Estimation of directional spectra from the maximum entropy principle

ABSTRACT

The problem of the estimation of a directional

function is the same as the determination of the

probability density function for random variables from

a limited amount of information. The concept of

entropy is introduced as an index of uncertainty of the directional function. The maximum entropy principle is

employed for the formulation of a new procedure to

estimate the directional function for the case of three quantity measurement. The validity and accuracy of the

new method is examined by means of numerical

simulations and some field data analyses. The results show that the proposed method recognizes well not only unimodal seas but also bimodal ones with the highest

resolution among the currently available methods of

directional spectral estimation.

NOINCLATURE

C1(f) :

Co-spectrum

f

: Frequency

c(elf) : Directional function for specific

frequency f

¡j : Entropy

H(k.o) : Transfer function for several quantities related to wave motions

k Wavenumber

Q1(f) : Quadrature-spectrum

s(f) : Power spectrum

s(k,a) Wavenumber-frequency spectrum

s(f.0) : Directional spectrum Lagrange! s multipliers

O : Wave propagation direction O : Angular frequency

mn(C) : Cross-spectrum among the measured quantities TC*USCHE wavEnsrrELT Çaboratof%Um 1100f SChOePShYd101fl0cha0a Ñchlof MekeiWeg 2,2828 CD Deift

ia

ESTIMATION OF DIRECTIONAL SPECTRA FROM THE MAXIMUM ENTROPY PRINCIPLE

K. Kobune and N. Hashimoto Port and Harbour Research Institute

Ministry of Transport Yokosuka, Japan

INTRODUCTION

For the estimation of directional wave spectra,

several methods have been proposed and improved. The

direct Fourier transformation method, the parametric

methods (the earliest of which is due to

Longuet-Higgins et.al. ), the maximum likelihood method2 and the extended maximum likelihood method are presently employed for wave data analyses.

However, these methods still have shortcomings such as the underestimation about the directional peaks and the leakage of the wave energy into neighbouring directional bands.

The directional resolution in the estimation can

be improved by increasing the number of quantities of the wave motion in the simultaneous measurements, for instance, by operating a large number of wave gauges or a cloverleaf buoy which was developed by Nistuyasü

et.al. '

to measure the curvature of the water surface

as well as the water level and the slope of

the

surface. For practical measurements, however, it is

disadvantageous to measure many quantities because of the cost of the measurements and the analyses.

Another attempt to improve the directional

resolution has been made by fitting an appropriate

directional function, such as the circular normal

distribution, for instance, to the measured one. The

method may be useful to determine the predominant wave direction of random seas. But, because it assumes a

directional function a priori, it may yield biased

estimates for the seas having different directional

spectra from the assumed one.

The problem of the estimation of a directional

function is the same as the determination of the

probability density function of random variables from a limited amount of information. The concept of entropy has successfully been employed in the latter problem. By introducing the entropy as an index of uncertainty of the directional function, the authors formulate the

procedure to estimate the directional function on the

basis of the maximum entropy principle. Although the method may be expanded for arbitrary combinations of

di8cussi0fl in this report is limited for the case of

three quantity measurement, such as the combination of

tte water

level and two components of horizontal

,0100ity or that of the water level and the two

of surface slope, because three quantity

measu1m1t is supposed to be the simplest and

therí'°'° the most practical one for field directional

yave observation.

The accuracy of the proposed method is examined by means of numerical simulations and field wave data analyses in this report.

