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International Shipbuilding Progress 61 (2014) 203–223 203

DOI 10.3233/ISP-140113

IOS Press

Limits to the extent of the spatio-temporal domain for

deterministic wave prediction

Peter Naaijen a,∗, Karsten Trulsen band Elise Blondel-Couprie c

aDepartment of Ship Hydromechanics, Delft University of Technology, Delft, The Netherlands

bDepartment of Mathematics, University of Oslo, Oslo, Norway

cAllseas Engineering BV, Delft, The Netherlands

Received 18 September 2012

Revised 22 July 2014

Accepted 3 September 2014

We discuss the spatio-temporal domain, here referred to as the predictable zone, in which waves can be

predicted deterministically based on an observation in a limited spatial or temporal domain. A key issue

is whether the group or phase speed of the observed waves governs the extent of the predictable zone.

We have addressed this issue again using linear wave theory on both computer-generated synthetic wave

ﬁelds and laboratory experimental observations. We ﬁnd that the group speed adequately indicates the

predictable zone for forecasting horizons relevant for offshore and maritime applications.

Keywords: Wave prediction, deterministic, phase resolved

1. Introduction

It has been common practice for many years to assess operability of offshore oper-

ations, critical with regard to vessel motions in waves, by considering statistical prop-

erties like signiﬁcant motion amplitudes or most probable maximum values. These

are typical properties related to the sea surface elevation described as a stochastic

process. With the development of various remote surface elevation sensors like lidar

[2,9] and X-band radar [18], there has been recent interest in considering the surface

elevation from a deterministic point of view. It has been shown that in principle, for

time scales in the order of tens of seconds, it is feasible to accurately predict wave

elevation and related behavior like vessel motions in a deterministic way: For long-

crested waves Morris et al. [11] and Edgar et al. [8] reported on prediction accuracy

*Corresponding author: Peter Naaijen, Department of Ship Hydromechanics, Delft University

of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. Tel.: +31 15 2781570; E-mail:

p.naaijen@tudelft.nl.

0020-868X/14/$27.50 ©2014 – IOS Press and the authors. All rights reserved

204 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

of wave elevation computed with a linear wave model. They showed the effect of

optimal truncation of input data for the initialization of the model and the effect of

the shape of the wave spectrum and water depth on the prediction horizon. Also

using a linear wave model for long-crested waves, Naaijen and Huijsmans [14] con-

sidered the prediction accuracy using experimental data. Trulsen and Stansberg [21],

Trulsen [20] and Shemer et al. [19] showed results using nonlinear wave models with

experimental data of bi-chromatic and irregular long-crested waves. Successful pre-

diction of wave elevation and wave induced ship motions in short crested seas from

experimental data was reported by Naaijen et al. [16] and extended to the predic-

tion of wave drift forces by Naaijen and Huijsmans [15]. For both studies the two-

dimensional representation of the wave ﬁeld was obtained by correlating a limited

number of input time traces of the wave elevation, recorded at a sparse set of loca-

tions, as suggested by Zhang et al. [23] and Janssen et al. [10]. A different approach,

more directly related to the analysis method of X-band radar images of the surface

elevation was used in [13] and [5], relying on 3D FFT techniques, widely accepted

for retrieving statistical seas state properties from nautical radar and [17] using an

alternative which is shown to be more suitable when deterministic (phase-resolved)

wave sensing from nautical radar is aimed for.

When applying deterministic prediction, it is crucial to be well aware of its limi-

tations. Concerning these limitations, the following distinction can be made:

(1) Given an observation of the wave elevation in a limited domain in space or time,

the associated spatio-temporal zone where an accurate prediction can be made

is limited and will depend on the wave spectrum. This zone will be referred

to as the theoretical predictable zone and will be explained in detail hereafter.

A good understanding of this limitation is important to enable an efﬁcient wave

sensor positioning with regard to the target location and an adequate interpre-

tation of predictions.

(2) Depending on the applied wave model and the method of initialization using

available observation data, the accuracy of the prediction itself will be limited.

