Preliminary results of some stereophotographic sorties flow over the sea surface

10JAN.1979 E: ._..4 y - t i J4

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lab. vi. SheepbouWk

Technische Hoges

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OVER THE SEA SURFACE

L,H. Holthuijsen

.Delf t University of Technology, The Netherlands

SYNOPSIS

Preliminary results are presented of a stùdy which is concerned with the directional characteristics of wind generated waves. The basic ap-proach adopted was to measure the actual sea surface elevation as a function of horizontal coordinates by méans of stereophotogrammetric

techniques. The surface representations thus obtained have been Fourier

transformed to estimate two-dimensional _{wave number spectra.}

Basic considerations concerning the photogrammetrical process, the

tranformation rules and the statistical significance _{of the results}

are described.

The required stereo photographs have been obtained during photographic

missions which were carried _{out in 1973 and 1976 off the island of Sylt}

(Germany) and off the coast of Holland. _{So far three two-dimensional}

spectra, each from a different flight, have been calculated. The sea and weather conditions during these flights are briefly stated. The wind

direction in these flights was off-shore.

Frequency spectra computed from the observed wave number spectra are compared with a hindcasted frequency spectrum and an observed frequency spectrum. The agreement is reasonable but some discrepancy needs to be

resolved.

For two of the three observations the directional distribution of the

wave energy is strongly asymmetrical around the wind direction. _This

asymmetry seems to correspond to asymmetry in _{the up-wind coast line.}

From the observed spectra a directional spreading parameter has been computed as a function of wave number. The results in normalized form agree well with published data. The absolute values of the spreading parameter for two spectra are within 30% of the anticipated values. For

the third spectrum the values were almost five times too large but a

2

In one of the spectra some indications of bi-modality around the wind dIrection have been observed in the directional distribution function near the peak of the spectrum.

1. INTRODUCTION

Observations of the two-dimensional spectrum of wind generated waves are

relatively few and are mostly based on methods with rather _poor

direc-tional resolution. The techniques which are used for the observations may be based on such systems as a sparse wave gauge array (e.g. Panicker

and Borgman, 1970) or a buoy capable of detecting directional

characte-ristics of the sea surface (e.g. Longuet-Higgins et al., ]963). The few detailed observations which have been públished were based on other

techniques such as high-frequency radio-wave backscatter (e.g. _Tyler

et al., 1974), analysis of the sea surface brightness (e.g. Stilwell,

1969, Sugimori _{1975) or stereophotography (e.g. Cote et al., 1960).}

These provided information with a high directional resolution but the analysis of the results in terms of wave characteristics has not been

very extensive.

The Deif t University of Technology and the Ministry _{of Public Works in}

the Netherlands have developed a system based on stereophotography which

monitors the instantaneous _{sea surface elevation as a function of}

hori-zontal coordinates. It has been used in this and other studies and it

is anticipated that it will also be used in future _{studies of wave}

phe-nomena such as wave transformation in the surf _{zone or wave patterns}

around marine structures.

The present study, which is a jointeffortof the University _{and the}

Ministry, is aimed at observing and interpreting two-dimensional spec-tra of wind generated waves in a variety of atmospheric conditions. The study is primarily directed towards the evaluation of the shape

charac-teristics of the directional _{energy distribution of the waves.}

For this study a few hundred stereo pictures have been taken since 1973 and the analysis has recently started. The results reported here are

preliminary in that the number of analyzed pictures _{is only a fraction}

of the total and in that the interpretation of these pictures has not as yet been completed.

The spectra which are presented here have been calculated from three

sets of pictures, each containing ten stereo pairs. _{These sets were}

cho-sen on two bases. One is the photographic quality which was judged by

photogrammetric experts, the other is the scientific _{interest. In this}

stage of the study it was felt that wave fields _{generated by off-shore}

winds would be of most interest because the boundary conditions are well defined. Also, results of past investigations of wave generation

(Hassel-mann et al., 1973, Hasselmanri et al., 1976) _{suggests that observations}

The first set of pictures which was analyzed was taken in September ₁₉₇₃

during almost _{idealtt off-shore wind conditions in the area just west}

of the German island of Sylt. These observations were carried out in

the framework of an international oceanographic project known _{as the}

Joint North Sea Wave Project (JONSWAP) which is concerned with the

Stu-dy of wave generation and prediction. A variety of articles directly related to JONSWAP has been published and more are being prepared for

publication. Some references are: Hasselmann et al., 1973, Spiess, 1975,

Hasse et al., 1977 and Hiihnerfuss et al., 1978.

The two other sets of pictures were taken in March and November 1976 in the area west of Holland near the town of Noordwijk, also in off-shore wind conditions.

Wave observations at sea level during the first and last flights are

available and these have been used for comparison with the

stereophoto-grammetric results.

2. STEREOPHOTOGRAMETRY OF THE SEA SURFACE

When an object is photographed from two slightly different positions,

the imagery in the two pictures will also be slightly different. The

differences depend upon the geomtry of the object. _{By measuring the}

differences, the elevation of the surface relative to an arbitrary plane

of reference can be determined. The conventional technique _{of analysis}

requires human interpretation of the pictures and complicated

stereosco-pic viewing devices. More advanced procedures, which have only recently been developed, use a computer to carry out a correlation between the

images to arrive at the same results (e.g. Crawley, 1975).

In conventiànal geodetic aerial survey the pictures are taken vertical-ly from an airplane in sequence and the interval is chosen such that

the pictures overlap iñ the area directly under the line of f1ight An

obvious condition is that the object does nót change between the expo-sures. In land survey this poses no problem since the ground surface does not move. The sea surface, however, changes very rapidly. To limit

the distortions between two succesive pictures _{to an acceptable level,}

they should be taken within an interval of 1 - 5 ms. The airplane cannot

possibly fly from one required point of photography to the other within

this time lapse. The consequence is that _{not one but two cameras are}

needed which take the pictures "simultaneously", _{that is, within an}

interval of I - 5 ms and that two aircraft are needed to position the

two cameras.

Apart from these technical differences in obtaining the _{stereo pairs,}

the methods and procedures used in this study are standard in geodetic survey and they have been used in the past by various oceanographic in-vestigators. A well publicized effort is the Stereo Wave Observation Project (SWOP, Cote et al., 1960) and the present system is essentially a revised version of the system used in SWOP.

4

It will suffice here to comment only briefly on the operational system.

