Research ArticleComputational Simulation and Modeling of Freak WavesBased on Longuet-Higgins Model and Its ElectromagneticScattering Calculation
Gengkun Wu 1 Chuanxi Liu2 and Yongquan Liang1
1College of Computer Science and Engineering Shandong University of Science and Technology Qingdao 266590Shandong China2College of Intelligent Equipment Shandong University of Science and Technology Tairsquoan 271000 Shandong China
Correspondence should be addressed to Gengkun Wu wugengkunsdusteducn
Received 5 April 2020 Revised 6 July 2020 Accepted 27 July 2020 Published 17 August 2020
Academic Editor Marcio Eisencraft
Copyright copy 2020 GengkunWu et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In order to solve the weak nonlinear problem in the simulation of strong nonlinear freak waves an improved phase modulationmethod is proposed based on the Longuet-Higgins model and the comparative experiments of wave spectrum in this paperExperiments show that this method can simulate the freak waves at fixed time and fixed space coordinates In addition bycomparing the target wave spectrum and the freak wave measured in Tokai of Japan from the perspective of B-F instability andspectral peakedness it is proved that the waveform of the simulated freak waves can not only maintain the spectral structure of thetarget ocean wave spectrum but also accord with the statistical characteristics of the wave sequences)en based on the Kirchhoffapproximation method and the modified Two-Scale Method the electromagnetic scattering model of the simulated freak waves isestablished and the normalized radar cross section (NRCS) of the freak waves and their background sea surfaces is analyzed )ecalculation results show that the NRCS of the freak waves is usually smaller than their large-scale background sea surfaces It canbe concluded that when the neighborhood NRCS difference is less than or equal to minus 12 dB we can determine where the freakwaves are
1 Introduction
Freak waves are high and steep waves in the ocean )eduration of freak waves is very short but the contingencyand great destructiveness are extremely threatening toshipping and marine engineering structures )ereforethe study of freak waves has attracted more and moreattention [1] )e occurrence mechanism and engineeringprediction of freak waves have become a hot researchtopic in the field of physical oceanography and ship hy-drodynamics [2] However the reasons for the occurrenceof the freak waves are still unclear their occurrence hasmany uncertainties and they are difficult to observe by thefixed point marine buoys shore based radars or opticalsensors It has been confirmed that the formationmechanism of freak waves is divided into linear mecha-nism and nonlinear mechanism but the specific reasons
are still being explored Freak waves occur suddenly havea short duration and have a wide range of time and spacein the global ocean )erefore they are very difficult torecord Because most phenomena of the freak wavescannot be observed which means the data cannot becollected actually it is very important to use the tech-nology of mobile radar and satellite remote sensing tostudy and simulate the freak waves Synthetic ApertureRadar (SAR) is capable of capturing high-resolutionmicrowave images )e microwaves are highly penetrativeand can work under any weather conditions For the safetyof maritime navigation and offshore platforms it is ofgreat significance to explore the physical mechanism ofthe occurrence evolution and extinction of freak waves[3ndash6] Moreover the use of SAR to monitor and predictfreak waves is a disaster reduction technology that shouldbe valued [6]
HindawiComplexityVolume 2020 Article ID 2727681 14 pageshttpsdoiorg10115520202727681
Given the difficulty of using SAR to observe the freakwaves more and more scientists are beginning to pay at-tention to numerical simulation methods [7 8] Now nu-merical simulations of freak waves in deep water are mainlybased on linear superposition methods or cubic nonlinearSchrodinger equation Based on nonlinear wave equation ofwave modulation instability we can study the occurrencemechanism of freak waves [9ndash11] However due to the largeamount of computation this method is not easy to apply inengineering At the same time it is very difficult to controlthe time and space conditions in the simulation of freakwaves [12ndash15] )e method based on the Longuet-Higginsmodel is effective in simulating the freak waves in thelaboratory which is simple and practical [16] Lawtonsimulated the freak waves to form the random initial phaseof waves through artificial intervention which was an in-efficient method and we could not control the generationtime and place of freak waves [17] Pei simulated therecorded freak waves using the three wave trainsrsquo super-position model [18] Kriebel simulated the freak waves usingthe double wave superposition model consisting of a basicrandom wave and a linear superposition of transient wave[19] Liu proposed a new efficient method by modifying theLonguet-Higgins model which greatly improved the simi-larity between the simulated freak waves and the targetspectrum structures [20] In the early practical applicationbased on the numerical simulation of one- and two-di-mensional space freak waves we have calculated and ana-lyzed the electromagnetic scattering coefficient of freakwaves to study the formation mechanism remote sensingrecognition and other related characteristics [21] Based onthe Two-Scale Method (TSM) and the Harger distributionsurface Franceschetti proposed two kinds of sea surface SARsimulator models [22] however this research method hasgreat limitations because of its failure to fully consider thestrong non-Gaussian statistical characteristics and velocitybunching effect of the height distribution of freak waves [2324] In recent years freak waves have attracted much at-tention because of their potential for serious damage toshipping and offshore structures In practice people paymore attention to the possibility of predicting the occurrenceof freak waves while most of the previous research on freakwaves has focused on the mechanism of freak waves )emodeling flowchart that explains the main goal the appliedmethods the intermediate steps and the obtained results isshown in Figure 1
In this work an improved phase modulation method forsimulating freak waves is developed based on the Longuet-Higgins model and the comparative results betweenJONSWAP [25] spectrum and Elfouhaily [26] spectrum andan electromagnetic scattering model of the simulated freakwaves is established )e random phase correction methodand its special application in freak wave simulation arepresented in Section 2 According to the research conclu-sions the backscattering model of the simulated freak waves
is developed and the results of the simulation were com-pared with the freak wave measured in Tokai of Japan inSection 3 from the perspective of B-F instability and spectralpeakedness In Section 4 the scattering calculation resultsare analyzed and the experimental conclusions are sum-marized Moreover a feature identification method of freakwave from its background sea surface is proposed Finallybased on the normalization method of Z-score the influenceof the deviation coefficient a1sima4 on the height of the freakwaves is measured in this work
2 Numerical Simulation Model of Freak WavesBased on the Longuet-Higgins Model
21 Research on Random PhaseModulationMethod Based onthe Longuet-Higgins Model Marine shipping and oceanengineering structures are terribly threatened by the freakwaves which leads to many maritime accidents Becausemeasuring the data of freak waves is difficult and themethod of laboratory simulation is expensive it is neces-sary to study the occurrence and evolution characteristicsof freak waves through numerical simulation Based on theLonguet-Higgins model the simulation of normal randomwaves can be realized without freak waves [27] Howeverthe numerical calculation of the freak wave and its back-ground waves is simulated by correcting the random phasein this work )e fixed point of the wave equation can beexpressed by superposition of a large number of randomcosine waves [28]
H(l t) 1113944M
i1ai cos kil minus ωit + θi( 1113857 (1)
In (1) t is the time course of wave M is the sum of wavenumbers and l is the distance from the simulated wave aiki ωi and θi are the amplitude wave number of the i waveangular frequency and random initial phase of the com-position wave When we simulate the conventional randomwave the initial phase of the wave components is evenlydistributed in (0 2π) In order to simulate the freak waves inthe random wave series we need to focus the energy of thewaveform Normally the method can be realized byadjusting the initial phase of the part composition wave Ifthe modulation process is not reasonable the statisticalcharacteristics of the numerical simulation of random wavesequence are not in conformity with the statistical charac-teristics of the natural wave and the structure of the wavespectrum can be changed)erefore in this work we use thefollowing method to realize the simulation of random wavesequences of freak waves Supposing that the freak waves aregenerated at the position of l lc and at the time t tc wemodulate θi to make Hi(lc tc) a positive value so when wesimulate wave superposition the wave height increases Werewrite (1) into the following synthetic waves containingfreak waves and normal waves
