Kalman filtering of vessel motions for ocean wave directional ...

Kalman filtering of vessel motions for ocean wave directional 

spectrum estimation 

Ricardo Pascoal, C. Guedes Soares 

Centre for Marine Technology and Engineering (CENTEC), Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal 

article info 

Article history: 

Received 20 May 2008 

Accepted 18 January 2009 

Available online 31 January 2009 

Keywords: 

Harmonic detection 

Kalman filtering 

Spectral estimation 

Hydrodynamics 

1. Introduction 

abstract 

Spectral estimation using, as data, the measurements from 

inexpensive shipborne sensors has become a topic of renewed 

interest. Publications have already dealt with several approaches 

to solve this estimation problem, which is recognized to be an 

important one for advisory support, mission planning, guidance 

and control systems. 

Most of the previously published procedures estimate the 

wave spectral densities by minimizing an error measure built 

upon cross-spectral densities. Spectral densities are, in this case, 

available from two sources. One source is to estimate them 

directly from the data and the second source is to calculate them 

using the estimated wave spectrum. 

Generally, the formulations for estimation may be classified as 

parametric and non-parametric, depending on whether or not a 

parametric formula exists to constrain the spectral shape. Under 

these classifications there exist various methods and there are 

also several procedures in use to estimate the data cross-spectral 

densities from the motion data. These procedures may be 

generally classified as based on fast Fourier transform (FFT) and 

Auto-Regressive Moving Average. If the formulation is nonparametric, 

constraints on the smoothness of the estimate are 

applied by invoking prior distributions and Bayesian theory 

(Tannuri et al., 2001; Nielsen, 2005; Saito et al., 2000; Iseki and 

Ohtsu, 2000) or directly through a functional (Pascoal et al., 2007; 

Pascoal and Guedes Soares, 2008). There are also cases without 

Corresponding author. 

E-mail address: guedess@mar.ist.utl.pt (C. Guedes Soares). 

0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.oceaneng.2009.01.013 

ARTICLE IN PRESS 

Ocean Engineering 36 (2009) 477–488 

Contents lists available at ScienceDirect 

Ocean Engineering 

journal homepage: www.elsevier.com/locate/oceaneng 

This paper proposes a high-speed iterative procedure for estimating the ocean wave directional 

spectrum from vessel motion data. It uses as input data, the measurements from motion sensors that 

are commonly available on dynamically positioned vessels and which may easily be installed on any 

ship. Because the necessary sensors are relatively inexpensive or may already be installed, it becomes an 

ideal solution to provide initial estimates to offline estimation procedures and to give spectral updates 

under quickly changing weather conditions. The Kalman filtering algorithm, for iterative harmonic 

detection, and frequency domain vessel response data are used in the estimation procedure. The results 

and conclusions are still based on synthesized data, but very promising. 

& 2009 Elsevier Ltd. All rights reserved. 

applied smoothing, such as Waals et al. (2002), and which are 

closely related to the present work. 

In case the non-parametric formulation is used, then spectral 

density positiveness constraint must be applied to the spectral 

amplitudes. Publications which invoke Bayesian theory, to date, 

have enforced the constraint through the transformation e x , with x 

being found from the minimization. Those which use the error 

functional enforce this directly as a domain constraint or 

alternatively as x 2 . The reason for the latter transformation is to 

reduce the size of the search domain from the whole real line to 

just positive or negative real values. 

Presently the computational bottleneck is the error minimization 

procedure, which requires iterative methods. Furthermore, 

depending on the necessary frequency resolution and crossspectral 

estimation procedure, several minutes of data may be 

required to provide stable results. Though time-varying vector 

auto-regression has been proposed in Iseki and Terada (2003) for 

real-time estimation of motion cross spectra, the bottleneck 

persists since estimation currently takes from 3 to 8 min (Pascoal 

et al., 2007; Nielsen, 2006). This is provided that the vessel has not 

pulled too fast a maneuver, because this may jeopardize the FFT 

based estimates. Because, for most purposes, a sea state is taken 

to be stationary for periods of 20 min, providing estimates in 

3–8 min can be considered real-time with respect to the sea state 

parameters. On the other hand, if there is a wish to know about 

single wave information or to update information under fast 

changing vessel heading or weather, then updates need to be 

made available at shorter intervals. 