Actually two independent systems were built. One is based on Hasseiblad

cameras and has been described in detail elsewhere (Holthuijsen et al., 1974). The other is an almost exact copy of that system except that the Hasseiblad cameras were replaced by UNK cameras of Jenoptik

which are superior in optical and metrical aspects. The Hasseiblad

tem was used for the observations in the area off Sylt and the UMK

sys-tem. was used in the area off the coast of Holland. Synchronization of

the cameras was achieved by using a radio signal that triggered a

corn-manU pulse which was manipulated electronically in such a way that it

complied with the timing characteristics of the receiving camera. The

synchronization error for the Hasselbiad system was less than 1 ms for

all of the analyzed stereo pairs and forthe ¡JMK system the synchroni-zation error was less than 5 ms. To position the cameras two Alouette III helicopters were used. These helicopters had a drop-door over which

the cameras could mounted. The distance between the helicopters was

estimated during the flight through a range finder which was imposed on the viewer of a third camera which looked from one helicopter to the other. It took a picture of the other helicopter every time the

down--ward looking cameras were activated. From these photographs the dis-tance between the helicopters could be computed and the scale of

photo-graphy could be determined.

The specification for the helicopter formation during a photcgraphic

sortie were largely based on photogrammetric requirements. Only the al-titude was based on the anticipated sea state since the noise and reso-lution in the spectrum are directly related to the altitude of

photo-graphy. The upper limit of the altitude was based on noise

considera-tions. The standard deviation of the..rneasurement _{error is estimated to}

be 0.03% of the altitude (HoLthuijsenet al., 1973) Taking

anoise-signal variance ratio of 1:10 as an acceptable upper.iimit, it can be

shown that the altitude should be less than 1000 times the standard

deviation of the instantaneous sea surface elevation (or 250 times the

significant wave height). The lower limit of the altitude is directly related to the resolution. If a resolution in the spectrum is required

equivalent to of the peak wave number oi better , it appears that for the Hasseiblad system the altitude should be higher than 6.7 times the reciprocal of the peak wave number. For the UMK system the factor is

4.0. For most htyoungtt sea states these _{upper and lower limits are not}

in conflict. The final choice of the altitude was confined to multiples

of 250 ft for the pilott s convenience.

The size of the sea surface covered in stereo in _{one stereo pair is}

usually too small to produce sufficient data for a reliable estimate

of the two-dimensional spectrum. To increase the _{amount of data more}

pictures were taken in sequence with a space - interval sufficiently

large to ensure photography of non-overlapping _{sea areas. The}

corres-ponding time interval between the _{exposures would be typically between}

4 s and 20 s (depending on camera type, ground speed _{and altitude).}

The.pictures can in principle be analyzed with the recently developed,

fully automated processes. The facilities however were not available

for the present study and the conventional technique was used. In the

three-dimensional space which is reproduced in the stereoscopic viewing

devices a righthanded system of coordinates was defined with the y-axis

in the direction of flight and with the z-axis upward. During the

ana-lysis the sea surface was read at a square grid with spacing Ex =

which was chosen such that aliasing in the spectrum would be limited

to only a fraction of the total wave variance. For each stereo pair the

analysis was carried Out in a square field as large as possible and

the elevations were determined relative to an arbitrary plane of

rafe-.rence. In the subsequent numerical analysis the linear trend was

remo-ved through a least-squares analysis. The fields obtained from a series

of stereo pairs were initially arbitrary in shape but fairly close to

a

rectangle. Later they were clipped or extended to a square of one

coumion size of L .L

as required in the spectral analysis. Sections

where no stereo nfrmation was available (mainly in the areas of

extension were filled with zeros.

3. TRANSFORMATION AND STATISTICAL SIGNIFICANCE

The sea surface data from the stereophotogrammetric analysis were

Fou-r.jer transformed to estimate the two-dimensional wave number spectrum

(k-spectrum). To insect the directional characteristics as a function

of wave number, the k-spectrum was transformed

o the wave-number,

di-rection space to produce the k,O-spectruxn. The k-spectrum was also

trans-formed to the frequency domain.

3.1. The It-spectrum

The definition adopted here for the two-dimensional wavenuinber

spec-trum E(k) is given by equations 1, 2 and 3.

Edt)

H(Ì)

(1)

where

e

_dxl

H() = Ii

h()

-i2Trk.x

-- 2

and

A=ffd

(3)

and <> denotes ensemble averaging. Observations of h(x) were available

from the stereo analysis in a number of square fields and these fields

were considered to be realizations of the ensemble. They were Fourier

transformed with a multi-dimensional multi-radix FFT procedure

(Single-ton, 1969) and the final estimates were obtained by averaging the

re-suits over the .available realizations. The sea surface data were not

tapered and the spectral estimates vere not convolved and consequently

the spectral estimates are ttrawlt estimates. In analogy with time series

analyss (e.g. Bendat and Piersol, 1971) the reliability is _represented

by a X -distribution with 2n degrees of freedom, where n is the number

of f ie'ds. The resolution denoted by ¿k .Lìk _{is on the order of (L}

.X

y

3.2. The k,8-spectrum

The transc5rmation of the i-spectrum to the k,O-spectrum Ls formally

given by equation 4.

E(k,O) _{= E()IJ1I}

₍₄₎

where k magnitude of O = orientation of and where the Jacobian

J1 = _{k. Computing the values of E(k40) at a regular grid in the}

k30-plane requires the estimation of E(k) _{at corresponding values of k.}

This was done by bi-linear interpolation of E(k) _{at the proper values}

of .k (see Fig. 1).

The directional resolution can be estimated by considering _theangular

distance betwen two neighbouring4 independent estimates of E(k) on a

circle in the k-plane centred in k = . On this circle with arbitrary

radius k, approximately 2irk/Ak independent estimates of E(k) are

avai-lable and the directional increment between these estimates in _radians

is ¿k/k. This would be a fair approximation of the directional

resolu-tion if all pictures were oriented in the _{same direction. But actually}

the orientation is a random variable due to the helicopter motion during

the sortie. The directional bandwidth to be added will be on the order

of twice the standard deviatibn (a0) of the helicopter yaw. The final

expression for the directional resolution (tO) is given in equation 5.

