Numerical simulation of a random sea: a common error and its effect upon wave group statistics

TECHNISCHE UNIVERSTEIT LaboratorIum voor ScheepthydromeChafllca

Archief

Mekolweg 2. 2628 CD Deift

Numerical simulation of a random

sea: a coMìTh1t73°151

error and its effect upon s'ave groLI) statistics

M. J. TUCKER

InstIlute ('f O.canocrap/uc Sciences, crossiiavs, 7thnwn. Somerset, UK

P. G. CFIALLENOR and D. J. T. CARTER

ins dru te of Oc'anogìap1iic Sciences, lVortu lev. Godainiing, Surrey, LK

A commonly used method of simulating ocean waves from a specified frequency spectrum is shown to be incorrect. The method consists of adding numerous sine _{curves with random phases: and the} error arises from assuming that the amplitudes of these component sine waves are deterministic, when they are in fact random variables. Methods of using random amplitudes are described and only one is found to he satisfactory. In this method the number of randonivalues simulated - and then trans-formed with an inverse FFT - equals the required number of simulated data points. So simulation in the frequency domain can only give relatively short runs; it is necessary to work in the time domain if arbitrarily long runs are required.

Errors in wave group statistics derived from the incorrect _{simulation method are discussed and} related to discrepancies reported between groupiness in simulated data and ocean measurements.

INTRODUCTION

This paper reports a minor detective story. lt started when two of us employed a commonly-used method for the numerical simulation of ocean waves in order to examine some statistical properties which could not be calculated theoretically. The sea surface is generally assumed to be

Gaussian, and our method of simulation consisted

of summing a finite number of Fourier components to obtain the surface elevation (t) as a function of time, i.e.:

N

(t) = e,,cos(2irf,,t + (1)

n=j

where f,, are fixed frequencies, e,, _{are fixed amplitudes} determined from the assumed energy spectrum, and 5,, are random phases.

We happened to calculate the variances of the simulated records and noticed that these values varied less than the1 should'(according to the theory, Tucker'). On looking into the reason for this, it turned out that the method of

simula-tion is inherently incorrect. The method, assuming

deter-ministic values for e,, (n _{= 1. . - . , N), does not model a}

random Gaussian surface, except in the limit N - oo, as noted by Rice2 and explained below.

Thus the use of equation (1) in wave theory with the conditions

that N -

_{- e.g. Cartwright and}

Longuet-Higgins3_{- is correct, but for wave simulation with finite}

N - e.g. Goda.4 Hurwitz and Neal.5 Rye and Lervik,b _and

Haver and Moan7_{- is incorrect. lt is not clear which wave}

statistics simulated in this way are sciously in _{error, but} slatislics of wave iroups are certaiiìl affected, and it is

possible that some rather surprisingresults on wave group

Pc Ju,ic J )'SL(lç,l I, CIOt' J(Iflt. I 9(4.

118

_{-l'piid Oeca;: ]',ir:'j;, ¡ 'S..}

';

f\,'

lengths quoted in the literature (see references) could be due to faulty simulation.

An alternative method for numerical simulation is to

generate random numbers with a Gaussian probability distribution and then to pass these through a linear filter whose frequency response is calculated to give the correct spectrum. We are aware of no fundamental problems with this approach, but it _{is expensive in computer time} com-pared with frequency-domain synthesis.

REPRESENTATION OF A GAUSSIAN PROCESS IN TIlE FREQUENCY DOMAIN

For simplicity only a one-dimensional measurement, i.e. the

elevation _{(t) of the sea surface above still water lèvel at a}

fixed point, will be considered. However, _{the same}

prin-ciples apply to

a _{directional wave simulation. In this} paper it is assumed that the sea surface can be represented as a stationary Gaussian process with zeromean.

A finite wave record taken over an interval i = - T/2 to t = + T/2 from such a random process _{can he represented} by a Fourier series. If _{(r) consists of N values}

rn sampled

at discrete times t,,, with intervals t, then: *

IV/2 rn= (a,,cos27rfnt,,,+bnsin2irì'ntm) (2) ,z=0 N/ 2 e,,cos(2rf,,t,,, + ç5,,) ₍₃₎ 1= O

1or th nrcnt purpose it i much ciinplcr tn work oit!; the rJ

r:, t hir t ian w i iIi t ti _{cnn:plc'. nrnhcr rcrrescnti,n of E nouer} tranror,n. a i;ct ;vitli a one-sided spectrum. SU;.