The effort here has been to propose a spectral estimation 

procedure that presents some important benefits when compared 

to existing ones, the most outstanding ones are with respect to

478 

Nomenclature 

conc(a, u) concatenates a, u times (used for building large 

vectors from a stencil vector) 

diag u ( ) takes the diagonal from a matrix into vector form and 

vice-versa, u superscript indicates number of times 

the operator is repeated 

E[ ] expectation operator 

Hjlm complex-valued transfer function for frequency j, 

heading l and sensor, m 

Hs significant wave height calculated from zeroeth 

I 

spectral moment 

identity matrix 

number of headings in the discretization 

ny 

speed of estimation and fusing data from several sources. This 

work is a consequence of the objectives that have been defined in 

a previous work Pascoal et al. (2007), to further study the zero 

speed case and produce a faster spectral estimator. In the previous 

work the reduction of computational time has been achieved 

mostly at the cost of eliminating the frequency smoothing 

constraint. Here this is taken to further extent and a prototype, 

high speed, Kalman filter-based wave spectral estimator is 

proposed. 

The procedure may be termed faster than real-time because it 

can provide many spectral estimates before sea state stationarity 

ceases to be a good hypothesis. An update is available once every 

five seconds on a 2.4 GHz Dual Core processor running National 

Instruments s LabVIEW Real-Time engine. In this setup, the 

formulation is non-parametric and a major difference with respect 

to all other existing formulations is that cross-spectral calculations 

are not performed and the wave amplitudes are estimated 

directly. 

Currently no frequency or directional smoothing constraints 

exist, but since they can be cast into linear form through use of 

finite differences, it is possible that the same filter may be used at 

the cost of increased CPU time. The presented formulation is for 

zero speed of advance. Since sensors perceive the encounter 

frequency, for a given value of speed of advance the equations 

must be adapted in order to take this frequency shift into account. 

This is not considered in the scope of this paper and though it may 

appear simple, in practice, as presented in Nielsen (2007), it may 

actually turn out to be a very difficult task. 

The results with synthesized data are very promising; however, 

more research is needed to verify and improve the present 

proposal, including comparisons with model test data and field 

data. 

Previously existing proposals of spectral estimation procedures 

which use vessel motion records are not intended to take into 

account the merging of data from additional sources, such as 

available radar spectral estimates or other sensor data. The 

procedures based on Bayesian models were up to now the ones 

closest to providing such capability, because the probabilistic 

setting enables designers to specify measures of trust and 

variability along with rules to adapt the initial estimates. 

The Kalman filter framework lends itself to the possibility of 

including data from many sources and it may be found to improve 

the estimation capabilities at hand. 

2. Formulation 

Spectral estimation has been performed using Kalman filter on 

a rotating reference frame. This type of reference frame and 


R. Pascoal, C. Guedes Soares / Ocean Engineering 36 (2009) 477–488 

nf nr 

rjk Tp 

number of harmonics to estimate 

number of measured responses 

measurement from jth sensor at instant k 

spectral peak period 

X0 superscript as written indicates transpose 

X states, in this case the real and imaginary parts of 

wave amplitude 

dtt Kronecker delta 

xk state white noise vector at the kth instant 

Xt state noise covariance matrix 

yk measurement noise vector at the kth instant 

Yt measurement noise covariance matrix 

¯y 

P 

t 

mean direction input to the spreading function 

estimation error covariance matrix 

filtering is used to improve harmonic suppression in electrical 

networks, through active circuit designs (Liu, 1998; Rechka et al., 

2003). In order to import these procedures to wave spectral 

estimation using vessel motions, the vessel’s hydrodynamics have 

to be considered and the measurement equations augmented 

accordingly. 

Knowledge about the frequency domain vessel responses is 

assumed here, as in all prior publications, to be the most accurate 

which can be made available in the form of first-order transfer 

functions. 

