Wave Directional Spectra Measurements by Small Arrays in Lake Ontario

J. Great Lakes Res. 18(3):489-506 Internat. Assoc. Great Lakes Res., 1992

WAVE DIRECTIONAL SPECTRA MEASUREMENTS BY SMALL ARRAYS IN LAKE ONTARIO

Ioannis K. Tsanis and Francois P. Brissette Department of Civil Engineering McMaster University Hamilton, Ontario L8S 4L7 ABSTRACT. Wave records collected during the fall period of 1987 from a wave gauge array at the National Water Research Institute's tower on Lake Ontario are used to obtain detailed directional information of the wave field. Two direct Fourier transform and three maximum likelihood methods are used for calculation of directional wave spectra. Comparison of these techniques with simulated data shows that the convolutive maximum likelihood method (CMLM) performs better for the wave gauge array used in this experiment. The CMLM is used to obtain the directional wave spectra in the cases of onshore and offshore wind, and offshore wind in the presence of an onshore swell in Lake Ontario. INDEX WORDS: Wave gauge array, wave period, wave direction, Lake Ontario.

the structures are expected during storms where the spectra are not symmetrical and of an ill defined functional form. In a review of advances in the study of wind waves, Barnett and Kenyon (1975) noted that at the time Ursell (1956) reviewed the field of wind waves, field observations relevant to wave generation and dissipation in lakes and oceans were very nearly nonexistent and they were (in 1975) simply very scarce. Although not entirely true today, theoretical ideas still seem to be ahead of experimental testing. Donelan et al. (1985) pointed out that previous field studies have been hampered by a lack of spatial resolution in both the wave and wind fields and especially a lack of wave directional information. Recognizing this problem amongst others, a 5-year international research project entitled "Deep Water Wave Breaking and WaveTurbulence Interaction" funded by the Panel on Energy, Research and Development (PERD) of Environment Canada was undertaken to study the physical processes at the air-water interface (Tsanis and Donelan 1987). One of the goals of this study was the detailed determination of the directional wave spectrum in different environmental conditions, i.e., fetch, wind direction (fetch gradient), and wind speed. The energy distribution of a sea state is usually expressed as a function of wave frequency and

INTRODUCTION An accurate description of a sea state is essential to many ocean endeavors such as wave forecasting, satellite surveillance, shore protection, and design of marine structures and vehicles. It is also important in the studies of upper mixed layer dynamics. From the viewpoint of exploration and use of offshore resources (in particular petroleum) the required knowledge of the surface structure of random seas has applications in (a) engineering design, (b) operational efficiency and safety, and (c) environmental protection. Application (a) is improved by advanced hindcasting techniques and accurate wave load calculations on offshore platforms, and application (b) requires good wave forecasting techniques, while application (c) is improved by advance knowledge of the fate of pollutants which eventually will aid in regulation development. The limited use of directional spectra by the offshore industry has been partly due to the lack of well defined directional spectra models for storm sea states and efficient computer programs for complete structural analysis using the directional spectrum. In addition to this, a cosine spreading function that is accepted by design engineers (Berge 1981) for computation with directional spectra is not correct because the extreme forces on

489

490

TSANIS and BRISSETTE

wave propagation direction. It is easy to obtain the frequency spectrum, because the record of the water surface at one point is sufficient, but in order to obtain detailed directional information, wave records from a large number of points are needed. In practice, only several simultaneous wave records are possible and it is important to obtain them in such a way so as to be able to make the most accurate estimate. Wave gauge arrays (Oakley and Lozow 1977) are often used for estimation of wave directional spectra. The purpose of this paper is to address the above question by using the pitch-roll and cloverleaf buoy parametric techniques, the maximum likelihood method (MLM), the iterative MLM (IMLM), and the new convolutive MLM (CMLM) for directional spectra measured by small wave gauge arrays. Comparison of the results from the above techniques is accomplished by using monochromatic waves at different frequencies and predetermined directional distributions. The best performing method is then used to describe the wave directional spectra in Lake Ontario under different environmental conditions. The results of this study will be important to offshore activities, e.g., exploration and use of offshore resources, for petroleum production. It will also be a useful and practical tool for engineers and scientists involved in offshore petroleum activities and concerned with engineering design, operational efficiency, safety, and protection of the environment. In addition scientists and limnologists will also benefit because the results of this study will lead to improved hind/forecasting methods. EXPERIMENTAL APPARATUS The field data were collected from the National Water Research Institute's (NWRI's) tower on Lake Ontario during a 3-year period (1985-1987) for the "Deep Water Wave Breaking and WaveTurbulence Interaction" project (Tsanis and Donelan 1987). The tower is in 12 m of water, 1.1 km off the beach at the west end of Lake Ontario, see Figure 1. In the tower's location the shoreline is straight and the bottom slope is gentle. The tides, seiches, and wind set-up can change the water level by an amount less than 0.1 m and the windinduced currents are typically less than 10 cm/s. The location of the tower makes possible fetches from 1.1 km for the prevailing west winds up to 300 km for east winds. Every year, Lake Ontario

Lake Ontario tower ~Okm

t

~2gi~ o

I 2 3 km

Om

N

?

