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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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