The incident waves’ in phase and quadrature components are 

the state variables and thus there will be intrinsic information 

about the phase angles. For each frequency there exist ny 2 

states. The hypothesis is that the states have constant or very 

slowly varying values when compared to the filter dynamics, and 

thus the transition matrix is identity. Consequently (e.g. Liu, 1998; 

Rechka et al., 2003), the state equation is 

Xkþ1 ¼ Xk þ xk, (1) 

i.e. the state at the next time instant, Xk+1, is the same as the 

present (very slow variation), Xk, except possibly for some 

variation due to the state noise xk. The measurement equation, 

for each sensor, m, is supposed to model what happens at the 

available sensors and is of the form (Liu, 1998; Rechka et al., 2003) 

rmk ¼ Cmk Xk þ yk, (2) 

where rmk is an available response at the kth time instant, Cmk 

the measurement matrix for the mth sensor and yk the measurement 

noise. The measurement matrix is responsible for taking 

the states and producing a response (output) that is to be 

compared with one which is available for measurement, while 

the measurement noise corresponds to the sensor’s limited 

capabilities. 

This formulation is possible because a response can be 

approximated by using a finite number of harmonics, each of 

which has amplitude determined from the complex wave 

amplitude and adequate transfer function. Any of the responses 

may be obtained from the Fourier representation 

r ¼ Re Xn f Xn y 

Hjm ðX2j 1;m þ ffiffiffiffiffiffiffi p 

1X2j;mÞðcosðojtÞþ 

ffiffiffiffiffiffiffi 

0 

1 

p 

@ 

1 sinðojtÞÞA, 

j¼1 m¼1 

which, after collecting states, becomes 

r ¼ Xn f 

Xn y 

ðReðHjmÞ cosðotÞ ImðHjmÞ sinðotÞÞ X2j 1;m 

j¼1 m¼1 

X n f 

Xn y 

ðImðHjmÞ cosðotÞþReðHjmÞ sinðotÞÞ X2j;m. (3) 

j¼1 m¼1

The Re(Hjm) stands for real part of the transfer function from 

wave to the desired response, indexed for each frequency and 

direction. 

The summation in Eq. (3) can be conveniently cast into an 

augmented matrix vector product. The matrix will be the timevarying 

measurement matrix for discrete time and all the states 

are cast into a single vector. The matrix is understood to be one 

line for each sensor, containing all frequencies and headings. This 

matrix has dimension nr by ny 2 nf. 

Because the measurement matrix includes the vessel response 

functions, it is particular to the proposed procedure. In Eq. (4), a 

part of this matrix is shown for the jth frequency, kth time instant, 

lth response and mth direction, respectively. To build the full 

matrix, these submatrices have to be concatenated, leaving k as 

the only free index. 

C jklm ¼ ReðH jlmÞ cosðo jkTÞ ImðH jlmÞ sinðo jkTÞ 

ImðH jlmÞ cosðo jkTÞ ReðH jlmÞ sinðo jkTÞ 0 . (4) 

Any errors in the hydrodynamics, unmodeled dynamics and 

actual sensor noises have been collected into the measurement 

noise term. This term should posses the properties of being 

Gaussian, zero mean white noise and of being uncorrelated with 

the state noise. Because of the complexity of the real ocean and 

vessel responses, such assumptions are hardly possible to verify 

beforehand and only a real application can point out if there will 

be additional problems to solve. 

It is very difficult in this case, if not impossible, to determine 

beforehand the ideal size of the measurement vector. This 

happens because the measurement matrix is time-varying and 

there is no control over the excitation imposed by the environment. 

One certain thing is that, due to kinematic constraints, the 

number of independent variables that can be measured is much 

less than the number of states to be estimated. Thus, if one does 

not take adequate action, this leads to singular or badly 

conditioned observability matrix, in which case the state 

estimates will either fail to be determined or will be very erratic 

and without physical meaning. The ability to determine correct 

transfer functions has an influence on this as well (e.g. Nielsen, 

2007) and some are more sensitive to modeling error than others. 

In order to stabilize the state estimates, and increase the rank 

of the observability matrix, the measurement equation for a given 

instant is built upon measurements of the current time step and 

additional lagged measurements. The number of time instants to 

consider depends on the number of headings and frequencies 

under estimation, and should be tuned, in the field, for a particular 

vessel and sensor locations. 

The covariance matrices are conventional and written as E[xt, 

xt] ¼ Xtdtt for the state noise, E[yt, yt] ¼ Ytdtt for the measurement 

noise, and E[Xˆ t Xt, (Xˆ t Xt) 0 ] ¼ P t for the state error. The 

capital Greek letters stand for a matrix and dtt for the Kronecker 

delta. The delta being zero unless subscripts are equal, in which 

Table 1 

Simulated sea states and their identification for further reference. 