41-1 J 87184/02011 sOS $02.no © 1984 CML PubItcatonc

where: j;, = (nV t)= ,,/T; b0, h\/2 = O; _{, 0N/2=}O. Also

= T/2 is the Nyquist frequency.

Note that the number of values of a,, plus the number of values of h,, equals N. the number of input values of 1). a,, is the nican and is one of the 1V values. thouth we shall assunte it to hc zero.

This representation of

r,,, repeats itself alter a time

T = ¡ /f, where ¿.f is the frequency separation of the har-monics, and is therefore valid only in the interval O ' t T.

For a finite Af equation (2) can thus only represent a

sample taken from the infinite time history of (t), and hence the spectrum of (2) is not the same as that of the whole signal. There is a clear distinction between S(f), th spectrum of the process of which.the record we have is a sample, and E(f,,) which is the spectrum of the individual sample, and which is defined 6y

E1J,,) f= (a + b,)/2 =

If, for example, we were trying to estimate S(f) from a real

wave record, then E(J) would be our raw estimate of it, but subject to a large sampling variance. Values of E(f,,) taken from different intervals when averaged will converge

on SU).

If T is long so that the f,, are densely packed, then_EUH) can be averaged over adjacent values of f;, within a small but finite frequency range so that as T - 00, _{this average}

E(i) - S(j). In the limit, the result would be the same if equation (I) were used, which is why it is possible to _use equation (1) for analytical purposes so long as T is taken to inflnity at a suitable stage in the argument.

From the theory of Gaussian processes, given for

ex-ample by Koopmans,8 ía,, and b,, are independent random variables chosen from a Gaussian distribution with common

variance a =S(f,,)f,,,using sf,, to include the general

case where f is not constant.

Since a,, and b,, are independent, their joint probability p(a,,, b,,) is given by the product of their individual prob-abilities, so that:

p(a,,, b,,) da,, db,,

I-[a1

i _L,2

-

exp , I

expl ---lda,,dh,,

2iw,, 2a i \/21r0,,

L 2ui

I

(c + hz)]

da,, db,,

2a

[

2a

= ;- exp

Changing to polar co-ordinates:

= r,, cos

= e,, sin ,,

[c1

p(c,,, 0,,) de,, dÓ,, = -- exp e,, de,, dØ,,

ro,

L _oj

dO,,

(,

=.----expj

1dc,,

_ir u

L.UJ

Therefore e,, has a Rayleia.h distribution with an rms value

of v'2,j,, (=\/[2s)\J;,J)c:uid the

,, are independent

wit h uniform distribution ori (O. ir).

lt follows that lias a negative exponential distribu-tinti with mean value of a, i.e. S(j,)

I, (and a mode of

zero).

Numerical simulation o/a random sca: M. J. Tucker. P. G. Cliallenor and D. J. i Guter The common error in wave simulation is to assume that the e,, are not random hut that:

e,, = v'S(J,,) AI,,

The ra ndom nature of t lie record is t hen thought t o arise from the raridorri _,,, i.e. front sittiply selecting the tuh:uis of the Cortiporlent sine curves at random. i hie use of

equa-tion (4) in the time history given by equaequa-tion (3) will

always produce a spectrum equal to S(f and titus some of the randomness of the real wave system will be 1ot.

DIGITAL SIMULATION

Random _{waves frequently need to be simulated, in order} to solve problems which are intractable analytically. The solutions may be obtained physically, in a wave tank, or numerically from digital output. Either way often involves a digital simulation procedure in the frequency domain, in which a number of sine curves are summed to producean

approximate Gaussian process. There are essentially two methods for carrying out this procedure. In both methods the amplitudes of the component sine curves have to he derived from the spectrum to he simulated. If they are

given deterministic values then the simulated process is not Gaussian; for instance the sample variance is bounded and essentially tIte sanie for all simulations.