The ﬁrst issue has been raised in various publications. In [8,11] and [14] the phase

speed was assumed to govern the predictable zone. Abusedra and Belmont [1] specif-

ically explain why they believe the phase speed governs the predictable zone. On the

other hand Blondel et al. [3], Naaijen and Huijsmans [15], Dannenberg et al. [7] and

Blondel and Naaijen [5] use the group velocity as the governing velocity concerning

predictability. More speciﬁc explanations and observations that support this latter

point of view can be found in [22] and [16].

The aim of this paper is to address this topic again and provide further explanation

of predictability in space and time, supported by numerical simulations and exper-

imental data, thus providing clarity on which wave speed governs predictability. In

order to do this, irregular long-crested waves are considered.

The second issue is not addressed in this paper.

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 205

2. Approach

An observation of the sea surface in a limited spatial, temporal or spatio-temporal

domain is not sufﬁcient to predict the sea surface elevation in the entire ocean for all

time. The limited domain where a prediction can be expected to correlate well with

the true surface elevation will be referred to here as the ‘theoretical predictable zone’.

The concept of theoretical predictable zone will be explained in detail in the section

on predictability, following a brief description of a linear, long-crested wave model

and its initialization from an observed time trace of the surface elevation. The section

on predictability employs the method of stationary phase to explain the relevant wave

speed governing the theoretical predictable zone. This theory will then be veriﬁed by

means of numerical simulations, using both synthetic and experimental data, using

the following approach:

Long-crested linear irregular waves with a given wave spectrum are synthesized in

a spatio-temporal domain (x,t). A time trace of these synthesized waves at one spe-

ciﬁc location xis then used to initialize a linear wave model as described in the next

section. With the wave model, the surface elevation is computed in the entire domain

(x,t) and compared to the synthesized waves. The difference between the computed

(predicted) and the synthesized wave elevation will be quantiﬁed and referred to as

the ‘practical’ prediction error. Additionally, a similar procedure is followed using

experimental data: long-crested waves are generated in a basin, and measured by a

number of probes. The wave elevation at one probe is used to initialize the wave

model which will then compute the wave elevation at the remaining probes. At each

of these remaining probes the ‘practical’ prediction error can be determined. The

theoretical predictable zone indicates where in space and time an ‘accurate’ predic-

tion is possible. Comparing this to the practical prediction error obtained from the

actual simulations will provide insight in the adequacy of the chosen deﬁnition of

the theoretical predictability.

3. A linear model for propagation of long-crested waves

In the following it is assumed that all waves are long-crested and traveling in

the same direction. Under these assumptions, prediction of the wave elevation with a

linear wave model is straightforward and has been addressed several times (Morris et

al. [11], Trulsen and Stansberg [21], Blondel et al. [4], Clauss et al. [6], Naaijen and

Huijsmans [14] to name a few). We will consider the case where the measurement

used as input for the prediction is a time trace of the surface elevation observed at a

ﬁxed location. Then the linear wave model used can be expressed as

η(x,t)=Re∞

0

ˆη(ω)ei(k(ω)x−ωt)dω,(1)

206 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

where ηis the wave elevation, ωis the angular frequency, ˆη(ω) is the Fourier trans-

form of the time series of surface elevation at a reference location, and k(ω)isthe

wavenumber related to ωaccording to the linear dispersion relation.

A discrete form of this integral can be written

η(x,t)=ReM/2

m=0

˜ηmei(kmx−ωmt),(2)

where the complex amplitudes ˜ηmcan be obtained from a Discrete Fourier Trans-

form (DFT) of a time trace of duration T.Hereωm=2πm

T,kmis related to ωm

through the linear dispersion relation, and Mis the number of samples in the time

trace. Having obtained the discrete complex amplitudes, they can be used in Eq. (2)

to compute the prediction for the surface elevation at any required location and time

(x,t).

4. Predictability

4.1. Predictable zone

The limited predictable zone afforded by Eqs (1) or (2) has been discussed before

(e.g. [1,3,5,7,11,14,15,22]). This theory is brieﬂy revisited here, see Fig. 1.

Suppose the wave elevation is measured during the time interval from Oto Tat

location x0.Tis the duration of the measurement. The question is where in space

and time this measurement is useful for prediction of the surface elevation. To ad-

dress this question, we ﬁrst make the trivial observation that each individual term in

Eq. (2) is a sinusoidal oscillation uniform throughout space and time, with crests and

troughs propagating with the phase speed, but with no otherwise localizable infor-

mation in space or time. Localizable information appears through the superposition

Fig. 1. Construction of predictable zone.

