Real time forecasting for real time ship motion predictions

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Date Author Address

June 2008

Peter Naaijen & René Huljsmans Deift University of Technology Ship Hydromechanics Laboratory

Mekeiweg 2, 26282 CD Deift

TUDeift

Deift University of Tethnoiogy

Real time wave forecasting for real time ship

motion predictions

by

Ir. Peter Naaljen & Prof.dr.ir. R.H.M. Huijsmans

Report No. 1582P

₂₀₀₈

Published: Proceedings of the ASME 27"' International Conference on Offshore Mechanics and Arctic Engineering, OMAE2008, Estoril, Portugal, ISBN: O-7918-3821-8

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ABSTRACT

This paper presents results of a validation study into a linear

short term wave and ship motion prediction model for long

crested waves. Model experiments have been carriedout during

which wave elevations were measured at various distances

down stream of the wave maker simultaneously. Comparison between predicted and measured wave elevation are presented

for 6 different wave conditions. The theoretical

relation

between spectral content of an irregular long crested wave

system and optimal prediction distance for a desired prediction time is explained and validated. It appears that predictionscan

be extended further into the future than expected based _{on this}

theoretical relation.

INTRODUCTION

Within the offshore industry there are various operations for

which a motion-prediction based decision-support systemcan

be beneficial: Top-side installations (liftingor float-over), LNG

connecting and helicopter/automatic UAV landing are examples of operations for which safety and operability can be increased if a reliable prediction of the vessel motions were available. In 2006 an international Joint Industry Project called Onboard Wave and Motion Estimation (OWME) was launched to develop, test and demonstrate a practical system to predict quiescent periods of ship and platformmotions some 60 seconds in advance. The approach of the system is based on measuring the wave field around the vessel by means ofan

Xband radar having a reach of appr. 2 km.

Within the project Delft University of Technology (DUT) has been commissioned to provide a wave propagation model which

uses the measured wave field to make a prediction of the vessel's motions. It was agreed, for the sake of minimum

calculation time, robustness and maintainability and due to the

uncertainty of the accuracy of the input wave measurement

Proceedings of the ASME 27th InternatIonal Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal

OMA E2008-57804

REAL TIME WAVE FORECASTING FOR REAL TIME SHIP MOTION PREDICTIONS

Ir. Peter Naaljen (DeIft University of

Technology) Prof. dr. Ir. René Huljsmans (Delft University of_Technology)

(which has a large effect on the accuracy of higher order

prediction models) to use linear wave theory.

This paper presents a first study on the accuracy ofa linear long

crested wave propagation model for various wave conditions.

An optimization study for the distance between measurement

and prediction location, depending on the desired forecast time has been carried out and restiltsare presented.

EXPERIMENTS

The experiments with scale 1 30 were carried out at towing tank #1 of the DUT Ship Hydromechanics Laboratory, having the

following particulars:

Length: 150m Width: 4.Om

Depth: 2.41 m

During the experiments, waves were created by the wave maker

consisting of one flap hinged at the bottom of the towing tank.

Alongside the towing tank, down'stream' of the wave maker, an

array of 9 wave probes was installed. See Figure 1 for the

schematic experimental set-up.

83,49

9

'./ov9r'iclker _Beoch

Figure 1, probe positionsexperimental set-up

The position of the wave probes was chosen such that for each wave condition, at-least one probe was positioned at a distance from the reference probe 1 that would theoretically just allowa

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120 s

prediction within the 'predictable

area' based on

theoretical spectrum of the wave condition. (For the swell

condition, the limited length of the basin allowed only a 90 s)

The words 'predictable area' are explained in detail in the

theory paragraph.

The campaign existed of 6 measurements of about_{2 hours full}

scale duration with different wave conditions. 5 Jonswap

spectra weretested and one typical swell spectrum:

e 1, Properties of tested waveconditions (asobserved)

Before sampling, low-pass analogue filteringwas applied with a

cut-off frequency of 10Hz.