2 Complexity
H(l t) h1(l t) + h2(l t)
h1(l t) 1113944
M1
l1ai cos kil minus ωit + θi( 1113857
h2(l t) 1113944M
lM1+1ai cos kil minus ωit + θi( 1113857
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
We assume that the second part of the wave h2(l t) willproduce the freak waves at a predetermined position at thistime we should modulate θi to make Hi(lc tc) a positivevalue When kilc minus ωitc ge 0 we make P int[(kilcminus ωitc)2π]
and Pge 0 and we modulate θi(0lt θi lt 2π) after which wecan get the conclusions that minus (π2)lt (kilc minus ωitc minus 2Pπ +
θi)lt (π2) and cos(kilc minus ωitc minus 2Pπ + θi)gt 0 At this timeHi(lc tc)gt 0 and h2(l t)gt 0 we modulate the value of θi asfollows in this paper
When the value of kilc minus ωitc minus 2Pπ + θi is in therange of (0 π2) the range of random values θi is(3π2 2π)
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π2 π) the range of random values θi is (π 3π2]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π 3π2) the range of random values θi is (π2 π]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (3π2 2π) the range of random values θi is (0 π2]
By the same way we can deduce the value of θi when itsatisfies the condition of kilc minus ωitc lt 0
Based on the abovementioned random phase modula-tion method the research goal can be achieved )e originalwave height is artificially divided into two parts randomsuperposition and positive superposition and then thesimulation of the freak wave is realized)e advantage of thismethod is that different wave height simulations can beachieved by controlling the ratio of the two parts of thesuperimposed wave during the experiments
22Numerical SimulationofFreakWavesandResultAnalysis)e actual sea surface is often an unsteady sea surface causedby complex environments such as swells )e classic JONS-WAP spectrum [12] corrects the gravity wave area on the basisof the traditional PM spectrum so that it contains the unsteadysea spectrum Its power spectrum is shown as follows
JONSWAPspectrum
Elfouhailyspectrum
Freakwave in
tokai
B-F instability
Spectralpeakedness
Wave spectrummodels
Longuet-Higginsmodel
Phasemodulation
method
1D freak wavesand their
background waves
NormalizedmeasurementKA and TSM
NRCS ofbackground
waves
NRCS offreak waves
Identificationmethod
Figure 1 )e modeling flowchart
Complexity 3
S(k) 1k4B
JONl
1k4
aJ
2LPMJp (3)
In (3) BJONl is the directionless curvature spectrum l is
the wavelength of the gravitational wave LPM is close to thePM shape spectrum parameter andLPM exp[minus (5k2p)(4k2)) Jpis the peak enhancement fac-tor Jp cτ In addition aJ 0076 1113957X
minus 022 1113957X k0xkp k0Ω2c k0 (gu)210 and x are wind zones in units ofmWhen the corresponding values of Ωc are 084 10 and 20they represent fully developed sea surface mature sea sur-face and developing sea surface respectively
Elfouhaily proposed a joint spectrum function based onPM spectrum JONSWAP spectrum and Apel spectrumand its power spectrum is defined as follows [12]
S(k) BL + BH( 1113857
k4 (4)
In (4) BL and BH respectively represent the low-fre-quency nondirectional curvature spectrum corresponding tothe gravity wave and the high-frequency nondirectionalcurvature spectrum corresponding to the capillary wave)elow-frequency curvature spectrum BL satisfies the followingform
BL(k) αp
2c kp1113872 1113873
c(k)Fp (5)
Here
αp α0Ω
kp gΩ2
u210
c(k) g 1+k2k2m( )
k1113876 1113877
12
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
In the equations above αp is the generalized P-Kequilibrium zone parameter in the low-frequency wavenumber range c(kp) is the phase velocity corresponding tothe peak of the spectrum u10 is the wind speed at 10m abovethe sea and Ω U10c(kp) is the inverse wave age c(k) isthe phase velocity kp is the wave number distributed in thepeak of the spectral domain km
ρωgτω
1113968asymp 370ra dm
wherein ρω is the density of seawater τω is the surfacetension of seawater and g is the acceleration of gravitySubsequently Elfouhaily further proposed thatαp 60 times 10minus 3
Ω
radic and based on the dimensionless pa-
rameters kkp and Ω we can further determine the long-wave edge effect function Fp
In the comparative experiment a simplified scatteringmodel is used to analyze the power characteristicsδ0(ρ) πk2|R|2(q2q4z)P(sx sy) where sx and sy representthe sea surface slope in different directions P is the seasurface slope density function k is the wave number R is theFresnel scattering coefficient and q represents the scatteringvector Based on the control variable method the scatteredpower of the JONSWAP spectrum and that of the Elfouhaily
spectrum are compared to normalize the power waveformpeak value and the slope change rate under different windspeeds and wind conditions wherein the receiver height is45 km the satellite elevation angle is 30deg the wind directionis 0deg the wind speed varies from 6ms to 20ms and thefetches varies from 10 km to 19 km )e numerical resultsare shown in Figures 2 and 3
Comparing the numerical results in Figure 2 it can beseen that as the wind speed becomes larger the peaks of theJONSWAP spectrum and the Elfouhaily spectrum graduallydecrease the delay slope of the retardation gradually in-creases and the effect of medium and low wind speeds isobvious At the same time the numerical results in Figure 3show that with the increase of the fetch the peak and delayslope of the JONSWAP spectrum show regular changeswhile the Elfouhaily spectrum is not sensitive to the fetch
In this paper the JONSWAP spectrum is used as the targetspectrum [12] which means that the parameter ai in (1) alwayscomplies with the JONSWAP spectrum )e time series offreak waves can be simulated when the distance from thesimulated wave is x 0 in Figures 4 and 5 Similarly the spaceseries of freak waves can be simulated when t 0 in Figure 6
When the depth of water is 43m the effective waveheight is 510m the spectral peak period is 12s the spectralelevation factor is 320 the wave number is 200 themodulation wave number is 160 the spectrum rangechanges from 0 to 032 and the time tc is 100s or 200s thesimulation of the time series of freak waves is illustrated inFigures 4 and 5
According to the definition of the freak waves the heightof freak wave Hj should meet the following conditionsa1 HjHs ge 2 a2 HjHjminus 1 ge 2 a3 HjHj+1 ge 2 anda4 ηjHj ge 065 wherein ηj is the crest height of freakwaves corresponding to the horizontal line Hs is the ef-fective wave height and Hjminus 1 and Hj+1 are the wave heightsof adjacent waves before and after the deformed wave a1 a2a3 and a4 are characteristic parameters of freak waves [2829] )e characteristic statistics of wave duration are carriedout using the method of positive and reverse zero-crossingcounting and the characteristic parameters of the extremewaves are shown in Tables 1 and 2
)e effective wave height Hs is 376m Compared withthe input parameter the relative error is less than 5 FromTable 1 and Table 2 we can see that all of the parametersabove meet the definition of the freak wave and the freakwave is generated at the scheduled time which proves thevalidity of this model Wave time history spectrum and thetarget spectrum are compared in Figure 7 and the com-parison results between wave height distribution and Ray-leigh distribution are shown in Figure 8
)e results shown in Figure 6 indicate that the simulatedwave spectrum keeps the structure of the target spectrumand the spectral peak frequency is very similar to that of thetarget spectrum Figure 7 shows that the normalized cu-mulative probability distribution of wave height of thesimulated data agrees well with the Rayleigh distribution)e results show that the simulation results meet the re-quirements of random waveforms and the simulationmethod proposed in this paper is effective
4 Complexity
3 Electromagnetic Scattering CalculationModel of Simulated Freak Waves
31 Research on Backscattering Model Based on KA Methodand TSM )e Two-Scale Method (TSM) is developed on
the basis of Kirchhoff approximation (KA) method adaptedto the large-scale sea surface and the small perturbationmethod adapted to the small-scale sea surface [30 31]Firstly the scattering coefficients of small-scale sea surfaceare calculated by the perturbation theory secondly the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash38
ndash36
ndash34
ndash32
ndash3
ndash28
ndash26Peak power spectrum
Wind speed (ms)
Peak
pow
er sp
ectr
um (d
B)
JONSWAPElfouhaily
(a)
5 7 9 11 13 15 17 19 21ndash5
ndash45
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1Delay slope of power spectrum
Wind speed (ms)
Dela
y slo
peJONSWAPElfouhaily
(b)
Figure 2 Comparison of peak power and delay slope of power spectrums under different wind speeds
JONSWAPElfouhaily
10 11 12 13 14 15 16 17 18 19ndash4
ndash39
ndash38
ndash37
ndash36
ndash35
ndash34
ndash33
ndash32
ndash31Peak power spectrum
Fetches (km)
Peak
pow
er sp
ectr
um (d
B)
(a)
JONSWAPElfouhaily
9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0Delay slope of power spectrum
Fetches (km)
Dela
y slo
pe
(b)
Figure 3 Comparison of peak power and delay slope of power spectrums under different fetches
Complexity 5
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