¿O ¿k/k + 2a0 ₍₅₎

The resolution in k will be on the order of the increment between

₊

es-- -I -I

timates of E(k) in the k-plane which is L = L

X y

The reliabilty of the estimates of E(k,e) can again be expressed in

terms of a X -distribution but the number of degrees of freedom is not

uniformly distributed over the k,0-plane. It constitutes an undulating function due to te fact that the estimated value of E(k,0) is based on four values of E(k) which are usually not equally weighted in the given interpolation technique. They are only equally weighted when a

transfor-med _{ridpoint in the k,0-plane coincides with the centre of a mesh in}

the k-plane. In that case the number of degrees of freedom _{for E(k,O)}

is four times the number of degrees of freedom for each individual es-timate of E(k). This is the upper extreme of the undulating function.

The lower extreme ocurs when a transformed k,0 gridpoint coincides _with

a gridpoint in the k-plane. Then the number of degrees of freedom of

an individual estimate of E(i). The values of the two extremes are 8n

and 2n respectively.

3.3. The f-spectrum

The f-spectrum is determined by integrating the f,0-spectrum over the

range (O,iî) and multiplying the result by two. The-operation is given by quation 6.

'T

E(f) = 2

f

o

The f,0-spectrum has been computed from the -spectrum. The

relation-ship to transform from wave number vector to frequency is based on the

linear dispersion relation for deep water corrected for currents. This

expression and the transformation are given in equations 7, 8 and 9.

f (gk/27r) + k.V (7)

E(f,0) =E()jJ2I.

= fi

cos (o

-

is the current vector and Vand O _{are its magnitude and orientation.}

To determine the values of ECk) theCsame procedure as described above was used.. The resulting spectrum is the frequency spectrum as observed in a point stationary with respect to:the sea bottom. This was done so as to be able to compare the results with measurements carried out

with anchored buoys.

Expressions for the approximate resolution (M) and numbér of _degrees

of freedom (N) are given by equations 1O and 11. I

(IO)

2 2 271 g

N8n'T

(II)

4. DESCRIPTION OF THE SITES AND THE WEATHER CONDITIONS

Maps of the areas off Sylt and off Noordwijk and two bottom profiles are given in Fig. 2, 3, 4 and 5. It may be noted that both areas are

similar in general appearance but an important difference seems to

be that the coast near Sylt recedes sharply North and South of the is-land and is strongly asymmetric with respect to the off-shore direction, whereas the coast near Noordwijk is more continuous and symmetric. For

both sites the water is effectively deep for waves generated by an of

f-shore wind.

The sortie in the area west of Sylt was car±ied out during the field

operations of JONSWAP in 1973, on September 18th, at 17:30 hr (local).

the JONSWAP operations of 1973 and also give results of meteorological observations from ships, buoys and balloons in the area. According to this information the windspeed and direction prior to the flight had been fairly constant for one day. Since the wind was almost perfectly

off-shore the situation was classified as an "idealt' generation case.

In the two hours prior to the flight the windspeed and direction at station 8 (see Fig. 4), at 10 m elevation was approximately 13 m/s and

1100

respectively. The direction is only a few degrees off the "ideal" off-shore direction of 107

The weather during this flight was poor for photographic operations and all pictures which were taken were under-exposed, in spite of the best possible photographic measures. Pictures were taken over six stations of JONSWAP, including active wave monitoring stations 5, 7 and 9 (see

Fig. 4). The frequency spectra observed at these stations are given in

Fig. 6 and they may be used for a direct comparison with the results of stereo observations over these stations. But in selecting the pictures

for preliminary investigation preference was given to photographic

qua-lity rather than availabiqua-lity of groundtruth information and it appeared that the best pictures were taken over station 10, which was otherwise inactive during the flight.

The frequency spectrum at station 10 was estimated with a "hindcast"

procedure based on the JONSWAP parameter relationships (Hasselmann et

al., 1973). The "hindcast" was attempted for stations 5, 7 and 9 with

the observed windspeed of 13 in/s but the results (Fig. 6) were rather poor, although they seemed consistent with the statistical variation

in the observations of J0NSWAP. The agreement improved when a windspeed

of 15 rn/s was used (Fig. 7). This was the windspeed estimated just prior

to the.;f light. Since this fictitious windspeed produced more realistic

results, in particular for station 9 wñich was the nearest to station 10

it was used for the "hindeast" at station 10. The resulting _{spectrum is}

given in Fig. 8, the comparison with the stereophotographic results will be discussed in paragraph 5.

The second and third set of pictures. to be analysed were chosen from the pictures obtained in the area off Noordwijk. The main reason to chose

from these pictures rather than from the pictures taken off Sylt was

that the results of the sortie just described indicated that the data

were influenced by the asymmetry of the coastline of Sylt. The coast near Noordwijk is more symmetric for off-shore wind directions. The

information on the atmospheric conditions during these flights was based

on standard synoptical observations which were received

through

office of the Royal Netherlands Meteorological Institute. In addition a cup-anemometer and a wind-cone were available at an observation tower located 9.5 km off-shore from Noordwijk (see Fig. 3).

The second sortie (the sequence refers to the sequence of analysis, not the time sequence of the flights) wasfiown in off-shore wind cânditions on November 12th, 1976 at 13:05 hr (local). From the synoptical

obser-vations it was found that the wind was rather weak over the entire North Sea and the wind in the area of observation was mainly caused by a weak and fairly large low pressure area over central France. Synoptical ob-servations in the coastal region 25 km North and 8 km South of Noordwijk

indicated windspeeds gf 4,5 rn/s and 4,0 rn/s respectively and wind direc-tions of 100 and 160 respectively. The wind observation at the

plat-form was carried out at 23 m above mean sea level. Averaged over the duration of the photographic operations (about 40 min.), the observed

windspeed was 16,4 rn/s and the directions just prior and just after the

flight were agproximately 140 . The "ideal" 6ff-shore direction would

have been 120 . To estimate the windspeed at 10 m elevation, the

obser-ved value was corrected. The correction for the bulk of the tower, is known from wind-tunnel tests, and the windspeed was extrapola9d using

a logarithmic windprofile with dragcoefficient c = 1.5 x 10 . The

resulting windspeed is 6.0 m/s. The corrections or the wind direction

are marginal and well within the error of observation.

Duriflg this flight pictures were taken over the observation tower and

at locations 30 km and 50 km from the coast (see Fig. 3). The pictures -taken 30 km off-shore seemed to contain sufficient stereo information

to obtain a relatively high directional resolution and these were chosen for preliminary investigation.

