laurence zsu hsin chuang^ and chia chuen

Directional characteristics of winter waves off

Taishi coast of Taiwan

Laurence Zsu Hsin Chuang^ and Chia Chuen

Coastal Ocean Monitoring Center, National Cheng Kung University,Tainan, Taiwan, R.O.C.EMail: [email protected]

Department of Hydraulics & Ocean Engineering, National Cheng KungUniversity, Tainan, Taiwan, R.O.C.EMail: [email protected]. tw

Abstract

A spatial array of wave gauges installed on an observation platform has beendesigned and arranged to measure the local features of directional wave offTaishi coast of Taiwan. A new method, named the Bayesian ParameterEstimation Method (BPEM), was then developed and adopted to determine themain direction and the directional spreading parameter of directional spectra.Also, the Mitsuyasu's empirical formula of directional energy spreading wasjustified to be representative of wave field. It was shown that the bigger thewaves age, the more rapid decrease in spreading. Nevertheless, the majorportion of the wave energy, which is around the peak frequency, may not becompletely related to the local wind condition. The relation between theparameter S^ and the wave age, as suggested by Mitsuyasu's formula, may beimproper in this area. The field data were shown that the wave steepness ismore related to S^ than wave age does. In general, the values of S^ decreaseas the values of the wave steepness increase.

1 Introduction

The dramatic growth of land demands for industry occurred over the past decadein Taiwan. Therefore, the Industry Bureau has made great efforts on reclaimingland from the sea. Among them, the Yulin reclamation industrial area is thelargest one. The total area of Yulin reclamation industrial area in the design is

Transactions on the Built Environment vol 40 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

254 Coastal Engineering and Marina Developments

10580 hectares. It is obvious that the construction of reclamation project willaffect the sea states and cause topographical changes nearby, or may arousedisaster. The Coastal Ocean Monitoring Center of National Cheng KungUniversity was then entrusted with the duty of collecting wave information inthis area. In consideration of severe weather and sea conditions, a spatial arrayof wave gauges installed on an observation platform was designed and arrangedto measure the local features of directional wave off Taishi coast of Yulincounty.

In estimation of wave directional information, several methods have beendeveloped for analyzing the directional spectrum. Since the directionalspectrum is expressed as a product of the directional spreading functionand the frequency spectrum and can be separately computed for eachfrequency, in this paper only directional spreading function will bediscussed. A new method, named the Bayesian Parameters EstimationMethod (BPEM), for estimating parameters of directional spreading function ofrandom waves from an array of wave probes was developed [1]. The method isthen applied to field data observed at Taishi station. Also, the Mitsuyasu'sempirical formula for directional spreading function is compared to the analyzeddata. For engineering application a modified Mitsuyasu's formula is finallyderived in consideration of the stage of wave developments and the wavesteepness.

2 A method for estimating directional distribution

Instead of solving cross-spectral density matrix in conventional directionalspectra analysis methods, the BPEM is to approach a chosen directionalspreading function by fitting its parameters to a criteria derived from a Bayesianviewpoint. The BPEM could be considered as a regression analysis to find themaximum joint probability of parameters, which best approximates the observeddata. On applying transfer function to directional spectra the method can alsobe applied to the directional wave analysis based on an arbitrary mixedinstruments array measurement. In comparison, the BPEM is formulated in thesame manner as the Bayesian Approach Method (BAM) [2], i.e., consideringerrors in the cross-spectra. However, instead of using complicated theoreticalmethodology and time-consuming iterative calculations in BAM, the simplicityand efficiency of the implement make the BPEM easy to apply for in-situanalysis. It also simplifies the sequences and lowers the possible accumulatederrors induced by traditional procedures, in which conventional directionalspectrum analysis methods are performed and then the estimated directionalspreading distributions are fitted to a chosen spreading model to have itscorresponding parameters.

The validity of proposed method was examined through numericalsimulations of various directional sea states, and its application to a practicaldirectional spectra analysis was also discussed with field wave data [1]. It wasshown that the method could be applied to the directional wave analysis on thebasis of arbitrary types and numbers of wave probes. There are theoreticalreasons for assuming that BPEM is superior to traditional methods when only


Coastal Engineering and Marina Developments 255

three (even only two) physical elements are available. Our previous researchresults verified this assumption [1]. This is very useful for in-site observations,because the simultaneous measurements of many wave properties are verydifficult for technical and financial supports. The results also have shown thatthe method is not sensitive to the layout of the sensor array and the spacingbetween the gages. However, the results may be affected by the number ofsegments of directional spreading function, if the number of partition is small,say K< 50 and K<4Q for an array of two probes and three probes, respectively.Also, the BPEM was compared with the BAM and the Finite Fourier SeriesMethod (FFSM) and was shown to be a powerful tool for directional waveanalysis [1]. As for the comparisons from measured data, the BPEM generallyyielded smoother and more continuous directional spreading for differentfrequencies than the BAM and narrower directional spreading than the FFSM,but all methods gave nearly identical main direction information.