Given the difficulty of using SAR to observe the freakwaves more and more scientists are beginning to pay at-tention to numerical simulation methods [7 8] Now nu-merical simulations of freak waves in deep water are mainlybased on linear superposition methods or cubic nonlinearSchrodinger equation Based on nonlinear wave equation ofwave modulation instability we can study the occurrencemechanism of freak waves [9ndash11] However due to the largeamount of computation this method is not easy to apply inengineering At the same time it is very difficult to controlthe time and space conditions in the simulation of freakwaves [12ndash15] )e method based on the Longuet-Higginsmodel is effective in simulating the freak waves in thelaboratory which is simple and practical [16] Lawtonsimulated the freak waves to form the random initial phaseof waves through artificial intervention which was an in-efficient method and we could not control the generationtime and place of freak waves [17] Pei simulated therecorded freak waves using the three wave trainsrsquo super-position model [18] Kriebel simulated the freak waves usingthe double wave superposition model consisting of a basicrandom wave and a linear superposition of transient wave[19] Liu proposed a new efficient method by modifying theLonguet-Higgins model which greatly improved the simi-larity between the simulated freak waves and the targetspectrum structures [20] In the early practical applicationbased on the numerical simulation of one- and two-di-mensional space freak waves we have calculated and ana-lyzed the electromagnetic scattering coefficient of freakwaves to study the formation mechanism remote sensingrecognition and other related characteristics [21] Based onthe Two-Scale Method (TSM) and the Harger distributionsurface Franceschetti proposed two kinds of sea surface SARsimulator models [22] however this research method hasgreat limitations because of its failure to fully consider thestrong non-Gaussian statistical characteristics and velocitybunching effect of the height distribution of freak waves [2324] In recent years freak waves have attracted much at-tention because of their potential for serious damage toshipping and offshore structures In practice people paymore attention to the possibility of predicting the occurrenceof freak waves while most of the previous research on freakwaves has focused on the mechanism of freak waves )emodeling flowchart that explains the main goal the appliedmethods the intermediate steps and the obtained results isshown in Figure 1
In this work an improved phase modulation method forsimulating freak waves is developed based on the Longuet-Higgins model and the comparative results betweenJONSWAP [25] spectrum and Elfouhaily [26] spectrum andan electromagnetic scattering model of the simulated freakwaves is established )e random phase correction methodand its special application in freak wave simulation arepresented in Section 2 According to the research conclu-sions the backscattering model of the simulated freak waves
is developed and the results of the simulation were com-pared with the freak wave measured in Tokai of Japan inSection 3 from the perspective of B-F instability and spectralpeakedness In Section 4 the scattering calculation resultsare analyzed and the experimental conclusions are sum-marized Moreover a feature identification method of freakwave from its background sea surface is proposed Finallybased on the normalization method of Z-score the influenceof the deviation coefficient a1sima4 on the height of the freakwaves is measured in this work
2 Numerical Simulation Model of Freak WavesBased on the Longuet-Higgins Model
21 Research on Random PhaseModulationMethod Based onthe Longuet-Higgins Model Marine shipping and oceanengineering structures are terribly threatened by the freakwaves which leads to many maritime accidents Becausemeasuring the data of freak waves is difficult and themethod of laboratory simulation is expensive it is neces-sary to study the occurrence and evolution characteristicsof freak waves through numerical simulation Based on theLonguet-Higgins model the simulation of normal randomwaves can be realized without freak waves [27] Howeverthe numerical calculation of the freak wave and its back-ground waves is simulated by correcting the random phasein this work )e fixed point of the wave equation can beexpressed by superposition of a large number of randomcosine waves [28]
H(l t) 1113944M
i1ai cos kil minus ωit + θi( 1113857 (1)
In (1) t is the time course of wave M is the sum of wavenumbers and l is the distance from the simulated wave aiki ωi and θi are the amplitude wave number of the i waveangular frequency and random initial phase of the com-position wave When we simulate the conventional randomwave the initial phase of the wave components is evenlydistributed in (0 2π) In order to simulate the freak waves inthe random wave series we need to focus the energy of thewaveform Normally the method can be realized byadjusting the initial phase of the part composition wave Ifthe modulation process is not reasonable the statisticalcharacteristics of the numerical simulation of random wavesequence are not in conformity with the statistical charac-teristics of the natural wave and the structure of the wavespectrum can be changed)erefore in this work we use thefollowing method to realize the simulation of random wavesequences of freak waves Supposing that the freak waves aregenerated at the position of l lc and at the time t tc wemodulate θi to make Hi(lc tc) a positive value so when wesimulate wave superposition the wave height increases Werewrite (1) into the following synthetic waves containingfreak waves and normal waves
2 Complexity
H(l t) h1(l t) + h2(l t)
h1(l t) 1113944
M1
l1ai cos kil minus ωit + θi( 1113857
h2(l t) 1113944M
lM1+1ai cos kil minus ωit + θi( 1113857
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
We assume that the second part of the wave h2(l t) willproduce the freak waves at a predetermined position at thistime we should modulate θi to make Hi(lc tc) a positivevalue When kilc minus ωitc ge 0 we make P int[(kilcminus ωitc)2π]
and Pge 0 and we modulate θi(0lt θi lt 2π) after which wecan get the conclusions that minus (π2)lt (kilc minus ωitc minus 2Pπ +
θi)lt (π2) and cos(kilc minus ωitc minus 2Pπ + θi)gt 0 At this timeHi(lc tc)gt 0 and h2(l t)gt 0 we modulate the value of θi asfollows in this paper
When the value of kilc minus ωitc minus 2Pπ + θi is in therange of (0 π2) the range of random values θi is(3π2 2π)
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π2 π) the range of random values θi is (π 3π2]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π 3π2) the range of random values θi is (π2 π]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (3π2 2π) the range of random values θi is (0 π2]
By the same way we can deduce the value of θi when itsatisfies the condition of kilc minus ωitc lt 0
Based on the abovementioned random phase modula-tion method the research goal can be achieved )e originalwave height is artificially divided into two parts randomsuperposition and positive superposition and then thesimulation of the freak wave is realized)e advantage of thismethod is that different wave height simulations can beachieved by controlling the ratio of the two parts of thesuperimposed wave during the experiments
22Numerical SimulationofFreakWavesandResultAnalysis)e actual sea surface is often an unsteady sea surface causedby complex environments such as swells )e classic JONS-WAP spectrum [12] corrects the gravity wave area on the basisof the traditional PM spectrum so that it contains the unsteadysea spectrum Its power spectrum is shown as follows
JONSWAPspectrum
Elfouhailyspectrum
Freakwave in
tokai
B-F instability
Spectralpeakedness
Wave spectrummodels
Longuet-Higginsmodel
Phasemodulation
method
1D freak wavesand their
background waves
NormalizedmeasurementKA and TSM
NRCS ofbackground
waves
NRCS offreak waves
Identificationmethod
Figure 1 )e modeling flowchart
Complexity 3
S(k) 1k4B
JONl
1k4
aJ
2LPMJp (3)
In (3) BJONl is the directionless curvature spectrum l is
the wavelength of the gravitational wave LPM is close to thePM shape spectrum parameter andLPM exp[minus (5k2p)(4k2)) Jpis the peak enhancement fac-tor Jp cτ In addition aJ 0076 1113957X
minus 022 1113957X k0xkp k0Ω2c k0 (gu)210 and x are wind zones in units ofmWhen the corresponding values of Ωc are 084 10 and 20they represent fully developed sea surface mature sea sur-face and developing sea surface respectively
Elfouhaily proposed a joint spectrum function based onPM spectrum JONSWAP spectrum and Apel spectrumand its power spectrum is defined as follows [12]
S(k) BL + BH( 1113857
k4 (4)
In (4) BL and BH respectively represent the low-fre-quency nondirectional curvature spectrum corresponding tothe gravity wave and the high-frequency nondirectionalcurvature spectrum corresponding to the capillary wave)elow-frequency curvature spectrum BL satisfies the followingform
BL(k) αp
2c kp1113872 1113873
c(k)Fp (5)
Here
αp α0Ω
kp gΩ2
u210
c(k) g 1+k2k2m( )
k1113876 1113877
12
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
In the equations above αp is the generalized P-Kequilibrium zone parameter in the low-frequency wavenumber range c(kp) is the phase velocity corresponding tothe peak of the spectrum u10 is the wind speed at 10m abovethe sea and Ω U10c(kp) is the inverse wave age c(k) isthe phase velocity kp is the wave number distributed in thepeak of the spectral domain km
ρωgτω
1113968asymp 370ra dm
wherein ρω is the density of seawater τω is the surfacetension of seawater and g is the acceleration of gravitySubsequently Elfouhaily further proposed thatαp 60 times 10minus 3