1.ULATION OF DIRECTIONAL FUNCTION ESTIMATE

The relation between the cross-spectrum among the measured quantities and the wave power spectrum as a fmction of wave number vector and angular frequency is

- expressed in a genera]. form as follows = H,(k, u) H,(k, a)

(1) x exp - ik(x-x,,)) S(h, a)dk

where X, is the vector of the location of the sensors

utilized for the measurement, k is the wave number vector, O is the angular frequency, F1(k,a) is the

transfer function of m-th quantity such as slope of the

water surface, pressure and water particle velocity,

and the superscript * denotes the complex conjugate. The transfer function Hm(k.0) can be expressed in the following form

H..(k,a)=(s8)-(sin8)-h,,.(k,u) (2)

where O denotes the direction of a component wave having a wave number k measured counterclockwise from

the reference direction in the polar coordinates, and

hm(k.0) is the transfer function expressed in the same coodinates. The values of h , a and 8 _{in Eq.(2) for}

various quantities are listed in Table 1.

When the simultaneous measurement of several

quantities are performed at the same location, i.e.

x _{, such as the measurement utilizing a pitch} and roll buoy or a directional current meter, Eq.(1) _is rewritten after the conversion of the coodinates from -(k,a) to (f,e) as below:

2.,(f)

A.(f) h(f) 3(1)

Jo G(OIf) (cosO) (sinO) dO

a..+a ß.+ß (3)

where S(f) is the frequency spectrum, G(OIf)

_{is the}

directional function for specific frequency f, and it

is assumed that the directional

spectrum S(f.0)

is

expressed by Eq.(4).

S(f. 8) =S(f)G(81 f)

For the measurement of three quantities, i.e. the

Water level ri and the two components of the surface Slope r and T1) , Eq.(3) is reduced to the following form J G(8I

(i=O. 1.. 4)

a(0)=. i ai(8)=cos O as(8)=sin O a5(8)-=cos 28 a4(8)=sirs 28 ß= i QI2(1) - kC11(f) Q,s( f) kC,,(f) Cm(f) -C33(f) fis-2C,3(f) - kC,1(f)

where C1(f) and Qj(f) are the co-spectrum and the

quadrature-spectrum respectively. _{The combination of}

three wave quantities can be chosen arbitrarily, and

similar relations to Eq.(5) and (6) will be derived. By definition, the directional function G(OIf)

does not take negative value and should satisfy the

following relation

rG(oIf)de=1 ₍₇₎

Table 1. Transfer function from small _amplitude wave theory.

k: wave number, o: angular frequency, d: water depth, z: elevation from the bottom, p: fluid density, g: gravitational acceleration.

MEASURED

QUANTITY SYi150L h(k,c) a B

Water surface

elevation

00

Excess pressure p _g cosh kz

00

cosh kd Vertical water

surface velocity -io O O

Vertical water

surface acceleration o o

Surface slope (X) ik _o

Surface slope (Y) ny ik O i

Water particle

velocity (X) u cosh kz_{sinh kd} i o

Water particle

velocity (Y) cosh kz_{sinh kd} o i

Water particle

velocity (Z) w ₁₀ sinh kz _{o o}

Therefore, the directional function is regarded as a

probability density function defined between O and 2IT Then, the entropy fot the dirèctional function is given by Eq.(8)

NU}RICAL SOLUTION OF LAGRANGE' S MULTIPLIERS

The Lagrange's multipliers A aiie calculated by

the Newton-Raphson method for multiple variables.

Let where xexp(-± 2ta(8)d8 (15) BL=Ç xexp(-±lkak(8)}dO (16)

k and k+1 denote the k-th and (k1)-th iteration

respectively

Eq.(14) is the linear equation with respect to

C and can be solved for Sien álues of Xj. For the

first. step, j are calculated for the initial values of

.A°O (j=1,...,4) byEq.(11) and (15).

Then, the

values, of the muitipliersXj are ca'lculatedb Eq.(16).

The residuals Çj of the second step are Obtained by Eq.(14) from Aj1 . The same pocedure is repeated until the residuals C become small enough to get. the solution of . The present method is abbreviated as ME? hereafter.

EXAMINATION OF ACCURACY OF ESTIMATE

The validity of. MEP fo the estimation of directional functions is examined by numerical

simulation. The directional function employed In the exaèinatlon is a cosine type function given by Eq.(17).

