The use of a filter to sort out directions in a short crested gaussian sea surface

ii(x,y,t)=1 (xcosetysinc9)-iit+

1KM)] Ejiti,ed

AI

This is the Gaussian case of the Lebesgue energy integral for short-crested wave systems, Where' the elevation of the free surface at any point and time is a function of the energy spectrum

1A2 0)112, the spectral frequency AL = 27r/T, gravity g, direction 0, space coordinates, x and y, time t, and an arbitrary term which assigns the phases [kit ( 0)].

It is possible to rewrite Was a limit of partial sums if a network is 'constructed over the pu 0 coordinate system and the mesh of the network is allowed to go to zero. The result is

7?(X44

= ln

A 25 "0

5 2

_\

1 6

R 421to.zAv,11.4

THE USE OF A FILTER TO SORT OUT DIRECTIONS IN A ,SHORT-CRESTED 'GAUSSIAN SEA SURFACE

Wilbur Marks

Abstract--The method of stereo pairs may be used to obtain wave records which characterize the sea surface completely at any given place. The three-dimensional aerial photograph thus obtained is divided into 2N +1 wave records spaced Ay ft apart. The 2N + 1

wave records are combined statistically into one wave record, and the energy spectrum for spectral components within a certain angular band is then computed. A filter function which depends on N and ,tly is derived from the averaging process. The filter function operates on the energy spectrum in such a way that, for proper choice of N and Ay, the waves whose spectral directions are any other than that which is desired are eliminated. The energy spectrumwhich results is composed of the energy of those waves which travel in the desired direction. A sample example illustrates this process.

IntroductionThe sea surface is believed to be made up of a sum of an infinite number of in-finitesimally high sine waves traveling in different directions with different frequencies added to gether in random phase [PIERSON, 1952]. This surface, at any point, has a certain amount of en-ergy associated with it, and this enen-ergy is distributed among the component waves according to frequency and direction. It is this distribution of energy and the particular direction of travel of each of the component waves which defines the two-dimensional energy spectrum [A2(ii,0)12,

where = 21t/T is the spectral frequency and 0 is the direction of propagation. This is the prop-erty of the ocean surface which must be measured.

The mathematical expression which appears best to characterize this sea surface is

2n

]

--I-4*r

t

758

In order to compute [A2 (.1., 8)]2, it is first necessary to obtain a wave record which describes the short-crestedness of the sea surface in the area of interest. Since one problem of importance is to forecast the propagation of waves out of a generating area, it would be best to make the ob-servations at the edge of the storm over an area where the sea is fully developed. In addition, if a record is to fulfill all the requirements requested of it, there must be some mechanism which yields information about the direction of travel of the spectral components, so that the complete form of the two-dimensional energy spectrum can be computed.,

The problem therefore resolves itself '''to' two distinct parts: (1) What is the best way to record ocean waves at the edge of a wave generating area,, on the shortest notice, that will yield

Transactions American Geophysical Unicin. Volume 35,, Number 5, October 1954, .

(2) (1)

(

, N/k;fal q. 414_ Aka, 7i, cS, 1;:V. gio

Lb.

v. Scheepsbouwkunde

Technische. Elzps:thool

Deilt

[Oceanography] DIRECTION FILTER IN SEA SURFACE 759

information about the distribution of energy among the spectral components according to frequency and direction? (2) What is the best method of analysis of this wave record?

Energy spectra as a function of frequency alone have been computed from available wave rec-ords [PIERSON and MARKS, 19521, and theoretical calculations have been made which describe the state of the sea at various times and places with respect to the distribution of energy among the spectral frequencies, but little has been done about the distribution over 6. In lieu of actual wave records from a wave-generating area at sea, energy spectra have been assumed at the edge

of such an area of regular shape [PIERSON, 19521. NEUMANN [1953] has derived theoretical wave spectra as a function of g. and the wind velocity. Theoretical results have explained many features of the sea surface such as the narrowing of the frequency band width with distance from the storm, the distribution of energy at a given time and place, the decrease in wave height with distance, the increase in the period of swell, and other features which have and have not been observed [PIER-SON, 1952].

Since no actual wave records have ever been obtained at the edge of a generating area to cor-roborate the far-reaching implications of the theory and to determine the exact form of [A2 (p,6)]2, it seems apparent that such wave records should be obtained as soon as possible.