Ω
radic and based on the dimensionless pa-
rameters kkp and Ω we can further determine the long-wave edge effect function Fp
In the comparative experiment a simplified scatteringmodel is used to analyze the power characteristicsδ0(ρ) πk2|R|2(q2q4z)P(sx sy) where sx and sy representthe sea surface slope in different directions P is the seasurface slope density function k is the wave number R is theFresnel scattering coefficient and q represents the scatteringvector Based on the control variable method the scatteredpower of the JONSWAP spectrum and that of the Elfouhaily
spectrum are compared to normalize the power waveformpeak value and the slope change rate under different windspeeds and wind conditions wherein the receiver height is45 km the satellite elevation angle is 30deg the wind directionis 0deg the wind speed varies from 6ms to 20ms and thefetches varies from 10 km to 19 km )e numerical resultsare shown in Figures 2 and 3
Comparing the numerical results in Figure 2 it can beseen that as the wind speed becomes larger the peaks of theJONSWAP spectrum and the Elfouhaily spectrum graduallydecrease the delay slope of the retardation gradually in-creases and the effect of medium and low wind speeds isobvious At the same time the numerical results in Figure 3show that with the increase of the fetch the peak and delayslope of the JONSWAP spectrum show regular changeswhile the Elfouhaily spectrum is not sensitive to the fetch
In this paper the JONSWAP spectrum is used as the targetspectrum [12] which means that the parameter ai in (1) alwayscomplies with the JONSWAP spectrum )e time series offreak waves can be simulated when the distance from thesimulated wave is x 0 in Figures 4 and 5 Similarly the spaceseries of freak waves can be simulated when t 0 in Figure 6
When the depth of water is 43m the effective waveheight is 510m the spectral peak period is 12s the spectralelevation factor is 320 the wave number is 200 themodulation wave number is 160 the spectrum rangechanges from 0 to 032 and the time tc is 100s or 200s thesimulation of the time series of freak waves is illustrated inFigures 4 and 5
According to the definition of the freak waves the heightof freak wave Hj should meet the following conditionsa1 HjHs ge 2 a2 HjHjminus 1 ge 2 a3 HjHj+1 ge 2 anda4 ηjHj ge 065 wherein ηj is the crest height of freakwaves corresponding to the horizontal line Hs is the ef-fective wave height and Hjminus 1 and Hj+1 are the wave heightsof adjacent waves before and after the deformed wave a1 a2a3 and a4 are characteristic parameters of freak waves [2829] )e characteristic statistics of wave duration are carriedout using the method of positive and reverse zero-crossingcounting and the characteristic parameters of the extremewaves are shown in Tables 1 and 2
)e effective wave height Hs is 376m Compared withthe input parameter the relative error is less than 5 FromTable 1 and Table 2 we can see that all of the parametersabove meet the definition of the freak wave and the freakwave is generated at the scheduled time which proves thevalidity of this model Wave time history spectrum and thetarget spectrum are compared in Figure 7 and the com-parison results between wave height distribution and Ray-leigh distribution are shown in Figure 8
)e results shown in Figure 6 indicate that the simulatedwave spectrum keeps the structure of the target spectrumand the spectral peak frequency is very similar to that of thetarget spectrum Figure 7 shows that the normalized cu-mulative probability distribution of wave height of thesimulated data agrees well with the Rayleigh distribution)e results show that the simulation results meet the re-quirements of random waveforms and the simulationmethod proposed in this paper is effective
4 Complexity
3 Electromagnetic Scattering CalculationModel of Simulated Freak Waves
31 Research on Backscattering Model Based on KA Methodand TSM )e Two-Scale Method (TSM) is developed on
the basis of Kirchhoff approximation (KA) method adaptedto the large-scale sea surface and the small perturbationmethod adapted to the small-scale sea surface [30 31]Firstly the scattering coefficients of small-scale sea surfaceare calculated by the perturbation theory secondly the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash38
ndash36
ndash34
ndash32
ndash3
ndash28
ndash26Peak power spectrum
Wind speed (ms)
Peak
pow
er sp
ectr
um (d
B)
JONSWAPElfouhaily
(a)
5 7 9 11 13 15 17 19 21ndash5
ndash45
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1Delay slope of power spectrum
Wind speed (ms)
Dela
y slo
peJONSWAPElfouhaily
(b)
Figure 2 Comparison of peak power and delay slope of power spectrums under different wind speeds
JONSWAPElfouhaily
10 11 12 13 14 15 16 17 18 19ndash4
ndash39
ndash38
ndash37
ndash36
ndash35
ndash34
ndash33
ndash32
ndash31Peak power spectrum
Fetches (km)
Peak
pow
er sp
ectr
um (d
B)
(a)
JONSWAPElfouhaily
9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0Delay slope of power spectrum
Fetches (km)
Dela
y slo
pe
(b)
Figure 3 Comparison of peak power and delay slope of power spectrums under different fetches
Complexity 5
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
H(l t) h1(l t) + h2(l t)
h1(l t) 1113944
M1
l1ai cos kil minus ωit + θi( 1113857
h2(l t) 1113944M
lM1+1ai cos kil minus ωit + θi( 1113857
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
We assume that the second part of the wave h2(l t) willproduce the freak waves at a predetermined position at thistime we should modulate θi to make Hi(lc tc) a positivevalue When kilc minus ωitc ge 0 we make P int[(kilcminus ωitc)2π]
and Pge 0 and we modulate θi(0lt θi lt 2π) after which wecan get the conclusions that minus (π2)lt (kilc minus ωitc minus 2Pπ +
θi)lt (π2) and cos(kilc minus ωitc minus 2Pπ + θi)gt 0 At this timeHi(lc tc)gt 0 and h2(l t)gt 0 we modulate the value of θi asfollows in this paper
When the value of kilc minus ωitc minus 2Pπ + θi is in therange of (0 π2) the range of random values θi is(3π2 2π)
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π2 π) the range of random values θi is (π 3π2]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (π 3π2) the range of random values θi is (π2 π]
When the value of kilc minus ωitc minus 2Pπ + θi is in the rangeof (3π2 2π) the range of random values θi is (0 π2]
By the same way we can deduce the value of θi when itsatisfies the condition of kilc minus ωitc lt 0
Based on the abovementioned random phase modula-tion method the research goal can be achieved )e originalwave height is artificially divided into two parts randomsuperposition and positive superposition and then thesimulation of the freak wave is realized)e advantage of thismethod is that different wave height simulations can beachieved by controlling the ratio of the two parts of thesuperimposed wave during the experiments
22Numerical SimulationofFreakWavesandResultAnalysis)e actual sea surface is often an unsteady sea surface causedby complex environments such as swells )e classic JONS-WAP spectrum [12] corrects the gravity wave area on the basisof the traditional PM spectrum so that it contains the unsteadysea spectrum Its power spectrum is shown as follows
JONSWAPspectrum
Elfouhailyspectrum
Freakwave in
tokai
B-F instability
Spectralpeakedness
Wave spectrummodels
Longuet-Higginsmodel
Phasemodulation
method
1D freak wavesand their
background waves
NormalizedmeasurementKA and TSM
NRCS ofbackground
waves
NRCS offreak waves
Identificationmethod
Figure 1 )e modeling flowchart
Complexity 3
S(k) 1k4B
JONl
1k4
aJ
2LPMJp (3)
In (3) BJONl is the directionless curvature spectrum l is
the wavelength of the gravitational wave LPM is close to thePM shape spectrum parameter andLPM exp[minus (5k2p)(4k2)) Jpis the peak enhancement fac-tor Jp cτ In addition aJ 0076 1113957X
minus 022 1113957X k0xkp k0Ω2c k0 (gu)210 and x are wind zones in units ofmWhen the corresponding values of Ωc are 084 10 and 20they represent fully developed sea surface mature sea sur-face and developing sea surface respectively
Elfouhaily proposed a joint spectrum function based onPM spectrum JONSWAP spectrum and Apel spectrumand its power spectrum is defined as follows [12]
S(k) BL + BH( 1113857
k4 (4)
In (4) BL and BH respectively represent the low-fre-quency nondirectional curvature spectrum corresponding tothe gravity wave and the high-frequency nondirectionalcurvature spectrum corresponding to the capillary wave)elow-frequency curvature spectrum BL satisfies the followingform
BL(k) αp
2c kp1113872 1113873
c(k)Fp (5)
Here
αp α0Ω
kp gΩ2
u210
c(k) g 1+k2k2m( )
k1113876 1113877
12
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
In the equations above αp is the generalized P-Kequilibrium zone parameter in the low-frequency wavenumber range c(kp) is the phase velocity corresponding tothe peak of the spectrum u10 is the wind speed at 10m abovethe sea and Ω U10c(kp) is the inverse wave age c(k) isthe phase velocity kp is the wave number distributed in thepeak of the spectral domain km
ρωgτω
1113968asymp 370ra dm
wherein ρω is the density of seawater τω is the surfacetension of seawater and g is the acceleration of gravitySubsequently Elfouhaily further proposed thatαp 60 times 10minus 3
Ω
radic and based on the dimensionless pa-
rameters kkp and Ω we can further determine the long-wave edge effect function Fp