S.(0)= Dcos''(!.ZL) (17)

When i=1 only, Eq.(17) yields a uriimodal

directional function, while a bimodal function is

formúlated by superposition of two unimodal directional functions, i e i=1 and 2, having different amplitudes

D _{, mean direction 8j} , and spreading parameter Sj The cross-spectra utilized to determine Aj in Eq.(11) are calculated through the numerical integration by

Eq.(1).

Figure 1 shows the comparison between the given directional function (true) and the estimated ones by

the method proposed by Longuet-Riggins et.a.l. (which is called LHM hereafter), the extended maximum likelihood

method (which is called EMLM hereafteÏ) and the

proposed method ( ME? ).

In Fig.1, the ordinate is

no.malized by dividing the valué of the dilectional

function by the maximum value of the true direction4.

function. For all the cases, it is seen that MEP

yields the closest estimate of the tx'ue directional function among these three methods. As to unimodal

directional functions, ME? yields a very accurate eàtlrnate for a directioñal spectrum with a sharp

directional peak. Especially for the case of S =5 and 20, there is no substantial difference between the true

directional function and the one estimated by MEP.

I

G(OIf) which maximizes the entropy H under the

According to the entropy theory, a function

constraint condition of Eq.(5) is the best estimate for

G(OIf) . The question is therefore to determine such function G(8jf) . This can be achieved by the use of

the method of Laangé's multipliers. For this

problem, an auxiliary function Eq.(9) with the unknown

parameteís Aj (Lagrange's multipliers) is set up.

L= -"G(of) tri G(61f)dO

+(2,- 1){1 - G(OIf)dO}. (9)

+

The estimate Of the directional function 5(8 If) which

maximize the entropy His obtained as the solution of

aL/oG(olf)=o ,i.e.

(lo)

Thus, the estimate 5(0 f) is given as an exponential function with a power expressed in a form of Fourier

series.

Substitution of G(01f) given by Eq.(1O) into

Eq.(5) yields the following nonlinear equations with respect to A.

ßt-a,(0))ecp {- 2iai(0)}d00

(1=1, ..,4)

The unknown parameter A1 (j=i,...,) are determined by solving Eq.(11) numerically, and Ai is calculated by

Eq.(12).

, _{In [ exp {}

_-

E 1iaAO)}dO] (12)

The procedure of the computation of the estimate G(61f) Is summarized as follows

Solve the nonlinear équations Eq.(ll) and get A Calculate Ao by Eq.(12).

Substitute these values of Aj into Eq.(1O) to get the estimate of the directional function G(elf)

Inôidentally, the estimate 5(8If) , when it is

calculated from Eq.(5) fo± i=O,1,2, results in the circular normal distribution which was proposed by

Borgaan

exp ocos(8-8,))/2rvl,(a) (13.)

where,

00 is the mean wave direction, bC ) is the

modified Bessel fünctioñ f the first kindòf order O

and a is an arbitrary constant.

the process of the numericál calculation of the arameter X, ,

it was observed that

i converged apidlY for S <22 even though the initial values of

were set to zero, büt Aj _{did not converge for the} yalues of S larger than 22.

1h order to obtain the

oiution of X for large S , the initial values were

oven a manner of trial and error. With appropriate a.iues, Àj converged for the values of S up to 40.

- Some examples of hi-directional seas are shown in yigs.2 and 3. In Fig.2, wind generated waves with

s1 =10 -and swell with S2- _{=100 are coming from}

different directions. The directional function of the eU has the peak value being one half of that of the

wind waves. The difference in the mean wave directions

j

denoted by6. It is seen that

LIDI does not detect

the existence of the swell (the emaïl peaks in the

figure). EMLM seems to recognize the existence of

the aveU, but the estimated direction of the swell is

not proper. MEP does not detect the swell when the

directional difference LO is small,

i.e. 8 =60°. However, for large 6 (120° and 180°), it estimates the

direction of the swell properly,

though it

overestimates the main peak brought by the wind

generated waves and underestimates the second peak caused by the swell. It is also seen that the estimate

by MEP shows the minimum leakage of the wave energy into neighbouring directional bands.