The first method is based on the theory given above, with ¿/,, constant, independent of n, so f, n _{/. Random} a,, and b,, are generated from a Gaussian distribution with yariance S(f,,) ¿f and the sum produced from equation (2). pr, more efficiently, an inverse Fast Fourier Transform is used to obtain the simulated record. (An alternative is to generate c from a Rayleigh distribution with mean square

value of 2S(f,,)Í.f and 6,, from a uniform distribution

(O, 2ff). This method is suggested by Cuong et al.,9 but to the present authors, it seems more laborious than using a,, and b,,.)

lt should perhaps be pointed out that in this method the number and values of the frequencies are determined by the

required time interval between record values and

the

length of record, T: and that we

can say nothing about what happens outside the period T.

Figure 1 a-d show spectra estimated from simulations of I 800 values at 0.5 s intervals from a Pierson-Moskowitz spectrum with a variance of I in2 _{so 900 frequencies were} used in the FFT. These estimates were smoothed with a Daniel window, i.e. replacing every ten adjacent points w

their arithnietic nican.

Tihie second method involves summation of components at frequencies with unequal intervals, so the period of the sine curves are not harmonically related and tIte series

repeats only ait er a long time, t lins enabling much loníier

series to he simulated than h' t lie first method for a given

number of frequencies. There are various ways nf selectinC

these frequencies, such as choosing it;, such thatS(.f,,)

i.

is constant. Sometimes a random element is included, _mr example by Goda.4 which introduces slight differences sample variance from run to run, hut they are much stn:tllc than would arise from a t rue Gaussianprocess.

1f tite only purpose of tite experiment is _{to derive} i

linear response function for a structure, thus lack of randnot

deviation in _{the sample spectrum E(t) from the inruor} spectrum S(f) can he an :tdvantaee : hut it is no

clear that workers realise tira t n tirer statistics can e

hi:uscd. _{brennan'0 discirsscs i his problem uuid :uv; ' ha} 'i lucre _{tue many unsolved iiucouctic:rl quesliotus Ci)uiuiC}

_'I

with constrained sirnulations, that is. with simulations which arc flot trtily random represen t a t lotis. \t hen

simm-lating waves for noii-linear problems und statistical work it is extremely dangerous to use the relationship of equa-tion (3).

These effects are illustrated by Figs. 2 and 3 which show examples of spectra calculated from simulations of 15 min

o 00 20 15 10 5 o 00 25 20 -10 o 00 25 15 10 O L I 00 0-1 ₀₂ Frequency (Hz)

Fi cure la-d. _{Examples of spectra ¡i-oui 15 min records} simulated frani a Prerso,i-Mosk 'oit: _{spectrum (shoii',i}

dashed) using iwmch rn a,, and b,, and_{au inierse FFT}

ìaran&Jm sea: 11. J. Tucker, 1'. G. Cha/lanar and D. J. T. ('crier

o-t _0-2

0-2

I 0 _{il pp/led Ocean Research, 1984, I'l. 6. Vo. 2}

25 20 E 115 G? 10 u o 00 25 20 15 10 5 o 00 25 20 15 lo 5 o 00 25 20 15 10 15 o 00

records at 0.5 s intervals from _{a Pierson-Moskowitz} spec-trum. with a variance of I rn2, using Goda's method to

obtain the loo component frequencies _{between 0.05 and} 0.1 lIz. In Fig. 2, the component amplitudes _{vere taken} deterniinisticallv from the Pierson-Moskowitz _spectrum: and the sample spectra show very small variation _{from i lie}

0'l 02

0'l 0-2

0-2

01

Frequency (Hz)

Figure 2e-d. _{Examples of spectra from 15 min records} siniulated from a Pierso,;-Mos/-oo'itz _{spectrum (shown} dashed) using Goda's method (with _{deterministic} ampli-tudes)

02

o-1

02

23 o 00 0l 25 20 15 10 o

Numerical simulation of a random sea: if. J. Tucker, _{P. G. Cha//enr'r and D. J. T. C'arter}

02

o i

09 01 ₀₂

Freauency (Hz)

Figure 3a-d. _{Examples of spectra from 15 i',iu records} simulated from a _{Pierso,i-1Ioskoit'irz spectrion (S/low,,} dashe&/ using Goda 's _{method wit/i Rat'/ci'h- iisîrthuted} amplitudes

Pierson-Moskowitz these

_{simulations are not from}

_a

random Gaussian process. In Fig. 3, the component ampli-tudes were obtained by sampling from a Rayleioh distribu-tion with rms given by the _{Pierson-Moskowitz spectrum:} the sample spectra exhibit considerable variation about the

Pierson-Moskowitz spectrum.