In the comparative experiment a simplified scatteringmodel is used to analyze the power characteristicsδ0(ρ) πk2|R|2(q2q4z)P(sx sy) where sx and sy representthe sea surface slope in different directions P is the seasurface slope density function k is the wave number R is theFresnel scattering coefficient and q represents the scatteringvector Based on the control variable method the scatteredpower of the JONSWAP spectrum and that of the Elfouhaily
spectrum are compared to normalize the power waveformpeak value and the slope change rate under different windspeeds and wind conditions wherein the receiver height is45 km the satellite elevation angle is 30deg the wind directionis 0deg the wind speed varies from 6ms to 20ms and thefetches varies from 10 km to 19 km )e numerical resultsare shown in Figures 2 and 3
Comparing the numerical results in Figure 2 it can beseen that as the wind speed becomes larger the peaks of theJONSWAP spectrum and the Elfouhaily spectrum graduallydecrease the delay slope of the retardation gradually in-creases and the effect of medium and low wind speeds isobvious At the same time the numerical results in Figure 3show that with the increase of the fetch the peak and delayslope of the JONSWAP spectrum show regular changeswhile the Elfouhaily spectrum is not sensitive to the fetch
In this paper the JONSWAP spectrum is used as the targetspectrum [12] which means that the parameter ai in (1) alwayscomplies with the JONSWAP spectrum )e time series offreak waves can be simulated when the distance from thesimulated wave is x 0 in Figures 4 and 5 Similarly the spaceseries of freak waves can be simulated when t 0 in Figure 6
When the depth of water is 43m the effective waveheight is 510m the spectral peak period is 12s the spectralelevation factor is 320 the wave number is 200 themodulation wave number is 160 the spectrum rangechanges from 0 to 032 and the time tc is 100s or 200s thesimulation of the time series of freak waves is illustrated inFigures 4 and 5
According to the definition of the freak waves the heightof freak wave Hj should meet the following conditionsa1 HjHs ge 2 a2 HjHjminus 1 ge 2 a3 HjHj+1 ge 2 anda4 ηjHj ge 065 wherein ηj is the crest height of freakwaves corresponding to the horizontal line Hs is the ef-fective wave height and Hjminus 1 and Hj+1 are the wave heightsof adjacent waves before and after the deformed wave a1 a2a3 and a4 are characteristic parameters of freak waves [2829] )e characteristic statistics of wave duration are carriedout using the method of positive and reverse zero-crossingcounting and the characteristic parameters of the extremewaves are shown in Tables 1 and 2
)e effective wave height Hs is 376m Compared withthe input parameter the relative error is less than 5 FromTable 1 and Table 2 we can see that all of the parametersabove meet the definition of the freak wave and the freakwave is generated at the scheduled time which proves thevalidity of this model Wave time history spectrum and thetarget spectrum are compared in Figure 7 and the com-parison results between wave height distribution and Ray-leigh distribution are shown in Figure 8
)e results shown in Figure 6 indicate that the simulatedwave spectrum keeps the structure of the target spectrumand the spectral peak frequency is very similar to that of thetarget spectrum Figure 7 shows that the normalized cu-mulative probability distribution of wave height of thesimulated data agrees well with the Rayleigh distribution)e results show that the simulation results meet the re-quirements of random waveforms and the simulationmethod proposed in this paper is effective
4 Complexity
3 Electromagnetic Scattering CalculationModel of Simulated Freak Waves
31 Research on Backscattering Model Based on KA Methodand TSM )e Two-Scale Method (TSM) is developed on
the basis of Kirchhoff approximation (KA) method adaptedto the large-scale sea surface and the small perturbationmethod adapted to the small-scale sea surface [30 31]Firstly the scattering coefficients of small-scale sea surfaceare calculated by the perturbation theory secondly the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash38
ndash36
ndash34
ndash32
ndash3
ndash28
ndash26Peak power spectrum
Wind speed (ms)
Peak
pow
er sp
ectr
um (d
B)
JONSWAPElfouhaily
(a)
5 7 9 11 13 15 17 19 21ndash5
ndash45
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1Delay slope of power spectrum
Wind speed (ms)
Dela
y slo
peJONSWAPElfouhaily
(b)
Figure 2 Comparison of peak power and delay slope of power spectrums under different wind speeds
JONSWAPElfouhaily
10 11 12 13 14 15 16 17 18 19ndash4
ndash39
ndash38
ndash37
ndash36
ndash35
ndash34
ndash33
ndash32
ndash31Peak power spectrum
Fetches (km)
Peak
pow
er sp
ectr
um (d
B)
(a)
JONSWAPElfouhaily
9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0Delay slope of power spectrum
Fetches (km)
Dela
y slo
pe
(b)
Figure 3 Comparison of peak power and delay slope of power spectrums under different fetches
Complexity 5
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
S(k) 1k4B
JONl
1k4
aJ
2LPMJp (3)
In (3) BJONl is the directionless curvature spectrum l is
the wavelength of the gravitational wave LPM is close to thePM shape spectrum parameter andLPM exp[minus (5k2p)(4k2)) Jpis the peak enhancement fac-tor Jp cτ In addition aJ 0076 1113957X
minus 022 1113957X k0xkp k0Ω2c k0 (gu)210 and x are wind zones in units ofmWhen the corresponding values of Ωc are 084 10 and 20they represent fully developed sea surface mature sea sur-face and developing sea surface respectively
Elfouhaily proposed a joint spectrum function based onPM spectrum JONSWAP spectrum and Apel spectrumand its power spectrum is defined as follows [12]
S(k) BL + BH( 1113857
k4 (4)
In (4) BL and BH respectively represent the low-fre-quency nondirectional curvature spectrum corresponding tothe gravity wave and the high-frequency nondirectionalcurvature spectrum corresponding to the capillary wave)elow-frequency curvature spectrum BL satisfies the followingform
BL(k) αp
2c kp1113872 1113873
c(k)Fp (5)
Here
αp α0Ω
kp gΩ2
u210
c(k) g 1+k2k2m( )
k1113876 1113877
12
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
In the equations above αp is the generalized P-Kequilibrium zone parameter in the low-frequency wavenumber range c(kp) is the phase velocity corresponding tothe peak of the spectrum u10 is the wind speed at 10m abovethe sea and Ω U10c(kp) is the inverse wave age c(k) isthe phase velocity kp is the wave number distributed in thepeak of the spectral domain km
ρωgτω
1113968asymp 370ra dm
wherein ρω is the density of seawater τω is the surfacetension of seawater and g is the acceleration of gravitySubsequently Elfouhaily further proposed thatαp 60 times 10minus 3
Ω
radic and based on the dimensionless pa-
rameters kkp and Ω we can further determine the long-wave edge effect function Fp
In the comparative experiment a simplified scatteringmodel is used to analyze the power characteristicsδ0(ρ) πk2|R|2(q2q4z)P(sx sy) where sx and sy representthe sea surface slope in different directions P is the seasurface slope density function k is the wave number R is theFresnel scattering coefficient and q represents the scatteringvector Based on the control variable method the scatteredpower of the JONSWAP spectrum and that of the Elfouhaily
spectrum are compared to normalize the power waveformpeak value and the slope change rate under different windspeeds and wind conditions wherein the receiver height is45 km the satellite elevation angle is 30deg the wind directionis 0deg the wind speed varies from 6ms to 20ms and thefetches varies from 10 km to 19 km )e numerical resultsare shown in Figures 2 and 3
Comparing the numerical results in Figure 2 it can beseen that as the wind speed becomes larger the peaks of theJONSWAP spectrum and the Elfouhaily spectrum graduallydecrease the delay slope of the retardation gradually in-creases and the effect of medium and low wind speeds isobvious At the same time the numerical results in Figure 3show that with the increase of the fetch the peak and delayslope of the JONSWAP spectrum show regular changeswhile the Elfouhaily spectrum is not sensitive to the fetch
In this paper the JONSWAP spectrum is used as the targetspectrum [12] which means that the parameter ai in (1) alwayscomplies with the JONSWAP spectrum )e time series offreak waves can be simulated when the distance from thesimulated wave is x 0 in Figures 4 and 5 Similarly the spaceseries of freak waves can be simulated when t 0 in Figure 6
When the depth of water is 43m the effective waveheight is 510m the spectral peak period is 12s the spectralelevation factor is 320 the wave number is 200 themodulation wave number is 160 the spectrum rangechanges from 0 to 032 and the time tc is 100s or 200s thesimulation of the time series of freak waves is illustrated inFigures 4 and 5
According to the definition of the freak waves the heightof freak wave Hj should meet the following conditionsa1 HjHs ge 2 a2 HjHjminus 1 ge 2 a3 HjHj+1 ge 2 anda4 ηjHj ge 065 wherein ηj is the crest height of freakwaves corresponding to the horizontal line Hs is the ef-fective wave height and Hjminus 1 and Hj+1 are the wave heightsof adjacent waves before and after the deformed wave a1 a2a3 and a4 are characteristic parameters of freak waves [2829] )e characteristic statistics of wave duration are carriedout using the method of positive and reverse zero-crossingcounting and the characteristic parameters of the extremewaves are shown in Tables 1 and 2