R. Pascoal, C. Guedes Soares / Ocean Engineering 36 (2009) 477–488 479 

case it becomes unity. These matrices will include a minus sign 

superscript, as in P t , whenever they belong to the prediction 

cycle, i.e. when only prior information is available. 

The standard prediction and update cycles are then written as 

(e.g. Liu, 1998; Rechka et al., 2003; Ding et al., 2006) 

K k ¼ S k C 0 kðC kS k C 0 k þ YÞ 1 , (5) 

^X k ¼ ^ X k þ K kðr k C k ^ Xk Þ, (6) 

S k ¼ðI K kC kÞS k ðI K kC kÞ 0 þ KYK 0 , (7) 

^X kþ1 ¼ ^ X k, (8) 

S kþ1 ¼ S k þ X. (9) 

The circumflex, as in Xˆ, stands for estimated values. K k is the 

filter gain and all variables have been defined in the preceding 

text. This gain is responsible for weighing the estimation errors 

into the new state estimates and, in an idealized situation, the 

above procedure is optimal with respect to any well posed 

optimality criterion (e.g. Tse and Athans, 1967) 

These standard Kalman filter equations have been tested. It 

was found, however, that they do not suit the present application. 

They assume good knowledge of the process and that 

steady state is reached after some time (the filter drops off 

and ceases to perform), but neither is the process completely 

known nor steady state guaranteed due to changes in vessel 

heading and sea state. Some type of adaptive capability has to be 

considered. 

Changing the filter gain in a sensible manner is the key to be 

optimized. A way to change the filter gains is by updating the 

measurement noise covariance matrix. Alternatively the state 

noise covariance may be changed, or even both. These options and 

the way to go about changing, or adapting, the estimates appear in 

several publications (e.g. Liu, 1998; Ding et al., 2006). 

The option chosen here has been to estimate the measurement 

noise covariance through the use of ensemble averages. Although 

other options were tested it is probably better, as a first step, to 

change only one of the matrices and that this matrix be the 

measurement noise. The reason for this is that the solution to this 

Table 2 

Added noise standard deviations. 

Sea state # Hs swell (m) Hs sea (m) Tp swell (s) Tp sea (s) ¯y swell (deg.) ¯y sea (deg.) 

1 5 0 15 – 45 – 

2 5 0 15 – 90 – 

3 5 0 15 – 180 – 

4 5 2 15 10 180 45 

5 0 2 – 10 – 45 

6 0 2 – 7 – 45 

7 0 2 – 7 – 180 

x¨ 2 CG 

x¨ 3 CG 

x 4 x 5 x 6 x¨ 3 bow 

s 0.18 0.18 0.016 0.016 0.016 0.18 0.18 

x¨ 3 stbd

480 

problem is constant at steady state. Additionally, to reduce some 

high-frequency behavior presented by the estimates, state 

averages have been established between three time instants. 

Possibly this filtering may also be introduced through an 

augmented Kalman filter. The ensemble for covariance estimation 

is composed of data from twenty time instants. 

Notice that, because instability was arising if all values were 

considered, only the diagonal terms, i.e. auto-correlation, have 

been kept. The main reason for instability of the state estimates 

during the adaptive stage was found to originate at the 

measurement covariance update. Stability of correlation estimates 

largely depends on the size of the estimation window, large 

windows being best but at conflict with the real-time requirement 

(e.g. Ding et al., 2006), and it was found that cross-correlation 

estimates were very unstable whilst auto-correlation presented a 

nice behavior. 

By eliminating the off-diagonal terms, the filter is informed 

that measurement noise is not correlated. It is hard to say if this 

will hold in practice, but at this point, due to problem complexity, 

the heuristic approach was chosen. The revised part of the 

formulation is as follows: 

^X kþ1 ¼ 1=3 ð^ Xk þ ^ Xk 1 þ ^ Xk 2Þ (10) 

Generated Spectral Contours, θ m = 45 45° 

N 

S 

0.3Hz 

0.2Hz 

0.1Hz 


W E 

Generated 

Hs = 5m; 0m 

Tp = 15s; 10s 


Estimated Shape at Time instant, t = 100 s 

N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

^Y ¼ diag 2 ðE½r k C k ^ Xk ; ðr k C k ^ Xk Þ 0 ŠÞ þ C kP k C 0 k (11) 

3. Sample application 

A typical 70-m-long vessel is considered. As already mentioned 

in most publications addressing the topic (e.g. Tannuri et al., 2001; 

Nielsen, 2005; Pascoal et al., 2007), vessel length is the single 

most important geometric characteristic. It dictates the frequency 

up to which head waves may be sensed with good accuracy, 

because the zero of heave transfer function occurs for a frequency 

corresponding closely to that of a wave with length equal to ship 

length (Pascoal et al., 2007) (deep water). In this case it is 

expected that the frequency at which problems arise is 0.94 rad/s, 

corresponding to a period of around 7 s. 

