fluids

Article
On the Deterministic Prediction of W ater W aves
Marco Klein 1, *, Matthias Dudek 2 , Günther F . Clauss 3 , Sören Ehlers 1 , Jasper Behrendt 4 ,
Norbert Hof fmann 4 and Miguel Onorato 5
1 Ship Structural Design and Analysis, Hambur g University of T echnology , 21073 Hamburg, Germany
2 Neue W arnow Design & T echnology GmbH, 18055 Rostock, Germany
3 Ocean Engineering, T echnische Universität Berlin, 10623 Berlin, Germany
4 Mechanical Engineering, Hamburg University of T echnology , 21073 Hamburg, Germany
5 Department of Physics, University of T urin, 10125 T orino, Italy
* Correspondence: mar [email protected]
Received: 19 December 2019; Accepted: 27 December 2019; Published: 7 January 2020
     
  

Abstract:
This paper discusses the potential of deterministic wave pr ediction as one basic module for
decision support of of fshor e operations. Ther efor e, methods of dif fer ent complexity—the linear wave
solution, the non-linear Schrödinger equation (NLSE) of two differ ent orders and the high-or der
spectral method (HOSM)—ar e pr esented in terms of applicability and limitations of use. For this
purpose, irregular sea states with varying parameters ar e addressed by numerical simulations as
well as model tests in the controlled envir onment of a seakeeping basin. The irr egular sea state
investigations focuses on JONSW AP spectra with varying wave steepness and enhancement factor .
In addition, the influence of the propagation distance as well as the for ecast horizon is discussed.
For the evaluation of the accuracy of the pr ediction, the surface similarity parameter is used, allowing
an exact, quantitative validation of the r esults. Based on the r esults, the pr os and cons of the dif fer ent
deterministic wave pr ediction methods ar e discussed. In conclusion, this paper shows that the
classical NLSE is not applicable for deterministic wave pr ediction of arbitrary irr egular sea states
compar ed to the linear solution. However , the application of the exact linear dispersion operator
within the linear dispersive part of the NLSE incr eased the accuracy of the pr ediction for small wave
steepness significantly . In addition, it is shown that non-linear deterministic wave pr ediction based
on second-or der NLSE as well as HOSM leads to a substantial improvement of the pr ediction quality
for moderate and steep irr egular wave trains in terms of individual waves and pr ediction distance,
with the HOSM pr oviding a high accuracy over a wider range of applications.
Keywords:
decision support system; non-linear wave pr ediction; high-or der spectral method;
second-or der non-linear Schrödinger equation; JONSW AP Spectrum
1. Introduction
Ships and of fshor e structur es are exposed to the sea, limiting the scope of application of of fshore
operations, fr om ef ficient and economic of fshor e operations in moderate sea states to r eliability as well
as survival in extr eme wave conditions. T o classify offshor e operating conditions, predefined limiting
criteria such as absolute or r elative motions ar e typically combined with a limiting characteristic wave
height or sea state based on stochastic analysis in the design pr ocess, allowing identifying feasible and
infeasible sea states, respectively . T ypically , these limiting sea states are further r educed in terms of
allowable significant wave heights by insur er and surveyor of of fshor e operations. This appr oach limits
the applicability of of fshor e operations strictly as most sea states (in particular in the transition area
between feasible and infeasible r egion of a scatter diagram) will featur e favourable wave sequences
allowing short-term of fshor e operations which are elapsed unused. Knowing the future in terms of
deterministic pr ediction of the encountering wave field and structur e motion a sufficient time span
Fluids 2020 , 5 , 9; doi:10.3390/fluids5010009 www .mdpi.com/journal/fluids

Fluids 2020 , 5 , 9 2 of 19
in advance would allow r educing the downtime as the critical part of an offshor e operation typically
corr esponds to a short time interval of the whole operation. In addition, a decision support system
(DSS) based on deterministic wave pr ediction can also detect critical wave gr oups in pr etended feasible
sea states incr easing the safety of complex operations. The impact of such a DSS on shipping as
warning system (e.g., parametric r oll, extr eme wave events, extr eme motions and structur e response)
should not be left unmentioned even though the focus may lie on of fshor e operations.
The established r esear ch on deterministic wave pr ediction comprises two indispensable
constituents: sea state r egistration and wave pr ediction. The deterministic structure motion pr ediction
based on the wave pr ediction is a further key element for a DSS but will not be discussed in detail
within this paper . For the registration of the futur e encountering wave field, the surface elevation has
to be measur ed at a certain distance fr om the pr ediction point (e.g., location of of fshor e operation).
Dif fer ent measur ement methods ar e available, from point measur ements (e.g., wave buoy) in time
domain to surface elevation snapshots taken fr om the ship’s X-band radar in the space domain. As the
focus of the paper lies on wave pr ediction exclusively , the following brief description of the sea state
r egistration methods are only intr oduced in order to support the appr oach presented in this paper
without any technical details.
Point measur ements ar e disadvantageous compar ed to a surface elevation snapshot as a
single point measur ement cannot provide information r egarding the dir ectionality of the sea state.
This pr oblem may be solved by using a large array of several point measur ement devices in order to
identify the dir ectionality of the wave components. However , as the availability of point measurement
devices in the open ocean ar e generally limited, this approach would denote to install such an array
prior to the of fshor e operation excluding a plug-and-play application. In addition, this approach is
impractical for DSS of ships or offshor e structur es with forwar d speed. A further drawback is the
principle of measur ement as point measur ements denote that the wave field is r ecorded at a fixed
location over a specific time. As a consequence, the prediction time (time between the moment at
which the pr ediction is available and the moment at which the pr edicted waves physically arrive at
the pr ediction location) is r educed by the r ecor ding time, independently of the computational time
needed for the used wave pr ediction method [ 1 – 4 ].
The ship’s X-band radars pr ovide the surface elevation for domains of several square kilometr es
including information on dir ectionality . Theor etically , the detected wave field can be available after
every r evolution of the radar antenna (<2 s), even though time is needed for the inversion of the radar
clutter (due to the pr esence of the waves) to the wave field. In contrast to single point measur ements,
wher e a suf ficient measuring time is needed for an accurate and suf ficiently long prediction, this time
lag is only limited by the numerical ef ficiency of the used algorithm. Unfortunately , the principle
of measur ements of X-band radars also yield a main drawback as the radar cannot detect wave
tr oughs behind (steep) wave cr ests. Thus, the accuracy is dependent on the wave inversion algorithm.
However , the advantages of using a ship’s X-band radar (plug-and-play , very fast wave registration
over a lar ge space domain, directionality of the wave field, and large pr ediction horizon) seems to
outweigh the disadvantages as two commerci al prediction systems ar e available from the companies
Applied Physical Sciences Corporation (Futur eW aves
TM
[
5
]) and Next Ocean
TM
[
6
]. Even this paper
focusses on wave pr ediction exclusively , the presented r esearch assumes surface elevation snapshots
(taken continuously by a ship boar d radar at gr eat distance ahead the operational ar ea) as input for the
deterministic wave pr ediction.
One principal point of the development process of deterministic wave pr ediction is the
implementation of very fast algorithms to obtain the pr edicted wave field in a sufficient time span
befor e the physical wave field arrives at the pr ediction location. Only a few methods ar e capable
due to the contrary specifications of very fast calculation time and high accuracy at once. The fastest
method but also simplest one is the linear theory , which has alr eady been applied for wave pr ediction
applications with pr omising findings [
2
,
7
–
15
]. A Shipboar d Routing Assistance system (SRA) based
on the continuous ship’s X-band radar measurements of the surr ounding seaway were pr esented

