Estimations of on-site directional wave spectra from measured ship ...

Marine Structures 19 (2006) 33–69
<strong>Estimations</strong> <strong>of</strong> on-site directional wave spectra from
measured ship responses
Abstract
ARTICLE IN PRESS
Ulrik Dam Nielsen
www.elsevier.com/locate/marstruc
Department <strong>of</strong> Mechanical Engineering, Technical University <strong>of</strong> Denmark, DK-2800 Lyngby, Denmark
Received 21 October 2005; received in revised form 29 May 2006; accepted 26 June 2006
In general, two main concepts can be applied to estimate the on-site directional wave spectrum on
the basis <strong>of</strong> ship response measurements: (1) a parametric method which assumes the wave spectrum
to be composed by parameterised wave spectra; or (2) a non-parametric method where the directional
wave spectrum is found directly as the values in a completely discretised frequency-directional
domain without a priori assumptions on the spectrum. The paper outlines the theory <strong>of</strong> these two
concepts, and it is shown how to deal with the speed-<strong>of</strong>-advance problem for operating ships. In
addition, the methods include an equivalence <strong>of</strong> energy in the governing equations and, as regards
the parametric concept, a frequency-dependent spreading <strong>of</strong> the waves is introduced.
The paper includes an extensive analysis <strong>of</strong> full-scale measurements for which the directional wave
spectra are estimated by the two ship response-based methods. Hence, comparisons are made
between these estimates and, moreover, the agreement with the corresponding directional wave
spectra produced by the wave radar system WAVEX is studied. The agreement between the two
methods is reasonable, as well is the agreement between the results <strong>of</strong> these methods and those <strong>of</strong>
WAVEX. It is difficult to propose one <strong>of</strong> the ship response-based methods in favour <strong>of</strong> the other,
since they perform equally well.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Measured ship responses; Response spectra; Directional wave spectra; Bayesian modelling; Parametric
modelling; Full-scale data; Wave radar
Fax: +45 4588 4325.
E-mail address: udn@mek.dtu.dk.
0951-8339/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.marstruc.2006.06.001

34
1. Introduction
1.1. Motivation
The operational safety <strong>of</strong> ships can be increased by use <strong>of</strong> in-service monitoring systems.
For such systems to be used efficiently, the on-site directional wave spectrum needs to be
estimated and updated continuously. Today, means <strong>of</strong> obtaining estimations <strong>of</strong> the
directional wave spectrum exist. Such means include moored wave rider buoys and current
meters, satellite measurements and wave radar systems. The latter two <strong>of</strong> these means do
not suffer from the problems related to the fixed position <strong>of</strong> a moored buoy or current
meters, but do, on the other hand, require complex computational hardware and have a
high initial cost, cf. Tannuri et al. [1], not to mention calibration and maintenance. For
this reason it is <strong>of</strong> interest to be able to estimate the wave spectrum from measured
ship responses, which are accessible from the sensor measurements done in in-service
monitoring system.
1.2. Existing work
In the literature there exist several papers which deal with the estimation <strong>of</strong> (directional)
wave spectra on the basis <strong>of</strong> measured ship responses. In general, these works can be split
in two groups or as being part <strong>of</strong> one <strong>of</strong> two concepts: (1) a concept where the wave
spectrum is estimated on the basis <strong>of</strong> parametric modelling, e.g. Hua and Palmquist [2],
Tannuri et al. [1] and the EC project HullMonþ [3]; or (2) a concept where the estimation
<strong>of</strong> the wave spectrum is based on non-parametric modelling, e.g. Iseki and Ohtsu [4], Iseki
and Terada [5], Waals et al. [6], Isobe et al. [7] and Nielsen [8]. Fundamentally, the two
concepts are similar in the sense that both methods in their foundations use linear spectral
analysis to set up conditions/equations which relate the measured ship responses—the one
hand side—with the wave energy spectrum through complex-valued transfer functions—
the other hand side. Thus, from this relation, the principle is to minimise, in the least
squares sense, the difference between the two sides, see Fig. 1.
Although Tannuri et al. [1], including Pascoal et al. [9], to some degree compare the two
estimation concepts with each other, the comparisons apply for numerical simulations and
for a ship not being underway, since these works do not consider the speed-<strong>of</strong>-advance
problem in the derived theory. Similarly, Benoit and Goasguen [10] compares different
directional wave analysis methods for a so-called ‘‘single-point’’ measuring system, which
record data from a three-displacements buoy, that is located at a fixed position.
The speed-<strong>of</strong>-advance problem, governed by the deep-water relationship between the
encounter and the wave frequency, leads in certain cases to the so-called triple-valued
function problem in beam, quartering and following seas, cf. Fig. 3(b). This problem needs
to be strictly incorporated in an estimation procedure based on measured ship responses.
In Iseki and Ohtsu [4] the elementary problem is dealt with and in that paper it is suggested
how to incorporate it.
1.3. Objectives
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
In the present paper, the theory <strong>of</strong> a parametric modelling procedure and a nonparametric
modelling procedure is derived, and it is outlined in detail how to take into

account a vessel being underway, independently if the estimation is performed by
parametric or non-parametric modelling. Furthermore, it is shown how to include
additional equations, based on Iseki and Terada [5], in both <strong>of</strong> the estimation concepts,
so that it is sought to secure the conservation <strong>of</strong> energy in the responses. The main
part <strong>of</strong> the present paper is devoted to the analysis <strong>of</strong> numerical and full-scale data. Thus,
on the basis <strong>of</strong> simulated and measured ship responses, the parametric modelling and
the non-parametric modelling procedure are compared. As regards the analysis <strong>of</strong>
full-scale data, comparisons with data from the wave radar system WAVEX have also
been conducted.
1.4. Responses
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 35
Fig. 1. The fundamental idea in the estimation <strong>of</strong> wave spectra based on measured ship responses. EC project
HullMon+ [3].
In principle, any type <strong>of</strong> response signal, obtained as time series, may be utilised in the
wave spectrum estimation as long as a linear complex-valued transfer function exists, that
is, can be calculated. However, it is believed that the highest rate <strong>of</strong> success is achieved
when global ship responses, such as pitch, roll, vertically wave-induced bending moment,
etc., are used. This means that response signals <strong>of</strong> e.g. pressure transducers in
the hull should not be considered in the estimation, as local effects may affect such a
response. It is important to emphasise that at least one <strong>of</strong> the ship responses must have
port/starboard asymmetry, otherwise the modelling procedures cannot differentiate
between port and starboard entering waves. This explains also the need <strong>of</strong> complexvalued
transfer functions, so that the amplitudes as well as the phase angles <strong>of</strong> the
responses are obtained.
Specifics on response signals to be used for the estimation <strong>of</strong> wave spectra can be found
in, among others, HullMon+ [3] and Nielsen [8]. Tannuri et al. [1], however, brings
an interesting issue to question and, thus, argues correctly that the roll motion, in
general, has a non-linear and resonant behaviour and, furthermore, that the roll
motion is extremely sensitive to load variations. Tannuri et al. [1] therefore proposes to
use the sway response instead <strong>of</strong> the roll, as the frequency response function <strong>of</strong> sway is less

36
sensitive to loading conditions. The present work deals primarily with results based on the
three responses {heave, roll, pitch}. Though, estimations have also been carried out with
the roll response replaced by the sway. Later, in Section 7, more comments are given on
this topic.
1.5. Organisation <strong>of</strong> the paper
The paper is organised as follows: In Sections 2–5 theoretical aspects <strong>of</strong> the modelling
procedures are treated. Section 6 deals with a numerical example which verifies
and compares the modelling procedures. Then in Section 7, full-scale data is analysed
and comparisons between the estimations <strong>of</strong> the two modelling procedures and
those <strong>of</strong> WAVEX are carried out. Finally, in Section 8, the major conclusions are
formulated.
2. General theory
In this section the general theory <strong>of</strong> the parametric and the non-parametric modelling
procedure are described. Hence, the section deals with facts and conditions which are
shared by the two concepts and, subsequently, two sections yield the specifics for the
individual concepts.
2.1. Fundamentals
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
From time series <strong>of</strong> measured ship responses, the response cross spectra, SijðoeÞ, can be
obtained by application <strong>of</strong> fast Fourier transformation (FFT) with appropriate smoothing
or by application <strong>of</strong> multivariate autoregressive (MAR) modelling, e.g. Akaike and
Nakagawa [11] and Nielsen [8]. Fig. 2 illustrates an example (data from [12]) <strong>of</strong>the
outcome <strong>of</strong> a cross spectral analysis conducted on three (simultaneously) measured ship
responses. The analysis is carried out by the MAR modelling and it should be noted that
only the diagonal spectra are real-valued. The measured response spectra are shown as
function <strong>of</strong> the encounter frequency oe. In this study, the ship responses are taken to be
stationary. This means that the encounter angle, b, and other operational parameters are
assumed to be constant during the time which forms the basis <strong>of</strong> the measured response
spectra.
The encounter angle b is defined as the heading <strong>of</strong> the ship relative to the waves and
b ¼ p corresponds to head waves. From Fig. 3(a) it is seen that when the ship course a and
the wave direction y are measured relative to some fixed direction, the encounter angle is
given by
b ¼ y a. (2.1)
For convenience, the fixed direction in Fig. 3(a) is taken to be coincident to the ship course
so that the wave direction is given relative to the ship course, that is, b ¼ y. This means
that the wave direction is coincident to the encounter angle.
On the assumption that the ship responses are stationary and linear with the incident
waves, the complex-valued transfer functions Fiðoe; bÞ and Fjðoe; bÞ for the ith and jth
responses yield the theoretical relationship between the ith and the jth components <strong>of</strong> the
cross spectra SijðoeÞ and the directional wave spectrum Eðoe; bÞ through the following

Power
Power
Power
6
4
2
x 10 -3
0
0 0.2 0.4
x 10-3
ARTICLE IN PRESS
pitch/pitch x 10-3
2
pitch/roll
0.02
pitch/ver.acc.
1
real
imag
0
0
0.02
roll/pitch
2
1
0
-1
-2
-3
0 0.2 0.4
ver.acc./pitch
0.02
0
-0.02
-0.04
-0.06
-0.08
0 0.2
Frequency [Hz]
0.4
-1
-2
-3
0 0.2 0.4
5
4
3
2
1
x 10-3
roll/roll
0
0 0.2
ver.acc./roll
0.4
0.02
0.01
0
-0.01
-0.02
0 0.2
Frequency [Hz]
0.4
0.04
0.06
0.08
0
0.02
0.01
0
0.01
roll/ver.acc.
0.02
0 0.2
ver.acc./ver.acc
1
0.5
0.2
0
0 0.2
Frequency [Hz]
Fig. 2. Cross spectral analysis <strong>of</strong> three measured ship responses: {pitch, roll, vertical acceleration at FP}. Data
from [12].
θ
α
U.D. Nielsen / Marine Structures 19 (2006) 33–69 37
(a) (b)
=30°
following
=0°
=60°
quatering
beam
=120°
bow
=150°
head
=180°
Fig. 3. (a) Ship course, a, and wave direction, y, relative to a fixed direction. (b) Definition <strong>of</strong> terms as regards the
encounter angles. Note that b is shown in degrees.
0.4
0.4
0.4