Wave observations at sea level were available from a wave gauge at the observation tower and from an accelerometer buoy at the location 30 km off-shore. The spectrum of the buoy is given in Fig. 9. It will be used for comparison with the stereophotographic results. During the flight some swell coming from south-westerly directions was observed

from the helicopters.

The third sortie was flown off Noordwijk on 23rd of March 1976 at 12:20 hr (local). The wind was rather weak over the entire North Sea and the direction varied from ENE off the Dutch coast to SSW off the Norwegian coast. This windfield was mainly caused by a fairly weak high pressure ridge over the North Sea and a low pressure area over central France. Synoptical observations at the same coastal stations as mentioned above

indicated windspeeds o 11.0 m/s and 8.0 rn/s respectively and wind

di-rections of 80 and 70 respectively. The corrected wind speed and

di-rectiofl at the observation tower (averaged over 20 min.) were 8.3 rn/s

and 70 . Since the "ideal" off-shore wind direction would have been 120

thg wind is slanting across the coast line at an angle of approximately

50 . Obviously this implies a strong asymmetry of. the coast line with

respect to the wind direction.

Pictures were taken over the observation tower and at locations 17km and 30 km off-shore. Since the pictures taken 17 km off-shore seemed to be the best, they were analyzed. Unfortunately no simultaneous wave observations in the area were available.

5. RESULTS

The values of a number of parameters relevant to the photograuimetric process are given in table 1. In view of the preceding paragraphs this

table is largely self-explanatory but a few parameters will be dis-cussed briefly.

'o

The altitudes of photography are based on anticipated significant wave heights and peak wave numbers. These were estimated by substituting

the windspeed and fetch in the JONSWAP parameter relationships

(Hassel-mann et al., 1973). For the sortie off Sylt the wind information _was

fairly good as it was based on ship observations in _{the area but for}

the sorties off Noordwijk this information was poorer, partly because no observations prior to the flights were available.

The helicopters were flying directly into the wind during the sortie off Sylt. During the second and third sortie they were fying with the

wind in the left respectively right _{rear quarter with 11 drift.}

Ten stereo pairs were taken in each sortie but one pair was rejected

from the set taken off Sylt because it covered _{too small an area.}

Using the sea suif ace information from the stereophotogrammetric

ana-lysis the three k-spectra were computed according to the procedures

des-cribed in paragraph 3. The results are presented in the form

ofcontour-line plots in Fig. 10, _{11 and 12. Some isolated regions in the k-plane}

have been indicated where the spectra are thought to be seriously

af-fected by noise. This noise is dealt with in the appendix. Values of relevant spectral parameters are given in table 2. For the determination

of the directional resolution, _{was estimated at 0.06 (c.f. Holthuijsen}

et al., 1974). TABLE i Photogrammetric parameters Sylt Sept. 1973 Noordwijk Nov. 1976 Noordwijk March 1976 altitude of photography 1500 ft 500 ft 500 ft orientation of helicopters 1100 _275° 3100

relativeto true North .

percentage of zeros added :. . 4% . 5% 2%

in stereo area

number of pictures 9 : 10 10

accepted for stereo analysis

stereo area per picture 220x220 m2 156x156 m

2.

--.

grid in x-plane .

2.

On closer inspection of the contourline plot of the _{spectvum of Sylt} wo wave fields can be identified: one coming from approximately 110

and one coming from approximately 155 . This is rather surprisiñg

be-cause neither the wind conditions nor the groundtruth information _gave

such indication. The swell in the second spectrum (off Noordwijk) _coming

from south-westerly directions was observed during the flight. It is

well seperated from the locally generated wind _{sea arid it will be}

largely ignored in the following discussion. The peak of the third spectrum is, surprisingly, coming from Northerly directions rather than

from Easterly directions, as may be anticipated from _{the wind direction.}

Instead of the k,O-spectra, the normalized directional distribution

functions have been plotted in Figs. 13, 14 and 15. The definition of

these functions is given by equation 12 and 13.

D(O;k) _- E(k,O) _{for O < O < ir (12)} ¿ E(k,O)dO

D(O;k) = O for r < O < 2ir (13)

This seemed to be more illustrative than a contourline plot of the

k,O-spectruxn when the interest is primarily _{for the directional}

charac-teristics. An evaluation of these functions will be given in the next

paragraph.

The f-spectrum has been computed from the 1-spectrum according to the

procedures described in paragraph 3. The result for _{the spectrum off}

Sylt is given in Fig. 8 along with the corresponding _{JONSWAP spectrum.}

The resolution is about 0.02 Hz near the peakfrequency, which is 0.165 Hz, and 0.01 Hz at twice the pak.frequency. The number of degrees of

(x)

peak wave number of locally generated wind sea

TABLE 2 Spectral parameters Sylt Sept. 1973 Noordwijk Nov. 1976 Noordwijk March, .1976

resolution in it-plane _Lin 2 (220x220)1 (156x156)1 c1:7ox17o)'

number of degrees of 18 20 20

freedom

peak wave number (km) Em1]. 0.0045 0.0641' 0.0265

directional resolution .

-atk= k

k= 3km

12

freedom for frequencies greater than 0.13 Hz is 250 or more. Considering the scatter in the original data set of JONSWAP and taking into account the resolution, it is concluded that the agreement between the two

spectra is fair.

4.

The frequency spectrum computed from the observed k-spectrum of the second sortie is plotted in Fig. 9 along with the frequency spectrum of the buQy. The resolution of the spectrum based on the stereo data is on the order of 0.02 Hz near the peak of the swell and 0.015 Hz near the peak of the locally generated wind sea. The number of degrees of freedom is 125 or more for frequencies greater than 0.10 Hz. For the spectrum of the buoy the resolution is about 0.02 Hz and the number of

degrees of freedom is about 48. V

The spectrum based on the stereo data seems to be shifted in energy

density. This ay have been caused by noise and to appreciate this

influence the k-spectrum was corected. The noise was assumed to be

uniformly _{istributed over the k-plane and the variance was estimated}

at 0.002 m (based on the anticipated measurement error of 0.03Z of the

altitude of photography, see paragraph 2).Accordingly _{a uniform noise}

level of 0.018 m4 was subtracted from the k-spectrum and the transfor-mation was carried out again. The differences were marginal compared with

the earlier results and the shift cannot be .xplained with the anti-cipated noise uniformly distributed over the k-plane. Further investi-gation is needed to resolve the remaining discrepancy.