In this application, measured signals were synthesized through 

conventional use of FFT, vessel transfer functions and JONSWAP 

spectral density with cos 2s (same as in Pascoal et al., 2007). The 

spectral peak intensification factor, g, and spreading factor, s, have 

been kept constant at a value of 2.0 (as in Pascoal et al., 2007). 

Other parameters are identified in Table 1. In this table, Hs stands 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

Fig. 1. Generated and evolution of estimated contour lines sea state 1. 

E

for significant wave height, Tp is the spectral peak period, ¯y is the 

mean direction and the additional terms ‘‘swell’’ and ‘‘sea’’ are 

with respect to the corresponding component of a single- or a 

double-peaked wave spectrum (Guedes Soares, 1984). The sea 

state # is used in the figure captions. 

A total of 10 min were generated at a sampling frequency of 

5 Hz and then downsampled to 1 Hz. This allows for synthesized 

low- and high-frequency content, which will become noise due to 

the downsampling. In all presented results, a total of seven time 

steps are considered at a given instant to build the measured 

signals. The time interval between these signals corresponds to a 

lag of two samples, thus the estimator may only start at time 14 s 

from data acquisition startup, at which point the measurement 

vector is completely available. 

Transfer functions have been calculated using a linear strip 

theory (Fathi and Hoff, 2004) but any source of hydrodynamic data 

which provides good estimates is applicable. The calculated 

transfer functions are aimed at simulating a technically simple 

and plausible sensor installation. These are sway and heave 

acceleration at the center of gravity, roll, pitch and yaw motion, 


Generated 

Hs = 5m; 0m 

Tp = 15s; 10s 

N 

S 

0.1Hz 



0.3Hz 

0.2Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

vertical acceleration at the bow (35 m from midship) and vertical 

acceleration at starboard (5 m from centerline) close to the section 

containing the center of gravity and at the same vertical position of 

the latter. They are identified, respectively, by the symbols x¨ 2 CG , x¨3 CG , 

x4, x5, x6, x¨ 3 bow and x¨3 stbd , with the dot indicating time derivative. The 

transfer functions have been calculated at 301 intervals and 

frequencies from 0.1 to 3.6 rad/s at 0.07 rad/s intervals. These 

values have been interpolated and extrapolated in order to provide 

synthesized signals which do not repeat in the 10 min at the rate of 

5 Hz, but for estimation purposes only values between 0.1 and 

1.15 rad/s have been used, which means that everything else will 

actually become part of the induced noise. This results in estimates 

at 12 headings and 15 frequencies, totaling 360 state variables. 

Measured motions from advanced sensor boxes are normally 

the result of sensor fusion. There are some commercially available 

sensor packages, inertial and heading reference units, which 

inside have accelerometers, angular rate sensors, flux compass 

and GPS, and provide accurate measurements by fusing data 

through Kalman filters of their own. Depending on available 

sensors, different measurement equations must be considered. If 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

Fig. 2. Generated and evolution of estimated contour lines sea state 2.

482 

the location onboard and type of sensors may be specified, then 

alternative sensor configurations should be analyzed and the best 

one, if any, is chosen. Care should be exercised if prior filtering or 

processing of the signals is needed, for instance to band-pass filter 

or to integrate acceleration and rate measurements, since these 

may jeopardize estimation (Nielsen, 2005). 

Since acceleration transfer functions have a slower roll off than 

motion, i.e. their amplitudes decay slower at high frequency, a 

combination of acceleration and displacement measurements is 

recommended in order to reduce overestimation at high 

frequency and provide good low-frequency estimates. This 

suggestion may produce conflict if good motion measurements 

are not available, which may be the case for heave if GPS readings 

are too noisy and present large biases, so, even if at first adding a 

sensor may sound beneficial, configurations should be tested in 

scenarios closest to reality. 

