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REAL TIME ESTIMATION OF SHIP MOTIONS IN SHORT CRESTED SEAS

P. Naaijen

Delft University of

Technology

Delft, The Netherlands

R.R.T. van Dijk

MARIN

Wageningen,

The Netherlands

R.H.M. Huijsmans

Delft University of

Technology

Delft, The Netherlands

A.A. El-Mouhandiz

MARIN

Wageningen

The Netherlands

ABSTRACT

The presented research is part of the development of an

onboard wave and motion estimation system that aims to

predict wave elevation and vessel motions some 60 - 120 s

ahead, using wave elevation measurements by means of X-

band radar.

In order to validate the prediction model, scale experiments

have been carried out in short crested waves for 3 different sea

states with varying directional spreading, during which wave

elevation and vessel motions were measured.

To compare predicted and measured wave elevation, three

wave probes were used at different distances from a large set of

wave probes that was used as input to the model . At one of the

prediction locations, also tests were performed to measure

vessel motions.

This setup allowed validation of a method that was used for

initializing the linear wave prediction and ship motion

prediction model.

Various observations and conclusions are presented concerning

optimal combinations of prediction model parameters, probe

set-up and sea state.

INTRODUCTION

Within an international joint industry project called OWME

(Onboard Wave and Motion Estimation) a system is being

developed which aims to predict ship motions some 60 seconds

ahead. The main purpose of such a system is to increase safety

and operability during offshore operations that are critical with

regard to vessel motions, e.g. top-site installation (float-over or

lifting), helicopter landing on floating vessel and LNG

offloading connection. Use is being made of newest wave

sensing techniques by means of X-band radar: The Ocean

Waves’ WAMOS II radar image processing software is capable

of providing real-time time traces of wave elevation at a large

number of locations.

This paper describes the validation of a model used to compute

a deterministic prediction of wave elevation and ship motion by

using remote wave elevation measurements in short crested

waves. Linear theory is used resulting in a very simple and

straightforward propagation model. The challenging part is

within the initialization of this model which is the main focus

of the present study.

To assess the accuracy of the prediction, extended model tests

have been carried out at the Maritime Research Institute

Netherlands (MARIN). During these experiments the 2

dimensional wave field was measured by using a large array of

wave probes. The measured wave field is used to predict wave

elevation and ship motion at various distant locations. The

predictions are validated by means of measurements of both

wave elevation and vessel motions at the prediction location.

EXPERIMENTS

Wave measurements

As mentioned the present study aims to predict wave elevation

and vessel motion in a deterministic way using measured time

traces of wave elevation at various locations. To validate the

model that will be described in the next paragraph, model

experiments at a scale of 1:70 were carried out at MARIN, the

Netherlands.

For a selection of 3 short crested wave conditions, wave

elevation was recorded by means of a wave probe array

existing of 10 x 10 wire-type wave gauges.

Figure 1, 10 x 10 wave gauge array

1 Copyright © 2009 by ASME

Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering

OMAE2009

May 31 - June 5, 2009, Honolulu, Hawaii, USA

OMAE2009-79366

Tests were repeated with the wave gauge array positioned at

different locations in the basin, thus obtaining a relatively large

number of wave measurements that could be used to optimize

and validate the prediction model. Figure 2 shows all wave

gauge array positions for which tests were performed. Positions

1, 2 and 3 are the locations for which predictions will be made

and compared with the measurements. The probes at the

remaining locations are used as input for the model. The main

wave direction is in positive X direction as indicated in the

figure.

Figure 2, wave gauge array positions

As the different tests were not performed simultaneously,

checks were carried out in order to confirm that different wave

tests are reproducible. Figure 3 shows a sample of two wave

measurements from different tests, measured at the same

location. As can be seen good reproducibility is obtained.