The frequency spectrum of the third sortie is given in fig. 16 but no

attempt has been made to compare this spectrum with a hindcasted

spec-trum because the relatively simple relationships for off-shore wind situations cannot be applied.

6. DISCUSSION 0F THE RESULTS

In the area off Sylt, where the wind was almost perfectly off-shore _and

fairly homogeneous and stationarjt, _{one would expect to find a frequency}

spectrum with a shape similar to the shape fOund earlier in JONSWAP.

Finding a JONSWAP-type spectrum _{in the conditions off Noordwijk seems}

to be less likely because the differences between the wind _bservations

at the coast and at the tower are fairly large and the wind may have

varied between the point of observation and the _{coast. In particular for}

the slanting wind conditions it is not obvious _{to find a JONSWAP-type}

spectrum due to the asymmetry in the coastline around the wind direction.

On the other hand, non-linear interactions in the _{spectrum may produce}

a JONSWAP-type spectrum, in spite of the asymmetry and the variations

Th. the windfield (Hasselmann,. et al., 1976).

From an inspection of Fig. 8 it can be concluded that the frequency spec-trum in the sortie off Sylt is indeed JONSWAP-like. The correspondence of the frequency spectra off Noordwijk with a JONSWAP-type spectrum has

not yet been investigated. V

13

For the Ì-spectra of the first two sorties one would expect to find

di-rectional distribution functions having some kind of standard shape, symmetrical about the mean direction although some skewness may be ex-pected in the observation off Noordwijk because the wind direction was not perfectly off-shore. For the third spectrum strong skewness may be

anticipated due to the slanting position of the coastline.

-,.

These expectations seem to be far from reality in the k-spectrum off Sylt. The directional distribution near the peak of the spectrum (see Fig. 13) is distinctly asymmetric with respect to the wind direction with

the highest peak at + 45 off the wind direction (155 from true North).

It is highly improbable that the wave generation mechanism would build a directional distribution as strongly asymmetrical as this. An expla-nation for this unexpected observation can perhaps be found through a detailed studyof the wind and wave fields, possibly using hindcasting

procedures. But in the context of this paper one can only speculate on

some possible causes. The source function is symmetrical, as is the ra-diative energy transfer, since bottom and current refraction is

virtual-TLy non-existent. It seems then:that the asymmetry stems from asymmetry

in the wind field or in the boundary conditions. As for the wind field, a cursory inspection of the large scale weather maps revealed no asym-metry. As for the boundary conditions, the coast of Sylt, rather than

the main-land coast was deemed Lo be relevant as up-wind boundary. This

was based on the expectation that the wave energy is propagating in a narrow angular sector around the wind direction (e.g. Hasselmann et al.,

1973).and since the coast of Sylt is rather symmetric it should not

cause asymmetry in the wave field. But the coast to the North and South of Sylt is strorly asymmetric. In fact, the distance to shore in the

direction of 155 _{(the direction of the highest peak) is almost 2.5}

times the distance to shore in the direction of 65 _{(the 'syrmetrical}

direction, see Fig. 4). If this asymmetry in the windward boundary is

indeed the cause, then it seems that the "ideal" generation _{cases of}

JONSWAP may be contaminated to some degree by asymmetric _oundary

con-ditions. Still, relating this conclusion to the observed k-spectrum is

largely spectilative as long as i _{is not substantiated with more data.}

In particular the shapes of the k-spectra at locations closer to shore

may give some clues.

The expectations regarding the directional distributions for the

local-ly generated wind sea off Noordwijk in the second sortie _{seem to be more}

realistic, at least in an overall sense (Fig. 14, for k > k _{). Any}

skewness is hard to identify through visual inspection of tL _{plots due}

to the small scale variations in the functions. These probably stem from

the statistical variability of the estimates. The swell peak (k = 0.3 km

to k = 0.6 k ) is unimodal and covers a narrow angular _{sector with a}

m o

half power width of about 35

The directional distribution functions of the spectrum in the third sortie seem to be strongly skewed for the lower wave numbers (Fig. 15,

visually. As for the main direction of thg energy distribution, it varies

almost monotonously from approximately 80 at higher wave numbers to

about O for the lowest wave numbers (see also Fig. 17). The energy of the higher wave numbers travels more or less in the wind direction but the main direction of the peak of the spectrum appears to be about loo

relative to true North, that is about 60 from the wind direction and

almost parallel to the coast. This seems to be the most remarkable fea-ture of this spectrum as one would expect to find on uniform main

direc-tion of 70 , considering the wind direction and the effects of

non-line-ar interactions (Hasselmann et al., 1976). Again, as with the spectrum off

Sylt, it is felt that the observed phenomenon is due to the asymmetry of the coastline around the wind direction.

Tosubstantiatethis preliminary conclusion qualitatively, a simplified

hindcasting model was implemented for homogeneous, stationary wind

fields, arbitrary coastlines and deep water. In this model, which is

basically the same as suggested by Seymoor (1977), the wave components from different directions are decoupled. In this version the parameter relationships from JONSWAP (Hasselmann et al., 1S73) were taken and the

suggestions o: Mitsuyasu et al. (1975) were used for the directional distribution function. When applied to the situation of the first and third sortie it did produce two-dimensional f,0-spectra which at least

qualitatively agreed with the so far unexpected main directions in. the observed k-spectra.

This seems to be in contradiction with the conclusions of Hasselmann et al. (1976) that the shape of the spectrum is fairly insensitive to variations in the wind field due to the non-linear interactions in the spectrum. It should be noted however that the distance to the coast,

in terms of wave lengths, seems to be rather short for the lower wave

numbers in the two spectra so that non-linear interactions may not have been sufficiently effective to overcome the influence of the geometry of the coastline. For the higher wave numbers the distance to shore is relatively long and the non-linear interactions may have produced the observed directional distribution functions which indeed seem to be hardly affected by the asymmetry of the coastline. The observations therefore may still be consistent with the theory of non-linear inter-actions and the conclusions of Hasselmann et al. (1976) if the relevant space and time scales are considered.

In an "ideal" generation case the directional distribution of the wave

energy is often approximated with a simple uniniodal function. The

ob-served situations may not have been "ideal" and some distribution

func-tions are distinctly multi-modal, but one such function, given in

equa-tion 14, has been fitted to the data. This was done mainly to compare

the results with published data.