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 207

of terms in Eq. (2), in the form of nonuniformities that we shall call wave packets and

that propagate with the group velocity. The ability to give a prediction is therefore a

matter of predicting the propagation of wave packets, which is given by classical ray

theory, or which is captured by the method of stationary phase applied to the integral

in Eq. (1). A wave packet within the interval OT, propagating with one particular

group velocity, will contribute to the wave elevation between sloped lines OA and

TB whose slope equals that particular group velocity, thus contributing to a predic-

tion between t1aand t1bat location x1. A second wave packet with a higher group

velocity, equal to the slope of line TA, can contribute to a prediction between t2aand

t2bat x1. If the mentioned group velocities are the highest and the lowest associated

with the wave spectrum, this leads to a triangular ‘predictable’ zone (OTA) indicated

indarkgrayinFig.1.

In a real-time application of wave prediction, obviously the trace OT has to be

acquired before it can be processed. So in the ideal case, neglecting computational

time, only the part of OTA on the right-hand side of Tcan be considered as ‘pre-

diction’. The remainder of OTA should be referred to as ‘hindcast’. However, this

distinction will not be made in the rest of this paper.

For any time tat any location xwe can sum up the relative amount of energy in

wave packets arriving from the time interval OT at x0. The result of the summation

can be interpreted as a predictability indicator

P(x,t)=ωh

ωlS(ω)dω

∞

0S(ω)dω,(3)

where S(ω) is the energy density spectrum of the wave elevation. ωland ωhare

the lowest and the highest possible frequencies, respectively, for which energy of

wave packets could be propagated from OT at x0to the target time and location x,t.

Frequencies ωland ωhfollow from the highest and lowest possible group velocities

respectively for which this is the case. These group velocities are given by

cgl =x−x0

t−T,(4)

cgh =x−x0

t(5)

and can be interpreted as follows: a wave packet within the interval OT at x0with a

group velocity higher than cgl will have passed location xalready at time t, while a

wave packet within the interval OT at x0with a group velocity lower than cgh would

not have arrived at location xyet at time t.

We have assumed that, since we are dealing with gravity waves, the group velocity

is a strictly decreasing function of frequency, and we have cgl >c

gh.Pdepends on

208 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

xand tdue to the dependence on cgl and cgh and the integration boundaries ωl

and ωh.

The predictability indicator Pproposed here is based on the wave spectrum S

within linear wave theory for a uniform and stationary medium, thus Sitselfisas-

sumed to be independent of xand t. In reality, these assumptions will not hold, and

the spectrum will evolve in space and time.

Following Wu [22], 1 −Pcan be interpreted as a prediction error indicator, quan-

tifying the relative amount of wave energy represented by the ‘unpredictable’ fre-

quencies. (Wu [22] uses √1−P,weuse1−Pinstead.)

A question that has been raised several times [1,16,22] is whether the slope of

the lines bounding the predictable zone is given by the group or the phase speed.

For gravity waves the phase speed exceeds group speed, meaning that the prediction

horizon (the furthest point in the future that can be predicted) would reach further

into the future for group speed. An answer to this question can be found from the

method of stationary phase as indicated by Abusedra and Belmont [1] and brieﬂy

discussed in the next paragraph, or it can be found from simulations as those pre-

sented in Section 5.

Figure 2 shows contour lines with intervals of 0.2 of 1 −Pbased on the wave

spectrum observed during one of the basin experiments carried out for this study

and assuming the group velocity determines ωland ωh. The slopes of the contour

lines depend on the contour level for 1 −P. This can be explained by considering

Fig. 1 again, where it was assumed that there exist a fastest and a slowest propa-

gation speed. It was assumed that the line TA bounds the dark gray area with 100%

predictability (where 1−P=0). On the right-hand side of TA fast propagating wave

packets originating from x0for t>Twill contribute to the surface elevation, and

1−P>0. On the right-hand side of the line TB, even the slowest propagating wave

packets from x0for t>Twill contribute to the surface elevation, and 1 −P=1.