Figure 3, comparison of measured wave

elevation during different tests at identical

locations

Concerning the wave conditions the choice was made only to

vary the amount of directional spreading. (The effect of wave

steepness was examined in earlier work. (Naaijen et al. [9])).

Tests were performed for a Jonswap spectrum with a peak

period Tp of 9.0 s, significant wave height Hs of 2.5 m and

peakedness factor γ of 3.3.

To include directional wave spreading the following spreading

function was used:

(

)

()

()()()

0

0

2

00

090 20

90

() cos

1

cos

,

s

s

DD

with

D

d

and

SSD

μ

μ

μμμ

μ

μμ

ωμ ω μ

+

−

=⋅ −

=

−

=⋅

∫

(1)

X Three different values for the spreading parameter s were used,

being 4, 10, and 50 corresponding to very short crested wind

waves, average wind waves and an average swell respectively.

Y(Measurements at position #5, #6, #10 and #11 as indicated in

Figure 2 were only performed for the tests with s=10.)

For practical reasons concerning the wave maker, the

directionality was cut off leaving the sector of -15 deg – 15 deg

for s=50, -30 deg – 30 deg for s=10 and -45 deg – 45 deg for

s=4.

Ship motion measurements

Apart from wave elevation measurements, motions of a model

of an offshore support vessel, without forward speed, located at

position #2 (see Figure 2) were recorded. The model, being

kept in position by a soft mooring system such that the relative

wave direction was 165 degrees, is depicted in Figure 4.

Figure 4, 1:70 model of offshore support

vessel

The main particulars of the vessel are:

LPP 94.2 [m] - Length between perpendiculars

LWL 101.5 [m] - Length on waterline

2 Copyright © 2009 by ASME

B 21.0 [m] - Breadth max.

TF 6.0 [m] - Draught fore

TA 6.0 [m] - Draught aft

Focusing on the prediction of vertical motions, mainly heave

and pitch motions were considered, of which the RAO’s were

determined using a linear 3D diffraction program. Additional

tests were carried out in irregular (white noise) long crested

waves for relative wave directions varying from bow to bow

quartering. Calculated RAO’s appeared to be in good

agreement with the RAO's determined from the mentioned

experiments.

00.5 11.5 22.5

0

0.5

1

1.5

2

1D Spectrum [m

2

/ s]

00.5 11.5 22.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω

[rad/s]

Pitch RAO [deg/m]

Heave RAO

[

m/m

]

heave 165 deg

pitch 165 deg

wave spectrum

Figure 5, 1D wave spectrum and RAO's for

relative wave direction of 165 deg.

Figure 5 shows the calculated heave and pitch RAO's together

with the one-dimensional wave spectrum.

PROPAGATION MODEL

The theoretical model used to describe the wave field is a linear

superposition of cosine waves with different frequencies

traveling in different directions:

()

() ()

()

/2 cos sin

11

nnj nm nj nm nm

NM itkx ky

j anm

nm

te

ωμ με

ζζ

−⋅ − +

==

⎧⎫

=ℜ ⋅

⎨⎬

⎩⎭

∑∑ (2)

where:

nm

real part

amplitude of frequency component in

propagation direction ,

frequency of component n

wave number of component n

, co-ordinates of location j

propagation direction of dire

anm

nm

n

n

jj

k

xy

ζ

μ

ω

μ

ℜ=

=

=

=

=

=

nm

ctional

component m for frequency n

initial phase angle of component nm

= number of directional components

per frequency

M

ε

=

From Fourier analysis, frequency components of the measured

time traces at J locations can be obtained, corresponding to the

following representation :

()

()