- Figure 3 shows the results for the bi-directional

seas caüsed by two wind generated wave groups with the directional difference O 60°,120°or 180°. The energy of the two wave groups are the same and the values of S of the two wave groups are 20.

It is obvious that

MEP yields the closest estimate of the true

bi-directional function among the three methods.

S2

.4° 'o

8 C4,,

EST-I-NATION OF DIRECTIONAL SPECTRA FROM FIELD DATA

The wave data analyzed herein were obtained at Niigata Port (37° 58'46°N, 139 °05'20°E) during the period from March 10 through 12 in 1984. The sensors employed were a ultrasonic wave gauge and a directional current meter, and these sensors were placed at 1.5m above the sea bed, where the water depth was 22.7m. The water level and the horizontal velocity components were measured for 20 minutes every two hours1 The

output analogue signals were digitized at the time

interval of O.5s.

Figure 4 shows an example of the estimate of directional functions for several frequencies for the

sea state observed at 2400 on March 11. In Fig.4, the

ordinates are the ratio of the directional function to

the maximum value of the directional function estimated by MEP for the respective frequencies.

It is seen that

MEP estimates the spectral peaks more distinctively

than other two methods.

Figure -5 shows the phase angle and the coherence for various frequency components between the three

quantities, n

, u and y calculated from the wave data obtained at the above mentioned observation.

Within the frequency range from 0.08 through 0.20 Hz,

the phase difference between n and u , and u and y are c-lose to ir (180°), and that between r and y

is

near to 00. _{In addition, the coherences between these}

quantities are larger than 0.5.

Thus, the relations between the behavior of the

three quantities see: to show the similar

characteristics described by the linear wave theory.

0.0 0 00 lOO -110

8 (j)

0.0 90 lID -lOO 8 -IO -00 O IO lOO 0 IO 8 c4,)

Fig.1 Examination by numerical Fig.2 Examination -by numerical Fig.3 Examination by numerical

simulation. simulation. simulation.

Although the coherences show small values for high

frequency range (f > 0.20) and for low frequency range

(f < 0.08), HEP seems to estimate distinctive

characteristics of the wave propagation direction, while the other two methods result in fairly broad

directional functions. This may indicates that HEP is

able to detect the information related to the wave

motion which is hidden in noises.

Figure 6 shows an example of the estimate of

directional spectra for the case where the sea state

might be bi-directional. The significant wave height was 2.63m. In Fig.6, the value of the directional spectrum is shown with the contour lines which are

drawn for every 1/15 of the range from logS(f,6)0 to the logarithm of the maximum value of the directional

spectrum (max(log S(f.e)) ). The directional spectra estimated by the three methods show similar patterns. However, it is clearly seen that the estimate of the

directional spectrum obtained by MEP show the best

resolution of these three methods.

CONCLUSION

Summing up the results of the examination of the

newly proposed method, the following is the major

conclusions.

1. The maximum entropy principle can be applied for

the estimation of the directional spectra of

random seas. 1.0 CC8 0.8 0. 0 CCB 0. 0 1.0 0. 0. MOP EHLH W HEP EHLH HEP ..LHM 1 -' .-.--N E S DIRECT! ON W N DIRECTION 1984 3 II 24 0. 1093750 -IHM 1984 3 II 24 1 0.0781260 INH 0. 0 N E DIRECTION ' 1984'3 11 I- 0.048 EMIM s 1.0 Cc9 0.1 1.0 0c0 0. 0 1.0 2' 6k 0(8) MOP HEP HEP EHLM

f

0.0-0.0 E 9 S V N DIRECTION N DIRECTION

When the new method utilizing the Maximum Entropy

Principle is applied for three-quantity

measurement of the random seas, the estimate of

the directional functions show better directional resolution than the Longuet-Higgins Method and the Extended Maximum Likelihood Method.