The initial estimates were maintained throughout and are as 

follows: 

^X 1 ¼ 0 (12) 


Generated 

Hs = 5m; 0m 

Tp = 15s; 10s 

N 

S 

0.3Hz 

0.2Hz 

0.1Hz 




N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

S1 ¼ 2 " # 

i i 0 

0 1ii 2 3 

0:05 

6 

0:05 

7 

6 7 

6 7 

6 0:005 7 

6 7 

6 7 

Y1 ¼ diag conc 6 0:005 7; 

6 7 

6 0:005 7 

6 7 

4 0:05 5 

0:05 

1 

7 nr 

0 0 

11 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

B B 

CC 

@ @ 

AA 

2 

(13) 

(14) 

X1 ¼ 0:1 " 

i 

0 

i 0 

0:6i # 

i 

(15) 

with i being half the length of X. The subscripts i i stand for a 

diagonal matrix which has size i by i, and for which the repeated 

value on the diagonal is the one shown, e.g., 25 5 is a 5 by 5 

matrix whose diagonal is filled with 2s. These initial values have 

been chosen because it is assumed that initially there is no 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


information on the states, there is a wish to update lower 

frequencies faster and high frequencies have more unmodeled 

dynamics and noise. 

3.1. Verification with noisy sensor measurements 

Multiple noisy measurements allow for increased rank of the 

measurement matrix, otherwise the maximum possible rank is 

reduced due to kinematic constraints. If the measurements were 

not noisy, and under small motion assumption, it would make no 

sense to have vertical acceleration measurements at the bow, 

heave acceleration and pitch motion because in that case there is 

linear dependence between the harmonic amplitudes. 

As mentioned, synthesized data has been obtained through 

inverse FFT. Normally distributed random sequences have been 

generated and added to the signals with standard deviations as 

given in Table 2. Notice that, to purposefully test an unfavorable 

situation, they are larger (ffi10 times the variance) than the 

estimated values given to the filter. 

W 

N 

S 



Generated 

Hs = 5m; 2m 

Tp = 15s; 10s 


0.3Hz 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

In Figs. 1–6 are presented the spectral density contour lines 

resulting from estimates of the six simulated sea states. The initial 

three minutes of estimation are presented, this time interval has 

been chosen because that is roughly how long it takes for current 

offline procedures to produce an estimate. Notice again that ten 

minutes have been synthesized such that low-frequency content 

can be included and increased reality obtained, but the filter is run 

only for three minutes. 

Further to the promising cases, there are those about which 

little can be done through use of inexpensive shipborne sensors. 

To prove the point about lack of observability of head waves with 

lengths close to vessel length, sea state 7 has been simulated and 

the very poor results are shown in Fig. 7. It is quite clear that the 

filter prefers to build energy at 1601 and 2101, instead of at 1801. 

Probably this problem may be solved through use of other sensors, 

such as external wave probes (e.g. laser, ultrasound and radar), 

but, as already mentioned by Nielsen (2005), the signal conditioning, 

installation and maintenance becomes much more 

complex and expensive. Furthermore, signals requiring complex 

conditioning may increase uncertainty and introduce false 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


484 

dynamics to a point which makes their use damaging to the 

estimation. 

It is interesting to analyse the evolution of filter gain and the 

error between estimated and generated spectrum. Due to the 

large size of the system, only macroscopic evolution is 

shown, and, in order to do so, the measures given in Eqs. (15) 

and (16) have been calculated. Eq. (15) is the sum of all 

absolute values of the element of the filter gain and thus 

stands for a total gain measure. Eq. (16) is a normalized energy 

density error between generated and estimated values (in the 

form of a Frobenius norm). A typical result of these equations is 

presented in Fig. 8, which corresponds to simulation of sea state 1. 

It is possible to see that the initial total gain is much larger 

than after the first measurement noise covariance update is 

available, indicating that the initial covariance estimates were 

not correct, and stabilizes after some time, as expected 

under stationary conditions. The error, on the other hand, is 

much more dynamic, expressing the energy moving around that 

has been presented through evolution of the contour lines. Since 

there is no smoothing, or energy calibration, it is natural that 


Generated 

Hs = 0m; 2m 

Tp = 15s; 10s 

N 

0.3Hz 

0.2Hz 

0.1Hz 

S 


N 

0.3Hz 



S 

0.2Hz 

0.1Hz 

errors are still large, consequence of peaky, unscaled, energy 

estimates. 