Fluids 2020 , 5 , 9 3 of 19
by Payer and Rathje [
7
]. The linear evolution of continuously measur ed surface elevation snapshots
using the ship’s X-band radar ar e also the basis for a decision support system for Computer Aided
Ship Handling (CASH) developed by Clauss et al. [
8
]. It was shown that this tool can predict the
encountering wave field as well as the structur e response fairly accurate for moderate long cr ested
sea states. Later , Clauss et al. [
12
] and Kosleck [
2
] enhanced this tool to pr edict the short cr ested
encountering wave field and the corr esponding 6DOF motion. T yson and Thornhill [
16
,
17
] applied
linear wave prediction using W aMoS II
TM
on full scale tests evaluating the pr edictive skills of the
linear methods. They showed that for short for ecast duration and small to moderate wave steepness,
the accuracy of the linear approach is suf ficient. In addition, Naaijen and Huijsmans [
9
,
11
] as well as
Naaijen et al. [
10
,
15
] applied linear wave evolution equations for r eal time wave prediction and ship
motion estimation in long as well as short cr ested waves concluding “that a 60 s accurate for ecast of
wave elevation is very well feasible for all consider ed wave conditions and motion pr edictions are
even mor e accurate”. Due to the simpleness and r obustness, the linear appr oach is an integral part of
commer cially available pr ediction system (e.g., Futur eW aves TM [ 13 , 14 ], Next Ocean TM [ 6 ]).
However , the linear approach implies uncertainties due to its str ong simplifications of the water
wave pr oblem. Non-linear ef fects become dominant with increasing wave steepness r educing the
accuracy of the linear appr oach significantly . Non-linear methods enable advanced simulations with
high accuracy but at the expense of computation time. One of the few non-linear and numerical
ef ficient methods suitable for wave pr ediction ar e the so-called envelope equations of the NLSE
framework [
18
–
21
]. Both the NLSE and the modified NLSE (MNLSE) were extensively experimentally
and numerically investigated in terms of non-linear wave evolution (e.g., [
22
–
26
] among others).
Even if the NLSE captur es r elevant non-linear phenomena and pr ovides good accuracy for sufficiently
narr ow spectra, it was generally shown that the NLSE is less suitable for the prediction of irr egular sea
states due to the spectral bandwidth constraint. The MNLSE on the other side pr ovides significant
better agr eement due to the br oadening of the bandwidth constraints (e.g., [ 22 – 26 ]).
Ruban [
27
] showed that the NLSE is suitable for the pr ediction of extr eme waves applying the
Gaussian variational ansatz to the NLSE in or der to obtain a semi-quantitative pr ediction of non-linear
spatio-temporal focussing. Farazmand and Sapsis [
28
] pr esented a r educed-or der pr ediction of r ogue
waves using MNLSE. Their pr ocedur e is based on the identification of elementary wave groups
(EWG) assuming that critical EWGs do not interact with other EWGs within the short-term pr ediction.
The measur ed irr egular sea state is decomposed into EWGs and the evolution and amplitude growth of
critical EWGs is determined by pr ecomputed EWGs based on MNLSE. Thus, the numerical effort for the
pr ediction of extr eme events is r educed to the proper decomposition of the sea state and identification
of the underlying, pr ecomputed EWG. Cousins et al. [
29
] showed a similar appr oach: a data-driven
pr ediction scheme for the pr ediction on extreme events. This approach can also divided into two
separate components. The first component provides the data base for the pr ediction of extr eme events,
is based on the MNLSE and can be applied prior any r eal world application. The basis ar e localized
wave gr oups which ar e numerically evolved for varying amplitudes and periods resulting in specific
gr oup amplification factors. The second component is the r eal-time extr eme wave pr edictor using field
measur ements. W ithin the measur ements, coherent wave gr oups are to be identified and the futur e
elevation is estimated fr om the data base. Again, the numerical ef fort is reduced to decomposition of
the irr egular wave sequence, identification of coher ent wave gr oups, and r esulting amplification.
The NLSE as well as MNLSE has also been used for deterministic wave prediction (e.g., [
26
,
30
]).
Simanesew et al. [
30
] applied both for the deterministic prediction of long- as well as short-cr ested
sea states in time domain. They showed that both the NLS and MLSN pr ovide sufficient accuracies
in long-cr ested sea states for short for ecast distances and horizons. For incr easing dir ectionality
the quality of the forecast decr eases significantly . Of particular note in this investigation is that
the linear dispersion behaviour is captur ed exactly by using the dispersion operator intr oduced by
T rulsen et al. ([ 21 ]).

Fluids 2020 , 5 , 9 4 of 19
An alternative non-linear method for wave pr ediction is the numerically ef ficient high-or der
spectral method (HOSM) [
31
–
33
]. W u [
1
] as well as Blondel et al. [
34
] applied the HOSM for the
deterministic r econstruction and pr ediction of non-linear wave fields. The basis of their work are
wave r egistrations based on one or several wave probes at specific locations. Thus, sophisticated
optimization pr ocedur es for the r econstruction of the wave field in space wer e used at the beginning
as snapshots in the space domain ar e r equir ed as initial condition for the HOSM. This time-consuming
r econstruction pr ocess hindered an ef fective application of the HOSM for wave prediction. However ,
taking advantage of the supposed drawback by using surface elevation snapshots continuously taken
by a ship’s X-band radar at gr eat distance as input, enables the use of the ef ficient and accurate
HOSM without time-consuming r econstruction of the wave field in space domain. Nevertheless, they
generally showed that long-time and lar ge-space simulations of non-linear sea state evolutions can be
performed accurately and ef ficiently with the HOSM. Clauss et al. [
35
] and Klein et al. [
36
] pr esented
that the HOSM pr edicts non-linear wave gr oup propagation very accurately . The applicability of the
HOSM for non-linear r eal-time pr ediction was shown by Köllisch et al. [
4
]. Desmar et al. [
37
] applied
the HOSM for the generation of refer ence wave snapshots in order to evaluate r econstruction methods
of dif fer ent complexity and the consequences on the pr ediction accuracy . This investigation showed
that non-linear methods for wave r econstruction (based on spatio-temporal optical measur ements) are
impr oving the accuracy of the r econstructed initial wave field and pr ediction.
This paper pr esents a comparative study on the accuracy of intended wave prediction methods of
dif fer ent complexity . The objective is the evaluation of the applicability of the utilised methods for
accurate deterministic wave pr ediction using irr egular sea states. The focus lies on the investigation of
the influence of the wave steepness, wave pr opagation distance and shape of the underlying spectrum.
T o be consistent with a possible r eal world application (input snapshots from ship’s X-band radar),
the numerical and experimental investigations are performed in space domain. Herein, the unavoidable
radar shadow ar ound the ship’s position is also modelled and the influence is evaluated. For the
experiment’s validation, a semi-experimental validation procedur e is introduced as measuring the
surface elevation in space domain is almost impractical in the controlled envir onment of a seakeeping
basin. The accuracy of the pr edictions is evaluated with the surface similarity parameter allowing an
exact, quantitative evaluation.
2. W ave Theory
W e now pr esent a brief description of the water wave pr oblem without attempting to be
compr ehensive, herein only those methods ar e presented which ar e relevant for this paper . At the
beginning the boundary value problem of potential flow theory is briefly intr oduced, followed by the
used methods pr esented in the or der of complexity , fr om linear wave theory via weakly non-linear
envelope equations to (fully) non-linear simulations. At the end, the fully non-linear numerical wave
tank waveTUB is additionally pr esented as waveTUB depict the key element for the semi-experimental
validation pr ocedur e. For this study , the water wave pr oblem is simplified to long-crested waves.
Assuming that the Newtonian fluid is incompr essible, inviscid and irrotational, the evolution
of long-cr ested waves (travelling in x-dir ection) can be mathematically described by the following
governing equations:
4 Φ = 0, (1)
Φ z − Φ x ζ x − ζ t = 0 on z = ζ ( x , t ) , (2)
g z + 1
2 ( O Φ ) 2 + Φ t = 0 on z = ζ ( x , t ) , (3)
Φ z = 0 on z = − d , (4)
with
4 ≡ ( ∂ 2 / ∂ x 2
,
∂ 2 / ∂ z 2 )
,
O ≡ ( ∂ / ∂ x
,
∂ / ∂ z )
and the subscripts r epr esent the corr esponding
derivation. The bottom and the unknown fr ee surface r epr esenting the boundaries of the fluid