38
integral equation, e.g. Bhattacharyya [13],
SijðoeÞ ¼
Z p
p
Fiðoe; bÞFjðoe; bÞEðoe; bÞ db (2.2)
with denoting the complex conjugate. It should be noted that the transfer functions are
written as functions <strong>of</strong> only the encounter angle and the encounter frequency, since the
implication <strong>of</strong> changing operational parameters is understood.
The wave spectrum is advantageously estimated in the wave frequency domain
for which reason the right-hand side <strong>of</strong> (2.2) is transformed into the wave frequency,
o, by introducing the deep-water relationship between the encounter and the wave
frequency,
oe ¼ o o 2 A; A ¼ V
cos b (2.3)
g
with V being the ship speed and g the acceleration <strong>of</strong> gravity. In beam, quartering
and following seas, cf. Fig. 3(b), (2.3) leads to up to three wave frequencies for
certain speeds and, hence, the integral in (2.2) must be split apart in order to do the
transformation.
2.2. Discretisation
K k=1
m=1
2
3
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
4
By discretising the wave field in K M points, with K being the number <strong>of</strong> encounter
angles, i.e. wave directions, and M being the number <strong>of</strong> wave frequencies, the directional
wave spectrum is thus to be estimated in K M points, see Fig. 4 which illustrates the
discretisation in a polar format. Specifically, the discretisation means that the wave
spectrum must be estimated in a set <strong>of</strong> prespecified wave frequencies. Hence, the wave
frequency as found by (2.3) is taken to be a weighted distribution between the two closest
specified wave frequencies, cf. Fig. 4. Consequently, the transfer function in (2.2) for the
calculated wave frequency o01 is found by linear interpolation
-1
Fðo01Þ ¼ e b FðomÞþea Fðomþ1Þ, (2.4)
where the weights are e b ¼ b=ða þ bÞ and ea ¼ a=ða þ bÞ.
2
M
3
5
4
01
a b
Fig. 4. Discretisation <strong>of</strong> the wave field, and weighted distribution <strong>of</strong> a calculated wave frequency between two
specified frequencies.
+1

From the above considerations, the fundamental equation, (2.2), can be written in the
following discretised version
SijðoeÞ ¼Db XK
ebFiðom; bkÞFjðom; bkÞEðom; bkÞ k¼1
dom
"
doe b¼bk þeaFiðomþ1; bkÞFjðomþ1; bkÞEðomþ1; bkÞ domþ1
#
ð2:5Þ
doe b¼bk with
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 39
mðoe; bkÞ; m 2½1; MŠ,
k ¼ 1 : K. ð2:6Þ
The calculation <strong>of</strong> the response cross spectra reveals that SijðoeÞ is Hermitian, i.e. Sij ¼ Sji,
see also Fig. 2. Hence, by application <strong>of</strong> matrix notation, (2.5) is transformed into a
multivariate model expression, e.g. Iseki and Terada [5] and Nielsen [8],
b ¼ Af. (2.7)
In the event <strong>of</strong> N ship responses being considered, that is, i ¼ j ¼f1; 2; ...; Ng, the
response vector, b, is given by
2 3
SiiðoeÞ
6
. 7
6 7
6 . 7
6 7
6 Re½SijðoeÞŠ 7
6 7
b ¼ 6 .
7
7;
sizeðbÞ ¼N
6 . 7
6 7
6 Im½SijðoeÞŠ 7
4 5
.
2
1. (2.8)
The wave spectrum is discretised as illustrated in Fig. 4 and the sequence, as regards
the wave frequency and the encounter angle, in which the spectrum is evaluated follows
from
2
3
Eðo1; b1Þ 6
Eðo1; b
7
6
2Þ 7
6
7
6 . 7
6 . 7
6
7
6
Eðo1; b
7
6
KÞ 7
6
7
6
f ¼ Eðo2; b1Þ 7
6
7;
6 . 7
6 . 7
6 .
7
6 . 7
6
7
6
4 EðoM; b 7
K 1Þ 5
EðoM; bKÞ sizeðfÞ ¼ðK MÞ 1. (2.9)

40
The coefficient matrix, A, is thus determined according to
2
6
A ¼ Db 6
4
2 dom
jFiðom; bkÞj doe b¼bk .
.
ReðFiðom; bkÞFjðom; bkÞÞ dom
doe b¼bk .
ImðFiðom; bkÞFjðom; bkÞÞ dom
doe b¼bk .
3
7
7;
sizeðAÞ ¼N
7
5
2
ðK MÞ.
(2.10)
As regards A, the elements must be calculated precisely in the same order as the wave
spectrum, so that the correct relation between encounter frequencies and wave frequencies
are secured. It should be noted that, although not shown in (2.10), the weights ea and e b need
to be included in the set-up <strong>of</strong> the coefficient matrix according to (2.4). Moreover, it should
be realised that A may contain a large amount <strong>of</strong> zero elements, e.g. Nielsen [8].
So far, only one encounter frequency has been considered. In order to include an interval
<strong>of</strong> encounter frequencies discretised as
oe ¼ l Doe; l ¼ 1; 2; 3; ...; L, (2.11)
the fundamental equation can readily be extended by introducing the following response
block-vector
eb ¼½b1 b2 ... bLŠ T , (2.12)
where bl ¼ bðl DoeÞ and with bl
sizeðebÞ ¼ðN
determined from (2.8). It should be noted that
2 LÞ 1.
The coefficient matrix follows correspondingly from the block-matrix
2 3
A1
6 A2
7
6 7
eA ¼ 6 . 7
4 . 5 ; sizeðe AÞ¼ðN 2 LÞ ðK MÞ (2.13)
AL
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
so that the multivariate model expression reads
eb ¼ e Af. (2.14)
As mentioned, e A is likely to contain a large amount <strong>of</strong> zero-elements. This is illustrated in
Fig. 5 which shows an arbitrary example <strong>of</strong> the coefficient matrix e A for a certain speed and
discretisation <strong>of</strong> encounter angles and frequencies.

2.3. Equivalence <strong>of</strong> energy
0
100
200
300
400
500
600
In addition to the equations given by (2.14), N equations can further be specified, where
N, accordingly, is the number <strong>of</strong> considered response signals. Thus, as mentioned by Iseki
and Terada [5], it can be taken into account that the amount <strong>of</strong> energy in the signals should
be equivalent independently <strong>of</strong> the energy being calculated from the measured cross
spectra, i.e. the left-hand side <strong>of</strong> (2.2), or the theoretically estimated cross spectra, i.e. the
right-hand side <strong>of</strong> (2.2). Mathematically, it can be argued that N2 additional equations
may actually be specified, since the cross spectral analysis <strong>of</strong> N responses, as mentioned,
leads to a total number <strong>of</strong> N2 different real and imaginary spectra. However, the complexvalued
cross spectra, that is, the real and the imaginary parts <strong>of</strong> the <strong>of</strong>f-diagonal cross
spectra can have both positive and negative values which means that the area under the
respective curve can be close to zero in some cases. For this reason, the diagonal spectra
are only considered and, hence, N conditions/equations are specified to secure the
equivalence <strong>of</strong> energy between the measured and the calculated responses. Consequently,
the following equations are also taken into account
Z
Z Z p
SiiðoeÞ doe ¼ jFiðo; bÞj 2 Eðo; bÞ db do; i ¼ 1; ...; N, (2.15)
p
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 41
0 100 200 300 400 500 600 700 800 900
nz = 22509
Fig. 5. Example <strong>of</strong> non-zero elements in the coefficient matrix. ‘nz’ is the total number <strong>of</strong> non-zero elements; the
matrix has a dimension <strong>of</strong> 657 900.
where it should be noted that the right-hand side is based on the wave frequency.
From a physical point <strong>of</strong> view, the equivalence <strong>of</strong> energy <strong>of</strong> the measured and the
estimated spectra is important. In the equation system (2.14), the equivalence, which is
expressed by (2.15), may therefore be weighted relatively heavily.

42
2.4. A least squares problem
In general, the directional wave spectrum, f ¼ Eðom; b kÞ can be sought in the least
squares sense by solving
min w 2 minke Af ebk 2
(2.16)
with w ¼k kbeing the L2 norm. Specifically, the solution is obtained by application <strong>of</strong>
one <strong>of</strong> two main concepts: (1) a non-parametric modelling concept denoted the Bayesian
method or (2) a parametric modelling concept denoted the Parametric method. In the
following, these concepts are dealt with. As regards the non-parametric modelling concept,
it should be noted that in the present work the term Bayesian is used to characterise this
concept, however, inverse theory, e.g. Dimri [14] and Tarantola [15], could also have been
used as the term, since inverse theory and Bayesian modelling are basically two sides <strong>of</strong> the
same coin, e.g. Nielsen [8]. In this relation reference may also be given to Herbers and
Guza [16].
3. Bayesian modelling
Without explicit assumptions on the directional wave spectrum, (2.16) expresses ðN 2
L þ NÞ equations from which K M unknowns, Eðom; b kÞ, are to be solved. For a
reasonable discretisation, say, K ¼ 18, M ¼ 30 and taking N ¼ 3 ship responses to form
the basis for the estimation, (2.16) is in general underdetermined. The number, L, <strong>of</strong>
encounter frequencies plays a role in the sense that more equations are established by
increasing L. However, new information put into the system in this way is <strong>of</strong> limited use
due to degeneration <strong>of</strong> the equation system. Therefore, additional equations must be
established by use <strong>of</strong> some kind <strong>of</strong> prior information. In the present model two sets <strong>of</strong> prior
information are introduced. These are: (1) the wave spectrum is assumed to be smoothly
changing with both frequency and direction, and (2) the wave spectrum is expected to have
negligible values for very low and high frequencies. Moreover, if at hand, information on
the wave direction can, theoretically, be included.
3.1. Basic assumptions
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
In addition, and beforehand <strong>of</strong> the introduction <strong>of</strong> prior information, two basic
assumptions are introduced to make the Bayesian modelling work. Firstly, as discussed by
Iseki and Ohtsu [4], the Bayesian modelling introduces a stochastic point <strong>of</strong> view as regards
the estimation <strong>of</strong> wave spectra. Thus, in order to facilitate the Bayesian modelling
procedure, cf. Akaike [17], the difference between the left- and the right-hand side <strong>of</strong> (2.14)
is taken as a white noise sequence vector w with zero mean and variance s2 . Secondly, to
avoid negative spectral estimates, a non-negativity constraint is applied to the wave
spectrum by use <strong>of</strong> a coordinate transformation
Eðom; bkÞ¼expðxk;mÞ, (3.1)
where it should be noted that the order <strong>of</strong> the indices has been changed.
Hence, the multivariate model expression (2.14) is written as
eb ¼ e A expðxÞþw. (3.2)

It is important to realise that the introduction <strong>of</strong> the non-negativity constraint leads to a
non-linear problem. This is undesirable for which reason a linearisation should be applied
to the coordinate transformation, e.g. Iseki and Terada [5] and Nielsen [8].
3.2. Prior information
As regards prior information (1), the condition is set up by requiring the second order
derivative <strong>of</strong> lnðEðo; bÞÞ to be minimum. Hence, by use <strong>of</strong> the L2 norm and introducing
finite differences, the equations are given from the minimisation <strong>of</strong> the two functionals
X M
X K
m¼1 k¼1
X K
XM 1
k¼1 m¼2
ðxk 1;m 2xk;m þ xkþ1;mÞ 2 ; ðx0;m ¼ xK;m; xKþ1;m ¼ x1;mÞ, (3.3)
ðxk;m 1 2xk;m þ xk;mþ1Þ 2 . (3.4)
Mathematically, this condition seeks to secure that the argument <strong>of</strong> the non-negativity
constraint is piecewise constant, which implies that the wave spectrum is smooth due to the
mathematical properties <strong>of</strong> the exponential function.
To deal with prior information (2), the sum <strong>of</strong> the power is minimised for o ! 0 and
o !1. Mathematically, this is done by minimising the functional
X K
k¼1
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 43
½ðxk;1 s0Þ 2 þðxk;M s0Þ 2 Š, (3.5)
where s0 denotes a prespecified small value. It should be noted that the indices (here ‘1’ and
‘M’) representing the wave frequency may be extended to include a larger interval, if needed,
in practical computations. The reason or necessity for the introduction <strong>of</strong> this prior is related
to practical computations as, if not introduced, problems are likely to arise due to the
frequency response functions being zero (or close to) at the frequency boundaries.
With respect to information on the wave direction, knowledge can, in principle, be
included in a similar manner as (3.5); however, with both k and m as running indices. Thus,
the limits on k and m must be specified in accordance with the wave direction(s) and
frequencies wherein the wave energy should be minimised. Information on the direction <strong>of</strong>
sea-waves (i.e. the on-site wind generated waves) can <strong>of</strong>ten be deduced from knowledge <strong>of</strong>
the wind direction, excluding information about swell-waves. An evident problem in this
context is therefore that with a known wind direction, which gives a good indication about
the direction <strong>of</strong> the sea-waves, the minimisation <strong>of</strong> wave energy applied to a certain, so to
speak, opposite domain <strong>of</strong> the frequency-directional range excludes swells (as well as seas)
from this domain. The present suggestion <strong>of</strong> including information on the wave direction
by minimisation <strong>of</strong> the wave energy in a certain domain is, thus, interesting from a
theoretical point <strong>of</strong> view, however, in practical applications notable problems exist. One
approach to overcome the problem may be to restrict the minimisation to frequencies
above some limit. Still, this approach is not considered to be optimal and to end the
discussion, it can be concluded that instead <strong>of</strong> doing the minimisation for directions where
the waves, according to the wind direction, do not come from, it would be desirable if
directions, coincident to the wind direction, were given more ‘‘weight’’ compared to other