K total 

k ¼ sumðjK kjÞ. (16) 

Error ¼ðsumððSgen SestÞ 2 ÞðsumðS 2 

gen ÞÞ 1 Þ 0:5 

(17) 

Because the focus has been chosen to be on the ability to 

determine the directional and frequency distribution of 

unscaled energy density, parameters of the estimated sea states 

are not determined as standard procedure. If the relative 

distribution is correct and the absolute value is not, then values 

may be scaled to produce the correct cumulative energy in a way 

similar to what is done for radar measurements (Seemann et al., 

1999). 

Examples on what may be done with the raw spectral estimate, 

i.e. the unsmoothed output of the estimator, and to give an 

indication of the parameters that may be extracted from the noisy 

non-parametric form, the procedure proposed in Pascoal and 

Guedes Soares (2008), which consists of parametric fitting, has 


N 

0.3Hz 

0.2Hz 

0.1Hz 

S 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 



Generated 

Hs = 0m; 2m 

Tp = 15s; 7s 


Generated 

Hs = 0m; 2m 

Tp = 15s; 7s 

N 

S 

N 

S 

0.1Hz 


0.3Hz 

0.2Hz 


N 

0.3Hz 

0.3Hz 

0.2Hz 

0.1Hz 


S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

Fig. 6. Generated and evolution of estimated contour lines sea state 6. 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

Fig. 7. Generated and evolution of estimated contour lines sea state 7. Simulation for lack of observability. 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz

486 

Total Gain Measure 

Relative Error [-] 

4000 

3000 

2000 

1000 

Kalman Filter Total Gain Measure 

0 

0 20 40 60 80 100 120 140 160 180 

time [s] 

10 0 

Relative Error Evolution 

10 

0 20 40 60 80 100 120 140 160 180 

-1 

time [s] 

Fig. 8. Evolution of total filter gain and estimation error for sea state 1. 

Generated Power Spectrum, mean θ = 45 45 

W 

W 

Generated 

Hs = 5m; 0m 

Tp = 15s; 7s 

s = 2.0; 2.0 

N 

S 

N 

S 

0.4Hz 

0.2Hz 

Estimated Parameterized Spectrum, mean θ = 50.1 41.9 

Estimated 

Hs = 3.4m; 2.7m 

Tp = 15s; 11s 

s = 2; 2 

0.4Hz 

0.2Hz 



E 

E 

S (ω) [m 2 s] 

S (ω) [m 2 s] 

Fig. 9. Sea state 1 after direct parametric fit of Jonswap spectra. 

been applied. The mentioned procedure consists of fitting 

Jonswap spectra to, at most, two wave systems, and borrows 

from that suggested by Guedes Soares (1984). The result of this is 

presented in Figs. 9 and 10. 

The complete absence of smoothing makes fitting very difficult 

and tends to induce underestimation of the fitted Hs. The Hs error 

is 13% for the unimodal case of Fig. 9 and 30% for the bimodal case 

of Fig. 10, respectively. The value of Hs calculated from the raw 

spectrum of the bimodal case is actually less, at 20%, but the fit is 

in the sense of least-squares on the spectral ordinates (Pascoal 

and Guedes Soares, 2008). 

Peaky raw spectra may also give rise to inexistent peaks which 

become clearly visible in the fitted scalar spectrum. This is 

shown in Fig. 9, in which the generated spectrum is unimodal but 

the estimated is bimodal. The larger estimated peak is at the 

correct frequency, but with lower energy than that of the 

generated spectrum, and a second, non-existent, secondary peak 

is estimated at 0.6 rad/s. This also happens with the other 

estimators that lack smoothing, because there is a tendency to 

concentrate energy at peaks with large peak intensification and 

spreading (explained in Pascoal et al., 2007; Pascoal and Guedes 

Soares, 2008 and clearly visible in the results of Waals et al., 

2002). 