Fluids 2020 , 5 , 9 5 of 19
domain. Equation ( 4 ) pr esents the boundary condition at the bottom which is consider ed to be
level, rigid and impermeable. For the dynamic boundary condition at the fr ee surface (Equation ( 3 )),
the dynamic pr essure is defined to be constant equalling the atmospheric pr essure. The kinematic
boundary condition at the fr ee surface (Equation ( 2 )) defines that particles, being element of the fr ee
surface at r est, must not leave the free surface in the pr esence of waves. The kinematic as well as
dynamic boundary condition have to be fulfilled at the unknown fr ee surface
z = ζ ( x
,
t )
complicating
the solution of the boundary value pr oblem significantly .
2.1. Linear W ave Theory
Li ne ar wa ve t heo ry c an be d ed uce d fr om th e Ca uch y pr o bl em ( Equ at ion s ( 1 )–( 4 ) ) by ap pl yin g
pe rt urb at ion t he ory f or t he un kn own p ot ent ia l and s ur fac e el eva ti on. In a dd it ion , th e bou nd ary v al ue
pr ob l em a t th e unk no wn su rf ace c an b e app r ox im at e by T ay lo r se ri es ex pa nsi on . F o r St ok es wa ve t heo ry ,
th e pe rtu rb ati on p ara me ter i s r el at ed t o the w av e ste ep nes s
e = ζ a k
. T ru nca ti on of t he s er ie s exp an sio n
at o r de r
O ( e 1 )
(a ss umi ng t hat t he w ave h ei ght i s si gni fi can tl y sma ll er co mp ar ed t o the w av e len gt hs)
r es ul ts i n a si mpl if ied b ou nda ry v alu e pr o bl em f or wh ic h the a na lyt ic al li ne ar so lu tio n ca n be de te rmi ne d.
Fo r ir r eg ul ar su rf ace e le v at io ns of l on g-c r es te d wa ves i n ti m e an d sp ace , th e s ol ut ion r ea ds
ζ ( x , t ) = ∑
n
ζ a n cos ( k n x − ω n t + φ n ) , (5)
with
ζ a n
,
k n
,
ω n
and
φ n
as amplitude, wave number , angular frequency and phase of the
n t h
component wave. The desir ed sea state is regar ded as superposition of independent harmonic
component waves, each with a particular amplitude, frequency and phase. Applying Equation ( 5 ),
a measur ed irr egular surface elevation can be transformed at any position in time and space enabling
a very fast and simple approach for deterministic wave pr ediction. The analytical basis for irregular
sea states is the Fourier transform. The angular frequency and wave number ar e coupled via the
dispersion r elation
ω n = q k n g tanh ( k n d ) , (6)
with water depth
d
and gravitational acceleration
g
. The linear description of the natural
seaway enables a simple handling of a complex process with widely acceptable r esults for
engineering applications.
2.2. Non-Linear Schrödinger-T ype Equations
The classical NLSE can be derived similarly to the linear theory by applying perturbation and
T aylor series expansions on the Cauchy pr oblem [
18
,
19
]. Assuming small amplitude waves and a
narr ow bandwidth spectrum, the perturbation parameters ar e the wave steepness
e = k c ζ a
and the
r elative bandwidth
µ = ∆ k / k c < <
1 (assuming
µ 2 ∝ e
). The T aylor series expansions about still water
level is intr oduced in or der to simplify the surface boundary conditions. Furthermor e, the boundary
value pr oblem is separated into the dif fer ent or ders
O ( e ( n ) )
and each or der is solved sequentially
fr om the lowest to the highest or der . T runcation of the perturbation expansion at or der
O ( e 3 )
leads
to the NLSE. The NLSE captur es some important aspects of non-linearity of water waves such
as modulation instability and pr ovides exact solutions enabling fundamental r esear ch on weakly
non-linear wave-wave interaction. However , the NLSE is limited in non-linearity magnitude and
spectral bandwidth hindering a diversified application as deterministic wave pr ediction method.
These limitations can be r educed by including higher or der terms of the asymptotic series expansion.
The next higher or der
O ( e 4 )
leads to the well known MNLSE [
20
,
21
]. Brinch-Nielsen and Johnson [
38
]
derived the MNLS for arbitrary water depth. However , in most of the cases, the MNLSE is applied
assuming infinite water depth and is typically referr ed as Dysthe equation. Recently , Slunyaev and
Pelinovsky [
39
] extended the Dysthe equation to the next or der (
O ( e 5 )
) r eferring to higher -or der
Dysthe equation.

