Howe, 2005].
An alternative to the energy conservation method is the acceleration method described
in section 7.1, and the main reason this method was not chose is the fact that it had not
been tested when this project was initialized. Though, it could be interesting to test if
this method could in fact be used with success on Ørsted data, since the method has been
tested on CHAMP data with success [Abt, 2004].
70

13. Final Considerations and Discussion
Determination of the Non-Gravitational Disturbing Potential.
In this project, it was chosen to model the non-gravitational potential, F , as linear in
time.
F = A · t + E0
This choice was based on figure 25, which shows an overall trend which corresponds
with the assumption that the non-gravitational potential decreases at a constant rate in
time.
It is clear though, that this is just an approximation since a complete model of F
should take e.g. satellite height, particle height, solar activity and cross section area of the
satellite into account. But it was not realistic to derive such a complicated model in this
project.
Two other models of the non-gravitational potential were tested in this thesis, namely
1. F = A · t + B · t 2 + E0
2. F (h)
In the first assumption the non-gravitational potential is a second order polynomial in
time, and in the second assumption, F , is a function of satellite height instead of time.
The latter is tested since the friction on the satellite depends on the particle density in the
atmosphere which again is a function of ellipsoid height, h. It therefore could be expected
that F and h are correlated.
When testing the first assumption, it was found that when fitting the data to a second
order polynomial, the second order term was very small such that:
F = A · t + B · t 2 + E0 ≈ A · t + E0
which brings us back to the assumption of F being linear in time.
When testing the second assumption, non-gravitational potential F was plotted as a
function of ellipsoid height instead of time, and the result is seen in figure 43.
It is seen that there is no clear correlation between F and h, and from this observation
it was concluded that this assumption would not be used in this thesis.
71

13. Final Considerations and Discussion
Figure 43: The non-gravtational potential F as a function of ellipsoid height h.
Calculation of the Spherical Harmonic Coefficients.
Two methods of calculating the spherical harmonic coefficients have been tested in this
project, namely method 1 using numerical integration described in section 10.1 and method
2 using the orthogonality properties of the Legendre polynomials, which was described in
section 10.2.
It was shown that equal results could be achieved using the two methods, when choosing
a number of grid point for the numerical integration method, which was sufficiently
large. But since the computation time of method 1 was found to be longer than method
2, the latter method was chosen for the further modeling.
The kind of grid which was defined for the interpolation using method 1 was shown in
section 10.1, figure 29. This kind of grid is simple and it fulfills the criteria for the Simpson’s
rule of integration stated in section 4.6.
The disadvantage of using this kind of grid is the fact that the grid point density
is largest at the poles, where there are no measurements, see figure 9. Method 1 could
probably be improved by using a more appropriate grid. But since the orthogonality
method was chosen for the modeling, this was not tested.
72

Collo.f90
13. Final Considerations and Discussion
The program collo.f90 was developed during this project for the purpose of determining
the gravitational potential model ORSTED05. The program outline was described in
section 11.1.
During the development of the program collo.f90, many problems arose and solutions
had to be found. These were described in section 11.2.
The multi-grid function was introduced in order to minimize the roundoff errors in the
program, by dividing the Earth into boxes containing a limited amount of data points
and executing the calculations in each box, see figure 33. The use of overlap was also
implemented in the multi-grid function, in order to avoid errors on the edges of the boxes
which stems from the fact, that data in neighboring boxes are assumed uncorrelated when
they are in fact correlated. If this problem should be eliminated, the box-size should be
chosen to be equal to the correlation length found from the covariance function of the
spherical harmonic coefficient in question. In this way, all correlated data points are used
when finding the solutions of each box. But a look at the covariance function of for example
¯C20 shows that the correlation length is so large that the purpose of using the multi-grid
function vanishes if choosing the box-size from this correlation length. Therefore this idea
was not implemented in collo.f90.
Figure 44: The covariance function of ¯ C20.
73

14. Conclusion
14 Conclusion
As described in section 12, an Ørsted gravity potential model, ORSTED05, has been
calculated. A test run of collo.f90 on EGM96 generated data, showed the errors which
were a result of the numerical calculations themselves.
The first term in the spherical harmonic series, GM, was successfully determined from
Ørsted data, in section 12.1. The found value was within 0.0216% of the known value.
The program collo.f90 was used to determine the following spherical coefficients from
the Ørsted data. The second term ¯ C20 was found with an accuracy of 0.03%. Higher
degree terms was found but with larger errors compared to EGM96. By comparing the
results with the EGM96 coefficients and the numerical errors found from the test run, it
was concluded that the ORSTED05 model should contain coefficients of degree up to 6.
This choice was based on the fact that the relative errors of the higher order coefficients
increased significantly.
All in all, it must be concluded that the result was satisfactory. Some gravity information
could indeed be derived from the Ørsted data, as it is clearly seen when comparing
figure 39 and 41.
If the program collo.f90 could be developed further, it is very likely that even more
gravity information could be derived from the Ørsted data.
74