High-altitude stereo photography--High-altitude stereo photography appears to be the best method available to determine the two-dimensional energy spectrum of the sea surface at the edge of a storm. Two photographs, made at a safe height, reduce the danger involved in making the records. In addition, the photographs obtained in this manner will eliminate the necessity for

mak-ing many runs over the sea surface as would, for example, be required by an altimeter type of wave recorder. This cuts down the time required for making the observations. The sea surface may be divided up into as many runs as desired and the records may be arranged in any pattern. A typical pattern of runs and a grid system are shown in Figure 1. This method has all the de-sired properties which were outlined in the introduction. The wave spectrum as a function of both g and 8 can be determined from the stereo aerial photographs. It is necessary to know the wind direction. A wave record as a function of time at a fixed point would be useful as a check of the results, but it would not be essential.

An alternate method has been described [PIERSON, 1952] on the basis of airborne recording. For a fixed direction 8*, the sea surface can be recorded as a function of distance along the line of flight x'. In addition, the free surface as a function of time can be recorded. Both of these functions are examples of stationary series, and both may be analyzed by the methods described in the literature [TUKEY, 1949; TUKEY and HAMMING, 1949]. Many different values of 6* can be chosen and a whole set of functions for different 8* can be found. It is necessary to determine

[A2 0)]2 from the stationary series thus observed. Pierson found that seven analyses of the form described by TUKEY and HAMMING [1949] would have to be carried out. The result would be 77 linear inhomogeneous equations with 77 unknowns which would need to be solved. This rep-resents an immense amount of work and it is enough to make an easier method worth seeking.

A theoretical discussion of a sample case will illustrate the usefulness of stereo pairs in power spectrum analysis.

Method of observation--A camera with a wide-angle lens, say 90°, can take a picture the di-mensions of which are twice the height of the plane. For example, at 5000 ft a sea composed en-tirely of waves with 20-sec period will yield a photograph which contains approximately 4.91 waves, equivalent to a wave record 1.64 min long. On the other hand, a sea composed entirely of three-second waves would result in a photograph which contains 208.15 waves, the equivalent of a wave record 10.41 min long. Since the sea at the edge of the storm has a wide band of spectral compo-nents, the range of wave periods recorded would probably include all periods from three to 20 sec, so that a photograph taken at 5000 ft would more likely yield a photograph equivalent to about six or seven minutes of wave record.

It is necessary to compute the energy spectrum at the edge of the generating area. Therefore, a point on the leading edge, which is free from angular spreading, should be selected as the place to take the stereo photographs. Once the place to be photographed is chosen, the weather conditions will determine whether it is feasible to make observations and at which height the observations

should be made.

Theoretical discussion of power spectrum analysis of stereo pairs--A photograph might be divided up into a grid of points (Fig. 1). Since one pair of photographs is not equivalent to 20 min

R. "12/142Atti °

Fig. 1--Diagram of runs to be marked off on stereo photographs; the x-axis is parallel to the dominant direction of the crests; the grid system is arranged for 0 = 15°

of wave record. (the desired length of record clue to compromise between a snail loss in accuracy and a large saving in computational labor), several pictures of the 'same' sea surface may betaken and spliced by statistical methods. (As long as the waves retain their basic features and the sta-tistical properties of the sea surface remain constant during the photographing process, the rec-ords obtained will all have the same properties and can be analyzed in the same manner.) The grid contains 2N.+1 horizontal rows of points spaced Ay ft apart. The points on each row, spaced

Ax ft apart, are read off the photograph directly as heights above an arbitrary datum plane situated below the lowest trough. Each column is averaged, and the result is one wave record where the points are the average values of the columns A, B C . . . The statistical methods of TUKEY [1949]

and TUKEY and HAMMING [1949] which describe the computation of energy spectra, in communi, -cation theory, have been adopted for ocean waves [PIERSON, 1952; PIERSON and MARKS, 1952].

From the averaged wave record, the energy spectrum can be computed.

Derivation of theoryIf the cosine term in (2) is expanded trigonometrically the result iS,

7(x,y,t)=ii

Iy

n.o rec.,

'Att;

^'TC.05Oir,vIL2n4.1 ..9,1+ty sin 92r4,]

sin11/11/#1 X cos_{02r4i-ilizn,,, t+ Yi GLI/744 , lizr4)} 4. :' 4. '

' " "' ''''' (31

9 [Trans. AGU, V. 35 -35], . 6 75 60

[Oceanography] DIRECTION FILTER IN SEA SURFACE 761

If the 2N +1 wave records, spaced Ay ft apart, are averaged over y point for point, the Wave rec-ord thus obtained may be written as

+N,5

EzcaxeT, sinRI: 2_(N-i)A y sinel a

.

dizal

(2N+1) sin[Ard ysirilij

I.e.