Fluids 2020 , 5 , 9 6 of 19
Generally , and not only with r egar d to this study , the water depth influence cannot be neglected
for deterministic wave pr ediction as of fshor e operations ar e conducted in arbitrary water depth.
For the NLSE, the derivation of the necessary coef ficients taking the water depth influence into account
can be found in Mei [
40
]. For the next higher or der , Sedletsky [
41
] derived a second-or der NLSE
(
O ( e 4 )
) for arbitrary water depth, which reduces to the classical Dysthe equation in the limit of
infinitely deep water . Based on this outcome, Slunyaev [
42
] derived the thir d-or der NLSE (
O ( e 5 )
) for
arbitrary water depth investigating the impact of the dif ferent non-linearity or ders on the modulation
instability . For this study on deterministic wave pr ediction, the NLSE up to the second-order is used in
the following ways.
Accor ding to Slunyaev [
42
], truncation of the perturbation expansion at or der
O ( e 4 )
leads to the
temporal evolution equation
i  ∂ A
∂ t + V ∂ A
∂ x  + β 1
∂ 2 A
∂ x 2 + α 1 | A | 2 A + i β 2
∂ 3 A
∂ x 3 + i α 21 | A | 2 ∂ A
∂ x + i α 22 A 2 ∂ A ∗
∂ x = 0, (7)
with
A
being the envelope of the wave sequence,
V
the gr oup velocity of the carrier wave,
the superscript
∗
denotes conjugate complex. The indices of the coef ficients
α
and
β
mark the specific
or der of the underlying harmonic (first index) as well as or der of expansion (second harmonic),
which is adopted fr om Slunyaev [
42
] in or der to be consistent with the pr esented equations for the
coef ficients (summarized in Appendix A ). The group velocity depends on the carrier wave number
k c
,
the corr esponding angular fr equency ω c = p k c g tanh ( k c d ) ) and the water depth d ,
V = ω c
2 k c  1 + 2 k c d
sinh ( 2 k c d )  . (8)
The first four terms in Equation ( 7 ) display the classical first or der NLSE and the underlying wave
field can be r econstructed as follows:
ζ = R e ( A exp ( i ω c t − i k c x ) ) . (9)
T runcation at or der
O ( e 4 )
r esults in Equation ( 7 ) r epr esenting the solution for the second order
NLSE (
NLSE 2
) equivalent to the MNLSE (and Dysthe equation for
k c d → ∞
). The reconstruction
formula r eads
ζ = r 01 | A | 2 + R e  A exp ( i ω c t − i k c x ) ) + R e ( r 21 A 2 exp ( 2 i ω c t − 2 i k c x )  , (10)
with contributions of the zer oth, first and second harmonics.
The dispersive behaviour of the waves is r epresented by the second term and all terms r elated
to
β n m
; the non-linear wave-wave interaction corr esponds to the
α n m
terms. The dispersive terms in
Equation ( 7 ) ar e the consequence of the T aylor series expansion of the linear dispersion r elation so that
the dispersion of the waves is repr esented in the vicinity of the underlying carrier wave number
k c
.
For incr easing or der , the obtained dispersion behaviour appr oximates to the full linear dispersion.
However , T rulsen et al. [ 21 ] introduced the full linear dispersion operator
ˆ
L = q ( k − k c ) g tanh ( ( k − k c ) d ) ) − ω c , (11)
wi th
k
r ep r ese nt ing a n ar ray o f wa ve nu mb ers a nd t he hi gh er or de r dis pe rsi ve t erm s ca n be ca lc ula te d by
ˆ
F − 1  ˆ
L ˆ
F { A }  = i V ∂ A
∂ x + β 1
∂ 2 A
∂ x 2 + i β 2
∂ 3 A
∂ x 3 + β 3
∂ 4 A
∂ x 4 + ..., (12)
with ˆ
F marks the Fourier transform and ˆ
F − 1 the inverse Fourier transform, i.e., the operation is done
in Fourier space by means of pseudo-spectral appr oach. Equation ( 12 ) takes terms beyond the highest

Fluids 2020 , 5 , 9 7 of 19
or der (
O ( e 4 )
) consider ed in this paper into account and is formally not consistent with the series
expansions truncated at specific or ders. Nevertheless, introducing Equation ( 12 ) into Equation ( 7 )
pr ovides the advantage that the linear dispersive behaviour is fully taken into account r esulting in
incr easing accuracy compar e to the classical T aylor series expansion of the linear dispersion relation
at each selected or der . Another advantage is more practical, as in particular for the determination
of higher or der derivatives via pseudo-spectral methods, the difficulties with numerical noise and
aliasing ef fects ar e r educed.
For this study , the two non-linear envelope equations resulting fr om Equation ( 7 ) are implemented
in or der to investigate the influence of the dif fer ent or ders on the wave pr ediction accuracy . For each
or der , the full dispersion operator (Equation ( 12 )) is applied. In addition, the T aylor series expansion
of the linear dispersion is also applied for the NLSE (first four terms in Equation ( 7 )) to investigate and
evaluate the benefit of applying the full dispersion operator . The classical NLSE simulations including
T aylor series expansion of the linear dispersion ar e performed by implementing the pseudo-spectral
split-step method. Hereby , the linear and non-linear part of the NLSE are determined separately .
The linear part is calculated in fr equency domain applying Fourier transform and the non-linear part in
time domain. For all other simulations including the full dispersion operator , the linear and non-linear
parts ar e determined together with a pseudo-spectral appr oach at each time step and advanced in time
with the midpoint finite-dif fer ence appr oximation. The carrier war e number
k c
necessary to run the
simulations is determined for each snapshot by applying
k c = R k · E ( k ) P d k
R E ( k ) P d k , (13)
with
E
the variance spectrum of the snapshot and
P =
5 as weighting factor to weight the result towar d
the visual peak [ 43 ].
2.3. High-Order Spectral Method
The HOS method was intr oduced independently by W est et al. [
31
] and Dommermuth and Y ue [
32
]
(cf. T anaka [
33
]). This pr ocedure enables the non-linear simulation of short-cr ested sea states and
takes all non-linear interactions, resonant and non-r esonant, into account. In addition, wave-current as
well as wave-bottom interactions can be considered. For our investigation, the numerical procedur e
pr esented in W est et al. [
31
] is implemented using a pseudo-spectral method. Specifically , all derivatives
r elated to the potential and surface elevation ar e determined in Fourier space assuming periodic
boundary conditions while the non-linear products ar e calculated in physical space. “The (near)
linear computational ef fort” as well as “exponential conver gence (...) are notable characteristics of the
computational ef ficacy of HOS methods” [ 44 ].
At the beginning, the Cauchy problem is converted to equations at the fr ee surface
Ψ ( x
,
t ) ≡
Φ ( x
,
ζ ( x
,
t )
,
t )
. Using chain rules, the free surface boundary conditions (Equations ( 2 ) and ( 3 )) can be
r ewritten as:
ζ t = − Ψ x ζ x + W ( 1 + ( ζ x ) 2 ) on z = ζ ( x , t ) , (14)
Ψ t = − g ζ − 1
2 ( Ψ x ) 2 + 1
2 W 2 ( 1 + ( ζ x ) 2 ) on z = ζ ( x , t ) , (15)
with W = Φ z | z = ζ as vertical velocity at the fr ee surface.
Using Equations ( 14 ) and ( 15 ) as fr ee surface boundary conditions, the boundary value pr oblem
is now exclusively r elated to the vertical velocity
W ( x
,
ζ ( x
,
t )
,
t )
(in space domain) and the solution for
W ( x
,
ζ ( x
,
t )
,
t )
in terms of
ζ ( x
,
t )
and
Ψ ( x
,
t )
can be determined by series expansion. The pr ocedur e
pr oposed by W est et al. [
31
] starts fr om the formal expression that the velocity potential
Ψ ( x
,
t )
and
vertical velocity
W ( x
,
ζ ( x
,
t )
,
t )
can be r epr esented as T aylor series expansion a
z =
0. Assuming that
Ψ ( x
,
t )
and
ζ ( x
,
t )
ar e quantities of
O ( ζ n )
,
φ ( x
,
t )
and
W ( x
,
t )
ar e expanded by perturbation series,