REFERENCES REFERENCES
References
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Sneeuw, N., Frommknecht, B., Oberndorfer, H., Peters, T., Rothacher, M., Rummel,
R., and Steigenberger, P. (2003a). A champ gravity field model from kinematic orbits
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[Jekeli, 1999] Jekeli, C. (1999). The determination of gravitational potential differences
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M. H., Wang, Y. M., Williamson, R. G., Pavlis, E. C., Rapp, R. H., and
Olson, T. R. (1998). Egm96 - the nasa gsfc and nima joint geopotential model.
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76

A Data Files
A.1 ECI file example
A. Data Files
The eci file contains 13 coulombs. Time is in UTC, positions and velocities are relative to an inertial reference system centered
in the Earth.
#1: satellite ID, #2: year, #3: month, #4: day, #5: hour, #6: minute, #7: second #8: x[km], #9: y[km], #10: z[km],
#11: Vx[km/s], #12: Vy[km/s], #13: Vz[km/s]
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13
92 1999 10 10 15 19 7.00 −3983.28270 441.90984 6020.27834 6.11989181 0.83137380 4.01909742
92 1999 10 10 15 26 34.00 −936.32142 754.30908 7130.21862 7.29508572 0.54584941 0.87326629
92 1999 10 10 15 26 35.00 −929.02711 754.85445 7131.08799 7.29606278 0.54506542 0.86576632
92 1999 10 10 15 29 37.00 407.22290 840.35887 7163.81869 7.34574325 0.39209955 −0.50729543
92 1999 10 10 15 29 47.00 480.69588 844.29333 7158.42934 7.34101518 0.38333734 −0.58268589
92 1999 10 10 15 29 57.00 554.08061 848.08429 7152.22797 7.33557524 0.37444586 −0.65802639
92 1999 10 10 15 30 7.00 627.40914 851.81300 7145.28661 7.32918350 0.36535724 −0.73368695
92 1999 10 10 15 30 17.00 700.65498 855.39499 7137.57624 7.32216730 0.35648691 −0.80897753
92 1999 10 10 15 30 27.00 773.84133 858.92222 7129.10386 7.31443313 0.34748437 −0.88403814
92 1999 10 10 15 30 37.00 846.95207 862.37157 7119.88547 7.30593231 0.33832164 −0.95901880
92 1999 10 10 15 30 47.00 919.95377 865.69513 7109.90308 7.29665938 0.32915586 −1.03405949
92 1999 10 10 15 30 57.00 992.87952 868.95237 7099.19768 7.28659732 0.32001301 −1.10895022
92 1999 10 10 15 31 7.00 1065.69333 872.10356 7087.73228 7.27578059 0.31085708 −1.18379099
92 1999 10 10 15 31 17.00 1138.38322 875.16309 7075.51586 7.26420487 0.30180251 −1.25846179
92 1999 10 10 15 31 27.00 1210.93610 878.06984 7062.49644 7.25190662 0.29246722 −1.33319264
92 1999 10 10 15 31 37.00 1283.44160 881.04045 7048.87401 7.23864238 0.28308958 −1.40790353
A.2 JPL file example
The eci file contains 13 coulombs. Time is in GPS-time, positions and velocities are relative to an Earth-fixed reference
system centered in the Earth. #1: satellite ID, #2: x[km], #3: y[km], #4: z[km], #5: Vx[km/s], #6: Vy[km/s] , #7:
Vz[km/s], #8: year, #9: month, #10: day, #11: hour, #12: minute, #13: second
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13
92 1043.59400 −3869.18100 6020.45900 −3.29200000 5.31719000 4.01878000 1999 10 10 15 19 20.00
92 −397.59030 −1134.56600 7130.24400 −3.03950000 6.72036000 0.87290000 1999 10 10 15 26 47.00
92 −400.62860 −1127.84600 7131.11300 −3.03811000 6.72198000 0.86540000 1999 10 10 15 26 48.00
92 −927.03960 115.01480 7163.77700 −2.72954000 6.89527000 −0.50766000 1999 10 10 15 29 50.00
92 −954.30270 183.99780 7158.38400 −2.70971000 6.89759000 −0.58305000 1999 10 10 15 30 0.00
92 −981.30140 252.98300 7152.17900 −2.68953000 6.89924000 −0.65839000 1999 10 10 15 30 10.00
92 −1008.12300 321.97430 7145.23400 −2.66886000 6.90002000 −0.73405000 1999 10 10 15 30 20.00
92 −1034.67900 390.97330 7137.52000 −2.64820000 6.90010000 −0.80934000 1999 10 10 15 30 30.00
92 −1061.06400 459.97220 7129.04400 −2.62719000 6.89951000 −0.88440000 1999 10 10 15 30 40.00
92 −1087.25100 528.96280 7119.82200 −2.60579000 6.89821000 −0.95938000 1999 10 10 15 30 50.00
92 −1113.18400 597.92830 7109.83600 −2.58415000 6.89614000 −1.03442000 1999 10 10 15 31 0.00
92 −1138.93000 666.88080 7099.12700 −2.56229000 6.89328000 −1.10931000 1999 10 10 15 31 10.00
92 −1164.44000 735.79840 7087.65800 −2.54019000 6.88967000 −1.18415000 1999 10 10 15 31 20.00
92 −1189.72400 804.66480 7075.43800 −2.51796000 6.88527000 −1.25882000 1999 10 10 15 31 30.00
92 −1214.72000 873.48640 7062.41500 −2.49525000 6.88023000 −1.33355000 1999 10 10 15 31 40.00
92 −1239.66200 942.27980 7048.78900 −2.47221000 6.87426000 −1.40826000 1999 10 10 15 31 50.00
A.3 RINEX file example
year month day hour minute second ok # satellites ID ID ID
02 2 20 8 20 50.0000000 0 2 1 21
sat 1 data −0.05148 −0.04040 28912088.24948 28912088.24940 139.00048
0.00040
sat 21 data −0.85148 −0.66340 28324541.45848 28324550.65340 256.00048
0.00040
02 2 20 8 21 0.0000000 0 2 1 21
sat 1 data 303265.83247 236311.03840 28969796.76447 28969796.76440 89.00047
0.00040
sat 21 data −20438.44348 −15926.05940 28320654.28048 28320646.97640 258.00048
0.00040
02 2 20 8 21 10.0000000 0 3 1 21 11
sat 1 data 606423.40747 472537.72040 29027486.58147 29027480.28440 55.00047
0.00040
sat 21 data −42479.17848 −33100.65840 28316462.55548 28316451.22740 262.00048
0.00040
sat 11 data −0.03748 −0.02940 28570978.87048 28570978.87040 308.00048
0.00040
02 2 20 8 21 20.0000000 0 2 21 11
sat 21 data −66133.20148 −51532.36540 28311963.94948 28311975.08940 260.00048
0.00040
sat 11 data 399865.47348 311583.48640 28647062.12348 28647062.12340 306.00048
0.00040
77