I lowcvcr. this second simulai inn process is not entirely sat isfaci ory. even vith random amplitudes. for two reasons.

First, _{t he true process wit h a coot itluous sped rum IS}

ereodic. i.e. an average trotii one realisation over a lone time is equivalent to an averace of an ensemble of shorter realisations. This is not tie case for the simulation _process with a linìe Spectrum. For example. if the process used to obtain any one of the spectra in Fig. 3 had been continued after 15 min for a further 1 5 min. the samplespectra from

the two 15 min records would have been essentially the sanie. Any such simulation technique that involves

summing a finite number of sine curves will produce _series that are not ergodic; so if_{any averaees are to be used, they} must be ensemble averages and not timeaverages. Thus the

apparent advantage of this method, that

_{it produces a} longer record for a given number of frequencies. is_illusory. Secondly, the distribution of sample variances is not known. In particular, we do not know whether the simula-tion process is giving the correct value for the variance of this distribution. We might expect it to be too large: for example for Fig. 3, N = 100 frequencies were used to

produce the 1800 values, compared with 900 _frequencies for Fig. 1, suggesting an increase in variance of about

3

-i.e. proportional to _{/7V. However, for Fig. i, many of tite} 900 frequencies liad very small amplitudes, _{while for} Fig. 3, using Godajs method, the frequencies_{were clustered} around tite spectral peak.

The variance of the variance of records_{simulated from a} Pierson-Moskowitz spectrums

as in

Figs.

1-3, can be

derived theoretically, Tucker.' For _{a record of 900s with}

a sampling rate of 0.5 Hz and a variance of i m2

it is

0.0100 m4. Results calculated from 40 simulations of this

process are as follows:

Thus the 15 min record produced _{from 100 frequencies,as} for Fig. 3, does not have the_{correct distribution of sample} 1variance: perhaps if 150 or 200 frequencies were used the variance of the variance might he nearer the correct value, but tlos could only be determined by repeated simulations. and even tIten the correct distribution is not assured.

WAVE GROUPS

A wave group in a random wave train is a sequence of cmi-secutive high waves, and (lie _{number of such waves may}

used ro define the

_{group Icnet li. Zero-up-crossing wave}

height is L'erterallv used, and the bevel defining a htigitwave

is often ti-te significant wave height of tite wave train. The distribution of group length _{or the 'groupiness of Ute} waves - cari have a profound effect upon tile forces on ships. moored structures arid sea defences: so it is

ito-port lilt when carrying out model _{test s to have a realistic} simulation of eroilpiness.

Some recent papers have _{compared croup lengths} simulated waves vitb, ocean records. arid have _generalI':

replu ted discrepancies. For exatnplc. (;d'

_{finds liai}

!?'OtI (.)' 'i,z /?rcuuci,. !",-1. I r', 'i,

\'s

.. 121 Mean sample variance (m2) sample Variance of variance (m4) Theory 1.0 _0.0100 FF1 simulation 1.0 188 _0.0118 (Fig. 1) Goda simulation 0.9994 _0.0006 (Fig ) Goda + Rayleigh 0.9852 _0.0254 (Fig.3) 00 01 ₀₂

Yiofa rwiJin 5c;: 11. J. 7uckcì observed mein group lengths arc often greater titan those in

ive trains simulated using his met hod. l3urcha r ti'

com-pares waves recorded off Cornwall and off Jutland to waves simulated with the same spect ra iii two laboratories using different methods, digitally filtered random noise and the summation of ten sine curves with amplitudes fixed by the spectra. Ile found that with the first niet hod t he group

lengths of observed and simulated wave trains agreed well when wave heigit was defined by zero-down-crossings) but with the second method agreement was poor, with too few

long runs in the simulated waves.

An alternative approach to defining group length is in terms of the wave envelope. The niore narrow-banded, the more slowly the e-rvelope varies, and the greater the mean

group length. A marked feature of the sample spectra in Fig. i is the pronounced peakedness or narrow-bandedness of some compared to those in Fig. 2. The lengths of wave

groups within these samples would be greater than

ex-pected from any samples in Fig. 2, which might well explain the observations that observed mean run lengths are greater than obtained from simulations with fixed

coni-ponent amplitudes (although the difference found by

Burcharth'2 between results using _{up-crossing and} down-crossing waves smests a significant non-linear effect in this case, which wss in shallow water).