[

cos

N 2 al/

2 coEl2n44( 6,y. sin 9 co s

Lw""

9

. (4).

where the sine terms drop out because they are equal in magnitude and opposite in sign. As ->

2n+2 42n and - 92r 0, the partial sum in (3) is, for all purposes, indistinguishable from the sea surface it represents. Therefore, the limit in (3) may be dropped.

The coefficient of the cosine term in (4) is expanded and the collected terms are substituted back into the equation. The result is

2' S1/1[6+1 intezr,q)

-Ei.(ii,e92,-,

11+1) sin 5 'u!'"' sin'd2r4

[411/7# A ,

!.. A. cos -fi204./ T (l-4211*/ V2,*/).1

9

The averaged record can also be' represented as

Leos

ejcase;

e ei;iL

=f

I ( 7

)

The integral of the right-hand side over the 60, coordinate' IS

yr, 2 2Ek ,(-9__)1/2 de. = (11.) a 2 co s ° 2 7.1. Cos 6. if-Er 7,, 43 4)4. L.444! (0 .412 cos 19Ait +

/ 48)

where the notation reverts again to the integral form and the complete coefficient of the cosine. term is brought under the radical sign.

Since the record is made at an instant of time, a transformation is made to a ,space coordinate system. The transformation is

42 cos eil

... . t7Y,

ev,, = 6.

where vo is the observed spectral wa4 number and Bo measures angular direction When once the c °ordinate axes are chosen with respect to the selected runs.

From (7) it is seen that

. 03) 2N.+1 71(x/g+J4Y'Ms'Z

r,ol

t

762 [Trans. AGU, V. 35 - 5] Therefore under this transformation, (6) becomes

Z

zr+1[N-1)Asflo-^60J

I do]

I Ea,

,

cN) 5)n BAVV,teLn

where [A2 ( v0, 00) 2 is written instead of [A2 (g vojcos Go)1/2 00)12 for simplicity. Eq. (9) is a function of x alone and thus the series of points in the x direction as averaged over y can be analyzed for the spectrum as a function of vo by the techniques given by PIERSON and MARKS [1952].

In (9), the energy spectrum [A2 (v0, 00)]2 is multiplied by

sin [(KJ-Day 74 eo] 2 ₍₁₀₎

(2N+I) sinrkLyi./. tan et]

This serves to filter out those portions of the energy in the spectrum which travel in all directions except those within a small angular range about Go. By varying Bo in (10), the energy which travels in any desired direction can be found.

Properties of the filterIt is this filter as given by (10) which will now be investigated in de-tail. Some properties of this function are: (1) It is an even function of 60. (2) It is everywhere positive. (3) It has its greatest maximum at the origin of orientation and it damps out very rapidly with eo. (4) It becomes a low-oscillating function with several minor maxima before it reaches a major maximum again. An example of this appears in Figure 2. (5) The number of minor maxima varies directly as N. (6) The angular distance between the major maxima depends on Ay.

Fig. 2--Sample filter function for N = 10, y = 125 ft

In order to show how this filter function works, consider the particular filter illustrated in Figure 2. This function multiplies that portion of [A2 (vo, 00)12 between + 10 by large values and the rest of the energy spectrum by very small values or zero up to 66' where another maximum occurs. If the value of [A2 ( vo, 80)]2 is small beyond 80 = +660, then the result is a filtered energy spectrum which contains only the energy which travels between 00 = +10°, appropriate to the par-ticular wave number v0. This same procedure may be used for all the vo, and the result is the distribution of energy among the vo which travels in a certain 00 band-width only. In other words, the energy spectrum has been filtered to allow only energy which travels nearly in the 00 = 0 direc-tion to be shown and all the energy which travels in other direcdirec-tions is suppressed.