/2

0

nj

Nit

j anj

n

te

ωε

ζζ

+

=

⎧

⎫

=ℜ

⎨

⎬

⎩⎭

∑%

%%

(3)

where:

index of location for which time trace is provided

index of frequency component

N=number of samples of considered time trace

, amplitude and phase angle for frequency component n

of measured

anj nj

j

n

ζε

=

=

=

%%

time trace at location j following from FFT

From a predefined number M of directional components to be

used in equation (2), the discrete wave directions μnm were

chosen as was suggested by Zhang [1]: the average directional

wave spectrum was determined from the measurements by

means of the MLM method. For each frequency the energy

content was examined and the direction of the most energetic

component was identified. Ignoring on both sides of this most

energetic direction a predefined small amount of wave energy,

a range of wave directions is obtained that is divided into M

segments whose center values are used for μnm. (The values of

μnm determined this way will be frequency dependent which is

why a double index nm is used.) Figure 6 shows an example of

discrete ωn - μnm combinations (marked with dots) determined

as described above for M=10, together with the underlying

two dimensional spectrum for the wave condition with s=10.

Figure 6, directional wave spectrum for s=10

and discrete components used in

representation

As an estimate for the wave component amplitudes ζnm, the

spectral values S(ω,μ) were converted to amplitudes:

()

2,

nm n m

Sdd

ζ

ωμ ωμ

= (4)

3 Copyright © 2009 by ASME

Where:

the band width of each of the M segments mentioned aboved

μ

=

This way the only unknowns left in equation (2) are the initial

phase angles εnm.

By assuming that the above 2D representation of a wave time

trace at location j (equation (2)) should equal the measured

trace represented by (equation (3)), the unknown initial phase

angles in (2) can be solved. This is done by considering a

frequency domain representation of both measurement and 2D

representation.

One method to solve the unknown phase angles is proposed by

Zhang [1]:

For each frequency, an error can be defined as follows:

() ()

()

cos sin

1

n j nm n j nm nm

nj Mikx ky

i

nj anj anm

m

ee

μμ

ε

ζζ

−⋅ − +

=

Δ= ⋅ − ⋅

∑

%

%

ε

(5)

The unknown initial phase angles can be solved by minimizing

a target function Rn defined by the sum over all J locations of

the absolute squared error:

{}

()

{}

(

22

2

11

JJ

njn jn jn

jj

R

==

=Δ= ℜΔ +ℑΔ

∑∑

)

(6)

Another way to solve the phase angles is by defining a matrix

vector equation, Ax=b as given in equation (7), and solving

this.

() ()

()

() ()

()

() ()

()

() ()

()

1111 1 1

11

11

cos sin cos sin

1

cos sin cos sin

1

1

nnnn nnMnnM

nJ n nJ n nJ nM nJ nM

nn

nM nJ

ikx ky ikx ky

an anM

ikx ky ikx ky

an anM

ii

an

ii

anJ

Ax b

ee

ee

ee

ee

μμ μ μ

μμ μ μ

εε

εε

ζζ

ζζ

ζ

ζ

−⋅ − −⋅ −

−⋅ − −⋅ −

⋅==

⎛⎞

⋅⋅

⎜⎟

⎜⎟

⋅

⎜⎟

⎜⎟

⎜⎟

⋅⋅

⎝⎠

⎛

⎛⎞ ⋅

⎜

⎜⎟

⎜

⎜⎟

=⎜

⎜⎟

⎜⎟ ⋅

⎝⎠

⎝

%

%

K

MO M

L

%

MM

%

⎞

⎟

⎟

⎟

⎜⎟

⎠

(7)

Applying a Singular Value Decomposition (SVD) on A as done

by Janssen et. al. [4] appeared to give the best results. All

results and conclusions mentioned in next paragraphs are based

on the latter method.

Having solved the phase angles from equation (2), the wave

elevation can be calculated at any time and any location by

substituting the desired values for t, xj and yj in the equation.

However, in order to ensure physical significance of the

prediction, t, xj and yj have to be chosen within certain limits.

This will be explained in more detail in the next paragraph.