2

FIG. 1.

elevation =IOOm

Map showing the location ofthe NWRI tower.

sees several episodes of higher than 10 m/s wind speed. Capacitance type wave gauges were used as sensing elements to provide information on the water surface elevation changes. The wave gauges were Teflon-coated wires 6 m in length with an overall diameter of 4.8 mm. Six wave gauges were arranged in a pentagon with one at the center as shown in Figure 2 (the distance between the central wave gauge and the others was 0.25 m) and were held taut with rubber shock cords. The actual location of the wave gauge array on the NWRI platform is shown in Figure 3 together with all the other sensors used during this study (Valdmanis et al. 1989). The array was symmetrical, resulting in smaller azimuthal differences in directional sensitivity, and its size was appropriate for the wave periods of interest, Le., 0.6 to 8.0 seconds. The wave gauges had a stable linear calibration (voltage versus water elevation) and were cleaned every week in situ from any algae and impurities that had accumulated on them. For each experimental run the rms of the water elevation r(x,y,z)

491

WAVE DIRECTIONAL SPECTRA MEASUREMENTS y

x

FIG. 2. The wave staff array with the numbering of the wave staff and the coordinate system orientation.

f is the mean water elevation, fi are the measurements of the water elevation, and N is the number of observations] for each wave gauge was evaluated for all six gauges and any gauge that differed from the average by more than 5010 was rejected. In all cases discussed herein no wave gauge was rejected. The wave gauge data, after filtration and amplification, were digitized on the tower and transmitted via a system of modems, a microcomputer, and an underwater cable to the data acquisition system in another microcomputer in a trailer on shore. The data were digitized at 20 Hz and the length of the different episodes was about 31 minutes. 0

WAVE DIRECTIONAL METHODS The wave directional spectrum (energy distribution in frequency and direction) is typically expressed as: (1) 5(f,0) = F(f) D(f,O) where f is the frequency, 0 the direction, F(f) is the frequency spectrum, and D(f,O) is the normalized directional spreading function. Different spreading functions have been given by different authors using a number of directional spectra techniques.

The most common spreading function used is the cos 2s(11z (O-Op» (Longuet-Higgins et al. 1963, Mitsuyasu et al. 1975, and Hasselman et al. 1980) where 0p is the mean wave direction. The polar representations of this spreading function appear to broaden too quickly with increasing (O-Op) where o is the mean direction at a given frequency. Many attempts based on pitch-roll heave buoy data to describe the behavior of the "s" parameter led to disagreement rather than consensus (Donelan et al. 1985). The failure of tqe various cosine distributions to model observed polar distributions led Donelan et al. (1985) to propose another spreading function, the sech2 ({3(O-Op». The hyperbolic secant squared shape is preserved and the width of the spectral spread is determined by the parameter (3. Donelan et al. (1985) showed that when the model spreading function is fitted to the half power points of the observed spreading function, that the wave energy in the peak direction is overestimated by 2070 for the sech 2({3(0-Op» distribution compared with 10010 for the COS2S(11z (O-Op» distribution. For this reason the sech2 ((3(O-Op» distribution was chosen as the appropriate spreading function to be used in the present study. Five different directional spectra techniques are described below. Two direct Fourier transform methods, i.e., pitch-roll and cloverleaf buoys, and three maximum likelihood methods (MLM) , i.e., MLM, convolutive MLM (CMLM), and iterative MLM (lMLM). (a) The Pitch-Roll Buoy (FB) method is the first direct Fourier transform method developed by Longuet-Higgins et al. (1963). The directional spectra can be determined by the information yielded by the motion of a buoy that measures the vertical displacement and angles of pitching and rolling. In the case of an array of wave gauges the water elevation f(x,y,t) and slope components fx(x,y,t) and fix,y,t) in x and y directions, respectively, can be evaluated (a minimum of three wave gauges is required). In the symmetrical six wave gauge array shown in Figure 2 the elevation of the wave gauge at the center of the pentagon f3(0,0,t) is used and the rest of the wave gauges are used for evaluating the slopes as follows (2a) r

~y

(0

°t) = 2:1 [ 2r(2,2) f f + 2 -

"

4

f 5- t 6 2r(2,5)

(2b)

492

TSANIS and BRISSETTE

LIGHTNING ROD

,

\

WA VEAIDER BUOY ANTENNA

•

METEOROLOGICAL SENSORS

'

/

\ DATA ACQUISITION SYSTEM HUT

FIG. 3.

MICROW AVE.. SC ATTEROMETERS

The Waves Platform with instruments during the 1987 field study.

where r(i,j) are the coordinates of the wave gauges (i = 1,2 for the x and y coordinates, respectively, and j = 1,6 for the wave gauges). Eqs. (2a) and (2b) in this form apply to this particular array. For different array configurations the slopes should be calculated accordingly. The co-spectra Cjj and the quadrature-spectra Qij of any pair of elevation and slopes are directly related to the Fourier coefficients an and b n as follows (Longuet-Higgins et al. 1963) a

0

a2

=

1 - C ll

al

11"

1 = - -2 1I"k

(C 22

-

= C33 )

1 1I"k

bl

QI2

b2

=

1 1I"k Q13

= ~C 1I"k2 23

(3)

where k = .,j(C22 + C 33 )/C ll is the wave number. At a given frequency f, an estimate S(f,O) of the directional spectrum is given by the first five

Fourier coefficients of the angular distribution of energy S(f,O)

=

:21 ao + !