CONCLUSIONS

The sea surface is generally assumed to be a representation of a random Gaussian process and a method commonly used for simulating ocean _{waves, with a given spectrum,} both in a wave tank and numerically, is based upon the assumption that such a process

_{can be obtained by the}

addition of a fi.nite number of sine _{waves with random} phase and with amplitudes fixed by the spectral values at

the specifled frequencies. This method does not simulate a random Gaussian process, and hence does not correctly simulate ocean waves. The resulting constrained simulation can be useful in certain circumstances - provided its

limita-tions are recognised.

One such limitation is that it incorrectly reproduces the distribution of the lengths of wave groups. In particular, theoretical consideratkms and measurements, for example

by Goda," indicate that the

_{mean length of the wave}

groups can be expected to be higher in the

_{ocean than} obtained from such simulations.

Some simulation methods involve summing components at frequencies with unequal intervals, so that long series can be generated without repeating from relatively _few

i 2 _{Applied Occaim Rc'scarciz. 1984.} 1 4. & Xn. 2

P. G. CIza(lc,i'r ait! D. J. T. carier

frequencies. However, even if component amplitudes are

chosen randoritIv front Rayleigh distributions, the dist ributicirt _{of tite sample variances between runs will}

not be correct, and the process is not ereodic.

In order to simulate a random Gaussian surface in the frequency dom-air,, it is necessary to generate Gaussian vziriahhes tor each sitie auch cosine term, with hic total number of variables equal to the number of data points

to be simulated

¿om to generate tite same number of Rayleigh random amplitudes and random phases). An

inverse Fast Fourier Transform titen gives, very elli-ciently.

the required simulation for the specified

run length.

\Vorking in the tinie domain, rather titan in the fre-quency domain. and simulating waves by filtering Gaussian white noise avoids these problems and enables arbitrarily long series to be generated, at time expense of greater com-puting COStS.

REFERENCES

Tucker. M. J. The analysis of finite-h erigth records of fluctuat-ing signals, Br. J. App!. Phvs. 1957, 8, 137'

2 _{Rice, S. O. The mathematical analysis of random noise. Bell}

System 7'ech. J.. 1944, 23,282; 1945, 24, 46. (Reprinted in:' Selected Papers on Noise and Stochastic Processes, \Vax, N.;

cd., Dover Publications, New York. 1954)

3 _{Cartwright, D. E. and Lomuet-Fliggins. M. S. The statistical}

distrit,ution of the maxima of a random function. Proc. R. Soc. Lond. A 1956, 237, 212

4 _{Goda, Y. Numerical experiments on wave statistics with}

spec-tral simulation, Report of the Port and Harbour Res. Inst.

1970.9(3), 3

5 _{Hurwitz, R. B. and Neal. E. Digital simulation ofa Gaussian}

seaway based on the random pItase model. David W. Taylor Naval Ship Res, and Der. Center Rpt 79/056, 1979

6 _{Rye, Il. and Lervik. E. Wave grouping studied by means of}

correlation techniques, Preprinr In!. Situp. on Hydrodynamics in Ocean Engineering, Trondheim, Norway, August 1981, I, 25

7 _{Haver, S. and Moan. T. On some uncertainties related to the}

short _{term stochastic modelling of ocean waves, Applied}

Ocean Research 1983, 5 (2), 93

8 _{Koopmans, L. H. The Spectral Analysis of Time Series, Aca-'}

tiemic Press, 1974g

9 _{Cuong, Il. T., Trocsch, A. W. and Birdsahl, T. G. The} genera-tion of digital random time series, Ocean Engineering 1982,

9 (6), 581

iO _{Borgman. L. E. Conditional simulation of ocean wave}

proper-ties, Proc. 17th _{conf, on coastal Engineering, Sydney,} Australia, Vol. 1, 1980, 318

11 Goda. Y. On wave groups, Proc. conf _{on Behaviour of Off'} s/tore Structures. Trondhcim, Norway, Vol. 1, 1976, 115 12 _{Burcharth, H. F. A comparison of nature waves and model}

waves with special reference to ivave grouping. coastal .hnginecring 1981,4, 303