2.1 Fundamental equations related to directional spectrum

The general relationship between the cross-power spectrum, O „„(/), for a pair

of arbitrary wave properties and the directional spectrum, G(/,<9) , is derived by

Isobe etal [3] as:

(1)

where /is frequency, k is wave number, /' is imaginary unit, A^and Ymn are thedistance between the m-ih and the /7-th wave properties at the coordinate (X, F),//„(/,#) is transfer function from the water surface elevation to the m-th wave

property, and //„ (/, 0) denotes the complex conjugate of H» (/, <9) .

To simplify the nomenclature in the equations, eqn (1) was rewritten byHashimoto [4] in the following form:

/(/)= r (/ ) l/) (' = 1'-' ) (2)

where TV is the number of equations, and

A, (/, 2) = #„ (/, 2)/U/, 0). cos& cos 0 + F_ sin 0}

/PF_(/) (3)

(4)

in which S(f) is frequency spectrum, D(9\f] is directional spreading function,

and W (f) is a weighting function introduced for normalizing and non-

dimensionalizing the errors of the cross-power spectra.

2.2 The rules of probability theory

The basic reasoning used in this work will be a straightforward application ofBayes' theorem [5]: denoting by P(A\E) the conditional probability thatproposition A is true, given that proposition B is true, Bayes' theorem is



(5)

In our problems, H is any hypothesis to be tested, D is the data, and / is the priorinformation. To construct the likelihood we take the difference between themodel H function and the data D.

/). = y(t. ) = //.+ e, i = 1,2,..., # (6)

If we knew the true signal, then this difference would be just the noise. Toderive this prior probability for the noise is a simple application of the centrallimit theorem of probability theory, i.e., the least informative prior probability ofthe noise has a Gaussian form as following one:

where a new parameter cr^ (the variance of the noise) has been added. Next weapply the product rule from probability theory to obtain the probability of a set ofnoise values { , &,,...,%} given by

(8)

Then the direct probability that we should obtain the data D, given theparameters, is proportional to the likelihood function:

f(D|W) = I(//) (2 2)f E%J 1 (D, -#,)2 1 (9)|_2cr w J

The usual way to proceed is to fit the sum in exponent. Finding the parametervalues, which minimize this sum, is called least squares. The equivalentprocedure of finding parameter values that maximize L(H) is called 'maximumlikelihood'. The maximum likelihood procedure is more general than leastsquares: it may be used when the likelihood is not Gaussian.

After substituting eqn (9) into eqn (5) and absorbing the influence of theterm P(D\I) into a normalization constant, the problem can now be simplified to

have the following relationship with respect to H for the given realized sampledata D.

P(H\DI) oc P(D\H I)P(H\I} = L(H)P(H\I) ( 1 0)

2.3 Formulation of directional spectral estimation

Firstly, it is assumed that the directional spreading function is expressedas a piecewise-constant function over the directional range from 0 to 2K (KA0 =2 71 ). Eqn (2) could then be approximated by the followingequation.

in which



i <*-'*« *****«, k = l,2,.,K (12)0 otherwise

and

G,(/) = Z)(|/) (13)

If the value K is large enough, the integral of the right hand side of eqn(11) is further approximated by eqn (14).

\l"h,(f,e)- i,(e)de = C%/,(/,0)- WW = h,(f,0,) 0 «„(/) (14)

Eqn (11) is then expressed as following one,

V,(/) = i>u(/>G,(/) / = 1,2 ...... ,N (15)k=\

For the convenience of the expression of the equations, the complex numbers arewritten in the following forms,

When the relation (eqn (15)) is applied to the observed data, the errorcontained in the data must be taken into account and so eqn (15) is modified toeqn (17) in consideration of the existence of the errors.

&=fa,,Gt+e, , = 1,2 ...... ,27V (17)

2.4 Unimodal directional spectrum

The unimodal directional spreading function employed is proposed by Longuet-Higgins et al [6] as following one:

6 = 1,2,...,AT (18)

For the given ^ ( / = 1,2 ...... ,2N ), the likelihood function of G* (k = 1,2,..., A:) and

cr* could be derived from eqn (9) and is expressed as

-- (19)

in which2#

2

_ D2 D+ C-(y4 -- ) (20)

(21)

R(S,0) = I^;«,. c o s " ( ) (22)



i=i ( -=1 f=i L ^ 2 J J

Substitution of the above into eqn (19) gives

mil f ^ r „ , ,(-4 (24)

The problem to be solved is to compute the probability of the parameters Sand 9 conditional on the data and the prior information, this is abbreviated as