6

2 3 (alcos 0 + b l sin 0) +

(a2cos 20 + b 2 sin 20)

(4)

which corresponds to a weighted average of the actual directional spectrum by a weighting function proportional to the cosine to the fourth power (cos4 (0-0/2») and has a rms width of 51 0 • In this method, the weighting function is independent of the frequency. (b) The CloverleafBuoy (eL) is the second direct Fourier transform method used by Cartwright and Smith (1964) and Mitsuyasu et al. (1975). The directional spectrum can be determined by the information yielded by the motion of a cloverleaf

WAVE DIRECTIONAL SPECTRA MEASUREMENTS

buoy. The cloverleaf buoy measures the vertical acceleration rll, the slopes rx, r y , and curvatures rxx, r yy' rXY of the wave surface r(x,y,t). For the case of a wave gauge array a minimum of six wave gauges is required for evaluation of the slopes and curvatures (the evaluation is also possible with less than six wave gauges but the curvatures must be obtained via an interpolation between the calculated slopes). A second order polynomial is used to describe the water surface as follows (Tsanis and Donelan 1989) r(x,y,t) = c1(t) + c2(t) x + c3(t) Y + cit) x2 (5) + cs(t) y2 + c6(t) xy Using Eq. (5) for the six wave gauges yields a system of six linear equations with six unknown coefficients ci(t), i = 1,6. The system of equations is solved and the coefficients cj(t), i = 1,6 are evaluated. The coefficients represent the elevation, slopes and curvatures of the water surface at the point (x,y) = (0,0) (Tsanis and Brissette 1991), i.e., ct(t)

= r(O,O,t)

c2(t)

= rx(O,O,t)

c3(t) = riO,O,t) 2c4(t) = rxx(O,O,t) c6(t)

rxiO,O,t)

(7e)

-fb (r(I,I) [rs+ r6- 2 r3J -2 r(I,5) [rl-r3J))/fe (7f)

where r(I,I) r(1,5) [r(1,5)-r(1,I)J

(8a)

fb = r(1,I) r(1,2) [r(1,2)-r(I,I)J

(8b)

fc = r(2,2) r(2,5) [r(1 ,2)-r(1 ,5)J

(8c)

fa

=

(8d) fe = r2(2,2) r(1,I) fa-r2(2,5) r(1,I) fb

(8e)

r(I,4) = r(I,2); r(2,4) = -r(2,2);

(8f)

r(I,5) = r(1,6); r(2,6) = -r(2,5);

(6a)

2cs(t) = ryy(O,O,t) =

493

(6b)

The values of the coefficients are:

The co-spectra Cij and the quadrature-spectra Qij of any pair of elevation, slopes, and curvatures are directly related to the Fourier coefficients a" and b o • The first five coefficients are identical to the ones in Eq. (3) while the last four coefficients are given as follows:

(7a) (7b) (7c)

(7d)

(9)

The angular distribution of the wave energy is evaluated by using the first nine Fourier coefficients. At a given frequency f, an estimate S(f,O) of the directional spectrum is given by the first nine terms of the Fourier series of the angular distribution of energy

494

TSANIS and BRISSETTE time series of the six wave gauges using a FFT routine. Eq. (12) using the inverse of C(f) gives the angular distribution of the wave energy which is real and positive.

S(f,O)

;~

(az cos 20 + b z sin 20) +

56 165 (a3 cos 30 + b 3 sin 30) +

~~ (a4 cos 40 +

b4 sin 40)

(10)

which corresponds to a weighted average of the actual directional spectrum by a weighting function approximately proportional to the cosine to the sixteenth power (COS 16 (0-0/2» which has arms width of 29° (Cartrwright and Smith 1964). In this method, as in the previous one, the weighting function is independent of frequency. (c) The Maximum Likelihood Method (MLM) was developed by Capon (1969) and was applied to an array of sensors for determining the properties of propagating waves. Jefferys et af. (1981) used the MLM to estimate the directional spectra from wave height measurements obtained by a wave gauge array. The derivation of the MLM is similar to Lacoss (1971). If a sea state can be represented by the summation of a number of monochromatic waves of power S(f,Oi) coming from directions 0i' with i = 1,N, then the true cross spectral density matrix is given in terms of S(f,Oj) in the frequency band near f by: N

Cjk(f) =

E

cj(f,OJ Ck*T (f,OJ S(f,OJ

(11)

i= 1

where N is the number of considered directions, cj(f, 0i) is the complex phase lag between the jth sensor and the origin for a wave of frequency f approaching from direction Oi. The energy incident from direction Oi is evaluated by minimizing the influence from all the other components. The minimization uses Lagrange multiplier theory and leads to an estimate of the energy in the plane wave

where K is a normalization factor, C-l(f) is the inverse of the cross spectral density matrix. For the case of the symmetrical array of Figure 2 the crossspectral density matrix C(f) is evaluated from the

(e) The Iterative Maximum Likelihood Method (IMLM) was developed by Pawka (1983) and applied to determine the island shadows in wave directional spectra. Oltman-Shay and Guza (1984) used the IMLM for point measurement systems such as the pitch and roll buoy and slope array. Krogstad et af. (1988) used the IMLM to obtain high-resolution directional spectra from horizontally mounted acoustic doppler current meters. The method is briefly outlined. Generally the cross-spectral density matrix C MLM reconstructed from the MLM estimate using Eq. (11) will not be equal to the observed cross-spectral density matrix. Krogstad et af. (1988) used a simple iterative scheme to solve this inconsistency

where SMLM is the MLM estimate, M(Sn) is the MLM estimate obtained from the reconstructed cross spectral density matrix using Sn, w is a relaxation parameter slightly above 1 (a value of w = 1.2 appears to yield convergence in about five iterations), and n is the iteration number. (d) Convolutive Maximum Likelihood Method (CMLM): The CMLM aims to improve the MLM estimates by taking into account the artificial directional spread imposed due to the array transfer function. Tests with monochromatic waves show an artificial directional spread that increases with decreasing wave number (Tsanis and Donelan 1989), due to the increasing ratio between the wavelength and the size of the array. This spread is different for each frequency and is defined as the window function W(f,OJ of the array. In order to reduce this artificial spreading a deconvolution process is necessary. The MLM directional estimate S(f,OJ is a convolution of the true directional spectrum with the window function of the array We(f,Oj,D(f,O» at each frequency f. The window function We is dependent on the form of the true directional spectrum, which is unknown in general. Instead we apply an approximate window W(f,Oj) derived from tests with monochromatic waves. The convolution integral is as follows