The energy spectrum is the same at any point in the 'same' sea. Let the coordinate axes (v., e) be set with respect to the energy spectrum and if [A2 ( v0, 00)]2 is multiplied by the filter func-tion for 130 = 0 throughout, then the energy which travels in this direcfunc-tion is determined. If eo re-mains equal to zero in [A2 ( v0, 80)]2 but is varied in (10) then this is the same as turning the filter to a new direction. That is, if 00 = 0 in [A2 (720, 00)]2 and if 00 = 30 in (8), then the energy which travels in the direction 300 to the original orientation is found. Consequently, it is also possible to focus the filter function in any direction so that only the energy traveling in that direction is evident. This is accomplished in practice by turning the grid system in Figure 1 through a given angular range in the stereo photographs. Also the 00 band-width, through which the energy travels, may be regulated. Since N determines the 80 band-width, the smaller the energy passage is made, the greater N becomes and therefore the more computational work must be done. Also Ay determines

(9)

_

_A

A

I.R _{{Sin[lcton e. ay(pi- 4)] 12}

F F

(1N+1)Sin N. ton e. ay (-i-)1

50 40 >0 20 20 30 50 GO 70 00 90

[Oceanography] DIRECTION FILTER IN SEA SURFACE 763 the place where the secondary maxima appear. When Ay is made larger, the secondary maxima are displaced further. However, there must be 2NAy ft of width on the photograph to allow for the 2N +1 wave records. Therefore, it is necessary that N and Ay be carefully selected to (1) provide a small angular band-width through which the energy travels; (2) displace the secondary maxima far enough to make their effect negligible; (3) minimize computation; and (4) make the aerial pho-tography practical.

Fig. 3--Three-dimensional view of the filter function for N = 10, y = 125 ft Figure 3 shows a three-dimensional view of a typical filter function. The filter has a high ridge in the center whichis constant in amplitude but variable in width. For large V, (small period), the width of the filter is small. As tio becomes smaller, the width of the filter becomes larger. This means, for example, that for four-second waves, the filter might allow all the energy from these waves in an angular range of 10° to pass through, while the energy in waves with 20-sec

periods might pass through a band-width of, say, 30°. Therefore, the filter is more reliable for low-period waves than for high-period waves. The same holds true for the series of smaller ridges which flank the larger ridge. The contribution of all the smaller ridges is very small compared to that of the large ridge, (This will be discussed.) On the wings of the figure appear a pair of high ridges which have the same amplitude as the ridge in the center of the figure. These flanking ridges are always kept in such a position that they multiply portions of the energy spectrum which are nearly zero. Therefore, the contribution from the pair of flanking high ridges is negligible.

A sample example--At this time, no stereo pairs have been made and analyzed. Instead, a theoretical energy spectrum will be operated on by a filter with certain characteristics in order to demonstrate the properties discussed above.

The conditions imposed on the filtering process are that for waves with a ten-second period, the energy passing through must be confined to B. = +10° and also that no secondary maxima should appear before 00=_{±40°. Investigation showed that N = 10 and Ay = 125 ft yielded the desired}

re-sults. This filter is seen in Figure 3.

The two-dimensional theoretical energy spectrum is given by

_29314,

c

Eco.

9d

f

0 otherwise

764 WILBUR MARKS [Trans. AGU, V. 35 - 5]

where v is the wind velocity and C is equal to 3.05 x 104 cm2 sec-5. It is assumed that all energy travels out of the generation area. The one-dimensional energy spectrum[NEUMANN, 1953] was modified by (cos13)2 on consideration of the results of ARTHUR [1949]. It is the best available estimate of the two-dimensional energy spectrum for a fully developed sea at a given wind velocity v. For different wind velocities, the spectrum willhave different shapes. In the ( vo, 60) coordi-nate system, the energy spectrum becomes

P

N,

where the Jacobian is again included to preserve the mapping of the transformation and where eo equals B. For the purpose of this study, a value for the wind velocity of 35 knots was chosen and the constants were substituted in (10). The result is

2

*,

C (cose)

Figure 4 shows a three-dimensional view of (13).

<

c' z

(12)

Fig. 4--Three-dimensional view of energy spectrum in the (V0, 80) coordinate system; the wind velocity is approximately 35 knots

(13) 2 ° 2 cos e- 6.05 (cose,02. 4 _

3 -6.05 110-4cose.

FP, ,0-31sm( 38x103, ton 13.]} (cose.) z e

sin[ tane.]

and a three-dimensional view of the filtered energy spectrum appears in Figure 5. This shows the amount of energy that travels in a band centered about the direction eo = 0.