44
directions in the wave field. Unfortunately, no method for such a procedure has been
developed. In future work it would therefore be interesting to investigate how to include
information about the wind direction. For now, however, this issue is not dealt with any
further.
3.3. Final equations and solution
In matrix notation all conditions, i.e. the prior information, may be given by kDx ck 2 ,
see Appendix A. Hence, by combination <strong>of</strong> the prior information and the governing
equation system (2.16), a well-conditioned least squares solution is found by minimising,
cf. Akaike [17],
ke A expðxÞ ebk 2 þ u 2 kDx ck 2
(3.6)
which is controlled by a so-called hyperparameter u.
According to the Bayesian modelling procedure, Akaike [17] transforms the minimisation
<strong>of</strong> (3.6) into the maximisation <strong>of</strong> the posterior distribution <strong>of</strong> x, ppostðxjs2 ; uÞ, which is
proportional to
1
2ps 2
ðN 2 LþNÞ=2
u 2
2ps 2
KM=2
exp
1
2s 2 ðke A expðxÞ ebk 2 þ u 2 kDx ck 2 Þ ,
where u is the hyperparameter that expresses the degree <strong>of</strong> smoothness in the solution <strong>of</strong> x.
For details, see e.g. Nielsen [8], Iseki and Terada [5] and Press et al. [18].
On the assumption that the hyperparameter, u, and the variance, s2 , are known,
ppostðxjs2 ; uÞ can be maximised by application <strong>of</strong> various techniques from linear algebra,
e.g. QR factorisation. In practical computations, the maximisation is done for a range <strong>of</strong>
known values <strong>of</strong> the hyperparameter (and the variance). To obtain the optimum value <strong>of</strong> u
and s2 , Akaike [17] proposed a criterion known as Akaike’s Bayesian information
criterion, abbreviated as ABIC. Thus, the determination <strong>of</strong> u and s is based on the
minimisation <strong>of</strong> ABIC, which in its general form is given by
Z
ABIC ¼ 2ln ppostðyj ...Þdy. (3.7)
Basically, this finishes the Bayesian modelling, although with most details on algebra and
numeric left out. Details can be found in Nielsen [8]. As regards to the treatment <strong>of</strong> the
prior distributions it seems relevant, though, to mention that since three functionals are
introduced, three different hyperparameters, as a matter <strong>of</strong> fact, ought to be considered.
However, as the computational cost in connection with the optimisation <strong>of</strong> the
hyperparameters is relatively large, compared to the final outcome <strong>of</strong> the directional
estimation when considering three different hyperparameters, only one hyperparameter is
dealt with. Especially, the computational cost is important in the study <strong>of</strong> practical
decision support systems. Therefore, the optimisation is done for only one hyperparameter.
4. Parametric modelling
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
Parameterised wave spectra, e.g. Goda [19], are typically considered reliable for
describing the variation with frequency <strong>of</strong> ocean wave spectra. Moreover, the angular

spread <strong>of</strong> wave spectra can be described by certain parameters, e.g. Longuet-Higgins et al.
[20] and Goda [19]. On this assumption, the wave spectrum to be estimated from Eq. (2.16)
is based on the following 10-parameter bimodal spectrum, e.g. Tannuri et al. [1] and
Hogben and Cobb [21],
Eðo; yÞ ¼ 1 X
4
2
i¼1
ððð4li þ 1Þ=4Þo 4 p;i Þli
GðliÞ
y ymean;i
2si cos
2
exp
H2 s;i
AðsiÞ
o4liþ1 4li þ 1
4
with Hs being the significant wave height, l is the shape parameter <strong>of</strong> the spectrum, ymean is
the mean wave direction, op is the angular peak frequency, and s represents the spreading
parameter.
AðsÞ ¼ 22s 1 G2ðs þ 1Þ
(4.2)
pGð2s þ 1Þ
is a constant introduced to normalise the area under the cos2s curve and G denotes the
gamma function.
The wave spectrum expressed by (4.1) considers basically a sea component ði ¼ 1Þ and a
swell component ði ¼ 2Þ, and on this basis it is, in theory, possible to model most ocean
wave spectra, e.g. Hogben and Cobb [21]. Hence, by application <strong>of</strong> (4.1), and taking the
wave direction to be coincident to the encounter angle, the solution <strong>of</strong> equation (2.16)
implies a non-linear optimisation problem from which the best-fit-values can be
determined.
4.1. Frequency-dependent spreading
From measurements, cf. Longuet-Higgins et al. [20] and Mitsuyasu et al. [22], the
spreading parameter s is found to vary with frequency. In the present Parametric method
the frequency dependency <strong>of</strong> the spreading parameter is included in the sense that s takes a
maximum value around the spectral peak frequency, whereas s decreases for frequencies
both lower and higher than the peak frequency. According to Goda [19] the spreading
parameter can basically be modelled as
(
s ¼ ceil½ðo=opÞ 5 smaxŠ; opop;
ceil½ðo=opÞ 2:5 smaxŠ; o4op;
where the peak value <strong>of</strong> s, denoted smax, has been introduced as the principal parameter.
Compared to the original proposal by Goda [19], ceil[ ] has been introduced in (4.3).
Thus, ceil[ ] rounds towards plus infinity and is a numerical technique utilised
here to stabilise the optimisation. As regards smax, the following values for engineering
applications have been recommended, cf. Goda [19],
smax ¼
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 45
8
><
10; wind waves;
25;
>:
75;
swell with short decay distance;
swell with long decay distance:
op;i
o
4
ð4:1Þ
(4.3)
(4.4)

46
4.2. Estimated parameters
In conclusion, the optimal wave spectrum estimated by the Parametric method is found
from the optimisation <strong>of</strong> the parameters
f Hs;1 l1 ymean;1 op;1 smax;1 Hs;2 l2 ymean;2 op;2 smax;2 g. (4.5)
As has been discussed, the Parametric modelling method assumes basically the wave
spectrum to be composed <strong>of</strong> a sea and a swell part. In case <strong>of</strong> unimodal seas, however, the
method should predict a negligible value for one <strong>of</strong> the significant wave heights (Hs;1 or
Hs;2). Later, this is verified by a numerical example.
5. Solution procedures
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
From the preceding it appears that both methods, the Bayesian method and the
Parametric method, in their fundamentals are conceptually similar and deviate only in the
sense <strong>of</strong> different (prior) assumptions about the wave spectrum to be estimated. In
practical computations, though, the solution procedures <strong>of</strong> the two methods are quite
different, since the Bayesian modelling in a number <strong>of</strong> iterations, based on QR
factorisation, solves an overdetermined linear equation system, whereas the Parametric
method yields the solution <strong>of</strong> a non-linear programming problem. This means that
the solution <strong>of</strong> the Bayesian modelling can be obtained by use <strong>of</strong> ‘standard’
numerical techniques, contrary to the Parametric modelling which requires the use
<strong>of</strong> a genetic optimisation algorithm, or a gradient search algorithm with careful input <strong>of</strong>
initial values <strong>of</strong> the optimisation parameters, the so-called search basin. In the present
work, the optimisation <strong>of</strong> the Parametric method is based on the latter <strong>of</strong> the two
algorithms.
When the basic equation system is set up, cf. (2.2), N different time series, or signals, are
considered. The standard deviation <strong>of</strong> these N signals will, in some cases, be <strong>of</strong> a very
different magnitude depending on the kind <strong>of</strong> signals being recorded. Hence, the solution
<strong>of</strong> the equation system will be affected by this fact, which may be reflected by a poor
agreement between the solution and the data for the signals with the relatively smallest
standard deviation. Conversely, the solutions corresponding to the signals with the
relatively largest standard deviation may approximate the data in a better way. In order to
meet this problem the equation system can be normalised. Thus, if it is remembered that
the cross spectra, in general, are complex with the real part Cij being the co-spectrum and
with the imaginary part Qij being the quadrature-spectrum, e.g. Price and Bishop [23], so
that
SijðoeÞ ¼CijðoeÞþiQijðoeÞ. (5.1)
Bendat and Piersol [24] suggests to normalise the cross spectra according to
bCijðoeÞ
CijðoeÞ
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2½SiiðoeÞSjjðoeÞþCijðoeÞ 2 QijðoeÞ 2 q ,
Š
Q
bQ
ijðoeÞ
ijðoeÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2 ½SiiðoeÞSjjðoeÞ CijðoeÞ 2 þ QijðoeÞ 2 q . ð5:2Þ
Š

The normalisation <strong>of</strong> the equation system has also the implication that the optimal value
<strong>of</strong> the hyperparameter, which is an important factor in the Bayesian modelling, assumes
almost the same magnitude in all cases for similar data, cf. Iseki and Terada [5].
6. Numerical example
As a means to verify the two estimation concepts, numerical simulations are conducted.
Thus, on the assumption <strong>of</strong> a perfect relationship between ship responses and a wave
spectrum through (given) complex-valued transfer functions, generated ship responses can
be used to estimate the underlying wave spectrum.
The numerical simulations dealt with are based on the ship responses fheave, roll, pitchg
and the complex-valued transfer functions are calculated by a three-dimensional time
domain code. The considered ship is a container ship with main dimensions as given in
Table 1, and in the following a speed <strong>of</strong> V ¼ 23 knots is assumed.
Until now both the wave and the encounter frequencies have been expressed in terms <strong>of</strong>
the angular frequency o ¼ 2pf . However, in practical computations it is <strong>of</strong>ten more
convenient to use the frequency f, and from now on the wave and the encounter
frequencies will be given as f and f e, respectively.
6.1. Simulation <strong>of</strong> response spectra
ARTICLE IN PRESS
As regards response spectra (<strong>of</strong> ship responses), simulations can be calculated on the
basis <strong>of</strong> Eq. (2.14) on the assumption <strong>of</strong> a theoretically known wave spectrum, including
the location <strong>of</strong> the spectral peak(s). Hence, for given operational conditions for a specific
ship, the response spectra are given by
bsimul ¼ A0f þ w0, (6.1)
where w0 represents a certain amount <strong>of</strong> white noise, so that real measurements are better
simulated. In the examples, the standard deviation <strong>of</strong> the white noise is set to 0.5% <strong>of</strong> the
standard deviation <strong>of</strong> the actual responses. Although f contains the wave spectrum as
function <strong>of</strong> the wave frequency, it should be realised that the multiplication by A0
transforms the response spectra into the encounter frequency domain.
In the example, four conditions, A, B, C and D, are studied, and in each case response
spectra are generated on the basis <strong>of</strong> a theoretically known wave spectrum which is
calculated from (4.1). Thus, the wave spectra <strong>of</strong> the four conditions are characterised by a
set <strong>of</strong> wave parameters which may be estimated on the basis <strong>of</strong> the two estimation
methodologies. It should be noted that Condition A corresponds to unimodal seas, see
Table 2, while Conditions B and C represent bimodal seas. Finally, Condition D is taken
Table 1
Main dimensions <strong>of</strong> the considered ship
U.D. Nielsen / Marine Structures 19 (2006) 33–69 47
Length, Lpp
275.0 m
Breadth, Bmld 40.0 m
Draught, T 12.0 m
Displacement 50,000 t