WAVE DIRECTIONAL SPECTRA MEASUREMENTS 7r

! Sc(f,1P) Wc(f, ()-1P)d1P

(14)

-7r

where Sc(f,()) is the true spectrum. Using 72 directions with 50 directional resolution, the above integral is evaluated numerically by solving a set of 72 linear equations for the 72 values of Sc(f,()) at each frequency. MODEL TESTS An FFT routine is used to transform the time series of the water surface elevation in Fourier space, where the cross-spectral density matrix between the water elevation, slopes, and curvatures can be evaluated. The different directional spectrum estimates were obtained using the program described by Tsanis and Brissette (1991) with the following values: Number of wave gauges = 6 (symmetrical six wave gauge array shown in Figure 2 with the distance between the wave gauge at the center and the others being 0.25 m) Number of directions = 72 Sampling rate = 20 Hz Spectral analysis = FFT method Number of samples in data block/FFT = 4,096 Number of blocks = 8 Frequency interval 8 X 20 / 4,096 = 0.039 Hz (averaging every 8 intervals is performed for spectral smoothing ). The performance of the FB, CL, MLM, IMLM, and CMLM techniques is tested with a monochromatic wave of unit amplitude at two typical frequencies of the swell (f = 0.115 Hz) and wind sea (f = 0.427 Hz), see Figures 4a & 4b. The directional estimates for the FB method were obtained using Eqs. (2a) and (2b). The same estimates can be obtained using Eqs. (7b) and (7c) but this is a more computationally expensive procedure. In fact the cross-spectra values based on the two above approaches differ by less than 2% and produce essentially the same directional spectrum S(f,()). The direct Fourier transform techniques, FB and CL, consistently overpredict the spreading at any frequency. Their behavior is satisfactory only if the wind sea or the swell have rms width greater than the rms width of the individual weighting functions, i.e., 51 0 and 29 0, respectively. The results from the CMLM are as expected delta functions

495

and for narrow distribution functions, the CMLM performs best. Wind waves are not long-crested or plane monochromatic but display a significant angular spread about the mean wave direction. Donelan et al. (1985) found that the spreading function followed a hyperbolic secant squared distribution. Therefore, a sech2 ({3«()-()p)) angular distribution, where {3 determines the width of the spectral spread, is used to test the performance of the MLM, IMLM, and CMLM. The cross power spectra density matrix (CPSD) was calculated from the sech2 ({3«()-()p)) distribution using Eq. (11). Figures 5a and 5b show a sech2 angular distribution with {3 = 1.24 (wide spread) at frequencies 0.115 Hz and 0.427 Hz, respectively. The CMLM directional spectral estimate is much better that the MLM estimate while the IMLM estimate although closer to the target spectrum shows spurious peaks. At narrower spread ({3 = 2.62), see Figures 6a and 6b the IMLM seems more stable but tends to overpredict the target estimate. The CMLM estimate is close to the target at low frequencies and somewhat lower at higher frequencies. Although the IMLM is attractive from the theoretical point of view, some problems are encountered with the convergence of the chosen iterative scheme. For low signal to noise ratios, the solution was found to oscillate after a few iterations, resulting in unrealistic negative values of the directional spectrum. A further problem was observed at low frequencies (f < 0.15 Hz) where the solution was found to converge toward a bimodal distribution with a small spurious peak 180 0 from the true direction of propagation, see Figure 7. It was found that five iterations was the best compromise for our particular array. It would insure convergence to a reasonable accuracy in most cases while avoiding numerical oscillations in all the test cases. Figures 8 and 9 show the behavior of the techniques FB, MLM, IMLM, and CMLM for a skewed and a bimodal input distributions, respectively. FB fails completely to predict the peak wave direction in both cases of skewed and bimodal distributions. In addition the FB estimate is symmetrical. The direct Fourier tranform method (FB) seems to return a symmetrical estimate centered on the centroid of the true distribution. The IMLM estimate has the tendency to return a bimodal distribution for the skewed input distribution case (Fig. 8a). The CMLM performs better at lower frequencies and gives similar results with IMLM at higher frequencies. Figure 9 shows the shortcom-

496

TSANIS and BRISSETTE

300

a MONOCHROMATIC WAVE

250 -

-;::;- 200

MONOCHROMATIC WAVE

f=0.115 hz

- - CMLM MLM - - - ~ IMLM FB _ ... _. CL

250

-;::;- 200 ..c

..c

"0'

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OJ "0

OJ "0

"N

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~

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f=0.427 hz

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100

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,, I

100

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180

."t---,-=-;."T-',....,...,.-,-..,.,.--,-t--

270

o

355

90

180

355

DIRECTION

DIRECTION

FIG. 4. Performance of the five wave directional methods for an input monochromatic wave at frequencies (a) f = 0.115 Hz and (b) f = 0.427 Hz.