Using the RAO’s for the ship motions (obtained from linear 3D

diffraction calculations) a ship motion prediction can be made

in a straight forward way:

()

() ()

()

/2 cos sin

11

nnj nmnj nm nmknm

NM itkx ky

kj anm knm

nm

x t RAO e

ωμ μεε

ζ

−⋅ − ++

==

⎛⎞

=ℜ ⋅ ⋅

⎜⎟

⎝⎠

∑∑

(8)

where:

ship motion in mode k (k = 1...6)

frequency and direction dependent response

amplitude operator for mode k

frequency and direction dependent phase angle

for mode k

kj

knm

knm

x

RAO

ε

=

=

=

PREDICTABILITY

Having described the propagation model in the previous

chapter, some attention is paid to the question how the

difference between real surface waves and our representation of

them effects the predictability of those surface waves.

Let's reconsider equation (2). If no restrictions are put to the

domain in space (xj, yj) and time (t) for which we consider this

representation to be valid, we practically assume it to be able to

describe the entire ocean for an unlimited period of time.

Obviously, this isn't a realistic assumption. The validity of the

representation as given in equation (2) will be limited in space

and time.

To discuss these limitations, the one-dimensional case is

revisited here briefly. The space-time diagram for a long

crested wave traveling in positive X-direction showing the

predictable zone is depicted in Figure 7.

X

,

g

low

ct

,

g

high

ct

X0+ΔX

D

Figure 7, predictable zone long crested waves

t

t0 t0 +Δt

X0 t0+D

4 Copyright © 2009 by ASME

In this diagram, introduced by Morris et al [7] and used as well

by Edgar et al [8] and Naaijen et al [9], the triangles indicate

the zone in space and time where the wave elevation can be

predicted or reconstructed using a recorded time trace of the

wave elevation which is represented by the horizontal line at

the base of the triangle.

The second horizontal line represents the prediction based on

this recorded time trace, ΔX away from the measurement

location, shifted Δt seconds ahead. Only its part within the

predictable zone (gray triangle) is supposed to be useful.

In the mentioned publications the slopes of the left and right

boundaries of the predictable zone were considered to equal the

phase velocity of the shortest and longest wave components

present in the recorded time trace. However, as explained by

Wu [10], it is not the phase velocity but the group velocity of

the shortest and longest wave components that governs the size

of the predictable zone. This also explains that it was observed

during the experiments by Naaijen et al [9] that predictions of

the wave elevation could be extended further into the future

than expected based on the predictable zone bounded by the

highest and lowest phase velocities: when applying a long

enough duration D of the recorded wave elevation, only the

fastest wave components will limit the prediction and as their

group velocity is lower than their phase velocity, the steepness

of the right-hand boundary is decreased, meaning that the

prediction can be extended further into the future.

The concept of the predictable zone can be extended for the

three-dimensional case. Considering the three-dimensional

wave field to be a superposition of wave components traveling

in different directions, a similar predictable zone diagram can

be constructed for one specific traveling direction. See Figure

8.

For any point in space and time, the wave elevation due to all

components traveling in the considered direction is a

superposition of all frequency components traveling in that

direction. Depending on which point in space and time is

considered, not all these components might originate from the

measurement. The highest and lowest group velocities of those

components just originating from the measurement for a given

prediction point in space and time (xp, yp, tp) can be defined as

respectively:

()

()

()

()

(

()

()

()

()

()

(

1

2

cos sin /

cos sin /

gp p p

gp p p

cxx yy t

cxx yy t

μμ

μμ

=− +− −

=− +−

%%

%%

)

)

D

(9)

Where a tilde denotes the measurement location.