48
Table 2
Comparison <strong>of</strong> wave parameters
as the sum <strong>of</strong> a unimodal and a bimodal spectrum, so that the wave spectrum <strong>of</strong> Condition
D exhibits three spectral peaks.
6.2. <strong>Estimations</strong>
ARTICLE IN PRESS
Cond. Peak 1 Peak 2 Total
Hs (m) T p (s) l ymean (deg.) smax Hs (m) T p (s) l ymean (deg.) smax Hs (m) T s (s)
A TRUE 2.00 11.0 1.00 225 25 – – – – – 2.00 8.54
BAY 1.95 10.5 – 225 – – – – – – 1.95 8.51
OPT 2.00 11.0 1.00 225 25 0.05 9.77 1.00 55 25 2.00 8.54
B TRUE 2.00 11.0 1.00 225 25 1.50 5.00 1.00 295 25 2.50 6.16
BAY – 10.7 – 225 – – 5.56 – 298 – 2.44 6.67
OPT 2.00 11.0 1.00 225 25 1.50 5.00 1.00 295 25 2.50 6.16
C TRUE 2.00 11.0 1.00 225 25 1.50 5.00 1.00 105 25 2.50 6.16
BAY – 10.3 – 225 – – 5.26 – 102 – 2.37 6.66
OPT 1.07 8.80 3.28 150 75 1.53 5.12 1.00 104 25 1.87 4.98
OPT a
2.00 11.0 1.00 225 25 1.50 5.00 1.00 105 25 2.50 6.16
D TRUE 2.00 11.0 1.00 225 25 1.50 5.00 1.00 105 25 3.20 6.92
2.00 11.0 1.00 5 25
BAY – 10.5 – 225 – – 4.76 – 105 – 3.15 7.32
– 10.5 – 5 –
OPT 1.91 11.4 1.00 334 10 1.97 5.00 1.00 113 25 2.75 5.71
a The initial search basin is changed.
U.D. Nielsen / Marine Structures 19 (2006) 33–69
Table 2 shows the outcome <strong>of</strong> the estimations <strong>of</strong> the four conditions in the form <strong>of</strong> the
resulting wave parameters. For each condition, the true wave parameters (TRUE) are
compared with those based on the Bayesian method (BAY) and the Parametric method
(OPT). It should be noted that the peak period T p is specified rather than the peak
frequency f p.
As regards the results <strong>of</strong> the Bayesian method, the significant wave height Hs, the shape
factor l and the spreading parameter smax have not been determined for the individual
peaks in case <strong>of</strong> bimodal seas, since this is not possible. The peak period T p and the mean
wave direction ymean, on the other hand, are based on a detailed visual inspection <strong>of</strong> each <strong>of</strong>
the peaks <strong>of</strong> the spectrum. Contrary, the Parametric modelling yields all the parameters <strong>of</strong>
the individual peaks, since the parameters are the actual solution <strong>of</strong> this method. The two
rightmost columns <strong>of</strong> Table 2 show the significant wave height and the mean wave period
<strong>of</strong> the ‘total’ spectrum. In general, these numbers are calculated by, e.g. Jensen [25],
Hs ¼ 4 ffiffiffiffiffiffi p
m0
(6.2)
and
T s ¼ m0
, (6.3)
m1

where the spectral moments are given as
Z 1
mn ¼ f
0
n
Z p
p
ARTICLE IN PRESS
Eðf ; bÞ db df . (6.4)
In addition to Table 2, Figs. 6–9 illustrate the comparison between the true wave
spectrum and the estimated wave spectra <strong>of</strong> the four conditions. In each figure the left plot
(denoted TRUE) is the true wave spectrum, whereas the middle and the right plots are the
estimates by the Bayesian method (denoted BAYESIAN) and the Parametric method
(denoted OPTIMISATION), respectively. With regards to the ship course, this is
coincident with the radial line <strong>of</strong> 0 , so that beam, quartering and following waves are
experienced for seas in the interval <strong>of</strong> 90–270 .
From the numbers in Table 2 and from Figs. 6–9 and with focus on Conditions A and B,
there is seen to be a good agreement between the true and the estimated wave parameters.
Indeed, it should be noted that the Parametric method (OPT) yields a unimodal spectrum
in Condition A, as the significant wave height <strong>of</strong> the one peak is negligible. As a matter <strong>of</strong>
fact the value <strong>of</strong> Hs ¼ 0:05 m is the lower limit used in the optimisation for the significant
wave height. As regards Condition C, the Bayesian method produces a good estimate for
270
300
240
330
210
TRUE
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
BAYESIAN
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
OPTIMISATION
Fig. 6. Contour plots <strong>of</strong> the true and the two estimated wave spectra <strong>of</strong> Condition A. The ship course is 0 and the
waves are shown as approaching.
270
300
240
330
210
TRUE
0
180
0.1
0.4
0.3
0.2
30
150
U.D. Nielsen / Marine Structures 19 (2006) 33–69 49
60
90 270
120
300
240
330
210
BAYESIAN
0
180
0.1
0.4
0.3
0.2
30
150
60
90 270
120
300
240
330
210
0
180
0.3
0.2
0.1
0.2
0.1
0.4
OPTIMISATION
Fig. 7. Contour plots <strong>of</strong> the true and the two estimated wave spectra <strong>of</strong> Condition B. The ship course is 0 and the
waves are shown as approaching.
0
180
0.4
0.3
30
150
30
150
60
120
60
120
90
90

50
270
270
300
240
300
240
210
330
210
330
TRUE
0
0
180
TRUE
180
0.1
0.3
0.2
0.1
0.4
0.3
0.2
0.4
30
150
30
150
60
90 270
120
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
60
90 270
120
300
240
300
240
330
210
330
210
BAYESIAN
BAYESIAN
0
180
the wave spectrum. The Parametric modelling, however, gives a poor result, since the
parameters <strong>of</strong> Peak 1 is incorrect, see also Fig. 8. The reason for the inconsistency is to be
found in the initial search basin used for the optimisation. Hence, by changing the search
basin, the wave parameters corresponding to the label OPT a are obtained and it is seen
that the wave parameters are then estimated correctly. Clearly, this indicate an important
issue, since the initial search basin for an optimal parametric modelling procedure needs
not to be changed, independently on the underlying conditions. In practical systems based
on parametric modelling it is, therefore, likely that a genetic search algorithm will be the
best choice for the optimisation. In this study, however, the optimisation is based on a
gradient-based search algorithm. This means that the initial search basin needs to be rather
comprehensive/detailed for which reason the parametric modelling is more time
consuming than the Bayesian modelling. In this numerical example the Parametric
method takes in the order <strong>of</strong> 8 min, whereas the Bayesian modelling takes in the order <strong>of</strong>
5 min (with MatLab 7.0 on a Pentium 2.0 GHz processor) for each condition.
Since the Parametric method is based on the summation <strong>of</strong> two parameterised wave
spectra it is obvious that the three spectral peaks <strong>of</strong> Condition D make a problem for the
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
OPTIMISATION
Fig. 9. Contour plots <strong>of</strong> the true and the two estimated wave spectra <strong>of</strong> Condition D. The ship course is 0 and
the waves are shown as approaching.
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
240
330
210
OPTIMISATION
Fig. 8. Contour plots <strong>of</strong> the true and the two estimated wave spectra <strong>of</strong> Condition C. The ship course is 0 and the
waves are shown as approaching.
300
0
180
0
180
0.3
0.2
0.1
0.4
0.3
0.2
0.1
0.4
30
150
30
150
60
120
60
90
120
90

Parametric method. It is seen that Peak 1, cf. Table 2, in the estimation <strong>of</strong> the Parametric
method represents waves which come from a direction somewhere in the middle <strong>of</strong> two <strong>of</strong>
the true spectral peaks, see also Fig. 9. Moreover, Peak 2 <strong>of</strong> the Parametric method
contains too much energy, although the direction <strong>of</strong> those waves is close to that <strong>of</strong> the third
true spectral peak. The Bayesian method, on the other hand, yields a wave spectrum which
estimates the true wave spectrum with good accuracy.
6.3. Concluding remarks
If the Bayesian and the Parametric method are to be compared based on the preceding
results, the Bayesian method seems as the best estimation procedure <strong>of</strong> the two. Thus, this
procedure is the least expensive as regards the computational cost. Furthermore, and most
important, the Bayesian method did estimate a wave spectrum in good agreement with the
true spectrum for all four conditions. This were also partly the case for the Parametric
method, however, Condition C revealed the importance <strong>of</strong> the initial search basin <strong>of</strong> the
optimisation, and Condition D showed that the Parametric modelling cannot, by any
means, estimate a wave spectrum with more than two spectral peaks, since the method is
based on the summation <strong>of</strong> two parameterised wave spectra.
It should be noted that there are different opinions about which concept is the best.
Tannuri et al. [1] performs also numerical tests, for zero speed though, and, contrary to the
present study, Tannuri et al. [1] concludes that a parametric approach is superior to a
Bayesian approach. The example seen in the present paper, however, indicates clearly that
both <strong>of</strong> the estimation concepts, based on the derived theory in Sections 2–4, are able to
produce reasonable estimates for directional wave spectra when numerical simulations are
studied.
7. Analysis <strong>of</strong> full-scale data
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 51
In this section full-scale motion measurements <strong>of</strong> a container ship are studied. The main
dimensions <strong>of</strong> the ship are listed in Table 1, and the ship has been in route on the Pacific
Ocean where all the data has been recorded. The data consists <strong>of</strong> nine sets: A; B; ...; I, each
<strong>of</strong> a duration <strong>of</strong> 15 min. The duration is taken in the middle <strong>of</strong> a 30 min period used for the
WAVEX estimation and during this time, the operational conditions were (nearly)
constant. The estimations are based on the three responses {heave, roll, pitch}, with
complex-valued transfer functions calculated by a three-dimensional time domain code.
Unfortunately, no visual observations were carried out to hold against the estimations.
It is important to emphasise that the complex-valued transfer functions are believed to
give a perfect relationship between the wave excitations and the ship responses, similar to
the numerical example studied in the previous section where the ‘measured’ responses were
generated by utilisation <strong>of</strong> the transfer functions. In reality, however, there is a certain
amount <strong>of</strong> uncertainty related to the transfer functions due to phenomena such as: not
perfectly controlled/known operational conditions (in particular, as regards the loading
condition), unknown noise phenomena in the response signals, the degree <strong>of</strong> non-linearity
between excitations and responses is difficult to predict about, relative to the size
<strong>of</strong> the ship the high-frequency wave components induce no significant motions,
etc. Consequently, the estimated directional wave spectrum, including its underlying
wave parameters, ought to be given with some uncertainty, although the estimation