1.2

1 .2

a

BETA=1.24

f=O.115 hz

BETA=1.24

- - TARGET CMLM ~ - - - MLM _._._-- IMLM

0.9

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f=0.427 hz

- - TARGET CMLM - - .. - MLM _._._-- IMLM

0.9 N

..c

..c

"0'

"0'

OJ "0

OJ "0

"N

"N

E 0.6 u

E 0.6

~

>-

>-

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0:: W

0::

Z

Z

w

w

0.3

0.3

o

90

180

270

355

o

90

180

270

355

FIG. 5. Performance of the five wave directional methods for an input sech 1(fJ f) distribution with (a) fJ = 0.115 Hz and (b) fJ = 1.24 at f = 0.427 Hz.

= 1.24 at f

DIRECTION

DIRECTION

497

WAVE DIRECTIONAL SPECTRA MEASUREMENTS 1.2

a

BETA=2.62

.,

b

f=0.115 hz

I

./ .,.

/

,

, ,

r· /

- - TARGET CMLM - - - - MLM _._._ .. IMLM

0.9 N

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w

<.)

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w

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180

90

270

355

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90

180

270

355

DIRECTION

DIRECTION

FIG. 6. Performance of the five wave directional methods for an input sech 2(fJ 8) distribution with (a) fJ = 2.62 at f = 0.1l5 Hz and (b) fJ = 2.62 at f = 0.427 Hz.

0.10

a

BETA=2.62 f=0.115 hz 20 iterations

0.08

I-IMLM - - - - TARGET

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I

£

'-..

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0.06

u

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0.04

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FIG. 7. peak.

90

180 DIRECTION

270

355

Convergence of the IMLM toward a bimodal

TSANIS and BRISSETTE

498

2

a

1.2 f=0.125 Hz

TARGET MLM CMLM IMLM FB

1.0

0.8

0.8

>-

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TARGET MLM CMLM IMLM FB

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f=O.4 Hz

>-

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b

C)

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W

0.4

0.4

02

02

00

00 90

CJ

180

270

6U

0

90

Dlf-;:ECTlmJ

180

no

360

DIRECTIOI\j

FIG. 8. Performance of the four wave directional methods for a skewed input distribution at frequencies (a) f 0.125 Hz and (b) f = 0.4 Hz.

12

1.2

a £=0.125 Hz

TARGET MLM CMLM IMLM

1.0

0.8

TARGET MLM CMLM - IMLM

0.8

f=0.25 Hz

FB

>-

>C)

C)

D:::

D:::

liJ

b

1.0

FB

=

06

w

z

Z

0.6

w

w

04

0.4

02

0.2

00

00 0

90

180 DIRECTION

270

360

0

90

180

270

360

DIRECTION

FIG. 9. Performance of the four wave directional methods for a bimodal input distribution at frequencies (a) f = 0.125 Hz and (b) f = 0.25 Hz.

WAVE DIRECTIONAL SPECTRA MEASUREMENTS

a

00

b

499

CMLM RB7022a

CMLM RB7022a

>~

u

z

W :::J

a

Lel

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LL

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136

1. 70 -j--,-,----,----"-,-,---,-----,1-,-,----,-----,-1--.- 1--1

o

90

180

270

360

DIRECTION

10

C 9 - - CMLM MLM IMLM

8

FB

7

CL

N

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6

u

"-

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E

5

>(')

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~ Cl'

w Z

w

3

2

o

90

180 DIRECTION

270

355

FIG. 10. Wave directional spectra for a 10 m/s northwest wind obtained with the CMLM (a) 3-D representation, (b) contour plot, and (c) directional distribution at peak frequency.

500

TSANIS and BRISSETTE

a

b

0.0

CMLM R87185b

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1.36

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90

180

270

360

DIRECTION

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z

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200

100

0 0

90

180

270

355

DIRECTION

FIG. 11. Wave directional spectra for a 14.5 m/s east wind obtained with the CMLM (a) 3-D representation, (b) contour plot, and (c) directional distribution at peak frequency.

FIG. 12. Wave directional spectra for a 12.5 m/s south-west wind in the presence of swell from the east obtained with the CMLM (a) 3-D representation, (b) contour plot, (c) directional distribution at swell peak frequency f = 0.115 Hz, (d) directional distribution at wind sea peak frequency f = 0.468 Hz, (e) directional spectra summed over all frequencies S(().

WAVE DIRECTIONAL SPECTRA MEASUREMENTS

ooF~;::

b

120

C

CMLM R8719Z

R87192

100

o 34 .~

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> 0.681

~

LL

CMLM MLM IMLM FB CL

f:::O.115 hz N

80

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l

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501

E

60

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1.02

w z w

40

1.36 20

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270

90

0

360

180

270

355

DIRECTION

DIRECTION

Fig. 12. (continued).

15

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9

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I

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e

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180 OIRECTION

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355

0

90

180 DIRECTION

270

355

TSANIS and BRISSETTE

502

normalized by the area under the spectrum

400

I Z 0

310

+71"

TH FORE HCAl

J

OBSERVED ......... WIND = WAVE <:>

<:>

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180

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360

WIND DIRECTION

FIG. 13. Mean wave direction at the centroid of the directional distribution at the spectral peak against mean wind direction. The solid line was obtained following Donelan's similarity considerations. The dotted line represents the line of perfect agreement.

ings of all the above techniques in that none of the techniques can predict very well the bimodal input distribution. However, because of the limitations (spurious peaks, arbitrary choice of number of iterations) and because the IMLM failed to prove to be clearly superior to the CMLM, the latter method was chosen for routine analysis of field data. The CMLM is not considered an optimum estimator but between the techniques discussed in this paper the CMLM seems to perform the best. FIELD DATA