6.05o 10-4coose.

Fps 0-3 {sr 138o10 ton0.11(cos e

sin (Z s103 tan e,i,1 v.

[Oceanography] DIRECTION FILTER IN SEA SURFACE 765

The filtered energy spectrum is

;n[38 x1031! tan1901

/8 9/

Fig. 5--Three-dimensional view of Fig. 6--Three-dimensional view of filtered energy spectrum; filter is filtered energy spectrum; filter is oriented in direction of Go = 0 oriented in direction of Bo = 100 If the energy that travels in a band centered about the direction Go = 100 is desired, then the filter is turned 10" and the result is seen in Figure 6. The amount of energy is much less than that which travels in the direction Elo = 0°. It is also expected that the energy should diminish to nearly zero as the filter is turned through 50 or 600.

Some of the energy passes through the minor maxima, and this amount must be calculated to provide some limits of confidence in the results. Computations have shown that the effect of the minor maxima is 1.7 pct of the total effect of the filter function for the observed wave number ap-propriate to a period of four seconds. For smaller wave numbers (larger periods) the effect of the secondary maxima is less.

The record which is actually analyzed results in a spectrum which is a function of vo alone as given by (9). It no longer shows any angular variation. The numerical analysis of the series of averaged points yields estimates of the energy in a series of v9 bands along the vo frequency axis, just as the analysis of a record as a function of time yields estimates of the energy in a given vo band.

For each vo band the angular variation of the filter as shown in Figure 2 can be replaced by a rectangular filter with an equivalent area under it. The angular spread varies as a function of vo. The energy obtained in the numerical analysis can then be spread out over the Bo direction, and an estimate of the average value of [A2 ( 00)12 over a small part of the vo, eo plane is thus obtained.

Fig. 7--Graphic illustration of the integration of the filtered energy !spectrum over Bo, for all vo,

lc

1

766 WILBUR MARKS

Figure 7 shows. a series of such areas as they would be located in the vo, 60 plane for a given numerical analysis, Figure 7 also shows the areas which would result upon turning the filter through 400. The entire plane can thus be

scanned by turning the filter, and the variation of [A2 (v0, 80)]2 over the entire vo, 80 plane can then be inferred. From this, the spectrum as a function of At and B can be computed, and the

problem of determining the energy spectrum of the sea surface as a function of frequency and direction is solved.

ConclusionIt is believed that the method of stereo pairs is a practical way to obtain data which will permit the determination of the two-dimensional energy spectrum of an ocean wave system at the edge of a wave-generating area. The amounts of energy which travel in different directions may be sorted out with the aid of the derived filter function.

Since the fact that the wave spectrum is two dimensional affects the theory of wave propaga-tion and of swell forecasting to a great extent, this paper points a way to the solution of this fundamental problem of wave theory:

AcknowledgmentsThe author wishes to thank the Office of Naval Research for their spon-sorship of this study under Contract Nonr 285 (03)1, administered by the Research Division at New York University. Grateful appreciation is ex-. tended to W. J. Pierson, Jr., Project Director, for his advice, criticism, and helpful discussion

-80 of the basic nature of the problem.

fieferences

ARTHUR, R. S., Variability in direction of wave travel in Ocean surface waves, An. N. Y. Acad. Sci.,V.51, pp. 511-522, 1949. NEUMANN, G., On ocean wave spectra and a new

method of forecasting wind-generated sea, Beach Erosion Board Tech. Memo43,42 pp* 1953.

PIERSON, W. J., JR., A unified mathematical theory for the analysis, propagation, and refraction of storm-generated ocean surface waves, pt. 1, N. Y. Univ.,1952.

PIERSON, W. J. JR., and W. MARKS, The power spectrum analysis of oeean Wave records:, Trans, Amer. Geophys. Union, v. 33, pp. 834-844, 1952.

TUKEY, J. W., The sampling theory of power spectrum estimates in Symposium on applications, of autocorrelation analysis to physical problems, Off. Naval Res., Wash., D. C., 1949.

TUKEY, I. W., and R. W. HAMMING, Measuring noise color; I, Bell Telephone Lab., Murray Hill,, N. I., 1949.

Department of Meteorology and Oceanography,, New York University,

New York, New York

(Manuscript received November 209,1953; presented at the 'Thirty-Fourth Annual Meeting, Washington, D.C., May 6, 1953; open for formal discussion until March 1,.1955..)