Therefore a ﬁxed value of 1 −Pbetween 0 and 1 should result in a contour line

with a slope between those of lines TA and TB. Deﬁning the theoretical predictable

Fig. 2. Contour plot of theoretical predictability 1−P(x,t) in spatio-temporal domain. (Colors are visible

in the online version of the article; http://dx.doi.org/10.3233/ISP-140113.)

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 209

Fig. 3. Schematic 1 −P(x,t) in spatio-temporal domain for group and phase speed.

zone as 1 −Pbeing smaller than a threshold value, it is obvious that the size and

shape of the predictable zone will depend on the chosen threshold value. As shown

below, contour plots of the practical prediction error from simulations show the same

behavior, the contour plots of the practical prediction error show different slopes for

different contour levels.

Similar simulations as described in Section 5.2 have been carried out by Abusedra

and Belmont [1], who examine plots in the (x,t)-plane indicating whether at a certain

point (x,t), the maximum error obtained from 1000 realizations is less than a given

value. Observing the slopes of the area where this is the case, as done in [1], is

equivalent to observing the slopes of contour lines of the practical prediction error

Err in Eq. (11) at a certain contour level. The slopes that will be found to be bounding

the zone where the practical prediction error Err (or the equivalent practical error

as deﬁned by Abusedra and Belmont [1]) is less than a threshold value will depend

on the threshold, as is the case for the slopes of contour lines of the theoretical

prediction error indicator 1 −Pas explained above. This is conﬁrmed by simulation

results described in Section 5, Fig. 6(b).

Further insight about the relevance of group or phase speed is provided by consid-

ering a cross section of a contour plot instead of the slope of the contours, see Fig. 3.

Schematic theoretical prediction zones, deﬁned as domains within which 1 −Pis

less than an arbitrary threshold value, are drawn for the case of group and phase

speed, indicated by OTAgand OTAp, respectively. Plotting the practical prediction

error found from simulations together with the two versions of the theoretical error

indicator 1 −Pand examining a cross section at the position indicated by the dashed

line would reveal clearly whether it is 1 −Pbased on group or phase speed that

better matches the obtained simulation results for the practical prediction error. Such

ﬁgures are presented in Section 5.

4.2. Method of stationary phase

The method of stationary phase provides an approximate evaluation of the integral

in Eq. (1) and suggests an answer to the question raised here. An explanation of the

method related to linear dispersive waves can be found in e.g. Murray [12], and was

also given by Abusedra and Belmont [1] and is summarized here.

210 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

Equation (1) describes the surface elevation η(x,t) as a superposition of linear

modes with wave numbers kand frequencies ωrelated by the linear dispersion rela-

tion ω=ω(k). Assuming a ﬁxed ratio x/t and examining the properties of the wave

ﬁeld as xand tbecome large, it is convenient to rewrite the complex exponential in

Eq. (1) as

ei(kx−ωt)=eit(kx/t−ω)=eitχ(k,x/t),(6)

where χis the so-called phase function. For large values of t, this exponential repre-

sents rapid oscillations that are such that in the integration in Eq. (1), the positive and

negative parts effectively cancel each other out, except in the vicinity of k0and ω0,

where the derivative of the phase function χwith respect to kis zero

∂χ

∂k (k0,x/t)=0(7)

and consequently we have the group velocity

cg(k0)≡∂ω

∂k (k0)=x/t. (8)

The meaning of this is that in the limiting case of large xand t, for an observer

moving at constant velocity x/t, only wave packets whose group velocity equals

x/t will signiﬁcantly contribute to Eqs (1) or (2). For a point (x1,t) in Fig. 1 to be

predictable, based on an observation at location x0between O and T, all lines that

can be drawn through the point, with slopes corresponding to energetic frequencies

in the wave spectrum, should cross the observation OT. This results in a predictable

subset of points (x,t) indicated by the dark gray triangle OTA in Fig. 1, with the

slopes of lines OA and TA being equal to the group velocities associated with the

highest and lowest frequency, respectively, at which signiﬁcant energy is present in

the observation.

With cgl and cgh in Eqs (4) and (5) being group velocities, the associated frequen-

cies are related by

cgl =1

2+klh

sinh(2klh)ωl

kl

(9)

and

cgh =1

2+khh

sinh(2khh)ωh

kh

, (10)

where his the water depth.