The applied digital sampling rate of the data

acquisition

amounted to 100 Hz (model scale) which corresponds to 18.3

Hz prototype scale.

High-pass digital filtering was applied to eliminate the possible effect of seishes in the towing tank.

THEORY

This paragraph describes the theory that is used to predict the

wave elevation at a certain downstream location by using an

upstream wave elevation measurement which is based on linear

wave modeling.

As done by Morris et al. [1] and Edgar et aI. [3], the linear

wave propagation modeling problem can be represented

schematically by a time-distance diagram, see Figure 2.

Consider a time trace of one dimensional dispersive _waves

satisfying the linear wave equation measured at location A. A

Fourier Transform of this

irregular time trace yields the

amplitude and phase angles of a limited number of regularwave

components that it contains. (In order to reduce the end-effects a tanh-shaped window function with steep slopes was applied to

the input trace before the Fourier transformation _{was carried}

out. This slightly improved the results.) For a sampled irregular

input trace at location A of N points length yields:

N/2

COA

(t)=

Coa.ne e10" (1)

This trace is indicated by the thick line OT at the lower side of

the leftmost triangle in Figure 2.

With the linear dispersion relation the wave number k ofeach

of the components can be determined and for the predicted

wave at location B at a future time:+4z can be written: N/2

COB

(t + t )=

COa.n

(x-x.4))

(2)

where the wave is supposed to propagate in positivex direction and location B has a larger x-co-ordinate than location A.

This trace is indicated by the thick line CE in Figure 2 resulting

from shifting OT by iM horizontally and by_{xB - xA vertically.}

As the overall aim of the project is to predict quiescentperiods

in the vessel motions, it's not the deterministic _{wave or motion} that we are interested in but rather it's envelope. Theenvelope

of the prediction can _{simply be determined from the}

deterministic prediction by taking the absolute value of its

Hilbert transform

Predictable area'

However, for physical significance of the forecastedtrace from

equation (2), At and xB - XA cannot be chosen freely: The

longest and shortest wave components found in the irregular measured trace OT determine the so-called predictable area in space and time where prediction is possible using trace OT The predictable area, represented by the triangular region OTB, is

bounded by the line OB of which the slope equals the phase velocity of the shortest wave components. At any location above OB, the shortest wave components contained by irregular input trace OThave not arrived yet. The other boundary of the predictable area is formed by line TB whose slope equals the phase velocity of the longest wave component. Atany location below TB, the longest wave components havepassed already.

Of the forecasted trace CE only the part JG is useful: The leftmost part (CH) is outside the predictable area so theshorter waves from OT haven't arrived at location x5 yet, while the part

H!, though being within the predictable area, is useless as it

represents no forecast but hindcast: obviously, the analysis of

input trace OT can be started only if it has been completely

acquired, which is at time T (= time I). The time required to do the analysis is represented by ¡J which means that the analyzed result is no sooner available than at time J. The rightmost part,

GH, is outside the predictable area as the longer

wave components from OT have passed already at location x8

type Significant Wave Height Hg Peak period T

(6)

'C

Minimal

foreca8t

Slovest Waves time

haven't arrived yet

Outside

rrethtable

FastostWaves

Loca!ion _L have passed

--already 750 XB600 450 E a, o 300 ti, V >< 150 O 100 200 300 400 t Is] 500 T 600 580 600 620 640 t [s] o 4.8 -2.4 700 'V

Figure 3, example of realization, condition Jonswap 3,

probe 3

Figure 3 shows a visualization of one simulation step for wave

condition 'Jonswap3' at wave probe #3. In the upper most figure a time - distance diagram similar to Figure 2 is presented with the same capitals indicating beginning of forecast, end of

predictable part etc. At OT the envelope of the input trace is

plotted. Underneath CE, both predicted and measured

envelopes are plotted and above CE the absolute difference

between them is shown. The vertical axis for these plotscan be found at the right hand side of the diagram.