Fluids 2020 , 5 , 9 8 of 19
with
ζ
as or dering parameter and
M = m +
1 is the order of appr oximation of non-linearity . Separating
the terms of each or der O ( ζ n ) yields,
φ ( 0 ) ( x , t ) = Ψ ( x , t ) , (16)
for the first or der . The next higher order solutions ar e obtained successively from the next lower
or der solution,
φ ( m ) ( x , t ) = −
m
∑
n = 1
ζ n
n !
∂ n
∂ z n φ ( m − n ) ( x , t ) . (17)
The vertical velocity W at the surface ζ is obtain by
W ( m ) ( x , t ) = −
m
∑
n = 0
ζ n
n !
∂ n + 1
∂ z n + 1 φ ( m − n ) ( x , t ) . (18)
Assuming periodic boundary conditions, the Fast Fourier T ransform (FFT) can be used for
determining
φ ( m ) ( x
,
t )
and its derivatives as well as
ζ x
. For arbitrary water depth, the wave-bottom
interaction on φ ( m ) ( x , t ) can be derived in Fourier space by
φ ( m ) ( x , t ) = 1
2 π
∞
Z
− ∞
ˆ
φ ( m ) ( k , t ) cosh ( | k | ( z + h ) ) e i k x d k , (19)
with ˆ
φ ( m ) ( k , t ) as Fourier coef ficient.
For this study , a low pass filter accor ding to W est et al. [
31
] is implemented to avoid Fourier space
aliasing for the higher or der terms and the series expansion was expanded up to the fourth order
(
M =
4). T o suppr ess high frequency contamination occurring for the highest waves, which can cause
numerical instabilities [
35
], an exponential damping term was additionally introduced in the Fourier
space. However , the procedur e is neither capable of simulating steep waves close to br eaking nor
handling wave br eaking ef fects. In this study , the fourth order Runge-Kutta-Gill method is applied to
advance the evolution equations in time.
2.4. W aveTUB
This potential theory solver was developed at the T echnische Universität Berlin (TUB) for the
simulation of non-linear wave propagation [
45
]. The two-dimensional, non-linear fr ee surface flow
pr oblem (Equations ( 1 )–( 4 )) is solved in time domain. The velocity potential is calculated in the entire
fluid domain with the Finite Element Method and at each time step a new boundary-fitted mesh is
cr eated. Based on the calculated velocity potential, the velocities at the free surface ar e determined by
second-or der dif fer ences. The fourth-or der Runge-Kutta formula is applied to develop the solution in
time domain. At one side of the numerical wave tank, a numerical beach is implemented by adding
artificial damping terms to the kinematic and dynamic free surface boundary condition enabling
long term simulations. On the other side, a moving wall is implemented for the generation of waves
enabling the simulation of piston-type, flap-type and double flap-type wave boards. A detailed
description of the theor etical backgr ound can be found in Steinhagen [
45
]. The established program
waveTUB showed a high accuracy over r ecent years for a multitude of dif fer ent tasks (cf. [ 45 – 47 ]).
3. Experimental Program And Results
The following study comprises investigation of irr egular sea states based on JONSW AP spectra
evaluating the influence of wave steepness, the shape of the underlying spectrum (in terms of
enhancement factor
γ
) and the wave pr opagation over lar ge distances. The above introduced intended
wave pr ediction methods ar e used and the accuracy is evaluated quantitatively . A realistic r eal world
application is simulated by taking eight consecutive snapshots per sea state (similar to a ship’s radar)

Fluids 2020 , 5 , 9 9 of 19
into account. In addition, the unavoidable radar shadow is also modelled and its influence on the
pr ediction accuracy is discussed. Following, all data are pr esented in full scale (model scale 1:75).
T able 1 pr esents the investigated sea states based on JONSW AP spectra with varying significant
wave height
H s
and enhancement factor
γ
. Therefor e, three dif ferent significant wave heights ar e
investigated based on pr eselected wave steepness
e = k p · H s / 2
marking small, moderate and steep
irr egular sea states. For each wave steepness, the enhancement factor
γ
is varied. A special feature of
this investigation is the implementation of an identical phase distribution (at the wave boar d) for all
investigated sea states. This allows an exclusive evaluation of the influence of wave steepness and
bandwidth of the underlying spectrum. Thus, differ ences in the evolution of the sea states (and thus
to the accuracy of the intended pr ediction methods) ar e exclusively r elated to the underlying wave
spectrum allowing conclusions on the ef fect of non-linear wave propagation due to wave steepness as
well as band width of the spectrum.
T able 1. Overview of the investigated irregular sea states
No. T p k p d H s e γ
1
8.58 s 4.1
1.83 m 0.05
1
2 3
3 6
4
3.66 m 0.1
1
5 3
6 6
7
5.48 m 0.15
1
8 3
9 6
The experiments ar e performed in the seakeeping basin of the Ocean Engineering Division
at TUB. The basin is 110 m long and 8 m wide with a measuring range of 90 m. The water depth is
1 m. The waves ar e generated by an fully computer controlled, electrically driven wave generator ,
which can be used in piston as well as flap-type mode. The implemented wave generation software
enables the generation of r egular waves, transient wave packages, irregular sea states as well as
tailor ed (critical) wave sequences. On the opposite side, a wave damping slope is installed to suppress
disturbing wave r eflections.
The accuracy of the differ ent wave prediction methods applied on the measur ed irregular
sea states is investigated with an semi-experimental appr oach. The afor ementioned real world
application—deterministic wave pr ediction based on ship’s X-band radar —serves as pr ototype.
Thus, surface elevation snapshots in space domain ar e used as input. In addition, fixed prediction
locations ar e defined at which the pr ediction in time domain is pr esented. The influence of the
radar shadow on the pr ediction accuracy is additionally investigated. T o avoid time-consuming and
expensive series of measur ements, which would be necessary to detect the surface elevation in space
domain in the seakeeping basin, a semi-experimental approach is intr oduced for the preparation of
the input snapshots. Measuring the surface elevation in space domain would denote that plenty of
successive, pointwise measur ements have to be conducted (with suitable calm down time in between)
to obtain the snapshot.
The semi-experimental appr oach enables significantly fewer measur ements by mor e validation
scenarios at the same time. The fully non-linear numerical wave tank waveTUB is applied for
the determination of the r equir ed surface elevation snapshots for the deterministic wave pr ediction.
The seakeeping basin, used for the experimental validation, can be modelled exactly with this numerical
wave tank (including the wave maker) resulting in identical boundary conditions at the wave boar d,