Denoting ω1 and ω2 as the corresponding wave frequencies and

ωlow and ωhigh as the frequencies of the shortest and longest

wave components that occur in the wave field, an error estimate

for the predicted wave elevation can be defined as the relative

amount of wave energy represented by those components at (xp,

yp, tp) that do not originate from the measurement:

μ

Figure 8, predictable zone for one directional

component of a short crested sea

()

()

()

()

()

2

1

12

2

0

2

0

,

,, 1

,

high

low

ppp

Sdd

Err x y t

Sdd

ωμ

π

ωμ

ω

π

ω

ωμ ω μ

ωμ ω μ

⎛⎞

⎜⎟

⎜⎟

=−

⎜⎟

⎜⎟

⎜⎟

⎝⎠

∫∫

∫∫

(10)

As described in the previous chapter multiple probe records are

used to find the representation of the wave field given by

equation (2). However, keeping in mind the limited

representing capabilities in space and time of this

representation, it can only represent a decomposition of one

probe record.

Imagine that for one specific direction the frequency

components in the three-dimensional wave representation

represent a decomposition of a measurement of length D at

location

(

)

,

x

y

%%.

Figure 8 shows that a measurement at location

(

)

,

pp

x

y can

only be represented by this decomposition between tb and te..

(The vertical axis in Figure 8 corresponds with the spatial

coordinate

x

μ

, parallel to the traveling direction of the

considered wave components.)

The short wave components present in the part before tb do not

originate from the location

(

,

)

x

y

%%. For any point in time of

the wave elevation time trace at

()

,

pp

x

y, the error Err as

x

(

)

(

)

(cos sin )xy

μ

,

g

low

ct

t

p

x

μ

x

μ

%

,

g

high

ct

Dte

tb

μ

=+

t0

5 Copyright © 2009 by ASME

defined in equation (10) can be determined. A representative

mean error value for the whole time trace at

()

,

pp

x

y can

then be defined as follows:

() (

0

0

1

,

tD

meanpp pp

t

Err x y Err x y t dt

D

+

=∫

)

,,

)

(11)

So when attempts are made to find a three-dimensional wave

field representation (equation(2)) that yields for a certain period

of time D at location

(

,

x

y

%% , only those parts of the

simultaneously measured time traces at surrounding probe

locations (denoted by

()

,

pp

x

y) should be used that are

within the predictable zone. The presented methods to find the

three-dimensional wave field representation are ignoring this

fact since they are frequency domain methods for which it is

not possible to take it into account in a straightforward way.

(Wu [10] describes a time domain method for solving the three-

dimensional representation for which it is possible to account

for limitations in the usable part of the time traces to be used.)

In the area where the wave probes were located, Errmean from

equation (11) is shown for the three wave conditions for

which experiments were carried out in Figure 10. Locations of

the frames containing the probes whose simultaneous

measurements were used for decomposing the wave field are

indicated by the squares. (The dots indicate the used probe

positions that appeared to give the optimal results for the

considered wave condition and prediction location #2.) As can

be seen the probe positions are chosen such that Errmean does

not exceed 0.1 for any of the conditions.

Figure 9, predictable zone for s=10 and

forecast-time of 60 s

Y

X

Y

X

Y

X

Figure 10,

Err

mean

for s=50, 10 and 4 and

available probe-frame positions

When calculating the values for Err according to equation (10)

for a prediction of the wave elevation at a given set of locations

(xp,yp) for a given moment in time tp , the optimal relative

positions of measurement and prediction can be determined.

For a forecast-time of 60 s and wave spreading s=10, the value

of Err is presented in a similar way as was done by Blondel

[11] in Figure 9.

α X

Y

6 Copyright © 2009 by ASME

As shown by Figure 9, prediction location #2, the location

where also the ship model was located, was chosen such that it

would be optimal for a 60 seconds prediction for the wave

condition with s=10.

RESULTS

Several aspects have an effect on the accuracy of the wave

prediction:

The setup of the probes whose measurements are used to solve

the system of equation (7) should be such that their in-

between distances are optimal to identify phase differences for

the whole frequency range that contains wave energy. As

shown by Voogt et al. [12], for one regular wave component

this optimal probe distance amounts to ¼ of the wave length.