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 211

As rightfully pointed out by Abusedra and Belmont [1], it is not obvious that for

any point in the triangle OTA, the magnitudes of xand tare sufﬁciently ‘large’. In

order to assess the required magnitudes of xand tfor the above analysis to be valid,

we will resort to numerical simulations and laboratory experiments.

5. Simulations

5.1. Procedure

We have employed Monte–Carlo simulations to verify the theory on predictability

using the model outlined in Section 3: a DFT of a time trace of length Tof the

wave elevation observed at one speciﬁc location, which could be the ﬁrst probe for

simulations of the experimental data, is used according to Eq. (2) to make predictions

of the wave elevation at other locations. The result can be compared with synthetic

waves and with measurements from experiments.

Figure 4 shows a sample time trace of predicted and true wave elevation in a time

window around t=t1aand at x=x1as indicated in Fig. 1. The prediction is based

on the observed time trace OT at x=x0. As can be seen, the agreement is less good

at the left-hand side of the ﬁgure, as expected, since at earlier times we are outside

the predictable zone where 1 −Pis increasing.

We deﬁne the mismatch between prediction ηand true wave elevation ηmas

Err(x,t)=(η(x,t)−ηm(x,t))2

2σ2, (11)

where ηis the predicted surface elevation, ηmis the measured or synthesized surface

elevation, σ2is the variance of the measured or synthesized wave elevation averaged

over all probes, · denotes ensemble average.

Fig. 4. Predicted and true wave elevation around left boundary of predictable zone. (Colors are visible in

the online version of the article; http://dx.doi.org/10.3233/ISP-140113.)

212 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

Fig. 5. Obtaining multiple realizations by partially overlapping time traces.

We notice that Err =0.5 in the case that we predict a ﬂat surface η(x,t)=0.

Also Err =1 if the prediction ηhas the same variance but is otherwise uncorrelated

with the measured or synthesized wave elevation ηm. For values Err 0.5wemay

consider the prediction as useless.

For experimental data we achieve an ‘ensemble’ by using partly overlapping time

traces, separated by Δt, see Fig. 5. The interval Δtwas chosen to optimise conver-

gence of the practical prediction error, this is discussed in the Appendix.

From a time trace of length Tat the reference location x0, we obtain a predic-

tion which is expected to be accurate within the theoretical predictable zone. The

theoretical predictable zone, which is the domain where 1 −Pis less than a chosen

threshold, can be determined based on group or phase speed. In case of phase speed,

the propagation velocities in Eqs (4) and (5) would have to be substituted by the

phase speeds

cpl =ωl

kl

(12)

and

cph =ωh

kh

.(13)

Both options are indicated by the triangles in Fig. 3. The practical prediction error

Err is computed for large x−x0, such that both of these theoretical prediction zones

become clearly discernible. This in order to observe whether the practical prediction

error Err conﬁrms the assumed theoretical predictable zone based on group or phase

speed.

5.2. Simulations using synthetic waves data

5.2.1. Synthetization of wave data

A linear wave model has been used to synthesize the wave data. In order to avoid

any systematic coincidence between the frequencies used in the generation of the

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 213

synthetic waves and the ones that are used in its analysis by means of a DFT, gen-

eration has been carried out using non-equidistantly spaced frequencies: the waves

are composed of Ncomponents, each representing an equal amount of wave energy.

N=2000 has been used in this study. Consequently the discrete frequencies are

spaced closer together near the peak of the spectrum and further apart at the low and

high frequency ends of the spectrum. Each realization of the wave ﬁeld is generated

using a different set of random phase angles, thus assuring independence between

the different realizations.

5.2.2. Results

Figure 6 shows contour plots of both the theoretical prediction error indicator,

1−Pbased on group velocity, and the practical prediction error as deﬁned in Eq. (11)

from simulations averaged over an ensemble of size 5000. The wave spectrum Swas

chosen identical with the spectrum observed during one of the basin experiments

mentioned in the next section, a JONSWAP spectrum with a signiﬁcant wave height

Hs=0.024 m and mean zero crossing period T2=0.87 s. Although the result for

Err has converged, as shown in the Appendix in Fig. 12, still an apparently random

variation of the error Err is observed. For this reason Err(x,t) has been smoothed

by a two dimensional ﬁlter in space and time before constructing the contour lines

of Err in Fig. 6(b).