Fluids 2020 , 5 , 9 10 of 19
i.e., pr oviding an identical starting point of the wave evolution for both tanks. Thus, the irregular sea
states ar e investigated in the seakeeping basin and r epr oduced with the numerical wave tank waveTUB
pr oviding the input wave snapshots for the deterministic wave pr ediction. The input snapshots based
on waveTUB as well as the pr ediction r esults ar e experimentally validated.
Figur e 1 pr esents the general scheme of this pr ocedur e showing the side view on the wave tank
(all data full scale). On the left hand side the wave maker and on the right hand side a wave damping
beach is installed/modelled. A surface elevation snapshot at a specific time is also indicated which is
the basis for the intended wave pr ediction methods. For the pointwise validation of the waveTUB
input snapshots as well as wave pr edictions by measur ements in time domain at fixed positions, four
wave gauges ar e installed during the experiments. The waveTUB repr oductions are compar ed with
the experiments near the wave maker at wave gauge 1 to draw conclusions on the quality of the
r epr oduction. The predicted wave sequences for thr ee differ ent pr ediction distances ar e validated at
the other thr ee wave gauges (2-4).
Figure 1.
General scheme of the applied semi-experimental procedur e for the irregular sea states
investigations including validation locations.
The waveTUB simulations ar e used to extract eight consecutive snapshots fr om each sea
state simulating consecutive snapshots taken fr om the ship’s radar of a real world application.
Each consecutive snapshot is used as input for the wave pr ediction for the three location (cf. Figure 1
wave gauges (2-4)). Figur e 2 presents exemplary the applied pr ocedure in detail for sea state 1.
The surface elevation snapshots (gr ey curves) taken fr om the waveTUB simulations ar e pr esented in
the left diagrams—the consecutive time steps ar e displayed fr om top to bottom. In dependence on
the r eal world application, the real world radar shadow close to an of fshore structur e is simulated
by modifying the waveTUB r esults. Assuming that wave gauge 2 is the location of the radar and a
r ealistic radar shadow of 500 m, the radar shadow is simulated by changing the surface elevation to
zer os in this ar ea (as well as behind wave gauge 2). The modified surface elevation snapshots are
shown as r ed curves in the left diagrams of Figur e 2 . In addition, the positions of the four wave gauges
used in the experiments ar e indicated by the black vertical lines.
The measur ed surface elevations (blue curves) in time domain at wave gauge 2 ar e exemplary
compar ed with the HOSM simulations (r ed curves) on the right hand side of Figur e 2 —the prediction
basis ar e the modified waveTUB input snapshots (r ed curves on the left hand side). Only the time span
wher e an accurate prediction can be accomplished theor etically is presented. This prediction r egion
marks the spatio-temporal domain which can be pr edicted based on known ranges in space and time
for certain sea states. The group velocity of the fastest and slowest wave gr oups within the sea state
define the pr ediction range—the slowest wave gr oup must have r eached the tar get location (position
of of fshor e structur e or in our study the positions of wave gauges 2-4) and the fastest wave group must
not have passed the target location (cf. [
1
–
3
,
48
]). For this study , the pr ediction r egion is determined for
each consecutive input snapshot separately but the variation for the differ ent consecutive snapshot
as well as sea states is mar ginal. Based on this, the average prediction time at wave gauge 2 is
appr oximately
t p t ≈
300 s (cf. r ed curves in Figur e 2 on the right hand side) with an average minimum
for ecast horizon time
t mi n f h ≈
90 s (time the slowest wave gr oup within the sea state needs to r each the
tar get location, i.e., starting point of the pr ediction range) and an average maximum for ecast horizon
t m a x f h ≈
390 s neglecting any computational time (wave gauge 3:
t p t ≈
257 s,
t mi n f h ≈
223 s and
t m a x f h ≈
480 s; wave gauge 4:
t p t ≈
213 s,
t mi n f h ≈
355 s and
t m a x f h ≈
568 s). Please note that these

Fluids 2020 , 5 , 9 11 of 19
values depend str ongly on the dominant wave length and the underlying wave spectrum, i.e., with
incr easing wave length the for ecast horizon will decr ease.
Figure 2. Detailed overview of the applied wave prediction pr ocedure exemplary for sea state 1.
The measur ed and calculated wave sequences at specific locations in time domain ar e evaluated
quantitatively applying the surface similarity parameter (SSP) [
49
]. The SSP repr esents “a quantitative
method to compar e temporal or spatial series in one dimension or temporal or spatial surfaces in two
dimensions” [
49
]. The SSP is a normalized error between two signals or surfaces written in terms of
Sobolev norms
S S P = ( R | ˆ
F f 1 ( k ) − ˆ
F f 2 ( k ) | 2 d k ) 1/2
R | ˆ
F f 1 ( k ) | 2 d k ) 1/2 + R | ˆ
F f 2 ( k ) | 2 d k ) 1/2 , (20)

Fluids 2020 , 5 , 9 12 of 19
with
ˆ
F ( k )
as Fourier transform of the two signals. The magnitude of the SSP varies between 0 (perfect
agr eement) and 1 (perfect disagr eement). One important advantage compared to other available
coef ficients is the inclusion of both the amplitude and the phase dif fer ence of two series or surfaces [
49
].
Figur es 3 and 4 pr esent the accuracy of the predicted wave sequences compar ed to the
measur ements for all investigated methods. For the sake of clarity , the results ar e divided into Figure 3
comparing all NLS-r elated findings and Figur e 4 comparing the accuracy of linear transformation,
second-or der NLSE and HOSM.
Figur e 3 pr esents the SSP of the waveTUB input snapshot (black plus), the classical
first-or der NLSE (orange curves), the first-order NLSE including full dispersion operator (NLSE
FD
,
magenta curves) as well as the second-or der NLSE (NLSE
2
, green curves) simulations for the
investigated sea states. The darker illustrated curves repr esent the prediction accuracy including the
radar shadow and the lighter illustrated curves display the accuracy without radar shadow . The dots
on the curves illustrate the positions of the three wave gauges. The SSP for the predicted sea states
r epr esents the mean value of the eight consecutive forecasts. The corr esponding variance for the
pr ediction with radar shadow is pr esented by the transpar ent ar eas ar ound the r espective curves (with
the same colour). The SSP for the waveTUB input snapshot is calculated for the whole time trace at
wave gauge 1. Each row pr esents sea states with constant wave steepness but increasing enhancement
factor fr om left to right and each column sea states with constant enhancement factor but increasing
wave steepness fr om top to bottom. The black vertical line marks the border between “radar ” input
snapshot (left) and the pr ediction zone (right).
Noteworthy at a first glance is the influence of the inclusion of the full dispersion operator within
the first-or der NLSE: the accuracy of the first-or der NLSE is significantly impr oved by taking the full
dispersion into account. Based on the underlying simplified assumptions of the NLSE, two main
r easons can be identified. On the one hand, the assumed narrow bandwidth of the wave spectr um is not
fulfilled for the investigated sea states. Thus, the assumption that the waves within the spectrum evolve
with the same gr oup velocity based on a dominant wave length is too inaccurate for deterministic wave
pr ediction of arbitrary irr egular sea states. This is clearly supported by the data shown in Figur e 3
top as the accuracy of the NLSE incr eases with incr easing enhancement factor (narr ower spectrum
bandwidth). On the other hand, the range of validity of the NLSE are small amplitude waves. As a
consequence, the gain in accuracy with increasing enhancement factor is less distinctive for the steepest
irr egular sea states. NLSE
FD
and NLSE
2
have the same accuracy for the smallest steepness. This shows,
that the NLSE can be applied for br oader wave spectra by taking the full dispersion into account.
W ith increasing wave steepness, the advantage of the NLSE
2
becomes clear . The accuracy of the
NLSE
FD
decr eases with incr easing steepness compar ed to the waveTUB input snapshot accuracy ,
wher eas the accuracy of the NLSE
2
is indeed also ef fected but not so distinct showing a better overall
performance also for the steepest cases.
Figur e 4 pr esents the SSP of the waveTUB input snapshot (black plus), the HOSM (r ed curves),
NLSE
2
(gr een curves) as well as linear (blue curves) simulations for the investigated sea states.
The diagram is arranged as the previous one, the darker illustrated curves r epresent the pr ediction
accuracy including the radar shadow and the lighter illustrated curves display the accuracy without
radar shadow . The dots on the curves illustrate the positions of the three wave gauges. The SSP for the
pr edicted sea states r epr esents the mean value of the eight consecutive for ecasts. The corresponding
variance for the pr ediction with radar shadow is presented by the transpar ent areas ar ound the
r espective curves (with the same colour). The SSP for the waveTUB input snapshot is calculated for
the whole time trace at wave gauge 1. Each row pr esents sea states with constant wave steepness but
incr easing enhancement factor fr om left to right and each column sea states with constant enhancement
factor but incr easing wave steepness fr om top to bottom. The black vertical line marks the bor der
between “radar ” input snapshot (left) and the pr ediction zone (right).