52
methodology in itself, on the assumption, works perfectly in theory and yields the true
directional wave spectrum. So far, though, nothing can be said about the related
uncertainty <strong>of</strong> the Bayesian and the Parametric method. As regards WAVEX, the technical
manual [26] considers the statistical uncertainty <strong>of</strong> the measurements. Thus, based on
theory assuming that the sea surface elevation is a Gaussian process, it is specified that the
statistical measurement error <strong>of</strong> the standard deviation <strong>of</strong> the significant wave height is
Hs ¼ 0:66 g T p
, (7.1)
2p Lx
where g is the acceleration <strong>of</strong> gravity, T p is the peak period <strong>of</strong> the wave spectrum, and Lx is
the image size. The uncertainty <strong>of</strong> the estimated results by WAVEX will not be dealt with
in detail. In the following, reference will, however, be given to the number in (7.1) when
comparisons are conducted. (The image size is assumed to be Lx ¼ 520 m [26]).
As a general comment to the recorded data it should be noted that none <strong>of</strong> the data sets
have been recorded during particularly light sea states. The reason for this is that the
recording system saves only entire time histories <strong>of</strong> the hundred so-called worst cases (with
a duration <strong>of</strong> 30 min) experienced within a month. Otherwise, only statistical values are
stored. The lowest possible sea states are chosen in order to minimise the influence <strong>of</strong> nonlinearity
between wave excitation and response.
Furthermore, it should be noted that the cross spectral analysis <strong>of</strong> the responses is
carried out by use <strong>of</strong> MAR modelling, e.g. Akaike and Nakagawa [11], by application <strong>of</strong>
the so-called stepwise least squares algorithm, see Neumaier and Schneider [27], for the
determination <strong>of</strong> the MAR coefficients. Finally, it should be mentioned that the author
himself has not carried out the calculations <strong>of</strong> the transfer functions, for which reason
sensitivity analyses <strong>of</strong> the transfer functions have not been conducted. The transfer
functions have been calculated by use <strong>of</strong> a three-dimensional time domain code.
7.1. Estimated wave parameters
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
The nine data sets are analysed by the Bayesian modelling and the Parametric modelling
procedure and the analyses lead to the wave parameters presented in Table 3, see also
Table 3
Comparison <strong>of</strong> wave parameters: the significant wave height Hs, the mean wave period T s, and the primary
(mean) wave direction ym
Data Hs (m) T s (s) ym (deg.)
BAY OPT WAV BAY OPT WAV BAY OPT WAV
A 2.7 2.5 3.0 ( 0.4) 7.5 7.2 6.9 200 220 280
B 2.5 2.3 2.5 ( 0.3) 7.1 7.3 6.8 240 240 225
C 2.3 1.7 2.6 ( 0.3) 7.1 7.4 6.5 240 240 240
D 2.4 1.4 2.5 ( 0.3) 7.0 8.3 6.6 160 130 115
E 3.6 3.4 4.9 ( 1.5) 8.2 11.3 9.2 0 5 330
F 4.0 3.4 4.8 ( 1.0) 7.8 8.6 8.6 180 145 180
G 3.3 3.5 4.5 ( 0.6) 10.2 10.7 8.7 280 265 60
H 3.6 3.7 4.4 ( 1.4) 9.0 9.5 8.8 280 295 35
I 5.7 7.4 7.2 ( 2.2) 13.1 13.0 10.8 185 180 190

270
300
240
330
210
BAYESIAN
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
ARTICLE IN PRESS
300
240
330
210
OPTIMISATION
0
180
Figs. 10–18. The significant wave height Hs is calculated according to (6.2), and the mean
wave period T s is found from (6.3). In the table, numbers are also given for the estimations
produced by WAVEX. Thus, results <strong>of</strong> the former two methods are denoted by BAY and
OPT, respectively, whereas parameters calculated from WAVEX are denoted by WAV. It
is seen that the uncertainty related to the significant wave height estimated by WAVEX is
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
WAVEX
Fig. 10. The estimated wave spectra <strong>of</strong> Data A. The ship course is 0 and the waves are shown as approaching.
270
300
240
330
210
BAYESIAN
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
OPTIMISATION
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
0
180
WAVEX
Fig. 11. The estimated wave spectra <strong>of</strong> Data B. The ship course is 0 and the waves are shown as approaching.
270
300
240
BAYESIAN
330
0
0.4
0.3
210
180
0.2
0.1
30
150
U.D. Nielsen / Marine Structures 19 (2006) 33–69 53
60
90 270
120
300
240
OPTIMISATION
330
0
0.4
0.3
210
180
0.2
0.1
30
150
60
90 270
120
0
180
0.3
0.2
0.1
0.3
0.2
0.1
0.4
0.4
WAVEX
0
0.4
Fig. 12. The estimated wave spectra <strong>of</strong> Data C. The ship course is 0 and the waves are shown as approaching.
300
240
330
210
180
0.3
0.2
0.1
30
150
30
150
30
150
60
120
60
120
60
90
120
90
90

54
270
270
300
240
330
210
BAYESIAN
0
180
0.3
0.2
0.1
0.4
30
150
60
90 270
120
ARTICLE IN PRESS
300
240
330
210
OPTIMISATION
0
180
included in the parenthesis (calculated by use <strong>of</strong> (7.1)). As regards the columns <strong>of</strong> the
primary wave direction ym, these numbers are relative in the sense that they are given
relative to the ship’s longitudinal axis. The definition follows from WAVEX, so that beam,
quartering and following waves are observed for directions in the interval <strong>of</strong> 90–270
(similarly to the numerical example). It should be noted that ym represents the mean
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
WAVEX
Fig. 14. The estimated wave spectra <strong>of</strong> Data E. The ship course is 0 and the waves are shown as approaching.
270
300
240
300
240
330
210
330
210
BAYESIAN
0
180
BAYESIAN
0
0.4
0.3
30
180
0.1
0.2
0.1
0.4
0.3
0.2
30
150
150
U.D. Nielsen / Marine Structures 19 (2006) 33–69
60
60
90 270
120
90 270
120
300
240
300
240
330
210
330
210
OPTIMISATION
0
180
OPTIMISATION
0
0.4
0.3
30
180
0.2
0.1
150
60
90 270
120
300
240
330
210
0
180
0.3
0.2
0.1
0.2
0.1
0.4
WAVEX
0
0.4
Fig. 15. The estimated wave spectra <strong>of</strong> Data F. The ship course is 0 and the waves are shown as approaching.
0.1
0.4
0.3
0.2
30
150
60
90 270
120
300
240
330
210
WAVEX
Fig. 13. The estimated wave spectra <strong>of</strong> Data D. The ship course is 0 and the waves are shown as approaching.
0
180
180
0.3
0.2
0.1
0.4
0.3
30
150
30
150
30
150
60
120
60
120
60
120
90
90
90

270
270
300
240
300
240
330
210
330
210
BAYESIAN
0
180
BAYESIAN
0
180
0.1
0.3
0.2
0.1
0.4
0.3
0.2
0.4
30
150
30
150
60
60
90 270
120
90 270
120
ARTICLE IN PRESS
300
240
300
240
330
210
330
210
OPTIMISATION
0
180
OPTIMISATION
0
180
direction <strong>of</strong> the primary spectral peak, which is the peak that contains the most energy in
the case <strong>of</strong> more peaks being present. Moreover, it should be mentioned that Table 4
contains all the specific parameters obtained from the Parametric method. In the table, S1
ði ¼ 1Þ and S2 ði ¼ 2Þ represent the two components <strong>of</strong> the individual wave parameters,
which control the parameterised wave spectrum, in accordance with expression (4.1).
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
WAVEX
Fig. 17. The estimated wave spectra <strong>of</strong> Data H. The ship course is 0 and the waves are shown as approaching.
0.3
0.2
0.1
0.4
30
150
60
90 270
120
300
240
330
210
WAVEX
Fig. 16. The estimated wave spectra <strong>of</strong> Data G. The ship course is 0 and the waves are shown as approaching.
270
300
240
330
210
BAYESIAN
0
180
0.1
0.4
0.3
0.2
30
150
U.D. Nielsen / Marine Structures 19 (2006) 33–69 55
60
90 270
120
300
240
330
210
OPTIMISATION
0
180
0.1
0.4
0.3
0.2
30
150
60
90 270
120
300
240
330
210
0
180
0
180
0.3
0.2
0.1
WAVEX
Fig. 18. The estimated wave spectra <strong>of</strong> Data I. The ship course is 0 and the waves are shown as approaching.
0
180
0.4
0.3
0.2
0.1
0.2
0.1
0.4
0.4
0.3
30
150
30
150
30
150
60
120
60
120
60
120
90
90
90

56
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
Table 4
Wave parameters from the Parametric method: the peak wave period T p, the significant wave height Hs, the
(mean) wave direction ym, the shape factor l, and the spreading parameter smax
Data T p (s) Hs (m) ym (deg.) l smax
S1 S2 S1 S2 S1 S2 S1 S2 S1 S2
A 7.7 9.1 1.7 1.7 220 295 4.0 1.0 10 75
B 9.1 7.7 1.2 1.9 295 240 1.0 4.0 25 75
C 16.5 7.6 0.4 1.7 295 240 1.0 4.0 25 75
D 16.7 7.3 0.7 1.2 55 130 1.0 4.0 10 25
E 14.3 12.5 0.4 3.4 135 5 1.0 4.0 25 75
F 14.2 7.6 1.9 2.8 130 145 4.0 3.7 10 10
G 10.0 16.7 2.8 2.2 265 0 1.6 4.0 75 75
H 14.4 8.3 2.7 2.6 295 250 2.5 4.0 75 75
I 14.4 20.0 7.0 2.4 180 305 4.0 2.8 25 75
For the individual wave parameters, S1 and S2 represent the two spectral peaks <strong>of</strong> the wave spectrum.
From a study <strong>of</strong> the numbers in Table 3 it is seen that, in general, there is a fair
agreement between the Bayesian and the Parametric method. However, notable differences
(in the relative sense) are observed for the significant wave height <strong>of</strong> Data C, D and I. This
goes also for the mean wave period <strong>of</strong> Data E and F. With regard to the primary wave
direction and the overall distribution <strong>of</strong> energy <strong>of</strong> the estimations <strong>of</strong> the two modelling
procedures, see Figs. 10–18, the consistency is reasonable. Especially, it seems that the
introduced frequency dependency, cf. (4.3), <strong>of</strong> the spreading parameter <strong>of</strong> the Parametric
modelling yields directional estimations which exhibit somewhat similar properties as seen
from the Bayesian estimations. At this stage it is worth to mention that, although not
shown in the present paper, the introduction <strong>of</strong> a frequency-dependent spreading leads
actually to results which are in better agreement with the Bayesian as well as the WAVEX
estimates compared to what is seen without the introduction <strong>of</strong> a frequency-dependent
spreading parameter; both with respect to the wave parameters and the directional
distribution <strong>of</strong> energy.
Compared to the wave parameters estimated by WAVEX, the parameters <strong>of</strong> the
Bayesian and the Parametric method deviate slightly with a trend towards better
agreement for Data A–D as opposed to Data E–I. However, with respect to the significant
wave height, the best agreement is, in general, found between the Bayesian method and
WAVEX. As regards the mean wave period, the agreement between WAVEX and the
Bayesian and the Parametric method is equally good, with reservation to some data (as
mentioned above). The general tendency <strong>of</strong> less good agreement between the significant
wave height <strong>of</strong> WAVEX and the two modelling procedures for Data E–I could be the
existence <strong>of</strong> severer sea states compared to the sea states <strong>of</strong> Data A–D. Thus, it is seen that
WAVEX and the Bayesian method as well as the Parametric method estimate substantially
higher values for Hs, when the last five sets in Table 3 are compared with the first four data
sets. Hence, the assumed linearity between the ship responses and the waves might be less
good in the last five cases, a fact which would influence the Bayesian and the parametric
results since linearity is fundamental for these methods to be reliable. On the other hand, it
is believed that due to the high inertia <strong>of</strong> the ship, the errors in the Bayesian and the

ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 57
Parametric estimations should decrease as the wave period increases. The reason is that the
ship acts as a filter and, therefore, the ship will filter the high-frequency wave components
which induce no significant motions. To judge the increasing deviations between the
Bayesian/parametric estimates and the WAVEX results observed for Data E–I compared
to Data A–D, it is thus important to keep the filtering aspect in mind as well.
Returning to the deviations in each individual set <strong>of</strong> data, cf. Table 3, it should be noted
that with respect to the significant wave height, WAVEX yields always results which are
larger than those obtained by the Bayesian and the Parametric method, in particular this
applies to Data E–I. The general trend <strong>of</strong> larger significant wave heights estimated by
WAVEX is difficult to explain since, on the face <strong>of</strong> it, it may be the Bayesian/Parametric
estimations which exhibit too little energy or it may be the WAVEX results which exhibit
too much energy. As will be seen in a later section, studies suggest that energy is lacking in
the response spectra when measurements and estimations based on the Bayesian/
Parametric results are compared. The same trend is, however, seen for the estimated
response spectra based on the WAVEX results, although the tendency is not as unique as
observed for the Bayesian/Parametric results. Therefore, more detailed investigations and
analyses would be required to settle on the discrepancy in the significant wave height
estimated by the Bayesian and the Parametric method and by WAVEX. In the end, this
would require additional knowledge <strong>of</strong> the on-site sea state in the given situation and,
unfortunately, this information cannot be attained. However, it should also be realised
that—taking the uncertainty <strong>of</strong> WAVEX into account—almost all <strong>of</strong> the cases, the
Bayesian as well as the Parametric method, are within the confidence range <strong>of</strong> WAVEX,
considering the statistical uncertainty.
As regards the mean wave period T s, Table 3 shows no consistent tendency in the
deviations between the Bayesian/Parametric estimates and the WAVEX estimates
and, in general, the magnitude <strong>of</strong> the differences must be considered acceptable from an
engineering point <strong>of</strong> view.
The primary (mean) wave direction ym, cf. Table 3, is found with reasonable consistency
when the results <strong>of</strong> the two modelling procedures are compared with those <strong>of</strong> WAVEX, see
also Figs. 10–18. However, both for Data G and H there seems to be a conflict whether
waves are approaching from one side or the other side <strong>of</strong> the ship. Thus, the Bayesian as
well as the Parametric estimations have waves approaching on the port side, whereas
WAVEX yields waves entering from the starboard side. It is by no means easy to give a
reason for this controversy since nothing is changed in the computational set-up <strong>of</strong> the two
data sets. An apparent and straightforward explanation for the discrepancy, though, could
be related to port/starboard symmetric frequency response functions used in the Bayesian/
Parametric estimations. However, as the roll response is used as one <strong>of</strong> the responses in the
present data analysis, the port/starboard asymmetry is reflected in the estimations. In this
context it should be remembered that by use <strong>of</strong> complex-valued transfer functions, the
amplitude as well as the phase <strong>of</strong> the responses are taken into account. For this reason the
controversy in the wave direction for Data G and Data H has also not to do with the phase
shift <strong>of</strong> the roll transfer function experienced at frequencies close to the natural roll
frequency in beam seas. An estimate, e.g. Lloyd [28], <strong>of</strong> the natural roll frequency <strong>of</strong> the
ship is found to be f r ’ 0:06 Hz ð T r ’ 15 s) and Fig. 19, which depicts the roll phase
angle, confirms, more or less, this value and illustrates the mentioned shift in phase.
It can therefore be excluded that the roll response as such has to do with the controversy
in the wave direction observed for the two data sets. For this reason it would be interesting

58
[deg.]
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
Roll Phase
15
10
5
0
-5
-10
-15
0 0.05 0.1 0.15 0.2
Wave Freq. [Hz]
Fig. 19. The phase angle <strong>of</strong> the roll transfer function for an encounter angle equal to 90 . The operational
conditions correspond to Data G and H.
Table 5
Estimated wave parameters based on the responses {heave, sway, pitch}
Data Hs (m) T s (s) ym (deg.)
BAY OPT BAY OPT BAY OPT
A 2.4 ( 11%) 2.2 ( 12%) 8.1 (8%) 7.6 (6%) 200 (0) 300 (80)
B 2.3 ( 8%) 1.9 ( 17%) 8.1 (14%) 6.7 ( 8%) 240 (0) 275 (35)
C 2.0 ( 13%) 1.1 ( 35%) 7.6 (9%) 6.5 ( 12%) 240 (0) 270 (30)
D 2.2 ( 8%) 1.2 ( 14%) 8.0 (14%) 8.4 (1%) 125 ( 35) 90 ( 40)
E 3.1 ( 14%) 3.5 (3%) 9.5 (16%) 10.0 ( 12%) 0 (0) 0 ( 5)
F 3.9 ( 3%) 3.5 (3%) 8.9 (14%) 11.5 (34%) 225 (45) 200 (55)
G 3.4 (3%) 3.7 (6%) 9.5 ( 7%) 11.4 (7%) 270 ( 10) 300 (35)
H 3.7 (3%) 4.1 (11%) 9.5 (6%) 9.0 ( 5%) 270 ( 10) 250 ( 45)
I 7.2 (26%) 8.5 (15%) 10.1 ( 23%) 12.5 ( 4%) 185 (0) 180 (0)
Deviation relative to the estimated parameters based on {heave, roll, pitch} is shown in parenthesis.
to do the Bayesian and the Parametric estimations with the roll response replaced by
another response, e.g. the sway response which is also an odd function and therefore it
exhibits port/starboard asymmetry as well. Such estimations, with the roll replaced by the
sway, have been conducted and, as seen from Table 5, the sway-based wave parameters
deviate only slightly compared to the estimations where the roll response is used. Similarly,
the directional distribution <strong>of</strong> energy is more or less identical, independently if the roll or
the sway response is used in the Bayesian/Parametric estimations, cf. Figs. B1–B5 in
Appendix B which show the directional wave spectra, when the roll response is replaced by
the sway. As regards Data G and H, the sway-based estimations do not specifically confirm
the roll-based Bayesian/Parametric estimations and nor do they specifically indicate results
in favour <strong>of</strong> the WAVEX estimations. Thus, it is seen that the sway-based Bayesian
estimations have waves approaching from the starboard side as well as the port side,
however, with the waves on the port side containing the most energy. Therefore, in
conclusion on the controversy in the wave direction <strong>of</strong> Data G and H, it cannot be
decisively stated whether waves in reality were entering on the port or the starboard side.
As mentioned in Section 1, Tannuri et al. [1] suggests to use the sway response instead <strong>of</strong>
the roll. The present paper will not argue against this proposal, however, to follow on the
0.25

topic <strong>of</strong> sway vs. roll, it should be noted that, based on the estimations carried out in the
present work, it seems hard to justify whether the one response should be proposed in
favour <strong>of</strong> the other.
The Parametric method is based on the summation <strong>of</strong> two parameterised wave spectra
and from the data sets considered here, it appears to be reasonable to use a number <strong>of</strong> two
spectra. Thus, it is seen, see Figs. 10–18, that in neither <strong>of</strong> the data sets no more than two
(fully separated) spectral peaks are estimated by the Bayesian method and by WAVEX.
Moreover, the Parametric modelling predicts (nearly) unimodal seas, cf. Table 4, for Data B,
C and I, which is consistent to the Bayesian and the WAVEX estimations. However, with
consideration to the directional wave spectrum <strong>of</strong> Data E, cf. Fig. 14, it should be noted that
both the Bayesian and the WAVEX estimations exhibit two adjoined spectral peaks, that is,
the peaks are evidently separated but very close to each other. This property is not captured
by the Parametric method <strong>of</strong> Data E, cf. Table 4. In general, the property <strong>of</strong> adjoining peaks
is hard to be handled by the Parametric method because <strong>of</strong> the prespecified shape <strong>of</strong> the
wave spectrum, frequency- as well as directional-wise. A fact which is also touched partly by
Pascoal et al. [9], which in relation to a parametric formulation/modelling state: ‘‘The
formulation has shown to lack enough information to allow multiple peaked spectra with
strong frequency overlap to be correctly estimated in a reasonable time span, thus multiple
sea or swell components are, for now, out <strong>of</strong> reach in this formulation.’’ Although Data E to
some extent justifies this statement, at least directional-wise, the present paper is not able to
give a conclusive accedence to the statement.
7.2. Response measurements
Based on the measured response spectrum, cf. Appendix C, for each <strong>of</strong> the considered
responses, heave, roll, and pitch, the variance <strong>of</strong> the individual responses has been
calculated for all the data sets. The results are seen in Table 6.
Correspondingly, the variances <strong>of</strong> the responses can be calculated by use <strong>of</strong> the
estimated wave spectrum. Hence, by calculation <strong>of</strong> the response spectrum in the wave
frequency domain (cf. (2.2))
SRðf Þ¼
Z p
p
ARTICLE IN PRESS
jFRðf ; bÞj 2 Eðf ; bÞ db (7.2)
Table 6
Measured variances, calculated from the response spectra
Data Heave ðm 2 Þ Roll ðrad 2 Þ Pitch ðrad 2 Þ
A 5:7 10 2
B 4:0 10 2
C 1:5 10 2
D 3:6 10 2
E 22 10 2
F 14 10 2
G 72 10 2
H 61 10 2
I 65 10 2
U.D. Nielsen / Marine Structures 19 (2006) 33–69 59
3:1 10 4
13 10 4
8:3 10 4
2:1 10 4
0:60 10 4
19 10 4
0:88 10 4
3:0 10 4
30 10 4
1:3 10 5
1:1 10 5
0:46 10 5
0:69 10 5
20 10 5
4:7 10 5
18 10 5
13 10 5
26 10 5

60
Table 7
Deviation on variances for the considered responses
the variance is given, or estimated, by
VarR ¼
Z 1
0
ARTICLE IN PRESS
Data Heave Roll Pitch
BAY (%) OPT (%) WAV (%) BAY (%) OPT (%) WAV (%) BAY (%) OPT (%) WAV (%)
A 26 5.6 1.6 5.2 20 38 22 13 40
B 15 3.8 37 13 40 63 15 9.3 20
C 4.9 3.1 10 14 37 36 14 3.5 33
D 16 6.6 25 4.2 11 49 30 10 75
E 21 6.4 86 22 40 9.6 73 49 14
F 6.8 6.1 47 3.8 0.5 83 66 53 0.5
G 26 17 34 2.0 9.6 19 79 24 40
H 19 5.2 8.5 7.6 0.2 27 62 20 7
I 21 4.8 16 2.7 5.9 362 70 37 39
Figures relative to the measured variance.
[m 2 ]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Heave variance
Measured
BAY
OPT
WAV
A B C D E F G H I
Data set
U.D. Nielsen / Marine Structures 19 (2006) 33–69
[rad 2 ]
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Measured
BAY
OPT
WAV
Roll variance
A B C D E F G H I
Data set
Fig. 20. Visualisation <strong>of</strong> Tables 6 and 7.
A B C D E F G H I
Data set
SRðf Þ df , (7.3)
where index R represents the specific response.
Table 7 shows the relative deviation between the estimated variance and the measured
variance (Table 6) <strong>of</strong> the individual responses for each data set, when the estimation is
based on the Bayesian method (BAY), the Parametric method (OPT) and WAVEX
(WAV), respectively. The results <strong>of</strong> Tables 6 and 7 are visualised graphically in Fig. 20
which shows comparisons <strong>of</strong> the variances by use <strong>of</strong> bar-diagrams.
As seen from Fig. 20 and the numbers in Table 7 it appears, first and foremost, that the
estimated variances based on the Bayesian and the Parametric method (almost)
consistently are less than the measured variances. This tendency is somewhat peculiar
and indicates that energy is lacking in those estimations. As regards to the estimations <strong>of</strong>
WAVEX, a similar trend is not seen. However, the estimations by WAVEX do not, in any
way, reflect a better agreement with the measurements. On the contrary, it is the variances
based on the Parametric estimations which generally deviate the least from the measured
variances. Likewise, it is seen that the Bayesian estimates in many cases are closer to the
measured variances compared to the estimations <strong>of</strong> WAVEX. An exception is the pitch
[rad 2 ]
2
1
0
x 10 -4 Pitch variance
Measured
BAY
OPT
WAV

ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 61
response, where the Bayesian estimations are particularly poor for Data E–I. In this
relation, it should be noted, though, that the deviations, in general, are significantly larger
for the last five data sets compared to Data A–D, when the heave and the pitch estimations
<strong>of</strong> the Bayesian and the Parametric method are studied. This fact could be related to the
severer sea states observed for Data E–I, cf. Table 3, which may mean a less good linear
relationship between the ship responses and the waves. Besides, it is interesting to note how
the heave and the pitch responses are markedly increased for all <strong>of</strong> the last five data sets,
whereas the roll response experiences only an increase for Data F and I. Both <strong>of</strong> these data
sets, F and I, are characterised by following (and quartering) seas according to the wave
spectra produced by all three estimation procedures, cf. Figs. 15 and 18. This indicates that
the roll motion is a response which is quite sensitive to the encounter angle between the
ship and the waves. Hence, in the respect <strong>of</strong> decision support for the specific ship, it is
evident that course changes are likely to be effective to decrease the roll response, in spite
<strong>of</strong> a severer sea state. This is to some extent seen by comparison with Data G and H, cf.
Figs. 16 and 17, which are characterised by beam and quartering seas (although with a
port/starboard controversy as discussed previously) and characterised by an energy
content <strong>of</strong> notable size. Still, the roll variances <strong>of</strong> Data G and H are amongst the lowest
recorded (and estimated) <strong>of</strong> the nine data sets.
Previously, it was argued that it is the variances estimated by the Parametric method
which, generally, deviate the least from the measured variances. Without further
explanation this statement may seem rather subjective. However, by assigning points to
the individual estimation procedures according to the value <strong>of</strong> the deviation between the
estimated and the measured variance <strong>of</strong> the respective responses, a kind <strong>of</strong> objective
argument can be stated. Thus, by consideration <strong>of</strong> the individual responses <strong>of</strong> each data set
in Table 7, 1 point is given to the method which has the smallest relative deviation on the
variance, and 2 and 3 points are given to the methods with the second smallest and the
largest relative deviations on the variances, respectively. Based on this allotment, Table 8 is
obtained and it is seen that, indeed, the Parametric modelling (OPT) is the method which
scores the least total points. Specifically, it is observed that the Bayesian method, on
average, yields the best agreement, when the roll response is considered. From the
viewpoint <strong>of</strong> the solution <strong>of</strong> the estimation procedures, it is not surprising that the smallest
relative deviations are found for the Parametric method and the Bayesian method. This is
so, because the estimated wave spectra <strong>of</strong> these methods are based on the factually
measured response spectra. For this reason, it could be anticipated that the agreement
between the measured and the estimated variances <strong>of</strong> the Bayesian (as well as the
Parametric) method would be better than what is seen from Table 7. As regards the
Bayesian method, the explanation, for this not being so, is believed to has to do with
the hyperparameter, u, because no matter whether the frequency response functions are
Table 8
Points assigned to the estimation methods for the nine cases A–I according to the agreement with the measured
variances <strong>of</strong> the respective responses (The less points, the better the agreement.)
Heave Roll Pitch Total
BAY 21 12 23 56
OPT 10 18 12 40
WAV 23 24 19 66

62
(completely) wrong, whether non-linearities in reality dominate the relation between
excitations and responses, whether the cross spectral analysis contains errors, etc., the
solution x, i.e. the estimated wave spectrum, is basically found by minimising
ke A expðxÞ ebk 2 þ u 2 kDx ck 2 . (7.4)
Hence, the solution, although maybe unstable and completely unrealistic, will in the best
possible way approximate the measured response spectra contained in the vector eb,by proper multiplication and with reservation about smoothing. Thus, the more smoothing,
i.e. the larger the hyperparameter, which is needed in accordance with ABIC, cf. (3.7), the
less good agreement can be expected between solution and data. And, apparently, the
studied data sets require some smoothing to be applied. From a more general point <strong>of</strong> view
it can be said that the fundamental equation system (2.14), which applies both to the
Bayesian and the Parametric method, without introduction <strong>of</strong> any assumptions on the
wave spectrum to be estimated, is underdetermined (or degenerate) in such a way that it
permits only fitting with solutions which lacks energy compared to the data.
Although the problem <strong>of</strong> lacking energy in the response spectra has been looked into
intensively, the latter explanation is, unfortunately, the only reason that can be given to
explain the lack <strong>of</strong> energy in the response spectra.
7.3. Prediction <strong>of</strong> sway response
ARTICLE IN PRESS
Since the sway response has also been measured, it should be interesting to see how well
the variance <strong>of</strong> this response can be predicted by use <strong>of</strong> the wave spectrum—estimated on
the basis <strong>of</strong> the heave, the roll and the pitch responses—and the transfer function <strong>of</strong> sway.
Table 9 shows the measured variance <strong>of</strong> sway for each <strong>of</strong> the data sets and, moreover, the
table lists the predicted values <strong>of</strong> the variance in case the wave spectrum is estimated by the
Bayesian method, the Parametric method and by WAVEX, respectively. As it appears
from the table the agreement between the measured and the predicted values is far from
impressive. However, the latter five data sets reveal an agreement which is markedly
improved compared to the agreement observed for Data A–D; this applies (almost)
Table 9
Comparison between the measured sway variance and the estimated sway variances on the basis <strong>of</strong> the wave
spectrum estimated from the responses {heave, roll, pitch}
Data Sway ð10 2 m 2 Þ
U.D. Nielsen / Marine Structures 19 (2006) 33–69
Measured BAY OPT WAV
A 2:44 11:5 (376%) 4.89 (101%) 8.22 (241%)
B 3:17 11.5 (263%) 3.95 (27%) 6.39 (105%)
C 1:25 11.2 (872%) 3.42 (198%) 11.2 (873%)
D 2:35 13.2 (470%) 2.59 (12%) 3.04 (31%)
E 1:53 0.82 ( 46%) 0.28 ( 81%) 1.81 (20%)
F 20:9 23.4 (17%) 20.4 (2.1%) 20.8 (3.7%)
G 19:2 17.1 ( 10%) 9.07 ( 52%) 5.35 ( 72%)
H 24:1 13.5 ( 45%) 8.84 ( 64%) 7.04 ( 71%)
I 124 31.2 ( 75%) 20.7 ( 83%) 40.9 ( 67%)
Deviation relative to measured variance shown in parenthesis.

consistently to all the estimation procedures. Keeping in mind the severity <strong>of</strong> the sea states
for the individual sets <strong>of</strong> data, it is somewhat peculiar that the best agreement is found for
Data E–I.
The prediction <strong>of</strong> responses is <strong>of</strong> vital importance in the specific area <strong>of</strong> decision support
for safe navigation <strong>of</strong> ships which means that the above considerations on the prediction <strong>of</strong>
sway does not as such belong to the objective, or the topic, <strong>of</strong> the present work; the
estimation <strong>of</strong> wave spectra. However, it is interesting to see that the combined use <strong>of</strong> a
wave spectrum and a transfer function leads to relatively different results compared to
what is measured, although the estimated directional wave spectrum in the overall sense,
including the underlying wave parameters, is more or less similar in the cases, when the
spectrum is estimated from {heave, roll, pitch} and from {heave, sway, pitch}, cf. Table 5.
This fact illustrates the importance <strong>of</strong> reliable (complex-valued) transfer functions.
7.4. Bayesian vs. parametric modelling
Based on the discussions in the preceding sections on wave parameters and response
measurements, it is not straight forward to conclude, which estimation procedure is the
(most) correct. To draw such a conclusion knowledge <strong>of</strong> the absolute true wave spectrum
would be needed. Though, it should be noted that, in general, the Parametric modelling
and the Bayesian modelling method yield results which are in good agreement with each
other. Moreover, the results are comparable with the estimations by WAVEX, so that
the directional distribution <strong>of</strong> energy is nearly similar as well are the underlying wave
parameters. In the view <strong>of</strong> the produced results it is, therefore, difficult to propose the
Bayesian method in favour <strong>of</strong> the Parametric method, or vice versa. There are, however,
some general points which should also be addressed to get a more subtle picture when
judging the two modelling procedures.
For the analysed data sets, the computational time were always <strong>of</strong> the longest duration
for the Parametric method; in the order <strong>of</strong> 10 min compared to the order <strong>of</strong> 5 min for the
Bayesian method (with MatLab 7.0 on a Pentium 2.0 GHz processor).
The initial search basin is paramount and must be detailed for the Parametric method,
when a genetic search algorithm is not applied.
The number <strong>of</strong> spectral peaks is limited (to a certain number) in the Parametric method,
and in the event <strong>of</strong> strong frequency/directional overlap the individual spectral peaks
may not be ‘‘caught’’.
As regards the ABIC criterion in the Bayesian method, problems have been experienced
in the sense that it appears that the best solution was not obtained for the minimum
value <strong>of</strong> ABIC in certain cases; ABIC kept decreasing for decreasing hyperparameter
until the solution stopped converging. In such cases the ‘optimal’ solution must be
chosen manually.
8. Conclusions
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 63
The present work has described two methods to estimate the directional wave
energy spectrum on the basis <strong>of</strong> measured ship responses. The two concepts dealt with
are a parametric and a non-parametric method. The Parametric method is based on a
summation <strong>of</strong> a number <strong>of</strong> parameterised wave spectra, whereas the non-parametric

64
method, denoted the Bayesian method, estimates the directional wave spectrum in a
number <strong>of</strong> discretised points <strong>of</strong> the wave field, which is divided into a set <strong>of</strong> frequencies
and, as well, a set <strong>of</strong> directions. For both methods, linear spectral analysis is applied to the
motion measurements to set up equations from which the wave energy spectrum may be
determined.
The overall conclusions <strong>of</strong> the present work are
The Bayesian and the Parametric estimation method are capable <strong>of</strong> estimating
directional wave spectra from measured ship responses.
It is paramount that at least one <strong>of</strong> the responses has an asymmetric frequency response
function with respect to port/starboard entering waves. It is understood here that
complex-valued transfer functions are utilised, so that the amplitudes as well as the
phases <strong>of</strong> the responses are reflected by the frequency response function.
The uncertainty related to the transfer functions due to different physical phenomena is
not included in the methodologies.
The present analysis did not suggest to use the roll response in favour <strong>of</strong> the sway
response, or vice versa. However, in general, it may be noted that the sway response has
properties which may be preferred compared to the properties <strong>of</strong> the roll response, e.g.
Tannuri et al. [1].
Reasonable agreements have been observed for the Bayesian and the Parametric
estimation method when estimated wave parameters and directional distributions <strong>of</strong>
energy have been compared with results from a wave radar system.
It seems difficult to propose the one ship response-based method in favour <strong>of</strong> the other,
since they perform almost equally well as regards results (for the specific full-scale data
studied here). However, the Parametric method is the most time consuming (due to the
non-linear programming problem) and problems with frequency/directional overlap <strong>of</strong>
spectral peaks may influence the method badly. On the other hand, the Parametric
method yields a solution which, by definition, is smooth. In contrast, the smoothness
<strong>of</strong> the Bayesian method depends on the hyperparameter, <strong>of</strong> which the optimal
value is determined by the ABIC criterion and, in this relation, problems have been
encountered.
Acknowledgements
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
The author heartily thanks Associate Pr<strong>of</strong>essor Toshio Iseki, Tokyo University <strong>of</strong>
Marine Science and Technology, for his great willingness to share knowledge <strong>of</strong> related
research subjects. Moreover, the assistance and inspiration provided by Pr<strong>of</strong>essor Jørgen
Juncher Jensen, Technical University <strong>of</strong> Denmark, is highly appreciated.
The analysed full-scale data has been provided by Det Norske Veritas (DNV) and the
author would like to thank Dr. Bo Cerup Simonsen and Mr. Øyvind Lund-Johansen for
making the data available and for valuable discussions. Posthumous thanks go to
Mr. Andreas Aschim, DNV, for his work on the complex-valued transfer functions.
During the work, a stay at DNV was organised and for this, the European Network <strong>of</strong>
Excellence MARSTRUCT should be acknowledged for the financial support.
Finally, it should be mentioned that the overall work, which has lead to the paper, has
been funded by the Eureka project E!2097 MONITUS.