P(S,9\DI}. However, eqn (24) has two other parameters, A and a. In

this problem the two parameters are referred to as nuisance parameters,because the probability distribution that is to be calculated does notdepend on these parameters. To eliminate the nuisance parameters,Bayes' theorem was applied and then integrate out A to give

P(S, 0, o\D, I) = j>(S, 0, A, <r\D, I)dA <x j>(£, #, A, ff\I) • P(D\S, <9, A, cr, I)dA (25)

If we had prior information about the nuisance parameters then eqn (25)would be the place to incorporate that information into the calculation. Herewe assume no prior information about the parameter A and assign them a priorprobability, which indicates 'complete ignorance of a location parameter'. Thisprior is a uniform, flat, prior density; it is called an improper prior probabilitybecause it is not normalizable. The posterior probability density for parametersS,0,a is proportional to

r>2 ~n-~\\ (26)C JJ

The above equation is depend on the variance of the noise, howeverfrequently one has no independent knowledge of the noise. The noise variance0-2 then becomes a nuisance parameter. It could be eliminated in much thesame way as the parameter A was eliminated. However a is restricted topositive values and additionally it is a scale parameter, so that a Jeffreys prior [7],Ho-, is adopted. Multiplying eqn (26) by the Jeffreys prior and integratingover all positive values a give

(27)c

This is the posterior probability of the parameters S and 9 independent of theparameter A and the variance of the noise. By taking logarithm we have

P(S,9\DJ) oclog[p(S,0|/)]+- (l-2A01og(2A _^l)_logC| (28)21_ C J

If the prior probability P(S,9\I) of the parameters S and 9 is the same, a

criterion named Log Posterior Probability of Parameters (LPPP) couldbe defined as



LPPP = (27V-l)log(2A -—) + logC (29)

The most suitable values of the spreading parameter S and the mean direction 6are determined so that the LPPP is minimum.

„ Lungtung North '^Q 1 #1 gauge

-1-,?=

#3 gauge

Figure 1: Locations of COMC Figure 2: Plan view of the spatialoperational stations array of wave gauges

3 Analysis of field data

The Bayesian Parameters Estimation Method (BPEM) outlined in section 2 isthen applied to field data acquired at an observation platform, about 4 km off theTaishi coast, the middle-western coast of the Taiwan island. The station,which was designed, installed and maintained by Coastal Ocean MonitoringCenter, is located at 23°45'39"N and 120*09'10"E in a water depth of 15 meters,as shown in Figure 1. The local tidal range is about 3 meters. Thebathymetry change at the neighborhood of the platform is rather flat.There are four ultrasonic wave gages were mounted at fixed positionsextending from the four corners of platform deck, which has beeninstalled on a single pile as shown in Figure 2. The simultaneousmeasurement of four wave elevations is performed over 20 min. every 2hours at a sample rate of 5 Hz. The wave records as well as windrecords are immediately transmitted by a radio telemetering system to anearby coastal relay station for landline transmission to our researchcenter at Tainan, Taiwan.

3.1 Selecting and processing of sea states

The sea states are dominated by the prevailingly northeasterly monsoonin the Taiwan Strait during winter season. A population ofnortheasterly spectra was established by selecting all those spectra forwhich the wind and the mean wave direction were from the northeastduring the period from December, 1993 to February, 1994. From thispopulation a subgroup was selected on the basis that: (1) The significantwave height is larger than 1.4m. (2) The coherence function offrequency spectra measured between gauges is higher than 0.8. Thisselection process resulted in 64 groups of data.



3.2 Estimation of directional spreading function

The directional spectrum is expressed as a product of the directionalspreading function and the frequency spectrum, and is separatelycomputed for each frequency. In this paper, only directional spreadingfunction is examined. The true directional spreading function to beestimated is assumed to be an unimodal form of eqn (18). Then LPPPdefined as eqn (29) could be applied to determining the main directionand the directional spreading parameter of directional spectra.

Figure 3 gives the normalized spreading parameter, S/S ,

plotted against the normalized frequency,///^, for the selected wave

records. According to the data measured by a cloverleaf buoy, Mitsuyasu et al.[8] developed a relationship between 5 and frequency as eqn (30).

The lines with slopes of -2.5 and 5, representing eqn (30), are also shown inFigure 3. The results show the same tendency as previous observations thatdirectional concentration has maximum value at the spectral peak and decreasedat the both sides of spectral peak. However, as mentioned by Hasselmann et al.[9] that the maximum directional spreading parameter S didn't always happenat peak frequency for our collected field data. Also, the data scatter comparedto the Mitsuyasu' s formula indicates the complexity of S.