D(0) i r(s + 1) ,_

r(s + ) '' 2

ee

In this expression s is the spreading parameter and e is the mean

di-rection, both .of which may vary with k. The values fo e and s have 14

15

been computed using a least-squàres.technique. The results for O as

a function of wave number are given in Fig. 15. Noise in the spectra (see appendix) did influence these results and outliers had to be

iden-tified. As criteriôn for acceptation, the rate of change of O a1on the

wave number axis has been chosen. An accepted value of O should be

within 30° of its neighbouring values on the wave numbertmaxis. This is

equivalent to a rate of change of approximately 0.0024 in for the first

sortie, 0.0033 m for the second sortie and 0.0031 in for the third

sortie. This allows for slow but significant variations in O which is

required for instance in the spectrum of the third sortie.

TL

set of accepted values of O is also indicated in Fig. 17. The values

of s at the corresponding vhues of the wave number have been plotted in Fig. 18 in a fornat suitable fora comparison with data published by Mitsuyasu et al. (1975).

Mitsuyasu et aI. (1975) presented results of a number of measurements

(five) which were carried out with a cloverleaf buoy at several loca-tions around the Japanese islands. The observed wave fields were gene-rated by various types of wind fields, including on-shore and off-shore winds. It appears from the ratio of the wind speed and the phase speed

of the peak frequency of these observation that the state of develop-ment of the wave fields was rather advanced (the ratios ranging from

0.75 to 1.25). Based on the observed values of s, relationships in the

frequency domain were suggested. The relevant expressions have been transformed here to the wave number domain to produce equations 15 and

16.

for i>i

-i2.5

fork< I

(15)

S = 11.5 (16)

where = s/s and = k/k , s is the maximum value of s, k is the

peak wave nuiner, c _{is th phLe speed of the peak wave numLr and u}

is the wind speed. he data of Mitsuyasu et al. (1975) are probably ob-tained in situations where tidal currents were negligable and in the

above transformation the deep water linear relationship between f

requen-cy and wave number was used.

The expressions 15 and 16 are also plotted in Fig. 18 and the agreement

is fair, the scatter being on the same order of magnitude as the

scat-ter in the data of Mitsuvasu et al. (1975). The values of s computed

from the stereo data are 6.0 for the spectrum off Sylt, 5.0 for the first spectrum off Noordwijk. These are also in fair agreement with the values

suggested by Mitsuyasu et al. (1975) which are 4.6 and 6.1 respectively.

However, for the second spectrum off Noordwijk the observed value of s is 27.4 whereas the value following from expression 16 is 5.9. This is a very large discrepancy which is possibly due to the rather extreme asymmetry of the coastline around the wind direction where the suggest

16

tions of Mitsuyasu et al. (1.9751 naynotbeapplicable.

The above discussion concerned rather overall-characteristics, of the

directional distributions. It is planed to investigate these functions more in detail. For instance, in the k-spectrum off Sylt one aspect which will require closer study is the shape of the directional distri-bution near the peak of the sgectrum in asector around the wind

direc-tion. Two peaks at + and - 15 relative to the wind direction can be

identified and this phenomenon seems to be !'real!! in the sense that the

directional resolution seems sufficiently high (20 ) to resolve these

peaks in terms of statistical significance. The resonance theory of Phillips (1957) predicts a bimodal distribution for frequencies in the

initial stage of development, but the components around the peak have passed that stage and there is no relation with this theory. More relevant seem to be the theory and calculations of Hasselmann (1963), .Longuet-Higgins (1976) and Fox (1976) which produce a non-linear energy

transfer in wave number space with two lobes towards the lower wave numbers and two lobes towards the higher wave numbers. Fox (1976) noted

that this function resembles a "butterfly". Also the results of Tyler et al. (1976) who observed directional distributions of wind generated waves with high-frequency radio-wave backscatter may be of interest

since some of the distributions have a bimodal character around the mean direction.

7. CONCLUSIONS

Three two-dimensional wave number spectra have been computed from stereo-photographic data obtained inoff-shore wind conditions. The agreement with groundtruth information is reasonable but some discrepàncy needs

to be resolved.

The directional distribution of the wave energy near the peak of the first spectrum is strongly asymmetric. In the third spectrum the main direction of the waves differs appreciably from the wind direction. It

is speculated that these phenomena are due to asymmetry in the up-wind coastline. The directional distribution functions of the second spectrum are more symmetric and unimodal, at least in an overall sense.

A bimodality in a sector around the wind direction is observed near the peak of the first spectrum. This bimodality may be related to a multi-modal non-linear interaction in the spectrum.

The observed normalized directional spreading parameter as function of a normalized wave number is in fair agreement with published data. The absolute values are about 30% larger for the first spectrum and about 20% lower for the second spectrum. The values for the third spectrum are almost five times too large. This may be due to the rather extreme asym-metry of the coastline where a compariáànwith the published data may not be proper.

The results reported herein are preliminary. Additional analysis of available data is being carried out.

ACKNOWLEDGEMENTS

The helicopters were provided by the Royal Netherlands Ai,r Force and

they were flown by the Search and Rescue team of Soesterberg airbase (the Netherlands). This is gratefully acknowledged. Considerable support in terms of logistics, groundtruth data, meteorological observations etc. was received from colleagues in the framework of JONSWAP and this

- NOTATION

A area of spatial integration

Cm phase speed of component f

Cg group velocity

D(0) _{standard directional distribution function}

E(i) spectral density in k-space

E(k,O) spectral density in k,O-space

E(f,O) spectral density in f,O-space

E(f) spectral density in f-space

f frequncy

g acceleration due to gravity

h instantaneous surface elevation

H()

J Jacobian

k' wavenumber vector k = (k ,k )

_xy

k wavenumber, modulus of wavenumber vectOr

k _{wavenumber at peak of wavenumber vector spectrum of locally}

V generated wind sea

L _{dimension of area of analysis in x-direction}

L _{dimension of area of analysis in y-direction}

N number of degrees of freedom

n number of transformations

R boundary of spatial integration

s directional spreading parameter

s maximum value of s

dimensionless spreading parameter _s/sm

U windspeed at 10 m elevation

3

V,

V magnitude of V

+

x place vector x

= (x,y)

x,y,z -spatial coordinates

increment

O _{direction, orientation of wavenumber vector}

e orientation of tidal current

C

6 meàn direction .