Appendix A. Matrix formulation <strong>of</strong> prior distribution
The matrix D is composed <strong>of</strong> elements corresponding to (3.3)–(3.5). From the
organisation <strong>of</strong> x, D has the form
2 3
D1
6 7
D ¼ 4 D2 5; sizeðDÞ ¼ð2 K MÞ ðK MÞ. (A.1)
D3
Here D1 is composed as, cf. (3.3)
2
D1 6 0
6
D1 ¼ 6
4
0
D1 . . .
0
3
7
7;
7
5
sizeðD1Þ ¼ðK MÞ ðK MÞ (A.2)
0 D1 with
2 1 0 0 0 1
D1 ¼
1
0
. 2
1
1
2
0
1
. . .
0
0
0
0
. 2
3
6
4
7
7;
7
5
sizeðD1Þ¼K K. (A.3)
1 0 0 0 1 2
Regarding D2 this is determined from the following, cf. (3.4)
2
1 2 1 0
3
0
6 0
D2 ¼ 6
4
1 2 1
.
. .
7
5
0 1 2 1
; sizeðD2Þ ¼ðK M 2KÞ ðK MÞ, (A.4)
where
2
1 0
3
0
2
1 0
3
0
6 0
6
1 ¼ 6
4
1
.
. .
7
7;
7
5
2 ¼
6 0
6
2 6
4
1
.
. .
7
7,
7
5
ðA:5Þ
0 1
0 1
sizeðÞ¼size 1 ð 2Þ
¼ K K.
D3 is given by (3.5)
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 65
D3 ¼
1
0
0
0
0
0
0
1 ; sizeðD3Þ ¼2K ðK MÞ (A.6)
with 1 being the K K identity matrix as above.

66
Concerning the vector c in kDx ck2 2 3
0
, this is given as, cf. (3.5),
6 . 7
6 . 7
6 7
6
. 7
6 . 7
6 7
6 7
c ¼ 6 0 7;
6 7
6 s0
7
6 . 7
4 . 5
sizeðcÞ ¼ð2 K MÞ 1, (A.7)
s0
ARTICLE IN PRESS
where s0 fills the same number <strong>of</strong> rows which follows from D3.
Appendix B. Directional wave spectra—roll replaced by Sway
270
The estimated wave spectra for Data A–I are given in Figs. B1–B5.
330
BAYESIAN
0 0.4
30
0.3
300
0.2
0.1
60
270
240
300
240
210
180
BAYESIAN
0 0.4
330
30
0.3
0.2
0.1
150
90 270
120
60
90 270
120
U.D. Nielsen / Marine Structures 19 (2006) 33–69
300
240
300
240
330
210
OPTIMISATION
0.4
30
0.3
(A) (B)
OPTIMISATION
0 0.4
330
30
0.3
210
150
210
150
210
(C) 180
180
(D)
0
180
0.2
0.1
0.2
0.1
150
60
60
90
120
90
120
270
300
240
BAYESIAN
0 0.4
330
30
0.3
180
0.2
0.1
150
60
90 270
120
300
240
OPTIMISATION
0 0.4
330
30
0.3
Fig. B2. The estimated wave spectra <strong>of</strong> Data C and D. Based on the responses {heave, sway, pitch}. The ship
course is 0 and the waves are shown as approaching.
270
300
240
330
210
BAYESIAN
0
180
0.4
30
0.3
0.2
0.1
150
60
90 270
120
300
240
330
210
OPTIMISATION
210
0.2
0.1
180
0.4
30
0.3
Fig. B1. The estimated wave spectra <strong>of</strong> Data A and B. Based on the responses {heave, sway, pitch}. The ship
course is 0 and the waves are shown as approaching.
0
180
0.2
0.1
150
150
60
90
120
60
90
120

270
300
240
330
BAYESIAN
0
0.4
30
0.3
0.2
0.1
Appendix C. Measured response spectra
60
90 270
120
300
240
330
ARTICLE IN PRESS
OPTIMISATION
0.4
30
0.3
210
150
210
150
210
(E) 180
180
(F)
0
0.2
0.1
The measured response spectra for Data A–I are given in Figs. C1–C3.
60
90
120
270
300
240
330
BAYESIAN
0
180
0.4
30
0.3
0.2
0.1
150
60
90 270
120
300
240
330
210
OPTIMISATION
0
180
0.4
30
0.3
Fig. B3. The estimated wave spectra <strong>of</strong> Data E and F. Based on the responses {heave, sway, pitch}. The ship
course is 0 and the waves are shown as approaching.
330
BAYESIAN
0 0.4
30
0.3
300
0.2
0.1
60
270
240
210
180
150
90 270
120
300
240
330
210
OPTIMISATION
0.4
30
0.3
(G) (H)
0
180
0.2
0.1
150
60
90
120
270
300
240
330
210
BAYESIAN
0
180
0.4
30
0.3
0.2
0.1
150
60
90 270
120
300
240
330
210
0.2
0.1
0.2
0.1
150
OPTIMISATION
0.4
30
0.3
Fig. B4. The estimated wave spectra <strong>of</strong> Data G and H. Based on the responses {heave, sway, pitch}. The ship
course is 0 and the waves are shown as approaching.
270
300
240
330
210
U.D. Nielsen / Marine Structures 19 (2006) 33–69 67
BAYESIAN
0
180
0.4
30
0.3
0.2
0.1
150
60
120
90
270
300
240
330
210
OPTIMISATION
0.1
0.4 30
Fig. B5. The estimated wave spectra <strong>of</strong> Data I. Based on the responses {heave, sway, pitch}. The ship course is 0
and the waves are shown as approaching.
0
180
0.3
0.2
150
60
120
0
180
90
150
60
90
120
60
90
120

68
References
ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69
(A) (B) (C)
Fig. C1. The measured response spectra <strong>of</strong> {heave, roll, pitch} for Data A, B and C.
(D) (E) (F)
Fig. C2. The measured response spectra <strong>of</strong> {heave, roll, pitch} for Data D, E and F.
(G) (H) (I)
Fig. C3. The measured response spectra <strong>of</strong> {heave, roll, pitch} for Data G, H and I.
[1] Tannuri EA, Sparano JV, Simos AN, Da Cruz JJ. Estimating directional wave spectrum based on stationary
ship motion measurements. Appl Ocean Res 2003;25:243–61.
[2] Hua J, Palmquist M. Wave estimation through ship motion measurement. Technical Report, Naval
Architecture, Department <strong>of</strong> Vehicle Engineering, Royal Institute <strong>of</strong> Technology; 1994.
[3] Aschehoug M. Scientific paper on the sea state estimation methodology. Technical Report, SIREHNA,
France; 2003 [Paper prepared in the HullMonþ project].

ARTICLE IN PRESS
U.D. Nielsen / Marine Structures 19 (2006) 33–69 69
[4] Iseki T, Ohtsu K. Bayesian estimation <strong>of</strong> directional wave spectra based on ship motions. Control Eng Pract
2000;8:215–9.
[5] Iseki T, Terada D. Bayesian estimation <strong>of</strong> directional wave spectra for ship guidance systems. Int J Offshore
Polar Eng 2002;12:25–30.
[6] Waals OJ, Aalbers AB, Pinkster JA. Maximum likelihood method as a means to estimate the directional
wave spectrum and the mean wave drift force on a dynamically positioned vessel. Proceedings <strong>of</strong>
OMAE2002, Oslo, Norway; 2002.
[7] Isobe M, Kondo K, Horikawa K. Extension <strong>of</strong> MLM for estimating directional wave spectrum. Proceedings
<strong>of</strong> symposium on description and modeling <strong>of</strong> directional seas, vol. A-6; 1984.
[8] Nielsen UD. Estimation <strong>of</strong> directional wave spectra from measured ship responses. PhD thesis, Section <strong>of</strong>
Coastal, Maritime and Structural Engineering, Department <strong>of</strong> Mechanical Engineering, Technical University
<strong>of</strong> Denmark; May 2005.
[9] Pascoal R, Soares CG, Sørensen AJ. Ocean wave spectral estimation using vessel wave frequency motions.
Proceedings <strong>of</strong> OMAE2005, Halkidiki, Greece; 2005.
[10] Benoit M, Goasguen G. Comparative evaluation <strong>of</strong> directional wave analysis techniques applied to field
measurements. Proceedings <strong>of</strong> ninth international <strong>of</strong>fshore and polar engineering conference, Brest, France;
1999.
[11] Akaike H, Nakagawa T. Statistical analysis and control <strong>of</strong> dynamic systems. Tokyo: KTK Scientific
Publishers; 1988.
[12] Data recorded in Tateyama Bay, Japan. The data has kindly been <strong>of</strong>fered by the Tokyo University <strong>of</strong> Marine
Science and Technology. The data originates from the research and training ship Shioji-Maru.
[13] Bhattacharyya R. Dynamics <strong>of</strong> marine vehicles. New York: Wiley; 1978.
[14] Dimri V. Deconvolution and inverse theory: application to geophysical problems. Amsterdam: Elsevier;
1992.
[15] Tarantola A. Inverse problem theory: methods for data fitting and model parameter estimation. Amsterdam:
Elsevier; 1987.
[16] Herbers THC, Guza RT. Estimation <strong>of</strong> directional wave spectra from multicomponent observations. J Phys
Oceanogr 1990;20:1703–24.
[17] Akaike H. Likelihood and Bayes procedure. In: Bernado JM, De Groot MH, Lindley DU, Smith AFM,
editors. Bayesian Statistics. Valencia: University Press; 1980. p. 143–66.
[18] Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes in FORTRAN77: the art <strong>of</strong>
scientific computing, 2nd ed. Cambridge: Cambridge University Press; 1992.
[19] Goda Y. Random seas and design <strong>of</strong> maritime structures. Advanced series on ocean engineering, vol. 15.
Singapore: World Scientific; 2000.
[20] Longuet-Higgins MS, Cartwright DE, Smith ND. Observations <strong>of</strong> the directional spectrum <strong>of</strong> sea waves
using the motions <strong>of</strong> a floating buoy. Ocean Wave Spectra 1961; 111–36.
[21] Hogben N, Cobb FC. Parametric modelling <strong>of</strong> directional wave spectra. Proceedings <strong>of</strong> 18th <strong>of</strong>fshore
technology conference, Houston, Texas 1986.
[22] Mitsuyasu H, Tasai F, Suhara T, Mizuno S, Onkusu M, Honda T, et al. Observation <strong>of</strong> the directional
spectrum <strong>of</strong> ocean waves using a cloverleaf buoy. J Phys Oceanogr 1975;5:750–60.
[23] Price WG, Bishop RED. Probalistic theory <strong>of</strong> ship dynamics. London: Chapman & Hall; 1974.
[24] Bendat JS, Piersol AG. Random data—analysis and measurement procedures, 3rd ed. New York: Wiley;
2000.
[25] Jensen JJ. Load and global response <strong>of</strong> ships. Elsevier ocean engineering book series, vol. 4. Amsterdam:
Elsevier; 2001.
[26] WAVEX—principles <strong>of</strong> operation. Technical documentation by MIROS; 1998.
[27] Neumaier A, Schneider T. Estimation <strong>of</strong> parameters and eigenmodes <strong>of</strong> multivariate autoregressive models.
ACM Trans Math S<strong>of</strong>tware 2001;27(1):27–57.
[28] Lloyd ARJM. Seakeeping, 2nd ed. Chichester: Ellis Horwood; 1998.