The lower figure zooms in on the most relevant part of the

predicted trace representing the forecast, part JE. Both

predicted and measured wave elevation and envelope are shown.

The average results over 400 realizations, each of whose input and predicted trace was shifted 10 s. in time, is shown in figure 4. 3 _{Copyright © 2008 by ASME} 1. Duration D Computational J L time Time t

Figure 2, Wave propagation time-distance diagram The analysis of the measurement is done on a moving time

window of length D. The predictable area for thenext time step

is represented by the right triangular region in Figure 2. When the analysis of the first time step has been completed, which is

at time J (= time F), a useful forecast is available having _a

duration that equals JG. (As the analysis of thenext input time

trace (represented by the base of the rightmost triangle) can

start at F, and as it will again take computational time to obtain its result, the minimal time window into the future during the process of predicting will be of length JG - If.

Duration D and distance x5

-

XAcan be optimized in order to

minimize the prediction error for a certain required forecast time and wave condition. This has been examined using the

model test data and will be presented in the next paragraphs.

SDectral truncation

A validation of the linear propagation model was carried

out by comparing predicted and measured wave traces. The

wave measurement at the first wave probe (represented by ₍₄in equation (1)) was used as the input of the model and predictions

up to 140 s ahead were made at the locations of the other

probes. As in real life the wave measurement would be done by means of a radar having a significantly lower sample rate than

18.3 Hz, the input wave data was down-sampled to 1 Hz

(prototype scale).

In general, a long duration D appeared to be favorable in

terms of prediction accuracy. 512 s was considered as an

optimum as for longer durations the computation time increased significantly only resulting in marginal increase ofaccuracy.

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0.25

o

ç

E

Average error predicted envelope and determInistIc wave Disthnce: 600 m0 weed Input Dutplion: 511 1 Preqictlon:140 s20 15

'N44$444$4+$4$

w

_o-

t - Envelope - Deterministic

Figure 4, Average error deterministic wave and envelope, Jonswap 3, probe 3

The dotted and solid lines represent the normalized standard

deviation

of the

error

of the

predicted envelope and

deterministic wave respectively:

'M

aI(om (t)B (t))2

o(t)=

M Vrn=i Hsig 'M

(t)ÇOB

(t))2

cr

(t)

H3i5 Where:

= the standard deviation of theerror of the deterministic wave prediction. M is the number of realizations (400).

ÇO8.mand Ç, are the mth realization ofpredicted and

measured wave elevation at prediction location X8 respectively. is the significant wave height. Tildes indicate the envelope.

The dashed - dotted line in Figure 4 indicates the relative

amount of wave energy in the predicted trace that can originate from the input trace: In the upper graph of Figure 3, the left and

right hand side

sloped

lines OH and TG indicate

the

propagation of the shortest and longest Wave components found

in input trace OT respectively. As explained they bound the

predictable area. Apparently

for the shown example, the

distance between reference and prediction location is such that

the full predicted trace doesn't fit within the predictable area:

for part GE the longest wave components have passed the prediction site already. For this wave condition the distance

needs to be increased in order to make sure the whole

prediction fits within the predictable area. The dashed-dotted line in Figure 4 indicates how much of the wave content in the

measured input trace is present at the time and location of

prediction. The phase velocity of the longest wave component

that júst has passed at the prediction location at time T0, in

Figure 3 is:

COU, =(XB XA)i(TOU, T)

₍₅₎

Where:

c0, is

_{the phase velocity, XA and X8}

_{are the input} measurement and prediction location respectively. T and T0, are

start time of forecast part of predicted trace and considered

moment in time outside predictable trace respectively.

The corresponding wave frequency, W,, follows from the linear dispersion relation. All wave components in themeasured

input trace OT having frequencies lower thanW have passed the considered prediction location already at time T0,. The relative amount of wave energy that these components

represent, thespectral truncation, is the spectral area below w,, divided by the total spectral area.