REFERENCE S

Bendat, J.S. and A.G. Piersol (1971). RANDOM DATA: Analysis and Measure-ment Procedures, Wiley-Interscience, New York.

Briinimer, B., D. Heinrich, L. Kriigermeyer and D. Prim _{(1974). The}

Large-Scale Weather Features over the North Sea during the JONSWAP II

.

. Experiment, Berichte des Instituts fir Radiometeorologie und

Mari-time Meteorologie, Universität Hamburg, Institut der _{Frauenhof er}

Gesellschaft, 24.

Cote, L.J., J.0. Davis, W. Marks, R.J. McCough, E. Mehr, W.J. Pierson,

J.F. Ropek, G. Stephenson and R. Vetter (1960). The Directional

Spec-trum of a Wind-generated Sea as determined from Data obtained _by

the Stereo Wave Observation Project, Meteorological Papers, New York University, 2, No. 6.

Crawley, B.G. (1975). Automatic contouring on the Gestalt photomapper, testing and evaluation. American Society of Photogrammetry, Work-shop III, San Antonio, Texas, U.S.A.

Fox, M.J.H. (1 976). On the non-linear transfer of _{energy in the peak}

of a gravity-wave spectrum. II, Proceedings _{Royal Society of London,}

A. 348, p. 467.

Hasse, L., M. Grünewald and D.E. Hasselmann (1977). Field observations of flow above the waves, Preprint from the Proceedings of the

NATO-Symposium on "Turbulent Fluxes through the _{Sea Surface, Wave}

Dyna-mics and Prediction", Bendol, to be published by Plenum Press

(New York, London).

Hasselmann, K. (1963). On the non-linear energy transfer in a

gravity-wave spectrum. Part 3. Evaluation of the energy flux and _swell-sea

interaction for a Neumann spectrum, Journal of Fluid Mechanics,

15, p. 385.

Hasselmann, K., R.P. Barnett,E. Bouws, H. _{Canson, D.E. Cartwright,}

K. Enke, J.A. Ewing, H. Gienapp, D.E. _{Hasselmann, P. Krusemann, A.}

Meerburg, P. Muller, D.J. Olbers, K. Richter, _{W. Sell and H. Walden}

(1973). Measurements of Wind-Wave Growth _{and Swell Decay during the}

Joint North Sea Wave Project (JONSWAP), Ergnzungshef t zur Deutschen

Hydrographischen Zeitschrift, Reihe A (8°), no. 12.

Hasselmann, K., D.B. Ross, P. Müller and W. Sell (1976). A parametric

Wave Prediction Model, Journal of Physical Oceanography, 6, No. 2,

p. 200.

Holthuijsen, L.H., M. Tienstra and G.J. v.d. Viiet (1974).

Stereophoto-graphy of the Sea Surface, an Experiment, Proceedings _{of the}

Inter-national Symposium on Ocean Wave Measurement and Analysis, American

Society of Civil Engineers, _p. 153.

Hühnerfuss, J., W. Alpers and L. Jones (1978). Measurements at 13.9 GHz of the radar backscattening cross section of the North Sea covered

with an artificial surface film, _{to be published in Radio Science.}

Longuet-Higgins, M.S., D.E. Cartwright and N.D. Smith _(1963).

Obser-vations of the directional spectrum of sea waves using the motions of a floating buoy. In: Ocean wave spectra, pp. 111-132, Prentice Hall, Inc., New Jersey.

Longuet-Higgins, M.S. (1976). On the non-linear transfer of energy in the peak of a gravity-wave spectrum: a simplified model, Proceedings Royal Society of London, A. 347, p. 311.

Mitsuyasú, II., F. Tasai, T. Suhara, S. Mizuno, M. Ohkusu, T. Honda and

K. Rikiishi (1975). Observations of the Directional. Spectrum of

Ocean Waves Using a Cloverleaf Buoy, Journal of Physical

Oceano-graphy, 5, p. 750.

Panicker, N.N. and L.E. Borgman (1970). Directional Spectra from Wave Gage Arrays, Proceédings of the 12thInternational Conference on

Coastal Engineering, Washington D.C., p. 117.

Phillips, O.M. (1957). Onthe generation of waves by turbulent wind, Journal of Fluid Mechanics, 2, p. 417.

Singleton, R.C. (1960). An algorithm for computing the Mixed Radix Fast Fourier Transform, IEEE Transactions on Audio and

Electroa-coustics, AU-17, No. 2, p. 93.

Spiess, F.N. (1975). Joint North Sea Wave Project (JONSWAP) progress -an observer's report, Report ONRL-C-8-75, Office of Naval Research,

London.

Seymour, R.J. (1977). Estimating wave .generatin o restricted fetches,

Proceedings of the American Society of Civil Engineers, Journal of the Waterway, Port, Coastal and Ocean Division, WW2, paper

12924, p. 251.

Stiiwell, D. _{(1969). Directional Energy Spectra of the Sea from}

Photo-graphs, Journal of Geophysical Research, 74, No. 8, p. 1974.

Sugimori, Y. (1975). A study of the application of the holographic method to the determination of the directional spectrum of ocean waves, Deep-Sea Research, 22, p. 339.

Tyler, G.L., C.C. Teague, R.H. Stewart, A.M. Peterson, W.H. Hunk and J.W. Joy (1974). Wave directional spectra from synthetic aperture

ob-servations of radio scatter, Deep-Sea Research, 21, p. 989.

APPENDIX Noise

Inspection of contourline maps of the sea surface obtained from the

observation off Sylt revealed a dome-shaped distortion. This distortion is probably caused by the fact that the pictures could not be positioned in the stereoscopic viewing devices with the accuracy normally obtained with high grade pictures. When this positioning is not optimal, a

dome-shaped distortion is to be expectd. Unfortunately the _{exact distortion}

cannot be determined, but in the k-plane it seems to be well seperated

from the wave information (area no. I in Fig. 19) and the data in this

area was removed in the subsequent analysis.