)e effective wave height Hs is 376m Compared withthe input parameter the relative error is less than 5 FromTable 1 and Table 2 we can see that all of the parametersabove meet the definition of the freak wave and the freakwave is generated at the scheduled time which proves thevalidity of this model Wave time history spectrum and thetarget spectrum are compared in Figure 7 and the com-parison results between wave height distribution and Ray-leigh distribution are shown in Figure 8
)e results shown in Figure 6 indicate that the simulatedwave spectrum keeps the structure of the target spectrumand the spectral peak frequency is very similar to that of thetarget spectrum Figure 7 shows that the normalized cu-mulative probability distribution of wave height of thesimulated data agrees well with the Rayleigh distribution)e results show that the simulation results meet the re-quirements of random waveforms and the simulationmethod proposed in this paper is effective
4 Complexity
3 Electromagnetic Scattering CalculationModel of Simulated Freak Waves
31 Research on Backscattering Model Based on KA Methodand TSM )e Two-Scale Method (TSM) is developed on
the basis of Kirchhoff approximation (KA) method adaptedto the large-scale sea surface and the small perturbationmethod adapted to the small-scale sea surface [30 31]Firstly the scattering coefficients of small-scale sea surfaceare calculated by the perturbation theory secondly the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash38
ndash36
ndash34
ndash32
ndash3
ndash28
ndash26Peak power spectrum
Wind speed (ms)
Peak
pow
er sp
ectr
um (d
B)
JONSWAPElfouhaily
(a)
5 7 9 11 13 15 17 19 21ndash5
ndash45
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1Delay slope of power spectrum
Wind speed (ms)
Dela
y slo
peJONSWAPElfouhaily
(b)
Figure 2 Comparison of peak power and delay slope of power spectrums under different wind speeds
JONSWAPElfouhaily
10 11 12 13 14 15 16 17 18 19ndash4
ndash39
ndash38
ndash37
ndash36
ndash35
ndash34
ndash33
ndash32
ndash31Peak power spectrum
Fetches (km)
Peak
pow
er sp
ectr
um (d
B)
(a)
JONSWAPElfouhaily
9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0Delay slope of power spectrum
Fetches (km)
Dela
y slo
pe
(b)
Figure 3 Comparison of peak power and delay slope of power spectrums under different fetches
Complexity 5
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
3 Electromagnetic Scattering CalculationModel of Simulated Freak Waves
31 Research on Backscattering Model Based on KA Methodand TSM )e Two-Scale Method (TSM) is developed on
the basis of Kirchhoff approximation (KA) method adaptedto the large-scale sea surface and the small perturbationmethod adapted to the small-scale sea surface [30 31]Firstly the scattering coefficients of small-scale sea surfaceare calculated by the perturbation theory secondly the
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash38
ndash36
ndash34
ndash32
ndash3
ndash28
ndash26Peak power spectrum
Wind speed (ms)
Peak
pow
er sp
ectr
um (d
B)
JONSWAPElfouhaily
(a)
5 7 9 11 13 15 17 19 21ndash5
ndash45
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1Delay slope of power spectrum
Wind speed (ms)
Dela
y slo
peJONSWAPElfouhaily
(b)
Figure 2 Comparison of peak power and delay slope of power spectrums under different wind speeds
JONSWAPElfouhaily
10 11 12 13 14 15 16 17 18 19ndash4
ndash39
ndash38
ndash37
ndash36
ndash35
ndash34
ndash33
ndash32
ndash31Peak power spectrum
Fetches (km)
Peak
pow
er sp
ectr
um (d
B)
(a)
JONSWAPElfouhaily
9 10 11 12 13 14 15 16 17 18 19 20ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0Delay slope of power spectrum
Fetches (km)
Dela
y slo
pe
(b)
Figure 3 Comparison of peak power and delay slope of power spectrums under different fetches
Complexity 5
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)W
ave h
eigh
t (m
)Figure 4 Time simulation results of freak waves based on random phase modulation when tc 200s
0 100 200 300 400 500 600ndash5
0
5
10
15
t (s)
Wav
e hei
ght (
m)
Figure 5 Time simulation results of freak waves based on random phase modulation when tc 100s
0 100 200 300 400 500 600ndash6ndash3
0369
1215
FW
x (m)
Wav
e hei
ght (
m)
FW
(a)
0 100 200 300 400 500 600x (m)
ndash6
0
6
12
24BW
Wav
e hei
ght (
m)
BW
(b)
Figure 6 Comparison of the space sequence of freak waves simulated by phase modulation method and their background wave sequence
6 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
scattering coefficients of the mean sea surface are calculatedconsidering the slope distribution of the large scale andfinally the theory of the Two-Scale Method is used in thispaper When the incident plane is located in the x minus z spacethe backscattering coefficient is calculated as follows [32]
Here h means horizontal polarization and v meansvertical polarization θ is the incident angle zx and zy are theslopes of the rough surface along the x and y directionsP(zx zy) is the probability density function satisfying theslopes of the large-scale surfaces in different directionsθ0KAHH(θi) θ
0KAVV(θi) θ
0TSMHH(θi) and θ0TSMVV(θi) are the
backscattering coefficient results under different polariza-tion states σHH(θi
prime) and σVV(θiprime) are the backward scattering
coefficients of small-scale capillary waves in the horizontaland vertical polarization )e expressions are shown below[33]
θ0HH θiprime( 1113857 8k4i cos
2θiprime aHH1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
θ0VV θiprime( 1113857 8k4
i cos2θiprime aVV1113868111386811138681113868
11138681113868111386811138682W 2ki sin θi
prime 0( 1113857
1113946L
011139462π
0K
4S(KΘ)dKdΘK
1113971
ge μ
S(K) ai S(KΘ) S(K)f(KΘ)
P zx( 1113857 ki(1 minus R)sin θi P zy1113872 1113873 ki(1 + R)cos θi
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
Table 1 Statistical results of freak waves characteristic parametersbased on the upper and lower zero-point method when tc 200s
Figure 7 Comparison of the spectral structure of target powerspectrum and the spatiotemporal spectrum results of freak waves
Complexity 7
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
Here HH and VV represent different polarizationmodes f(KΘ) is the direction function μ is the incidentwavelength ki is the wave number of the incident wave andθi is the incident angle S(KΘ) is the two-dimensional seaspectrum)e simulation of the 2-D freak wave is realized byadding the direction function on the basis of the 1D powerspectrum to express the anisotropy of the energy distribu-tion aHH and aVV are the polarization amplitudes underhorizontal and vertical polarization In (8) KL and KS
represent the large-scale cutoff wave number and the small-scale cutoff wave number respectively Among them KL
determines the correction of the slope probability densityfunction in the TSM For a given wavelength the rough seasurface meets the condition KltKL which constitutes thelarge-scale part of the Two-scale Model whose scatteringcoefficient is calculated by the SPM method and it satisfiesthe condition of the first order perturbation approximationOn the other hand the small-scale rough sea surface shouldalso contain the spatial wave number which satisfies theBragg scattering condition of K KB 2ki sin θi We usethe condition of μ to determine the large-scale cutoff wavenumber However it is necessary to satisfy the condition ofKS ltKB 2ki sin θi and kiθsmall cos θi≪ 1 during the cal-culation of the scattering coefficient of small-scale rough seasurface and the boundary threshold μ is directly affected bythe wave number ki )e root mean square formula of thesmall-scale part is θ2small 1113938
+infinKS
11139382π0 S(KΘ)dKdΘ from
which we can calculate KS directly [34] Corresponding tothe spatial sequence simulation of freak waves based on (2)we set the parameter Θ to 0 which represents the wavedirection spectrum in θ2small
32 Analysis of Electromagnetic Scattering Results Based onthe simulation conditions of freak waves mentioned abovein this work we assumed that during the experiment the
one-dimensional freak wave appears at x 100m and theadjusted deformity ratio is 80)e simulation results of thebackground wave space sequence and the freak wave spacesequence are shown in Figure 6 BW means backgroundwave and FW means freak wave
During the calculation of the electromagnetic scatteringcoefficient of the sea surface the wind speed u10 is 14msbased on the actual data the radar operating frequency is118GHz the incident angle is 8938∘ the relative azimuthangle is 60∘ the polarization mode is VV the sea waterdielectric constant is 81 and the fetch is 10km Based on theparameters mentioned above the electromagnetic scatteringcoefficients of the freak waves in Figure 6 are calculated andthe results are shown in Figures 9 and 10
In Figures 9 and 10 the electromagnetic scatteringcharacteristics of freak waves and the background waves arecompared under the condition of VV polarization theNRCS of the freak waves varies periodically with the distanceof axis x and the scattering characteristics are similar to eachother When the NRCS of background waves reaches themaximum at the crest or reaches theminimum at the troughthe NRCS of freak waves is smaller than that of backgroundwaves )e maximum difference value minus 4712 dB appears atthe position 100m the freak wave is shown in Figure 6Comparing Figures 6 and 10 we can conclude that when theextreme wave appears in the one-dimensional simulated seasurface of background waves and freak waves the electro-magnetic scattering coefficient presents the nonsmoothtransition and instability obviously )is is because when weuse the Two-Scale Method to calculate the backscatteringcoefficient of sea surface the sea surface is divided into large-scale gravity waves and small-scale capillary waves artifi-cially However in fact the transition of the sea surface issmooth which indicates that the Two-Scale Method doesnot fully meet the physical significance In addition it isfound that in the vicinity of the extreme wave the