Fluids 2020 , 5 , 9 13 of 19
Figure 3.
Surface Similarity Parameter for the investigated sea states at the three wave gauges for
classical NLSE (orange curves), NLSE with full dispersion (magenta curves) as well as second-or der
NLSE (green curves) simulations in space domain. The darker illustrated curves r epr esent the
prediction accuracy including the radar shadow and the lighter illustrated curves display the accuracy
without radar shadow . The dots on the curves illustrate the positions of the three wave gauges. The SSP
of the waveTUB input snapshots is illustrated as black plus.
The first r ow (same small wave steepness and incr easing enhancement factor) shows that for small
wave steepness the HOSM, the NLSE
2
and linear r esults ar e in the same accuracy range. This indicates
that applying linear transformation for wave pr ediction of sea states with small wave steepness is
suf ficient and ther e is no need for mor e sophisticated methods. In addition, the three top diagrams
r eveal that the accuracy of the prediction r emains almost constant over a very large distance, i.e., 2025 m
or 18 dominant wave length fr om the beginning of the input snapshot and 4875 m or 43 dominant
wave length fr om the end of the input snapshot to wave gauge 4. The main r eason is the fact that
for sea states with small wave steepness non-linear ef fects are very small and these non-linear ef fects
evolve only in a lar ge time and space scale. However , comparing the diagrams of each column shows
clearly that with incr easing wave steepness (vertical from top to bottom) the accuracy of the linear
method decr eases wher eas the HOSM and NLSE
2
ar e also accurate for steeper sea states. For the
steepest investigated sea states, the HOSM prediction is still on the accuracy level of the waveTUB

Fluids 2020 , 5 , 9 14 of 19
input snapshot over very large distances, wher eas the NLSE
2
shows to be less accurate. This indicates
that the ar ea of application of a for ecast tool can be extended significantly by applying more complex
wave models. In this context, the HOSM seems to be even applicable for very steep sea states.
Figure 4.
Surface Similarity Parameter for the investigated sea states at the three wave gauges for
HOSM (red curves), second-or der NLSE (green curves) as well as linear (blue curves) simulations in
space domain. The darker illustrated curves repr esent the prediction accuracy including the radar
shadow and the lighter illustrated curves display the accuracy without radar shadow . The dots on the
curves illustrate the positions of the thr ee wave gauges. The SSP of the waveTUB input snapshots is
illustrated as black plus.
Comparing the SSP of NLSE
FD
(Figur e 3 ) and linear simulation (Figur e 4 ) shows no distinct trend.
For the smallest steepness, both methods yield the same accuracy (independently fr om enhancement
factor
γ
). W ith incr easing wave steepness, the accuracy of both methods is still similar with a marginal
tendency for better pr ediction with the NLSE FD .
The influence of the enhancement factor on the pr ediction accuracy of the HOSM, NLSE
2
and
linear simulations cannot be determined clearly . No tr end is visible for the irregular sea sates with the
smallest wave steepness. The discrepancy between linear transformation and HOSM simulation seems
to incr ease slightly for incr easing enhancement factor . Hereby , it remains unclear if this is caused by the
dif fer ent accuracies of the input snapshots or due the incr ease of the enhancement factor . The influence

Fluids 2020 , 5 , 9 15 of 19
of the radar shadow on the accuracy of the pr ediction does not show a clear trend. For most of the
cases, the influence of the radar shadow is negligible, otherwise the input snapshot without radar
shadow pr ovide a slightly impr ove as well as worsen of the accuracy .
Analysing the accuracy of the waveTUB input snapshot (black plus) r eveals that the accuracy of
the r epr oduction decr eases with incr easing wave steepness. In this context, the diagrams show that the
HOSM pr ediction based on the waveTUB input snapshot r emain almost on the same accuracy level
(of the waveTUB input snapshot) for all wave steepness. This illustrates clearly that the accuracy of a
r eal world wave pr ediction tool depends str ongly on the accuracy of the detected input wave sequence.
Strictly speaking, the wave pr ediction methods necessary for an accurate prediction ar e alr eady
available and not only the r esults of this investigation (cf. [
10
,
13
–
15
,
26
,
30
]) r eveals that depending
on the specific task very complex and numerically less ef ficient methods ar e not necessarily r equir ed
for a successful application. However , the detection of the surrounding wave field by applying
sophisticated wave inversion algorithm on X-band radar clutter is still challenging, but also the crucial
factor for a successful application of a wave for ecast tool. Hereby , the principle of measurements
of X-band radars yield a main drawback as the radar cannot detect wave tr oughs behind (steep)
wave cr ests. The main challenge hereby is, that most of the r elevant applications r egar ding wave
pr ediction and decision support ar e related to small wave steepness, wher e the intensity of the radar
clutter is significant smaller compar ed to mor e steeper sea states. Less intensity corr elates with
r educed accuracy , particularly in the far field of the radar r educing the wave prediction horizon.
Consequently , the fast linear method accurate enough for small wave steepness is significantly af fected
by the radar measuring accuracy . In addition, the vessel motions have to be measured accurately
(for cruising vessels) in or der to determine the position of the X-band radar as a critical prer equisite for
the wave inversion. Generally , also the measuring accuracy (radar clutter as well as vessel motion)
plays a major r ole for accurate wave prediction which can be divided in systematic and random err or .
The mentioned parameters r elevant for an accurate wave pr ediction are only summarized without any
detailed discussion as this is out of the scope of this paper .
4. Conclusions
This paper pr esents a numerical and experimental study on the applicability and limitations
of use of intended prediction methods of dif ferent complexities: linear wave solution, the NLSE,
NLSE
FD
, NLSE
2
, and the HOSM. The focus lies on the investigation of the prediction accuracy for
varying parameters such as wave steepness, enhancement factor of the JONSW AP spectrum and wave
pr opagation distance. For this purpose, irregular sea states with varying parameters ar e addressed.
The irr egular sea state investigations focusses on JONSW AP spectra with varying wave steepness
and enhancement factor . In addition, the influence of the propagation distance as well as the
for ecast horizon is discussed. The accuracy of the predictions ar e evaluated quantitatively by
the SSP . The results show that the linear method is suf ficient for the prediction of irr egular sea
states with small steepness with the same accuracy as the mor e complex methods. For increasing wave
steepness, non-linear ef fects ar e more dominant r esulting in a significant decreasing of the accuracy
of the linear method. The NLSE results have shown thr oughout that this method is not applicable
for deterministic wave pr ediction of arbitrary irr egular sea states compar ed to the other methods.
However , inclusion of the full dispersion operator resulted in a significant incr eased accuracy compared
to the classical first-or der NLSE. Both the NLSE
2
and HOSM r esults illustrate that deterministic wave
pr ediction can be extended to steeper waves by applying mor e complex wave models. Her eby the
HOSM shows a high accuracy also for the steepest investigated sea states over very large distances.
In this context, the HOSM prediction based on the waveTUB input snapshot r emains almost on the
same input snapshot accuracy level for all wave steepness.
The accuracy of the waveTUB repr oduction is throughout adequate but showed also that the
accuracy decr eases with incr easing wave steepness of the irr egular sea states. This illustrates clearly
that the accuracy of a r eal world wave for ecast tool depends str ongly on the accuracy of the detected