0.01 —0.3

f/fp4.0 < wave age < 6.4

1.7 < wave age < 2.4 1.0 < wave age < 1.4Figure 3: Normalized parameter SIS versus normalized frequency f/fp



It is then assumed that S might be related to wave age. Figure 3 showsthat the data scatter is reduced if the relationship between S and frequency isclassified into subgroups dependent on wave age. The linear regression is thenperformed on the data to have a modified Mitsuyasu's formula in considerationof the stage of wave developments. The bigger the waves age, the more rapiddecrease in spreading.

(///„)-"

(///„)-" ;/>/„

(///„)" ;/</,

4.0 1 < wave age < 6.3 5

,* 1.74 < wave age < 2.4

,* 1.04 < wave age < 1.4

(31-1)

(31-2)

(31-3)

Mitsuyasu et al. [8] also showed that the parameter S can be a functionof wave age, €„ /U. Based on the selected data sets, Figure 4 shows that the

relationship is scatter and the result is not so consistent with the empiricalequation derived by Mitsuyasu et al. [8]. Therefore, using the wave age maynot properly indicate the stage of monsoon-generated wave development in thisarea. Another parameter for indicating the stage of wave development is wavesteepness. Figure 5 gives the parameter S^ plotted against the wave steepnessin log-log plot. It shows that the values of S^ decreases as the wave steepnessincrease. The relationship between S^ and wave steepness could then beapproximately represented by an empirical equation by

(32)

The correlation coefficient between eqn (32) and the selected data is 0.78.

Cp/U, versus wave ageFigure 4:

4 Summary and conclusions

Figure 5: S^ versus wave steepness

It is the purpose of this study to introduce a new method, named the BayesianParameter Estimation Method (BPEM) and to discuss local features ofdirectional wave off Taishi coast in the winter monsoon season. A spatial array



of wave gauges installed on an observation platform has been arranged tomeasure the wave directionality and to evaluate the validity of Mitsuyasu'sempirical formula of directional energy spreading.

The Mitsuyasu's formula for directional spreading function was justified tobe representative of wave field. The normalized directional spreadingparameter SiS in the formula was derived and plotted as a function of/%.The results show the same tendency as previous observations that directionalconcentration has maximum value at the spectral peak and decreased at the bothsides of spectral peak. However, the data scatter compared to the Mitsuyasu'sformula indicates the complexity of S. It was then assumed and proved that Sshould be related to wave age. The linear regression was then performed on thedata to have a modified Mitsuyasu's formula in consideration of the stage ofwave developments. The bigger the wave age, the more rapid decrease inspreading. Nevertheless, the major portion of the wave energy, which is aroundthe peak frequency, may not be completely related to the local wind condition.The relation between the parameter S ax and the wave age, as suggested byMitsuyasu's formula, may be improper. The field data were shown that thewave steepness is more related to S^ than wave age does. In general, thevalues of 5^ decrease as the values of the wave steepness increase.

References

1. Lin, S.R., Chuang, L.Z.H. & Kao, C.C., The wave directional spectrumanalysis by the Bayesian parameter estimation method, Proc. 19* Conf. onOcean Engineering in Republic of China, pp. 17-24, 1997.

2. Hashimoto, N., Kobune, K. & Kameyama, Y., Estimation of directionalspectrum using the Bayesian approach and its application to field dataanalysis, Kept, of the P.H.R.L, Vol. 26, No. 5, 1987.

3. Isobe, M. & Horikawa, K., Extension of MLM for estimating directionalwave spectrum, Proc. Sympo. On Description and Modeling of DirectionalSeas, Paper No.A-6, 1984.

4. Hashimoto, N. & Nagai, T., Extension of the maximum entropy principlemethod for directional wave spectrum estimation, Proc. 25th Int. Conf. Oncoastal Eng. (ASCE\ pp.232-246, 1994.

5. Bretthorst, G.L., Bayesian spectrum analysis and parameter estimation,Maximum Entropy and Bayesian Methods in Science and Engineering, Vol.l,pp.75-145, 1988.

6. Longuet-Higgins, M.S., Cartwright, D.E. & Smith, N.D., Observations ofthe directional spectrum of sea wave using the motions of a floating buoy,Proc. Conf. Ocean Wave Spectra, Prentice-Hall Inc., pp. 111 -132, 1963.

7. Jeffreys, H., Theory of Probability, Oxford University Press, London,1961.

8. Mitsuyasu, H., Tasai, F., Suhara, T., Mizuno, S., Onkusu, M., Honda, T. &Rukiiski, K., Observations of the directional spectrum of ocean wavesusing a cloverleaf buoy, J. Phys. Oceanogr., Vol. 5, pp.751-761, 1975.

9. Hasselmann, D.E., Dunckel, M., & Ewing, J.A., Directional wave spectraobserved during JONS WAP 1973, J. Phys. Oceanogr., Vol. 10, pp. 1264-1280, 1980.


laurence zsu hsin chuang^ and chia chuen

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