The new method recognizes well not only unimodal

directional functions but also bimodal ones. The method yields the best estimate of the directional

functions of the above three methods, especially at the directional peaks, and the leakage of the

wave energy into the neighbouring directional

bands is minimal.

The new method is most useful to distinguish

bi-directional seas from unisodal ones. However, for

the bi-directional sea of which directional

difference between the two directional peaks is small ( dO =600), none of the three methods can

recognize the two directional peaks properly.

ACKNOWLEDGENTS

The authors wish to express their appreciation to

Dr. Yoshimi Goda, the Deputy Director General of the Port and Harbour Research Institute, for his helpful

advice on various matters during the study. The authors would also like to express heartfelt thanks to

the members of the First District Port Construction

Bureau, Ministry of Transport, for providing the field wave data which were analyzed herein.

1.0 1984 3 II 24 I 0.2031260 -IHM IHM E N E DIRECTION ¡904 3 11 24 8- 0. ¡718-700

/

1904 3 II 24 1 0. 1406260 - . -0.8 ¡.0 cB 0. 0. 0 1.0 0. 1 8. 0 ¡984 3 ¡"24 HEP I_ LHM EMLM HEP - EMLH -W N DIRECTION W W N DIRECTION E s 8984' 3 ii 24. 8 0.2959260 -E s

Fig.4 Directional functions estimated by various methods.

N

DIRECTION

RNCES

i ongüet-Higgins,M.S., D.E.Cartwright and N.D.Smit.h_{Observation of the directional Spectrum of sea} waves using the motion3 of a floating buoy, Ocean

Wave Spectra. Prentice Hall. Inc., New Jersey, pp111136. 1963.

2. Capon,.T. : High-resolution frequency-wavenumbez.

spectrum analysis, Proc. IEEE, Vol.57, No.8,

pp.i4O8-l4i8 1969.

. sobe,M., LKondo an4C.Ho±ikawa Extension of NLM

for estimating directionaJ wave spectrum, Symposium on Description and Modelling of Directional Seas, Paper No.A6,, 15p., 1984.

4 Miteuyasu,H.

et.al.

Observation of

the directional spectrum of ocean sea waves using a cloverleaf buoy, Jour. Physical Oceanography, Vol.5, No.4, pp.150-211, 1975.

5. Borginan,L.E.

Dirçtional spectral model for

design use for surface wave3, Hyd. Eng. Lab. Uùiv. of CaliforMa, Berkeley, EEL 1-12, 1969, 56p.

350 300 250 Lu Ct) 200 = A. loo 50 1.0 O 0. e

NIIGATAt4ISHI USW CYD 1984 21124

pill

J r I r I-t I l=i I I I I I I I I r t p I

00 0.04 0.08 0. Ï 0.16 0..20 0.12

FREQUENCY (CPS)

NIXGATAHISHI USW OVO

j-O- 14-U - H-V -6-- U-V 1984 31124 0.00 ob 0.08 0L2 OIS 0.20 0.23 FREQUENCY CCPS)

Fig.5 _{Phase angle and coherence between water level ri} and velocity components u and y.

2-C.) Lu a Ui

u-(0) MEP

0.266 NIIOAÏA-N1s141 0, 238-0. 211: 0.184- 0.157-0 129-0. 102-0; 075, 0.047 0.266 0. 239 0. 211 0. 184 0. 157 0. 129 0. 102 0.075 0. 047 S LHM 0.266 NIICATA-NISHI 0. 239 0.211 0. 184 0. 157 0. 129 0. 102 0.075 0.047 s s DIRECTION EM LM NI IGATA-NISNI f N DIRECTION 1984 3 12 14 1984 3 12 14

Fig.6 Directional spectra estimáted b' various nethods. (Hi-directional sea) 1984 2 12 14 W N DIRECTION E S