7ffc

percentagei . of spectral i'°° area inclUded j in -90 prediction lOO 200 300 400 time [s) 500 60 1.5 ₂₅ CU our _{w IrawsI}

Figure 5, Spectral truncation of wave spectrum

The relative amount of wave energy of the remaining wave

components (that don't have passed yet) is plotted in Figure 4 by the dashed-dotted line(vertical axis on right hand side of

Figure 4).

The theoretical relation between the mentioned relative amount

of waveenergy and the accuracy was strongly confirmedbythe

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this can result in a significant increase of the maximum forecast _{fetch for the long wave components is smaller resulting in a}

time. _{smaller error for those components.}

SHIP MOTIONS

As mentioned, the final purpose of the project in which this

study was carried out is to predict motion behavior of floating

structures.

Applying linear wave theory and linear motion - wave transfer functions, the step from wave prediction to motion prediction is

a simple and straight forward one. For any motion () the

predictioncan be written as:

N12

Ç8 (t + ¡it) =

Çøa.n IIj,n . e1 °' (6)

n=I

where: H is the complex transfer function of the motion for

mode j.

As no measurements of motions were carried out during the

measurement campaign, the ideal situation is assumed nowthat the motion transfer is perfect. This means amotion

'measurement' can be obtained by applying the transfer function to the measured wave at the prediction location.

Then, analogue to equations (3) and (4), thestandard deviation of the error of the motion can be written as:

'M

(t)Çj'ni

(t))2

oL.

(t) =

SDAJ

Where:

SDA1 is the significant double amplitude of the motion for mode J.

Depending on it's shape related to the wave spectrum, the

transfer function, acting as a fitter, can result in an increase or a decrease of the motion prediction error compared to the wave

prediction error.

Prediction simulations have been carried out for the heave and pitch motion in head waves of an offshore support vessel having

the following particulars:

Length: 106 m, Beam: 21 m Draft: 6.20 m

For both the heave and pitch transfer functions of the vessel

only the tail coincides with the frequency range where the wave energy is located. Figure 6 shows the absolute values of these

transfer functions (RAO's) together with one of the measured wave spectra. The fact that the joint frequencies are located at the low frequency range of the wave spectrum results in motion

prediction errors that are smaller than the errors of the wave

prediction. This might be explained by the fact that the relative (7)

Jonswap 4, probe 4, [m]

predicted measured

O)

FIgure 6, Wave spectrum and transfer functions of heave

and pitch (axes labels omitted for confidentlalily reasons)

a 0. 7711

580 600 620 640 660 680

Figure 7, typical sample of time trace of predicted and

measured wave elevation Jonswap 4, probe 4

hea Em]

1.2

1.2 I

580 600 620 640 660 680

Figure 8, sample of time trace of predicted and measured corresponding heave motion

0.5

-0.5

aO.i44l8

pitch [deg]

580 600 620 640 660 680

Figure 9, sample of time trace of predicted and measured corresponding pitch motion

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For that reason a poor wave predictioncan still result in a fairly

good motion prediction as is shown by theexample realizations of wave, heave and pitch prediction in Figure 7, Figure_{8 and}

Figure 9 respectively. The number in the top right corner of

each plot indicates the standard deviation of the prediction

error. (The thick vertical line indicates the theoretical end of the

predictable part as explained in the paragraph 'predictable

area'. The end of the shown trace is the end of the predictable part as it was found from the average result of 400realizations.)

RESULTS AND DISCUSSION

Predictable area

As mentioned already, the predictable area (as

it can be

expected from the explained theory) gives an indication for the

maximum prediction time that can be achieved for a certain

fetch. The experimental results however showed that

predictions can be extended significantly long beyond this

theoretical predictable area. Qualitatively it can be said that the

higher the

fetch over wave length

ratio, the further the

prediction can be extended beyond the predictable_{area. This is}

why the probe distances in the experimental set-up

were

actually larger than necessary for the aimed forecast times:

Except for the Jonswap 7 and Swell condition a forecast time of 60 s. appeared to be feasible at probe 3. (A mismatch between the theoretical Jonswap spectra and the observed spectra during

the experiments also partly explains the large difference between expected required probe distances on which the set-up,

was based and the found required distances from the

experiments.)