Although 1 −Pis not directly comparable to Err, we do observe that the two

regions outlined in Fig. 6(a) and (b) are both qualitatively and quantitatively quite

similar.

Figure 7 shows the practical prediction error Err against time at x=30 m, which

is a cross-section in time at the location indicated by the dashed horizontal line in

Fig. 6(b). Figure 7 also contains the theoretical prediction error indicator 1 −Pfor

Fig. 6. Contour plots of theoretical error indicator 1 −Pbased on group velocity and practical prediction

error Err from simulations with synthetic wave data. (a) Theoretical prediction error indicator 1−Pbased

on group velocity. (b) Practical prediction Error Err. (The dot is referred to in the Appendix.) (Colors are

visible in the online version of the article; http://dx.doi.org/10.3233/ISP-140113.)

214 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

Fig. 7. Prediction error against time at x=30 m. (Colors are visible in the online version of the article;

http://dx.doi.org/10.3233/ISP-140113.)

the location indicated by the dashed lines in Fig. 6. Two versions of 1 −Pare shown,

based on the group and phase speeds.

It is seen that 1 −Pbased on group velocity quite accurately matches the practical

prediction error Err while poor agreement is found for 1 −Pbased on phase speed.

The simulation results clearly indicate that it is the group velocity that governs the

predictable zone.

5.3. Simulations using experimental wave data

5.3.1. Experiments

We employ data of experiments in two different towing tanks, at the Ship

Hydromechanics Department at Delft University of Technology, The Netherlands

(TUD), and at Ecole Centrale Nantes, France (ECN). During these experiments, ir-

regular waves were generated and measured by an array of probes. Using the mea-

surement at the ﬁrst probe positioned closest to the wave maker, the amplitudes and

phase angles of the terms in Eq. (2) are obtained by DFT. Using the same formula

enables calculation of the wave elevation at the remaining probe locations at any re-

quired moment in time, which then can be compared to the measurements at these

probes, enabling computation of the practical prediction error Err. Figure 8 schemat-

ically depicts the experimental set up with the numbered dots indicating the probes.

The positions of the probes, which are not exactly equidistant, are listed in Table 1.

The water depth during the experiments amounts to 2.13 and 2.81 m for the TUD

and the ECN towing tank, respectively.

Characteristics for the experiments are listed in Table 2 where Hsis the signiﬁcant

wave height, Tpis the peak period, T2is the mean zero crossing period, and εis

average wave steepness

ε=√2kpσ, (14)

where kpis the peak wave number of the spectrum, and σis the standard deviation

of the surface elevation.

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 215

Fig. 8. Layout of probes in basin. The wavemaker (WM) is to the left.

Tab l e 1

Probe positions for TU Delft and EC Nantes experiments

Probe No. Position from WM ﬂap, TUD Position from WM ﬂap, ECN

(m) (m)

1 41.02 18.48

2 53.42 24.43

3 65.92 30.26

4 79.35 36.11

5 90.06 42.29

6 102.33 48.23

7 115.03 54.11

8 127.50 60.29

9 – 66.14

Tab l e 2

Summary of characteristics of wave conditions

Facility HsTpT2ε

(m) (s) (s)

TUD 0.025 0.923 0.885 0.040

ECN 0.031 2.008 1.523 0.011

Both experiments are supposed to correspond to a JONSWAP wave spectrum hav-

ing a peakedness factor γ=3.3.

The data in Table 2 corresponds to the values observed at the ﬁrst probe, i.e. the

reference probe that was used as input for the predictions. The main difference be-

tween the two experiments is the order of magnitude of the probe distances: for the

experiments carried out at TUD this is in the order of 10 peak wavelengths, while for

the ones conducted at ECN, the probes are positioned approximately 1 peak wave-

length apart.

5.3.2. Results

The error indicator 1 −Pand the actual error Err from simulations with the TUD

data are shown in Fig. 9. As was done for the simulations with synthetic data, cross

sections of the contours of Err and 1 −P, i.e. time traces of the practical prediction

error Err (left y-axis) and the theoretical error indicator 1−P(right y-axis), based on

group and phase speeds, are presented for probes 1 (observation probe), 2, 4 and 6.