ndash1 0 1 2 30
01
02
03
04
05
06
07
08
09
1
HHs
Nor
mal
ized
cum
ulat
ive p
roba
bilit
y di
strib
utio
n
Simulated distRayleigh dist
Figure 8 Comparison of normalized cumulative probability distribution of Rayleigh distribution and simulated freak waves
8 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60ndash50ndash40ndash30ndash20ndash10
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 9 NRCS comparison result of simulated freak waves and their background waves based on KA method
0 100 200 300 400 500 600x (m)
ndash525ndash45
ndash375ndash30
ndash225ndash15
ndash75
FW-NRCS
NRC
S (d
B)
FW-NRCS
(a)
0 100 200 300 400 500 600x (m)
ndash60
ndash45
ndash30
ndash15
0BW-NRCS
NRC
S (d
B)
BW-NRCS
(b)
Figure 10 NRCS comparison result of simulated freak waves and their background waves based on TSM
Complexity 9
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
electromagnetic scattering coefficient of the backgroundwaves and the freak waves changed nonsmoothly inFigures 6 and 10 because the incident angle is 8938∘ in thecalculation of electromagnetic scattering coefficient whichfails to fully consider shielding effect What is most im-portant is that the NRCS of freak waves is much smaller than
that of the background waves at the position of x 100 BWmeans background wave and FW means freak wave
According to the above simulation results the charac-teristic parameters of freak waves are analyzed when thewind speed changes from 6ms to 20ms and the experi-mental results are shown in Table 3 and 4 From the tables
Table 3 Standardized statistical results of characteristic parameters calculated by KA method for freak waves at different wind speeds
Figure 11 Contribution of normalized deformation to the difference of scattering coefficient
10 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
01 02 03 04 05 06 07 0810
20
30
40
50
60
70
80
Normalized mean of different malformation parameters
Diff
eren
ce d
istrib
utio
n of
NRC
S fo
r diff
eren
t sca
tterin
g m
odel
s
Simulated freak wavesTarget spectrum
Figure 12 Distribution of scattering coefficients of different normalized deformities
21 June 22 June 0400 23 June 24 June 25 June
14
16
18
2
22
24
26
28
Time series (D)
Nor
mal
ized
refe
renc
e val
ues o
f fre
quen
cy p
eake
dnes
s (dB
)
Tamurarsquos measured data in JapanQp of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
Accident time
Figure 13 Comparison of spectral peakedness of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
Complexity 11
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
we can find that when the wind speed is 14ms the char-acteristic parameters of a1 a2 a3 and a4 began to meet thecriteria of the freak waves At this time the difference inNRCS between FW and BW is minus 12 dB as the wind speedcontinues to increase the characteristic parameters meet thecriteria of freak waves stability and the difference in NRCSalso decreases steadily which means that the difference inNRCS can also be used as a criterion for the identification offreak waves and the decision threshold is minus 12 db
33 Comparative Study of the Numerical Calculation Modeland the Measured Data Finally based on the Z-scorenormalization method the effect of the deviation of thedeformity coefficient a1sima4 on the height of the freak wave ismeasured in this work )e formula is expressed as followsZif ainf sf wherein Zif is the standard value of thecharacteristic parameter of the freak wave ainf is themeasurement level of different measurement units whencalculating the average error mf is the average value ofdifferent measurement types and sf is the average absolutedeviation mf 1n(ai1f + ai2f+ middot middot middot +ainf ) sf 1n(|ai1fminus
mf| + |ai2f minus mf|+ middot middot middot +|ainf minus mf|) and the values of i are1 2 3 and 4
From Figure 11 it can be seen that the normalizedmalformed parameters have the same influence on thebackscattering coefficient and a1 has the greatest influenceon the NRCS of the simulated freak waves )e influences ofa2 a3 and a4 are small and the fitting accuracy of each ofthem is better Moreover when the normalized malformedparameters are distributed in the range of 050 sim 070 thebackscattering coefficient no longer increases At this timethe NRCS tends to be stable and reaches a critical value At
this time x 058 and y 4170 By analyzing the exper-imental result in Figure 12 it can be seen that the simulatedfreak wave is basically consistent with the target spectrum inthe distribution of scattering coefficients and both of themhave a critical smoothing phenomenon in the red region inthe range of 052ndash067
Based on the above numerical conclusions the effectiveindex of B-F instability (BFI) [34] is used in this work to testthe probability of occurrence of freak waves )is index isdirectly related to the quasi-four-wave resonance interactionand wave surface displacement deviation from the normaldistribution )e expression of the above factors is shownbelow [34]
Qp 2Mminus 20 1113946infin
0dσσ 1113946
2π
0F(σ θ)dθ1113890 1113891
2
BFI k0M120 Qp
2π
radic
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(9)
Here k0 is the wave number of the peak frequency of thespectrum M0 is the zero-order distance of the wave spec-trum Qp is a physical quantity describing the width of thewave spectrum F(σ θ) is the energy spectrum of thesimulated freak wave surfaces and θ reflects the energydistribution along different directions [35] On June 232008 Suwa Maru fishing boat carrying 20 fishermen sank inTokai of Japan [36] According to reports the sea conditionswere moderate and the wave height was about 2ndash3 meters atthat time )e result of the investigation is that the fishingboat is most likely to have encountered a freak wave A largespectral peakedness corresponds to a small spectral width)is paper uses ai[2] to calculate and analyze the spectralsharpness and BFI evolution over time based on the outputtwo-dimensional ocean wave spectrum Combined with the
21 June 22 June 0400 23 June 24 June 25 JuneTime series (D)
Tamurarsquos measured data in JapanBFI of freak wavesrsquo NRCSFreak waves base on Longuet-Higgins model
0
005
01
015
02
025
03
035
04
Nor
mal
ized
refe
renc
e val
ues o
f B-F
insta
bilit
y (d
B)
Accident time
Figure 14 Comparison of B-F instability of Longuet-Higgins simulation result and freak wave measured in Tokai Japan
12 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
comparative data of the freak wave measured in Tokai ofJapan (144∘minus 145∘E 35∘minus 36∘N04 0023JUNE(UTC)) theeffectiveness of the numerical simulation model and scat-tering calculation model proposed in the work is analyzedAmong them the wind speed at the accident site is about11ms and the wave height is about 350m [37] Figures 13and 14 show the results of spectral peakedness and B-Finstability (BFI) values of the freak wave at the point of timerespectively
From Figure 13 it could be obviously found that thespectral peakedness is near a minimum value at the time ofthe accident which is about 219 and the wave spectrum iswide at this time )is state indicates that the possibility offreak waves due to B-F instability at the accident site isextremely small As shown in Figure 14 the BFI value is nearthe minimum value at the time of the accident about 023which is far from the condition where B-F instability easilyoccurs In addition from the perspective of time changesthe wave height is in the process of rapid changes from largeto small )e measured wind speed is relatively stable in themeasured data and the wind speed has been maintained atabout 11ms for a long time at the time of the accidentcorresponding to a wave height of about 350m )e aboveconditions indicate that the wave was not in a pure windwave state at the time of the accident Compared with thesmooth transition coefficient of the target spectral scatteringmodel the freak wave simulation method based on Longuet-Higgins model is more in line with the variation law of seasurface which proves the effectiveness of the proposedmethod In addition it is found that the average absolutedeviation used to measure the influence of the deformedparameters and backscattering coefficient on the freakwaves is more robust than the traditional method in thecalculation process of NRCS )is method is more reflectiveon the characteristics of the data obtained by differentmethods Especially in the process of normalizing outliersthe traditional deformation coefficient is very sensitive butthe Z-score normalization can achieve better robustnessthrough the substitution method mentioned above How-ever its computational complexity is higher
4 Conclusion and Future Work