Annex B, Figure 11 shows sample time traces of predicted and

measured wave elevation, heave motion and pitch motion at

probe 3 for all Jonswap conditions and at probe 9 for the swell condition. Again the thick vertical lines indicate the theoretical

end of the predictable part as explained in the paragraph

'predictable area'. The end of the shown trace is the end of the predictable part as it was found from the average result of 400

realizations.

Accuracy

See Annex A, Figure 10. Following Trulsen [2] the standard

deviation of the wave prediction error (as defined in equation(4)

)

is plotted against e2 k,, . X being the non-dimensional

steepness squared times non-dimensional fetch where:

,J2H

non-dimensional steepness, e g

gT,,2

where = peak period of wave spectrum [si

(8)

where k,, = peak wave number of wave spectrum Erad/ml X = fetch Em]

As can be seen a fairly linear relation betweenerror (on the vertical axis) and the mentioned estimator (on the horizontal

axis) is found.

Of all points in Figure 10 that are marked with an arrow, indicating the corresponding wave condition, sanipletime traces of prediction and measurement can be found inAnnex B, Figure

11

It can be concluded that a 60 s accurate forecast of

wave elevation is very well feasible for all considered wave conditions Motion predictions are even more accurate.

ACKNOWLEDGMENTS

This paper is published by courtesy of all participants and

partners of the OWME-JIP for which they are gratefully

acknowledged.

NOMENCLATURE

REFERENCES

Moms E.L., ZienkewiczFLK., Belmont M.R. Short term forecasting of the sea surface shape,

International Shipbuilding Progress Vol. 45, No. 444, 1998

Trulsen K., Stansberg C.T.Spatial evolution of water

surface waves,_{Proc. Eleventh Intl Offshore & Polar}

Engng Conf. pp. 71-77, 2001

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 forecasEing of

the sea surface shape, _{International Shipbuilding}

Progress Vol. 47, No.45 1, 2000

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ANNEXA

ERROR OF PREDICTION AGAINST FETCH-STEEPNESS PARAMETER

Error Way Elevation

I I 125

49

Jonswap 7=_:. Jonswap

4___ 88

_{L 100 y 100}

4122

120 89 71 ₆₆

12 E90'Jonswap5N

27 Jonswap 6 Jonswap 3 79 :, 16 numbers indicate maximum forecast time

marker color Indicates wave

condition

114 o I I i 0 0.1 0.2 0.3 0.4 2 X

[-Figure 10, standard deviation of prediction_{error against non-dimensional steepness squared times non-dimensional fetch}

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0 -0.5 1540 1560 1580 1600 Jonswap 4, probe 3, [ml 0.5 O -0.5 "_i'

-a,O24

560 580 600 620

Jonswap 5, probe 3, Em]

1020 1040 1060 Jonswap6, probe 3, Ç[m] -2' -560 580 600 Jonswap 7, probe 3, [m] 2 ₁₇ O -2 510 520 530 540 550 Swell, probe 9, [ml

SAMPLE S OF PREDICTED AND MEASURED TIME TRACES

-0.1 0.5 O -0.5 o ANNEX B heave [m] 1540 1560 1580 1600 heave Em] 560 580 600 620 heave Em] 1000 1020 1040 1060 heave Em] hea Em] 600 620 640 t [s] 0.2 -0.2 1540 1560 1580 1600 pitch [deg] O pitch [deg]

a 0.08142

560 580 600 620 pitch [deg] 1020 1040 pitch [deg] 1060 2 510 520 530 540 550 pitch Ideg] 560 580 600 pitch [deg]

FIgure 11, typical samples of time traces of predicted and measured wave elevation, heave motion and pitch motion for all

Jonswap 3, probe 3, Em]

560 580 600

heave Em]

0.1

0.07591