It is also clear that directionality and peak period of the main 

peak are captured to good quantitative extent for single-peaked 

spectra (51 and 0 s errors for the main peak). However, just as 

9 

8 

7 

6 

5 

4 

3 

2 

Generated Scalar Power Spectrum 

Generated 

Hs = 5m; 0m 

Tp = 15s; 7s 

γ = 2.0; 2.0 

1 

0 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 

ω [rad/s] 

7 

6 

5 

4 

3 

2 

1 

Parameterized Scalar Power Spectrum 

Estimated 

Hs = 3.4m; 2.7m 

Tp = 15s; 11s 

γ = 4; 4 

0 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 

ω [rad/s]

Generated Power Spectrum, mean θ = 180 45 

N 0.4Hz 

W 

Generated 

Hs = 5m; 2m 

Tp = 15s; 10s 

s = 2.0; 2.0 

0.2Hz 

reported in Pascoal and Guedes Soares (2008) and mentioned 

herein, absolutely all existing estimators based on ship motions 

pull multiple strong spectral peaks together and produce 1801 

errors (so do radars in the absence of sufficiently good measurements, 

it is an ambiguity but leads to actual errors (Buckley 

(1999)). This is why the parameterized swell peak in Fig. 10, that 

was supposed to be at 1801, has been pulled towards the image of 

the wind sea peak at 2251, leading to a 241 error in directionality. 

There is still no universal solution to this problem but, as 

mentioned, other sensors, increased length of the measurement 

vectors, or heuristic approaches help in particular situations. This 

is, on its own, something worthwhile to investigate for a 

particular vessel. 

A test has also been performed on filter divergence, which is 

not intended to give a full picture because the number of 

interesting parameter combinations is huge. The most difficult 

situation presented herein is the combined spectrum of Fig. 4 

which has been chosen for the test. There is a tendency for 

spurious energy to appear where it really does not exist, the 

question to answer is if this energy grows and leads to filter 

divergence or if it has a behavior which is of no consequence 

to stability. The result of an 18 min simulation is presented in 

Fig. 11. The filter did not diverge but the estimate is quite noisy 

S 

Estimated Parameterized Spectrum, mean θ = 204 53.5 

N 0.4Hz 

W 

Estimated 

Hs = 3m; 2.4m 

Tp = 15s; 9.7s 

s = 1; 1 

S 

0.2Hz 



E 

E 

S (ω) [m 2 s] 

S (ω) [m 2 s] 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 

ω [rad/s] 

5 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

and errors build at 1801 sectors, thus leaving the motive for 

further research. 

4. Conclusions 

Generated Scalar Power Spectrum 

Generated 

Hs = 5m; 2m 

Tp = 15s; 10s 

γ = 2.0; 2.0 

Parameterized Scalar Power Spectrum 

Estimated 

Hs = 3m; 2.4m 

Tp = 15s; 9.7s 

γ = 4; 2.4 

0 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 

ω [rad/s] 

Fig. 10. Sea state 4 after direct parametric fit of Jonswap spectra. 

A new solution for the ocean wave spectral estimation has 

been presented. It uses measured vessel motions and zero speed 

of advance hydrodynamic data. The procedure uses a harmonic 

detection algorithm based on the Kalman filter. 

Tests have been performed using synthesized data and 

very promising results were obtained. Divergence problems were 

not identified under stationary conditions and there are still 

possibilities for improving performance of identifying spectral 

separation, reducing spread and increasing stability of the 

estimates. 

Since for these complex problems it is rare that direct 

application of theory to the real world immediately provides 

the expected results, this procedure needs to be tested 

and improved through use of basin data and later with field 

data.

488 

Generated Spectral Contours, θm = 180 45° 

N 

0.3Hz 

Acknowledgements 

The authors are grateful to Bruno Rodrigues, who has 

suggested model based estimation as a possible route of 

estimation of the spectrum in discussions some years ago. 

Thanks also go to the anonymous reviewers who have 

contributed to improve the readability of this paper. 

This work has been done within the project HANDLING 

WAVES—Decision Support System for Ship Operation in Rough 

Weather, which is partially funded by the European Commission 

through the programme Sustainable Surface Transport under 

contract TST5-CT-2006-031489. 

The first author has been funded by the Foundation for Science 

and Technology (Fundac-ão para a Ciência e a Tecnologia) under 

the Grant SFRH/BPD/42261/2007. 

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Generated 

Hs = 5m; 2m 

Tp = 15s; 10s 

S 

0.2Hz 

0.1Hz 


N 

0.3Hz 

S 

0.2Hz 

0.1Hz 

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N 

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0.1Hz

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