Three runs taken from two storm cases during the fall period of 1987 were analyzed using the CMLM for estimating wave directional spectra. Tables 1 and 2 give detailed information on the physical parameters during the storm such as wind speed and direction, wave direction, peak frequency and rms spread at peak frequency defined as the square root of the definite integral +71"

1 (O-Op)2S(f,O)dO

-71"

and significant wave height (defined as four times the rms of the water surface elevation). The first run is R87022a taken from the first storm case which is a case of a 10 mls north-west wind (303°) with a fetch of 1.5 km. This is a case of strongly forced waves (fp = 0.47 Hz, T p = 2.13 s, cp (wave celerity in deep water) = (g/271") Tp = 3.32 mis, wind speed U = 9.8 mis, inverse wave age U/cp = 2.95). The last two runs, R87185a and R87192a, were taken from the second storm which is a case of an onshore east wind (R87185a) with a fetch of 280 km and an offshore south west wind in the presence of an onshore swell (R87192a). The run R87185a is a case of under-developed long waves (fp = 0.18 Hz, Tp = 5.56 s, cp = 8.66 mis, U = 14.5 mis, and inverse wave age U/c p = 1.67). The run R87192a is a case of strongly forced waves (fp = 0.44 Hz, Tp = 2.27 s, cp = 1.56 mis, wind speed U = 12.5 mls and inverse wave age U/cp = 3.52) in the presence of a swell (fp = 0.115 Hz, Tp = 8.69 s, cp = 13.56 m/s). In each run, the difference in the variance of the six wave gauges is used as a correction factor to the wave time series. Application of this correction leads to one variance for all the wave gauges. Typical differences in these runs were found to be less than 1%. 3-D and contour plots (contours are 1 to 6 cm 21 Hz/deg.) of the wave directional spectra (R87022a) for all angles (0° -360°) and frequencies (0.02-1. 70 Hz) are given in Figures lOa and lOb. Figure We shows the directional wave energy distribution at the peak frequency (fp = 0.44 Hz) for the five wave directional methods. The spreading parameters {3 obtained from least squares fit of the secant squared for the MLM, CMLM, and IMLM estimates are 2.20,2.25, and 2.85. The wave direction (325°) differs from the wind direction (303°) by 22° because the waves tend to be aligned with the longer fetch (Donelan et al. 1985). Figures lla and lIb show a 3-D and contour plot (contours are 20, 100, 200, 300, 400, 500, 600 cm 2 /Hz/deg) respectively of the wave directional spectra for the run R87185a. Figure 11c shows the directional wave energy distribution at peak frequency (fp = 0.168 Hz) for the five wave directional methods. The parameters {3 for the MLM, CMLM, and IMLM estimates are 2.55, 3.20, and

WAVE DIRECTIONAL SPECTRA MEASUREMENTS

3.90. The wave direction (68°) differs from the wind direction (84°) by 16°. This difference is as before, due to the fact that the waves tend to be aligned with the longer fetch (Donelan et al. 1985). The final run is a case of an offshore wind blowing against an onshore swell and 3D and contour plots (contours are 2, 5, 10, 40, 60, 80 cm 2 /HzI deg.) of the wave directional spectra are given in Figures 12a and 12b. Figures 12c and 12d show the directional wave energy distribution at swell peak frequency (fp = 0.11 Hz) and at wind sea peak frequency (fp = 0.44 Hz) for the five wave directional methods. The directional spectrum S(f,O) shows the swell with relatively narrow spread at 55° and the somewhat broader wind sea centered at 210°, see Figure 12e. The parameters {3 for the MLM, CMLM, and IMLM estimates are 2.25, 2.25 and 2.85 for the wind sea and 2.50, 7.35, and 4.70 for the swell. The direction of the swell is aligned with the longer fetch (60°) while the wave direction for the wind sea and the wind direction coincide. Comparison of the propagation direction of the swell in run R87192a and long wave in run R87185a indicates a difference of 30° which is due to the fact that the long wave is forced closer to the wind direction.

DISCUSSION

When analyzing wave data to extract an estimate of the directional spectrum, the choice of the appropriate technique can be a difficult one. To a certain extent, that choice may be influenced by such factors as: instrumentation used, directional resolution needed, and frequency of interest. A modeller looking for the main direction of wave propagation does not need the same spectral representation needed by design engineers for accurate calculation of wave loads on offshore structures. For the wave gauge array used in this paper, the following points need to be outlined. The two direct Fourier transform methods presented in this paper generally yield smeared estimates of the true directional spectra, especially so for narrow distribution functions. Clearly, in these cases, the truncated fourier series is not a good approximation of the directional spectra. Both methods are nevertheless successful in identifying the main direction of travel of waves at each frequency if the distributions are symmetrical. These