In this work the practical problems of weakly nonlinearLonguet-Higgins model in the simulation of stronglynonlinear freak waves are fully considered Based on theLonguet-Higgins model and the comparative experiments ofwave spectrum a modified phase modulation method forsimulating freak waves is developed )e surface elevationsof some wave components at the preassigned place and timeare positive by modulating the corresponding random initialphases which enhances the total surface elevation andcauses a freak wave to be generated Comparative experi-ments with the measured data from Tokai of Japan show thatthe method can make the freak waves not only occur atspecified time and place but also conform to the statisticalcharacteristics of the wave sequence and keep consistentwith the target spectrum )e advantage is that the simu-lation of different wave heights can be achieved by
controlling the ratio of superimposed waves during theexperiment In addition this method can also make theinitial phase randomly distributed between 0 and 2π Anelectromagnetic backscattering model is established basedon KA method and TSM in this work and the numericalresults show that the NRCS of freak waves is much smallerthan that of the background waves )erefore the NRCS canbe used as the feature identification of freak waves especiallyconsidering that it is difficult to obtain the characteristicparameters of a1 a2 a3 and a4 in practical applications butit is relatively easy to obtain the NRCS difference betweenFW and BW from SAR images Based on the results of modeldata analysis we can draw the conclusion that when theNRCS difference of SAR image is less than or equal to -12 dBwe can determine the occurrence of the freak waves )eseresearch ideas provide a reference standard for early warningidentification of freak waves in practical engineeringapplications
Data Availability
No data were used to support this study
Conflicts of Interest
)e authors declare that they have no conflicts of interestregarding the publication of this paper
Acknowledgments
)is study was supported by the Shandong Key RampDProgram (Soft Science) Project under Grant no2019RKB01058 the Shandong Provincial Natural ScienceFoundation Grant no ZR2018BD029 the Scientific Re-search Foundation of Shandong University of Science andTechnology for Recruited Talents Grant no 2017RCJJ046the SDUST Research Fund the Training Program of theMajor Research Plan of the National Natural ScienceFoundation of China Research on Planting Structure Op-timization and Production Management Decision Tech-nology Based on Agricultural Big Data Grant no 91746104the National Key Research and Development Program ofChina Mine Internet of )ings of Cloud InteractionTechnology and Service Platform Grant no2017YFC0804406
References
[1] H D Zhang H D Shi and C Guedes Soares ldquoEvolutionaryproperties of mechanically generated deepwater extremewaves induced by nonlinear wave focusingrdquo Ocean Engi-neering vol 186 pp 1ndash11 2019
[2] S Son and P J Lynett ldquoNonlinear and dispersive free surfacewaves propagating over fluids with weak vertical and hori-zontal density variationrdquo Journal of Fluid Mechanics vol 748pp 221ndash240 2014
[3] B Guo H Dong and Y Fang ldquoSymmetry groups similarityreductions and conservation laws of the time-fractionalfujimoto-watanabe equation using lie symmetry analysismethodrdquo Complexity vol 2020 Article ID 4830684 9 pages2020
Complexity 13
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
14 Complexity
[4] N Li Z Zheng S Zhang Z Yu H Zheng and B Zheng ldquo)esynthesis of unpaired underwater images using a multistylegenerative adversarial networkrdquo IEEE Access vol 6pp 54241ndash54257 2018
[5] F Bonnefoy G Ducrozet D Le Touze and P Ferrant ldquoTimedomain simulation of nonlinear water waves using spectralmethodsrdquo Advances in Coastal and Ocean Engineeringvol 11 pp 129ndash164 2010
[6] B Guo H Dong and Y Fang ldquoLump solutions and inter-action solutions for the dimensionally reduced nonlinearevolution equationrdquo Complexity vol 2019 pp 1ndash9 2019
[7] T Xie T Shen W Perrie W Chen and K Hai-LanldquoElectromagnetic backscattering from freak waves in (1+1)-dimensional deep-waterrdquo Chinese Physics B vol 19 no 5pp 1ndash10 Article ID 054102 2010
[8] Li Wen-Long G Li-Xin M Xiao andW Liu ldquoModeling andelectromagnetic scattering from the overturning wave crestrdquoActa Physica Sinica vol 63 no 16 p 164102 2014
[9] Y Lin H Dong and Y Fang ldquoN-soliton solutions for theNLS-like equation and perturbation theory based on theriemann-hilbert problemrdquo Symmetry vol 11 no 6 p 8262019
[10] D Chalikov and A V Babanin ldquoComparison of linear andnonlinear extreme wave statisticsrdquo Acta Oceanologica Sinicavol 35 no 5 pp 99ndash105 2016
[11] J Lu Na Li S Zhang Z Yu H Zheng and B Zheng ldquoMulti-scale adversarial network for underwater image restorationrdquoOptics and Laser Technology vol 110 pp 105ndash113 2018
[12] G K Wu Y Q Liang and S H Xu ldquoNumerical compu-tational modeling of random rough sea surface based onjonswap spectrum and donelan directional functionrdquo Con-currency and Computation-Practice amp Experience 2019
[13] N Wang J Yu B Yang H Zheng and B Zheng ldquoVision-based in situ monitoring of plankton size spectra via aconvolutional neural networkrdquo IEEE Journal of OceanicEngineering 2018
[14] Y Liu S Gong and X Niu ldquoOn the sea surface wind fieldinversion models based on two wave spectrums using GNSS-Rrdquo Journal of Xi An University of Post and Telecommunica-tions vol 22 no 1 pp 106ndash110 2017
[15] H Lu D Wang Y Li et al ldquoCONet a cognitive oceannetworkrdquo IEEE Wireless Communications vol 26 no 3pp 90ndash96 2019
[16] Z Liu N Zhang and Y Yu ldquoAn efficient focusing model forgeneration of freak wavesrdquo Acta Oceanologica Sinica vol 30no 6 pp 19ndash26 2011
[17] M Sun Z Gu H Zheng B Zheng and J Watson ldquoUn-derwater wide-area layered light field for underwater detec-tionrdquo IEEE Access vol 6 pp 63915ndash63922 2018
[18] Y G Pei N C Zhang and Y Q Zhang ldquoEfficient generationof freak waves in laboratoryrdquo China Ocean Engineeringvol 21 no 3 pp 515ndash523 2007
[19] D L Kriebel ldquoEfficient simulation of extreme waves in arandom seardquo in Proceedings of the Abstract for Rogue waves2000 Workshops pp 1-2 Brest France 2000
[20] Z Q Liu N C Zhang Y X Yu and Y G Pei ldquo)e gen-eration of freak waves based on a modified phase modulationmethod I-)eory and validationrdquo Chinese Journal of Hy-drodynamics vol 25 no 3 pp 383ndash390 2010
[21] M G Di A Iodice and D Riccio ldquoClosed-form anisotropicpolarimetric two-scale model for fast evaluation of sea surfacebackscatteringrdquo IEEE Transactions on Geoscience and RemoteSensing vol 57 no 8 pp 6182ndash6194 2019
[22] G Franceschetti M Migliaccio and D Riccio ldquoOn oceanSAR raw signal simulationrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 36 no 1 pp 84ndash100 1998
[23] G Wu J Fan F Zhang and F Lu ldquoA semi-empirical modelof sea clutter based on zero memory nonlinearityrdquo IEEEAccess vol 7 no 1 pp 18125ndash18137 2019
[24] Z Zheng C Wang Z Yu H Zheng and B Zheng ldquoInstancemap based image synthesis with a denoising generativeadversarial networkrdquo IEEE Access vol 6 pp 33654ndash336652018
[25] K Hasselmann et al ldquoMeasurements of wind-wave growthand swell decay during the joint north sea wave Project(JONSWAP)rdquo Erganzungsheft zur Deutschen Hydro-graphischen Zeitschrift Reihe A vol A8 pp 1ndash95 1973
[26] T Elfouhaily et al ldquoA unified directional spectrum for longand short wind-driven wavesrdquo Journal of Geophysical Re-search vol 102 no C7 pp 781ndash796 1997
[27] X-L Zhou J Zhang H-J Lv J-J Chen and J-H WangldquoNumerical analysis on random wave-induced porous seabedresponserdquo Marine Georesources amp Geotechnology vol 36no 8 pp 974ndash985 2018
[28] G K Wu J B Song andW Fan ldquoElectromagnetic scatteringcharacteristics analysis of freak waves and characteristicsidentificationrdquo Acta Physica Sinica vol 66 no 13 pp 1ndash10Article ID 134302 2017
[29] W Guachamin-Acero and L Li ldquoMethodology for assess-ment of operational limits including uncertainties in wavespectral energy distribution for safe execution of marineoperationsrdquo Ocean Engineering vol 165 pp 184ndash193 2018
[30] A Khenchaf ldquoBistatic scattering and depolarization by ran-domly rough surfaces application to the natural rough sur-faces in X-bandrdquo Waves in Random Media vol 11 no 2pp 61ndash89 2001
[31] L Vaitilingom and A Khenchaf ldquoRadar cross sections of seaand ground clutter estimated by two scale model and smallslope approximation in HF-VHF bandsrdquo Progress In Elec-tromagnetics Research B vol 29 pp 311ndash338 2011
[32] G Wu J Fan F Zhang and T Wang ldquoComparative study ofTSM scattering characteristics based on improved Wenrsquosspectrum and semi-empirical modelsrdquo IEEE Access vol 7no 1 pp 24766ndash24774 2019
[33] LX Guo YH Wang and Z S Wu ldquoApplication of modifiedtwo-scale model for scattering from non-Gaussian sea sur-facerdquo Chinese Journal of Radio Science vol 22 no 2pp 212ndash218 2007
[34] P A E M Janssen and J Bidlot New Wave Parameters toCharacterize Freak Wave Conditions Research DeptMe-moR609PJ0387 ECMWF Reading UK 2003
[35] Y Goda ldquoNumerical experiments on wave statistics withspectral simulationrdquo Report of National Institute for Land andInfrastructure Management vol 9 pp 3ndash57 1970
[36] H Tamura T Waseda and Y Miyazawa ldquoFreakish sea stateand swell-windsea coupling numerical study of theSuwa-Maruincidentrdquo Geophysical Research Letters vol 36 no 1p L01607 2009
[37] S Liu Study of FreakWaves Based on Numerical OceanWavesSimulation pp 75ndash82 Ocean University of China QingdaoChina 2010