B. EGM96 coefficients
B EGM96 coefficients
EGM96 spherical harmonic coefficients of
degree n and order m up to (n,m)=(7,7)
n m Cnm
¯ ¯Snm
2 0 -0.484165371736E-03 0.000000000000E+00
2 1 -0.186987635955E-09 0.119528012031E-08
2 2 0.243914352398E-05 -0.140016683654E-05
3 0 0.957254173792E-06 0.000000000000E+00
3 1 0.202998882184E-05 0.248513158716E-06
3 2 0.904627768605E-06 -0.619025944205E-06
3 3 0.721072657057E-06 0.141435626958E-05
4 0 0.539873863789E-06 0.000000000000E+00
4 1 -0.536321616971E-06 -0.473440265853E-06
4 2 0.350694105785E-06 0.662671572540E-06
4 3 0.990771803829E-06 -0.200928369177E-06
4 4 -0.188560802735E-06 0.308853169333E-06
5 0 0.685323475630E-07 0.000000000000E+00
5 1 -0.621012128528E-07 -0.944226127525E-07
5 2 0.652438297612E-06 -0.323349612668E-06
5 3 -0.451955406071E-06 -0.214847190624E-06
5 4 -0.295301647654E-06 0.496658876769E-07
5 5 0.174971983203E-06 -0.669384278219E-06
6 0 -0.149957994714E-06 0.000000000000E+00
6 1 -0.760879384947E-07 0.262890545501E-07
6 2 0.481732442832E-07 -0.373728201347E-06
6 3 0.571730990516E-07 0.902694517163E-08
6 4 -0.862142660109E-07 -0.471408154267E-06
6 5 -0.267133325490E-06 -0.536488432483E-06
6 6 0.909789371450E-07 0.000000000000E+00
7 1 0.279872910488E-06 0.954336911867E-07
7 2 0.329743816488E-06 0.930667596042E-07
7 3 0.250398657706E-06 -0.217198608738E-06
7 4 -0.275114355257E-06 -0.123800392323E-06
7 5 0.193765507243E-08 0.177377719872E-07
7 6 -0.358856860645E-06 0.151789817739E-06
7 7 0.109185148045E-08 0.244415707993E-07
78

C Fortran programs
C. Fortran programs
This cd contains the three fortran programs rinexpass2.f90, stat2pot2.f90
and collo.f90, which have been developed for this project.
79