Therefore it was aimed to use a probe setup that provided for

all relevant frequencies in-between distances (projected in the

direction of each of the wave directions used in the

representation) of ¼ of the wave length.

The probe setup that appeared to give the best results for a

prediction at location #2 in waves with direction spreading of

s=10 is shown in Figure 9. For each of the 10 discrete wave

directions that were used at the peak frequency in the two-

dimensional representation, intermediate distances between the

used probes, projected in each of the 10 wave directions, were

determined. Figure 11 shows the 2D spectrum based on

wavelength / peak wavelength ratio. The dots indicate all the

available intermediate probe distances divided by ¼ peak

wavelength. This way of presenting should result in a high-

density of dots in the most energetic part of the spectrum for a

favorable probe set-up according to the statement above.

Figure 11, relative probe distances for optimal

setup s=10, prediction position #2

Another important variable that effects the accuracy to a great

extend is the number of directional components that is used in

the representation of the wave field. With a large number of

directional components, the directional spreading is better

covered. However, for a given probe set-up, choosing a too

large number of directional components dramatically decreases

the condition of system matrix A in equation (7) resulting in

poor predictions. For cases where the number of directional

components was too large, it was found that adding more probe

records to the equation (7) not necessarily improves the

prediction. The extra probe records have to add ‘information’ to

the system. Adding probes positioned in frame numbers 4 and

12 (Figure 2) for example did not improve the prediction for

any of the cases. When the requirement to the probe set-up as

stated above is fulfilled, the only way to improve the condition

of the matrix is to decrease the number of wave directions. The

set-ups shown in Figure 10, using 9 probes per frame as

indicated by the dots, were found to give the best results for

prediction at location #2. By extending the number of input

measurements by using all available probes i.e. 100 per frame,

no improvements were obtained.

Apart from the combination of probe set-up and number of

directional components, that determines the condition of the

matrix in (7), the combination of probe set-up and the

prediction location was found to have a significant impact. It

was observed that the use of extending the probe set-up in y-

direction is limited related to the amount of wave spreading and

the prediction location. See Figure 9. The angle α between the

dashed lines starting from prediction location #2 equals the

sector angle that bounds the directionality of the wave

condition: as mentioned, for practical reasons the directionality

was cut off at 30 and -30 degrees for this condition (s=10)

resulting in α = 60 deg. It was observed that using

measurement probes outside this angle did not improve the

accuracy. This observation was found to be consistent for all

combinations of prediction location and directional spreading

as far as the available measurements allowed to verify it. (For

the most short crested condition with s=4, measurements at

positions #5, #6, #10 and #11 that would enable confirmation

of this observation were not available.)

To assess the accuracy of the predictions an error value is

calculated that is defined as:

() () ()

()

2

*

1

*

0

1N

nn

ntt

N

Et m

ζζ

=

−

=∑ (12)

Where:

(

)

*

*

0

normalized prediction error

( ) realization of predicted wave elevation

( ) realization of measured wave elevation

total number of realizations

RMS of measured wave elevation

n

th

n

th

Et

tn

tn

N

m

ζ

ζ

=

=

=

=

=

Figure 12 shows the normalized error of the wave prediction

averaged over N =100 realizations at location 3, 2 and 1 for the

case with s=10. The vertical dashed lines indicate the

7 Copyright © 2009 by ASME

boundaries of the predictable zone in time. As expected the

error increases outside the predictable zone. The solid vertical

line indicates the boundary between hind cast and forecast.

Error values averaged over the predictable part of the predicted

traces are given in Annex A, Table 1. All values are based on an

average over 100 realizations.