216 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

Fig. 9. Prediction error and error indicator from TUD data. (a) Probe 6, x/λp=46.1. (b) Probe 4,

x/λp=28.8. (c) Probe 2, x/λp=9.3. (d) Probe 1, x/λp=0.0. (Colors are visible in the online

version of the article; http://dx.doi.org/10.3233/ISP-140113.)

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 217

The bottom ﬁgure represents the observation probe (probe 1) where an observation

of length 128 s was used as input for prediction at the remaining probes.

In order to make a visual inspection of the match between Err and 1 −Peasier,

the left y-axis corresponding to Err has been scaled such that the minimum and

maximum values of Err correspond to values of 0 and 1 for 1 −P, respectively.

This axis scaling is justiﬁed by the fact that 1 −Ponly gives a qualitative indication

of the expected prediction. 1 −Phas been determined according to a numerical

computation of Eq. (3) using the spectrum Sobserved at the observation probe. The

limiting value for Err is not exactly equal to 1 at each probe due to the fact that a

slightly different variance σ2was observed at different probes in the basin.

It is seen that the curves representing Err and 1 −Pbased on group velocity

coincide well, while the assumption of phase speed governing the predictable domain

is quite inadequate. From these ﬁgures we conclude that, at least for the propagation

distances employed in the TUD experiment, the boundaries of the predictable zone

are governed by the group velocity.

Recall that the method of stationary phase requires the propagation time and dis-

tance to be large. As indicated in Fig. 9 the smallest propagation distance, the dis-

tance from probe 1 to probe 2, amounts to 9.3 peak wave lengths. The presented

results using data from the TUD experiment do not indicate what conclusions could

be drawn for the minimum required propagation time and distance for the method of

stationary phase to be useful.

In order to conclude about smaller propagation distances, data from the ECN ex-

periment is considered. Here the distances between the probes, see Table 1, amounts

to approximately one peak wavelength only. Results are shown in Fig. 10. Again, the

bottom ﬁgure represents the observation probe (probe 1). An observation of length

76.8 s was used. Also for these rather limited propagation distances, it is seen that

the theoretical prediction error indicator based on group velocity shows signiﬁcantly

better agreement with the practical prediction error than the theoretical prediction

error indicator based on phase speed.

6. Prediction horizon and optimal prediction distance in practical application

of deterministic wave prediction

In order to relate the above results to the practical application of deterministic

wave prediction in maritime and offshore operations, we will here estimate the min-

imum required distance between observation site and prediction site for typical pre-

diction horizons of 30–120 s [7]. Therefore, the contour plots of theoretical pre-

dictability in Fig. 6 are considered again. On full-scale this ﬁgure corresponds to a

JONSWAP spectrum with a peak period of 5.0 s. The ideal distance between obser-

vation and prediction site would be such that:

•it is large enough for the prediction site to be inside the predictable zone implied

by the observation, and allowing a prediction some time into the future,

218 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

Fig. 10. Prediction error and error indicator from ECN data. (a) Probe 6, x/λp=4.8. (b) Probe 3,

x/λp=1.9. (c) Probe 2, x/λp=1.0. (d) Probe 1, x/λp=0.0. (Colors are visible in the online version

of the article; http://dx.doi.org/10.3233/ISP-140113.)

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 219

Fig. 11. Ideal distance and forecast horizon. (a) 1 −Pagainst distance for Forecast horizon =30 s.

(b) Ideal distance against forecast horizon.

•it is not so large that the accumulation of errors ruins the prediction.

In order to identify these ideal distances, vertical cross sections of the contour

plots of the theoretical predictability in Fig. 6(a) are considered: for a given required

prediction horizon, the ideal distance between observation and prediction is deﬁned

here as the distance from the observation to the closest point where 1 −Pis smaller

than a chosen threshold value of 0.01. Figure 11(a) shows a vertical cross-section

of the contour plot of 1 −Pfor a value of tthat corresponds to a 30 s forecast on

full-scale. The above deﬁned ideal distance is indicated by the circle in Fig. 11(a).