The other noise-affected areas are related to a phenomenon introduced

by the manner of scanning the _{ictures during the photogrammetrical}

process: the sea surface elevation at even-numbered lines (scanned in

positive y-direction) is systematically slightly too low, while the

Televation at odd-numbered lines (scanned in negative y-direction) are

systematically slightly too high. This effect has been _observed

ear-lier in the analysis of stereo photos of regular _{waves generated in a}

hydraulic laboratory. The principal wave length and direction of this

distortion correspond with the location of area no. 2 in Fig. 19,which

is the location of the Nyquistwavenumber in x-direction. _{This spectral}

information was removed from the _{spectra in the subsequent analysis.}

The noise in areas no. 3, 4 and 5 was labeled as such mainly because of

the deltatype behaviour of the directional distribution _{functions in}

these region; It is probably due to variations in the _{error introduced}

by the scanning and possibly also by "leakage" from _{area no. 2. In the}

k-spectrum off Sylt his noise was not removed. In the k-spectrum _off

Noordwijk the noise in the indicated region in Fig. 11 has been removed.

Fig. 1: Bi-linear interpolation _{in the k-plane}

Fig. 2: Sites of the field operations. _{The areas in the boxes}

are

shown enlarged in Fig. 3 and Fig. 4.

Fig. 3: The area of observation off Noordwijk. Locations of obser-vations indicated by dots.

Fig. 4: The area of observation off Sylt. Active wave monitoring stations and station of observation indicated by dots. Wind direction indicated by arrow

Fig. 5: Bottom profiles off Sylt _{(direction 287°) and off Noordwijk}

(direction 300 )

Fig. 6: Observed frequency _{spectra at stations 5, 7 and 9 and}

corres-ponding JONSWAP spectra for U = 13 rn/s.

Fig. 7: Observed frequency

spectra at stations 5, 7 and 9 and

corres-ponding JONSWAP spectra for U 15 in/s.

Fig. 8: Spectrum inferred from _{stereo data and corresponding JONSWAP}

spectrum for U = 15 mIs

Fig. 9.: Spectrum inferred from _{stereo data-and spectrum from:buoy} oeasurement.

Fig. _{IO: Contourline plot of it-spectrum}

off Sylt, Sept. 18th, 1973. Contourline interval equivalent to factor 2. Minor variations are dashed, shaded areas seriously affected by noise.

Orienta-tion of positive k -axis 110 , k -axis 200 _{from true North.}

Wind direction X

Fig. _{11: Contourline plot of i-spectruxn off Noordwijk, Nov. 12th, 1976.}

Contourline interval equivalent to factor 2. Minor variations are dashed, shaded areas seriously affected by noise. Orientation

of k -axis 275 _{, negative k -axis 185}

from true North. Wind

diretjon 1400. X

Fig. 12: Contourline plot _{of i-spectrurn off Noordwijk,}

March 23rd, 1976.

Contourline interval equivalent _{to factor 2. Minor variations}

are

dashed, shaded area seriously affected by noise. Orientation _of

k -axis 310°, k -axis 40° from _{true north. Wind direction 70°.}

y X

Fig. 13: Normalized directional

distribution functions of the k-spectrum

off Sylt, Sept. 18th, 1973. _{Directions are relative to}

true North.

Fig. 14: Normalized directional

distribution functions of the -spectrum

off Noordwijk, Nov. 12th, 1976. _{Directions are relative to true}

North. The peak wave number _{k is related to the locally}

generated wind sea.

Fig. 15: Normalized directional

distribution functions of the t-spectrum

off.Noordwijk, March 23rd, 1976. _{Directions are relative}

to true North.

Fig. 16: Spectrum inferred from stereo data of observation _{off Noordwijk,}

March 23rd, 1976.

Fig. 17: The mean direction of the waves relative _{to true North, as}

func-tion of wave number.

Fig. 18: The normalized spreading parameter as a function of the normalized wave number

E1

E5

+ gridpoint in k,O plane transformed

E2 It-plane (grid in t-plane E E6 E

is estimate of Edt) linearly interpolated between E1 and E2

E E5

'E0

₊

/

/

/

r-s--s

i

\

'(

1/

G.

If

25km

50km

/

ij

j'

30km

/

9.5km'/

/J

f

//

Noordwijk

/

(

/

/

J,

Ì

(_

/

(

'5.'.

---d

(

- - -. ,J

f

The Hague

f

-t7

V

Rotterdam

/

52e-/

__,

observation tower

60km

0km

20km

-

s-//

Sylt

Noordwijk

-10m

-20

-30

N

I

0.0-stat .9

Sylt

730918

JONSWAP spectrum

observed spectrum

17:51 start of record (duration 25 min.)

F1&

0.10 0.15 _{0.20 0.25 0.30}

frequency( Hz)

3.0 2.5 2.0 N

T

frequency (Hz)

Sy't

730918

JONSWAP spectrum

-observed spectrum

5.0 1. .0 _3.0

I

/

_/

/

_/

/

_'t

I'

/

I.

SyI.t

730918rn

JONSWAP spectrum

observed spectrum

(from stereo data)

0.30

Q35

I

Noordwijk 761112

,.'

/

I

/

/

I

buoy spectrum

observed spectrum

(from stereo dato)

0.10 _{0.20 0.30} 0.40 _0.50 f req ue ncy ( Hz

/

TI:::

D

,o.

kxky22Om)

'

-

'--t)

SyLt

730918

D resoLution area

KKy(17Omi'

c)

/_\

7/

r

to

k0.71km

k0.95km

&5 x103m1

k3i0km

k238km

_J

-k2.l6km

Sylt 73O918

3.33km '.

k3.57km

ttLT

ì3

40° 130° ?2km 44km B km 1.0 SCALE j. k2.00km k=2Á4 km k2ß7km k= 2ß9 km k=3.11 km k 3.78 km I i=4.00 km I -k:422km k&67km k= 538km k5.78km

Noordwijk 760323

f'.

-95° k0.2km k=0.6 k k=O.7 km kO.9km 1.01 SCALE 1850 _{275° kkm} b

1

-t--kt9km

- -..-.,.

k km

N

I

Noordwijk

760323,,

frequency spectrum

from stereo dato

i I L

0.10 OE20 _{0.30 0.40}

frequency (Hz)

210 180 150 120 90

60

o

__+-sweL1"

00000

0

++

i

kmtk

Sytt,730918Ak= 1/220m1

Noordwijk1761112,k1/156m1

Noordwjjk175O323k1,17O m1

+.

+

o

+0

1.0 0.3 0.6 o 2.5

+

+

o

M

0

Sylt

730918.

Noordwijk

+ Noordwijk

k/km

A

A

s

+

+