0100 200 300 400 500 600 700

0

0.5

1

1.5

E(t) [-]

0100 200 300 400 500 600 700

0

0.5

1

1.5

E(t) [-]

0100 200 300 400 500 600 700

0

0.5

1

1.5

t [s]

E(t) [-]

envelope

wave elevation

Figure 12, prediction error for locations 3,2

and 1 for s=10

Since for the applications of the onboard wave and motion

estimation system we are interested in the prediction of quiet

periods rather than in an exact deterministic prediction, also the

envelope of the deterministic prediction has been determined:

the predicted deterministic signal has been post processed by

taking the absolute value of its Hilbert transform. Its error,

which is defined similarly to the error of the wave elevation /

ship motion itself is in general significantly smaller.

For location #2, Figure 13 shows time traces of predicted and

measured wave elevation and heave and pitch motion for the

case with s=10. The maximum prediction time has been

determined from the measured mean two-dimensional

spectrum, allowing a value of Err as defined in equation (10)

of 0.1 and amounts to 78 s. The vertical solid line again

indicates the boundary between hind cast and forecast. The

rightmost vertical dashed red line indicates the end of the

predictable zone in time.

In Annex B, samples of time traces are shown for all wave

conditions at all prediction locations. For each prediction

location the most favorable probe set-up is plotted at the right

hand side of each of the corresponding time traces: The black

dots indicate the used probes, the grey cross with the circle

indicates the concerning prediction probe. The color indicates

the Err value as defined in equation (10) for the allowed

maximum forecast time that was aimed for (which is 30 s for

probe 3, 60 s for probe 2 and 120 s for probe 1). As can be seen

the chosen prediction locations match the intended forecast

time well in that sense that for all conditions they are

positioned in the predictable zone (where Err is app. 0.)

1850 1900 1950 2000 2050 2100

-2

0

2

ζ

[m]

1850 1900 1950 2000 2050 2100

-1

0

1

z [m]

measured

predicted

1850 1900 1950 2000 2050 2100

-5

0

5

θ

[deg]

t [s]

Figure 13, sample time traces of prediction

and measurement of wave elevation, heave

and pitch motion, location # 2, s=10

As can be seen from both the mean error values in Table 1 and

from the samples of the time traces in Figure 15, predictions for

s=4 are rather poor especially for locations 1 and 2. This is

caused by the fact that probe measurements at locations further

from the X-axes, that would be required for an optimal set-up

were not available for this condition.

ACKNOWLEDGMENTS

This paper is published by courtesy of all participants and

partners of the OWME-JIP for which they are gratefully

acknowledged.

Special thanks to MARIN for facilitating the experiments.

REFERENCES

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crested ocean waves: Part I. Theory and numerical

scheme, Applied Ocean Research 21, 1999

[2] Zhang et. al., Deterministic wave model for short-

crested ocean waves: Part II. Comparison wih

laboratory and field measurements, Applied Ocean

Research 21, 1999

[3] Prislin et. al., Deterministic decomposition of deep

water short-crested irregular gravity waves, Journal of

Geophysical Research, Vol. 102 No. C6, 1997

[4] Janssen et. al. Phase resolving analysis of

multidirectional wave trains, Proceedings of the Fourth

International Symposium Waves 2001, September 2-6,

2001, San Francisco, CA

[5] de Jong et. al. A phase resolving analysis technique for

short-crested wave fields

[6] Belmont et. al. The effect of statistically dependent

phases in short-term prediction of the sea: A simulation

study, International Shipbuilding Progress Vol. 51 no.4,

2004

8 Copyright © 2009 by ASME

[7] Morris, E.L., Zienkiewicz, H.K. & Belmont M.R.,

1998 Short term forecasting of the sea surface shape.

Int. Shipbuild. Proqr. 45, no.444, 383-400.

[8] Edgar D.R., Horwood J.M.K., Thurley R., Belmont

M.R. The effects of parameters on the maximum

prediction time possible in short term forecasting of the

sea surface shape, International Shipbuilding Progress

Vol. 47, No.451, 2000

[9] Naaijen P., Huijsmans R.H.M. 2008, Real time wave

forecasting for real time ship motion predictions, Proc.