503

techniques fail to predict the peak wave direction in cases of bimodal and skewed distributions. The main advantage of the methods is that they are simple to implement and do not require any lengthy computations such as matrix inversions or iterative calculations common to other methods. But for applications needing good directional resolution, these method are inadequate. The three Maximum Likelihood Methods presented in this paper all give excellent directional spectra estimates. With narrow distribution functions at low frequencies, the CMLM gives the best estimate which is expected because the window function is optimized for these cases. With wider distributions and higher frequencies the distinction between the MLM and CMLM estimates becomes more difficult. Overall the CMLM and IMLM outperform their MLM counterpart, at the expense of additional calculations. The IMLM is the most numerically "expensive" method, requiring the computation of a new MLM estimate at each iteration, while the CMLM requires one additional matrix inversion in order to solve the convolution integral. From a numerical point of view, the CMLM seems to be the best compromise. The additional calculations are not a problem for fast computers, but on slower machines they could become a burden in the case of routine data analysis. The two examples of storm cases revealed significant features of the wave structure under different environmental conditions. The results for the two storm cases are given in Tables 1 and 2. The first storm case involves a west wind up to 12.0 mls in intensity (short fetched waves). The second storm case involves an east wind up to 15.8 mls in intensity (long fetched waves) and a southwest wind up to 12.5 mls in intensity (short fetched waves) in the presence of a northeast swell. The wave direction differs from the wind direction by up to 50 degrees and is consistently from the northwest direction. Considering that the waves are developing along the entire upwind fetch, if the fetch gradient about the wind direction is large, one can expect the wave direction to be biased toward the longer fetch (Donelan 1980). Figure 13 shows the mean wave direction against mean wind direction. The dotted line indicates a perfect agreement between the wind and wave directions. The solid line was obtained by using similarity considerations (Donelan 1980) with fetch averaged over 30° about the wave approach direction. The symbols represent the results from the two storm cases by using the

504

TSANIS and BRISSETTE

TABLE 1. Run 87016 87017 87018 87019 87020 87021 87022 87023 87024

Wind and wave parameters during the 4-6 November 1987 storm event (#1).

segment

Julian date

GMT time

length (min)

U 12 (m/sec)

Wind dir (deg.)

Wave dir* (degrees)

(Hz)

RMS spread

Hs (m)

308 308 309 309 309 309 309 310 310

21.50 22.54 00.48 17.27 18.50 20.27 22.41 01.48 04.14

33 95 95 10 21 95 95 95 31

6.4 7.0 5.7 8.0 10.8 12.0 9.8 11.0 6.2

250 263 248 274 300 288 301 302 324

230 (325) 305 290 (270)

0.84 (0.7) 0.65 0.76 (0.80)

55* 32 44*

335 290 (320) 330 315 330

0.43 0.47 (0.42) 0.47 0.42 0.50

37 51* 40 33 36

0.12 0.12 0.10 0.21 0.35 0.39 0.34 0.31 0.23

a,b,c a,b,c a,b,c a,b,c a,b,c

jp*

All segments are 31 min. long (exc. 87020) giving at least 100 d.o.j. U12 -wind speed at 12 m,jp-peak frequency, Hs-sign. wave height-Wave parameters (CMLM) are for segment "a" of each run. *bimodal distribution at peak frequency. Values for secondary peaks are in parentheses.

TABLE 2.

Wind and wave parameters during the 15-16 December 1987 storm event (#2).

Run

segment

87185 87186 87187 87188 87189 87190 87191 87192 87193

a,b,c a,b a,b a,b,c a to h a to j

Julian date 349 349 349 349 349 349 349 350 350

GMT time

length U 12 (min) (m/sec)

11.54 93.0 13.31 29.0 14.02 48.5 14.53 66.0 16.01 84.0 20.18 35.0 20.56 250.0 02.18 38.5 04.31 320.0

14.4 15.8 15.7 14.0 14.1 9.5 8.8 12.5 11.0

Wind dir (deg.) 84 83 89 82 84 240 233 210 219

Wave dir swell (sea)

75 70 55 65

85 65 70 60 65 (250) (235) (210) (235)

jp (Hz) swell (sea)

RMS spr. swell (sea)

Hs (m)

0.18 0.16 0.175 0.175 0.175 0.12 (0.5) 0.115 (0.66) 0.115 (0.44) 0.125 (0.55)

37 27 37 28 37 41 (74) 32 (44) 42 (35) 53 (33)

1.68 2.00 2.18 2.37 2.51 1.88 1.11 0.53 0.30

All segments are 31 min. long (exc. 87186) giving at least 100 d.o.! U12 -wind speed at 12 m,jp-peak frequency, Hs-sign. wave height- Wave parameters (CMLM) are for segment "a" of each run.

CMLM method which are in general in good agreement with the similarity relation that includes both fetch and wind components. In the cases of southwest winds (210°-240°) the short fetched waves generated are propagated in the wind direction (second storm case - swell is present). In these cases the wind is blowing directly offshore from a straight shoreline, the wave directional distribution is unimodal, and the fetch gradient effect is minimum. In the cases of westerly winds (240°-290°) the waves reveal double peaks at different frequencies or asymmetrical wave directional distributions. The waves are propagated mainly from the wind and the longer fetch directions. The solid line in Figure 13 shows a sig-

nificant deviation from the dotted line in this wind direction range which indicates a significant fetch gradient effect. In the case 87016 the wave energy is divided in two parts and the waves are propagating at two different frequencies and directions. The waves that are propagating in the wind direction at a peak frequency fp = 0.84 Hz. The waves that are propagating in the longer fetch direction as expected have a smaller peak frequency f p = 0.70 Hz. The directional RMS spread at peak frequency is higher than the southwest wind cases (unimodal distribution) because of the bimodality of the distribution. Similar results may be deduced from the cases 87018 and 87021. In the case 87017 the wave directional distribution at peak frequency

WAVE DIRECTIONAL SPECTRA MEASUREMENTS

is unimodal but asymmetrical, the longer tail being toward the wind direction. In the cases of northwest winds (300°-330°) the wave directional distribution is unimodal and the direction at peak frequency is from the longer fetch direction. Finally, in the cases of strong easterly winds, the waves are forced and are not reaching the west end of the lake at the long fetch direction (55-60°) as previously thought. In the beginning of the storm (case 87185) the waves are duration limited and their direction coincides with the wind direction. Later on as the waves become fetch limited they propagate at some angle between the long fetch direction (60° -70°) and the wind direction (82°-89°). As soon the wind starts dying, the swell starts to realign itself with the longer fetch (Donelan et al. 1985). The ratio of wave energy that includes waves propagating in the wind direction to the total wave energy (the rest of the wave energy includes the waves that propagate in the longer fetch direction) as a function of wind speed and fetch cannot be quantified at this moment. More westerly (short fetched waves) and easterly (long fetched waves and swells) storm cases should be analyzed to statistically validate the above findings. CONCLUSIONS