The distance xis normalized by the peak wavelength in this ﬁgure. The observation

site is at x=0. Figure 11(b) shows the ideal observation distance for a range of

forecast horizons between 0 and 180 s. As can be seen, a forecast horizon of 30 s

requires an observation distance between 4 and 5 peak wavelengths, for which it has

been shown from the experiments in Fig. 10 that it is indeed the group velocity that

governs the predictable zone.

7. Conclusion

We have shown that for deterministic wave prediction applied on propagation dis-

tances and forecasting horizons of practical interest to offshore operations, the pre-

dictable zone is governed by the group velocity of the waves.

Acknowledgements

This research has been carried out within the research project ‘PROMISED Op-

erations’ (PRediction Of wave induced Motions and forces In Ship, offshorE and

Dredging Operations), funded by ‘Agency NL’, a department of the Dutch Ministry

220 P. Naaijen et al. / Limits to the extent of the spatio-temporal domain

of Economic Affairs, Agriculture and Innovation and co-funded by Delft Univer-

sity of Technology, University of Twente, Maritime Research Institute Netherlands,

Ocean Waves GmbH, Allseas, Heerema Marine Contractors, IHC. KT was funded

by the University of Oslo (UiO) and the Research Council of Norway (RCN) through

grants 177464/V30 and 214556/F20.

Appendix

Depending on the magnitude of Δt, the realizations constructed as shown in Fig. 5

will be dependent, thus the ensemble size required for convergence may be larger

than for independent realizations.

In order to assess the required ensemble size for independent realizations, we have

synthesized an ensemble of size 700 according to the spectrum of the TUD data

mentioned in Table 2. We have studied the convergence of the practical prediction

error Err at an arbitrary point in the sloped part of the predictable zone.

Figure 12 shows the average prediction error against the number of realizations for

independent realizations. It is anticipated from this ﬁgure that the result is sufﬁciently

converged after approximately 600 realizations. Additional examinations at other

locations within the spatio-temporal prediction zone led to the same conclusion.

Figure 13 shows the convergence in the case of partly overlapping sections of

one time trace of synthetic data generated as described in the previous paragraph.

The dashed lines in Fig. 12 and 13 indicate the value from the entire ensemble of

independent realizations in Fig. 12. Figure 13(b) shows results for increased values

of Δt. Considering some additional results for other times/locations, it was concluded

that the gain in the level of convergence from Δt=0.25 compared to Δt=2.5is

so limited that it justiﬁes the pragmatic approach of using Δt=2.5, resulting in an

ensemble size of 500. This level of convergence was considered to be sufﬁcient in

order to draw the conclusions within the scope of the presented paper.

Figure 13 shows the convergence in the case of partly overlapping time traces of

the experimental data. The dashed lines indicates the value obtained from averag-

ing over the entire ensemble obtained by using Δt=0.25 s, i.e. the last point of

Fig. 12. Prediction error against number of independent realizations.

P. Naaijen et al. / Limits to the extent of the spatio-temporal domain 221

Fig. 13. Prediction error against number of partly overlapping segments, synthetic data. (a) Δt=0.25 s

(1 sample). (b) Δt=2.5, 3.75, 5.0 s (10, 15, 20 samples). (Colors are visible in the online version of the

article; http://dx.doi.org/10.3233/ISP-140113.)

Fig. 14. Averaged prediction error against number of partly overlapping segments, experimental data.

(a) Δt=0.25 s (1 sample). (b) Δt=2.5, 3.75, 5.0 s (10, 15, 20 samples). (Colors are visible in the online

version of the article; http://dx.doi.org/10.3233/ISP-140113.)

Fig. 14(a). It is seen that using a Δtof 10 samples (corresponding to 2.5 s) and with

400 overlapping time traces, the same result is obtained as when using a Δtof 1 sam-

ple (0.25 s) using roughly 4000 overlapping time traces. We therefore conclude that

with the available experimental data, there is no gain in convergence from using a

smaller Δtthan 10 samples (2.5 s). For Δtof 15 or 20 samples (corresponding to

3.75 s or 5.0 s, respectively), small deviations from the optimally converged result

can be observed. Again, for several points in the spatio-temporal prediction zone sim-

ilar investigations were done all of which showed a reasonable level of convergence.

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