OMAE 2008

[10] Wu, G., Direct simulation and deterministic prediction

of large-scale nonlinear ocean wave-field, PhD Thesis,

M.I.T, 2004

[11] Blondel, E., Ducrozet, G., Bonnefoy, G., Ferrant, P.,

Deterministic reconstruction and prediction of non-

linear wave systems, 23rd IWWWFB, 2008, Jeju, Korea

[12] Voogt A., Bunnik T., Huijsmans R.H.M. Validation of

an analytical method to calculate wave setdown on

current Proc. OMAE 2005

9 Copyright © 2009 by ASME

ANNEX A

s [-] Error [-] Error Envelope [-] max pred [s]

prediction probe # wave heave pitch wave heave pitch

1 10 0,915 0,703 131

2 10 0,778 0,802 0,748 0,639 0,597 0,547 78

3 10 0,744 0,607 35

s [-] Error [-] Error Envelope [-] max pred [s]

prediction probe # wave heave pitch wave heave pitch

1 4 1,172 0,822 124

2 4 1,179 1,172 1,215 0,785 0,890 0,879 77

3 4 0,899 0,686 34

s [-] Error [-] Error Envelope [-] max pred [s]

prediction probe # wave heave pitch wave heave pitch

1 50 0,783 0,668 138

2 50 0,737 0,827 0,734 0,651 0,686 0,593 83

3 50 0,702 0,566 37

Table 1, Averaged Error values

10 Copyright © 2009 by ASME

ANNEX B

1600 1650 1700 1750 1800 1850 1900

-2

0

2

ζ [m]

measured

calculated

Time trace sample position 1, s=10, max prediction time =131 s, M=14

1850 1900 1950 2000 2050 2100

-2

0

2

ζ

[m]

1850 1900 1950 2000 2050 2100

-1

0

1

z [m]

2

measured

predicted

1850 1900 1950 2000 2050 2100

-5

0

5

θ

[deg]

t [s]

Time traces sample position 2, s=10, max prediction time =78 s, M=10

1680 1700 1720 1740 1760 1780 1800 1820 1840 1860 1880 1900

-2

0

2

ζ

[m]

measured

calculated

Time trace sample position 3, s=10, max prediction time =35 s, M=7

Figure 14, sample time traces for s=10

11 Copyright © 2009 by ASME

1600 1650 1700 1750 1800 1850 1900

-2

0

2

ζ

[m]

measured

calculated

Time trace sample position 1, s=4, max prediction time =124 s, M=8

1600 1650 1700 1750 1800 1850 1900

-2

0

2

ζ

[m]

1600 1650 1700 1750 1800 1850 1900

-1

0

1

z [m]

1600 1650 1700 1750 1800 1850 1900

-5

0

5

θ

[deg]

t [s]

Time traces sample position 2, s=4, max prediction time =77 s, M=12

di i f b 1(X ) 861 3 32)Y ( 69 6 8

1600 1650 1700 1750 1800 1850 1900

-2

0

2measured

ζ

[m]

calculated

Time trace sample position 3, s=4 max predict on, i time =34 s, M=9

Figure 15, sample time traces for s=4

12 Copyright © 2009 by ASME

1750 1800 1850 1900 1950 2000

-2

0

2

ζ

[m]

measured

calculated

Time trace sample position 1, s=50, max prediction time =138 s, M=7

1750 1800 1850 1900 1950 2000

-2

0

2

ζ

[m]

1750 1800 1850 1900 1950 2000

-1

0

1

z [m]

1750 1800 1850 1900 1950 2000

-5

0

5

θ

[deg]

Time traces sample position 2, s=50, max prediction time =83 s, M=7

1700 1750 1800 1850 1900 1950

-2

0

2

ζ

[m]

measured

calculated

Time trace sample position 3, s=50, max prediction time =37 s, M=4

Figure 16, sample time traces for s=50

13 Copyright © 2009 by ASME