The performance of five wave directional spectra methods is tested with monochromatic waves and various angular distributions. The maximum likelihood methods, MLM, IMLM, and CMLM, are superior to the direct Fourier transform methods. The CMLM is superior to the MLM and IMLM in the case of narrow distribution functions and superior to the MLM in various angular distributions at low frequencies. At high frequencies MLM and CMLM behave similarly. The IMLM, although attractive from a theoretical point of view, can display some problems with the convergence of the iterative scheme and may become unstable. Therefore, the CMLM was used to obtain the directional wave spectra in the cases of onshore and offshore wind and offshore wind in the presence of an onshore swell in Lake Ontario. The results are in general agreement with Donelan's similarity considerations except in cases of bimodal distributions. In these cases the waves are propagating in two directions, e.g., following the wind and the longer fetch. These results outline the potential of the CMLM in resolving directional seas and studying the evolution of wave fields.

505

ACKNOWLEDGMENTS

The authors gratefully acknowledge M. Donelan for his recommendations and input resulted in improving this paper, and D. Beesley and other members of the staff of the National Water Research Institute for their assistance in the data collection. The work reported herein was supported through Grant URFBS-142 of the Natural Sciences and Engineering Research Council of Canada. REFERENCES Barnett, T. P., and Kenyon, K. E. 1975. Recent advances in the study of wind waves. Rep. Prog. Phys. 38:667-729. Berge, B. 1981. Design of offshore structures: directional spectra. In Proc. Directional Wave Spectra Applications Conference, pp. 353-366. University of California, Berkley, California. Capon, J. 1969. High-resolution frequency-wavenumber spectrum analysis. Proc. IEEE 57: 1408-1418. Cartwright, D. E., and Smith, N. D. 1964. Buoy techniques for obtaining directional wave spectra. In Buoy Technology, pp. 112-121. Washington, D.C.: Marine Tech. Soc. Donelan, M. A. 1980. Similarity Theory applied to the forecasting of wave heights, periods and directions. In Proc. Canadian Coastal Conf., pp. 47-61. National Research Council, Canada. _ _ _ _ , Hamilton, J., and Hui, W. H. 1985. Directional spectra of wind-generated waves. Phil. Trans. R. Soc. Lond. A 315: 509-562. Hasselman, D. E., Dunckel, M., and Ewing, J. A. 1980. Directional wave spectra observed during JONSWAP 1973. J. Phys. Oceanogr. 10:1264-1280. Jefferys, E. R., Wateham, G. T., Ramsden N. A., and Platts, M. J. 1981. Measuring directional spectra with the MLM. In Proc. Directional Wave Spectra Applications Conference, pp. 203-219. University of California, Berkley, California. Krogstad, H. E., Gordon, R. L., and Miller, M. C. 1988. High-resolution directional spectra from horizontally mounted acoustic doppler current meters. Journal of Atmospheric and Oceanic Technology 5:340-352. Lacoss, P. 1971. Data adaptive spectral analysis methods. Geophysics 36(4): 661-675. Longuet-Higgins, M. S., Cartwright, D. E., and Smith, N. D. 1963. Observations of the directional spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, pp. 111-132. Prentice-Hall, New Jersey. Mitsuyasu, H., Tasai, E, Sabara, T., Mizuno, S., Okusu, M., Honda, T., and Rikiishi, K. 1975. Observation of the directional spectrum of ocean waves

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using a clover-leaf buoy." J. of Phys. Oceanogr. 5(4):169-181. Oakley, O. H., and Lozow, J. B. 1977. Directional spectra measurement by small arrays. In Proc. Offshore Technology Conference, pp. 155-166. Houston, Paper No. OTC 2745. Oltman-Shay, J., and Guza, R. T. 1984. A data adaptive ocean wave directional-spectrum estimator for pitch-roll type measurements. J. of Phys. Oceanogr. 14:1800-1810. Pawka, S. S. 1983. Island shadows in wave directional spectra. J. Geophys. Res. 88:2579-2591. Tsanis, I. K., and Brissette, T. 1991. A two-dimensional wave directional spectra program, International

Journal of Environmental Software 6(3):151-160. _ _ _ _ , and Donelan, M. A. 1987. The WAVES programme on the CCIW research tower. National Water Research Institute (NWRI) Report No. 87-65,

Research and Applications Branch, Canada Centre for Inland Waters. _ _ _ _ , and Donelan, M. A. 1989. Wave directional spectra in mixed seas. In 2nd International Workshop on Wave Hindcasting and Forecasting, pp. 311-321. Vancouver, British Columbia. Ursell, F. 1956. Wave generation by wind. In Surveys in Mechanics, pp. 216-249. London: Cambridge University Press. Valdmanis, J., Tsanis, I. K.,_ Donelan, M. A., and Desrosiers, R. J. 1989. The "WAVES" platform and research towers, buoys and instruments, Lake Ontario 1985 to 1987. 23rd Congress ofInternational Association for Hydraulic Research, Ottawa, Canada, 21-25 August 1989.

Submitted: 18 October 1991 Accepted: 24 June 1992