Fluids 2020 , 5 , 9 16 of 19
input wave sequence, i.e., assuming a radar-based wave detection, an accurate wave inversion
algorithm used to extract the deterministic wave field fr om the radar clutter is the crucial factor
for a successful application of a wave for ecast tool.
Author Contributions:
Conceptualization: G.F .C., M.K., S.E. and N.H.; methodology: G.F .C., M.K., M.O., S.E. and
N.H.; software: M.K., M.O. and J.B.; validation: M.D. and M.K.; formal analysis: M.K. and M.D.; investigation,
M.D. and M.K.; writing—original draft preparation: M.K.; writing—review and editing: M.K.; visualization: M.K.;
supervision: M.K.; project administration: G.F .C. and M.K.; funding acquisition: G.F .C. and M.K. All authors have
read and agr eed to the published version of the manuscript.
Funding:
This paper is published as a contribution to the joint resear ch project “PrOWOO”. The authors wish
to express their gratitude to the German Federal Ministry of Economic Af fairs and Energy (BMW i) and Pr oject
Management Jülich (PtJ) for funding and supporting the joint resear ch project (FKZ 03SX358A).
Acknowledgments:
G.F .C., M.K. and M.D. want to thank the project partner OceanW aveS. M.K. thanks Alexey
Slunyaev for fruitful discussions and sharing the subr outine for the calculation of the NLSE coefficients.
Conflicts of Interest:
The authors declare no conflict of inter est. The funders had no role in the design of the
study; in the collection, analyses, or interpr etation of data; in the writing of the manuscript, or in the decision to
publish the results.
Appendix A
Following the coef ficients r elevant for implementing Equation ( 7 ), which ar e adopted fr om
Slunyaev [ 42 ].
Appendix A.1. Coefficients of Non-linear Interaction T erms
χ 1 = 3 k 2
c σ 4 − 1
8 ω c σ 2 ,
χ 2 =  k c d − σ 2 + 3
k c σ + 1
k c  χ 1 + 3 k 2
c d ( σ 4 − 1 ) ( 3 σ 2 + 1 )
16 ω c σ 3 + 9 k c − σ 4 + 1
16 ω c σ 2 ,
Appendix A.2. Linear Dispersion Coefficients
β 1 = − 1
2 ∂ 2 ω
∂ k 2 , β 2 = − 1
6 ∂ 3 ω
∂ k 3 .
Appendix A.3. Coefficients of Wave-Induced Flow Components
γ 1 = k 2
c V ( σ 2 − 1 ) − 2 ω c k c
4 V 2
d
,
γ 2 = 2 V
V 2
d
γ 1 β 1 + k 2
c σ 2 − 1
4 V 2
d
β 1 + ω 2
c − k 2
c V 2 ( σ 2 + 1 )
4 ω c V 2
d
,
Appendix A.4. Coefficients in Equations
˜
ρ 11 = k 2
c σ 4 − 3
2 ( σ 2 + 1 ) χ 1 + k 4 − 5 σ 4 + 16 σ 2 − 3
16 ω c , ˜
ρ 12 = 2 ω c k c − k 2
c V ( σ 2 − 1 )
2 ω c , ˜
α 1 = ˜
ρ 11 + ˜
ρ 12 γ 1 ,
P 21 =  k 2
c d 2 − 4 σ 6 + 7 σ 4 − 2 σ 2 − 1
8 σ 2 + k c d 4 σ 4 − 9 σ 2 + 3
4 σ + − 4 σ 2 + 19
8  γ 1 + k 2
c − σ 4 + 3
2 ( σ 2 + 1 ) χ 2
+  k 2
c d − 3 σ 6 + 7 σ 4 − 9 σ 2 − 3
4 σ ( σ 2 + 1 ) + 3 k c σ 4 − 5
4 ( σ 2 + 1 )  χ 1 + k 4
c d 11 σ 6 − 23 σ 4 + 9 σ 2 + 3
16 ω c σ + k 3
c − 11 σ 4 + 40 σ 2 − 9
16 ω c ,
P 22 =  − k 2
c d 2 ( σ 2 − 1 ) 2
8 + k c d σ 4 − 5 σ 2 + 2
4 σ + − σ 2 + 8
8  γ 1 +  k 2
c d ( σ 2 − 1 ) ( σ 4 + 3 )
4 σ ( σ 2 + 1 ) − k c σ 4 + 3
4 ( σ 2 + 1 )  χ 1
+ k 4
c d − 3 σ 6 − 5 σ 4 + 11 σ 2 − 3
32 ω c σ + 3 k 3
c σ 4 − 1
32 ω c ,
˜
ρ 21 = P 21 + s β 1 γ 1 , ˜
ρ 22 = P 22 − s β 1 γ 1 , s = k 2
c σ 2 − 1
2 ω c , ˜
α 21 = ˜
ρ 21 − ˜
ρ 12 γ 2 , ˜
α 22 = ˜
ρ 22 + ˜
ρ 12 γ 2 ,
α 1 = ˜
α 1
λ 2 , α 21 = ˜
α 21
λ 2 , α 22 = ˜
α 22
λ 2 , V 2
d = g d − V 2 , λ = − ω
g , σ = tanh ( k c d ) ,

Fluids 2020 , 5 , 9 17 of 19
Appendix A.5. Coefficients Used in Constructing the Wave Field
r 01 = ˜
r 01
λ 2 , ˜
r 01 = σ k c k 2
c ( σ 2 − 1 ) + 4 V γ 1
4 ω 2
c , r 21 = − ˜
r 21
λ 2 , ˜
r 21 = 2 k c σ
ω c χ 1 + k 3
c σ ( − 3 σ 2 + 1 )
4 ω 2
c ,
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