PHYSICAL OPTICS

Reconstructing the statistical characteristics of the slope field of a sea surfaceby means of multichannel optical radar

A. Z. Zurabyan and V. K. Kachurin

NII FOOLIOS, S. I. Vavilov State Optical Institute All-Russia Scientific Center, St. Petersburg~Submitted April 25, 2002!Opticheski� Zhurnal70, 3–7 ~March 2003!

Relationships are obtained that make it possible to reconstruct the statistical characteristics of theslope field~the gradients of the prominences! of a sea surface from the results of a correlationanalysis of its image frames, recorded by means of optical radar operating in the active mode.Based on this, the possibilities of the method proposed earlier for measuring thecharacteristics of the ocean-atmosphere interface are substantially broadened. The method can beused when working in real time from aircraft or ships. ©2003 Optical Society of America

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INTRODUCTION

Data on the statistical characteristics of the shape ofocean-atmosphere interface in the gravitational-capillrange of scales~about 5 mm to 20 cm! are needed whensolving a number of scientific and technical problems. Thinclude the following: the formation of wind waves, nonlinear interactions of surface waves with each other and wthe flow inhomogeneities formed by internal waves, tdamping of waves by films consisting of surface-active sstances, interpretation of the data of spectrozonal probina marine medium from aircraft and spacecraft, reflectionradio and acoustic signals from the sea surface, and obsetion of underwater objects through the surface.

Optical methods are used to study the sea surfacecause they have the advantage that they combine high rlution with simplicity and because the interpretation of tresults of noncontact radar measurements lacks the ambity inherent to string wave-height recorders~the traditionalinstruments for observing a wind wave!.

The proposed work is a continuation of studies by stmembers of the S. I. Vavilov State Optical Institute.1–4 Theauthors developed and tested an optical method for meaing the shape characteristics of the sea surface, baseprobing it at the nadir with a narrow laser beam~about 3–5mm in diameter! from a moving ship.

Even though results were obtained that were of intein their scientific and applied aspects when the methodtested,1–4 the authors were faced with the need to develofurther. This especially applies to the need to increasemeasurement efficiency. It is substantially limited whensingle-channel device is used, because it takes a long timsample statistically representative implementations. A secdifficulty of using a single-channel radar is that the carrneeds to move fairly rapidly along given directions whmeasuring the spatial dependences of the surface charaistics. It must be hypothesized in this case that the seaface is ‘‘frozen,’’ in order to make it possible to neglect thtime variation of the functions being studied.3

This article discusses the more general case, in whic

147 J. Opt. Technol. 70 (3), March 2003 1070-9762/2003/030

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section of the sea surface~with dimensions of about 1.531.5 m) fairly large by comparison with the wavelengthsgravitational-capillary waves is probed at the nadir with laradiation ~light pulses with a width much smaller than thcharacteristic period of the capillary waves!. The statisticalcharacteristics of the field of surface slopes are reconstrufrom the results of an analysis of its image frames inflected light, obtained by means of the detector objectiThis transforms it from single-channel to multichannel opcal radar.

DERIVATION OF THE MAIN RELATIONSHIPS

A simplified diagram of the method is shown in Fig. 1We introduce a Cartesian coordinate system with

z50 plane coinciding with the unperturbed water surfaand with thez axis directed vertically upwards. We give thsurface equation in the formz5z(r), where r is a two-dimensional coordinate vector in thez50 plane~we are onlyinterested in the spatial dependence of the elevated surfafixed instants!.

Let the optical radar be installed at heightz0 above thesea surface. The surface is probed by a divergent light bdirected at the nadir. We shall describe the conventionalpendence of the power of the probe radiation by a Gausfunction with half-widthg. The back-reflected light is inci-dent on the detector objective~diameter 2c, focal lengthf )and is recorded by a photodetector array placed in the implane of the surface. The optical axes of the detector obtive and of the radiator are coincident~or close to coinci-dence!. After digitizing the analog signals of the photodetetors, the data are entered into the computer, andnecessary computations are carried out.

We shall assume that the surface of interest is stationand spatially homogeneous and that the mean absolute vof its Gaussian radius of curvatureR substantially exceedsthe wavelength of the probe radiation~the condition that thesurface unnevennesses are large! while the variance of thesurface slopes is small by comparison with unity:

147147-04$20.00 © 2003 The Optical Society of America

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^uRu&@l, ~1a!

^~¹z!2&!1 ~1b!

~the angle brackets denote the operation of averagingthe ensemble of surface implementations!.

Let us introduce limitations that are satisfied in practon the angular divergence of the light beam that probessurface, the detector aperture, and the distance to the sur

g,c

z0,uz~r!u

z0!1. ~2!

Because the condition given by inequality~1a! is satis-fied, the Kirchhoff approximation~the tangent-plane method!can be used when calculating the reflected field.5 Inequalities~1b! and ~2! make it possible to simplify the relationshipthat are used. Moreover, a consequence of inequality~1b! isthat multiple reflections of light from the surface can be nglected.

For the given geometry of the experiment~probing at thenadir!, we can neglect polarization effects in the reflectionthe light, as well as shading of individual sections of tsurface~since the angles of incidence and reflection ofrays being recorded are small!.

Starting from the relationship given in Ref. 5 for thscalar amplitude of the light field reflected by a rough sface, after simple transformations, the light field at the into the detector objective can be written as

U~r !5kVU0

4ipz02 E expH ik@A~z02z~r !!21~r2r!2

1A~z02z~r !!21r2#2r2

g2z02J dr. ~3!

Here U0 is the amplitude of the incident light wavek52p/l is the wave number of the laser radiation,r is atwo-dimensional coordinate vector in thez5z0 plane, andV

FIG. 1. Layout of the method.1—radiator, 2—beamsplitter,3—detectorobjective,4—photodetector array,5—processing unit.

148 J. Opt. Technol. 70 (3), March 2003

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is the reflection coefficient of the smooth water surface. Tintegration in Eq.~3! is carried out over thez50 plane.

Let us expand the exponential in Eq.~3! in series inpowers ofz(r)/z0 , limiting ourselves to the first two termof the series. Moreover, we carry out the same operation wtwo other small parametersr2/z0

2 and (r2r)2/z02. As a re-

sult, we get

U~r !5kU0V

4ipz02 E expH ikz0F2S 12

z~r!

z0D1

r2

2z02

1~r2r!2

g2z02 G2

r2

g2z02J dr. ~4!

The conditions required for going from Eq.~3! to Eq.~4!have the following form:

kz0g4@1, ~5a!

kg2z2

z0@1. ~5b!

The light field in the image plane of the unperturbed ssurface is connected in the following way with the fieldU(r )at the input of the detector objective:6

U~h!5k

2pz1E U~r !expS 2

r2

c2D3expF ikS ~r2h!2

2z12

r2

2 f D Gdr . ~6!

The integration in Eq.~6! is carried out over the entireinput planez5z0 , z1 is the distance from the objective to thimage plane of the unperturbed surface (1/z011/z151/f ),and h is a two-dimensional coordinate vector in the imaplane. To simplify the calculations, the entrance pupil of tdetector objective in Eq.~4! is described by a Gaussian funtion with half-width c.

After substituting Eq.~4! into Eq. ~6! and integratingover dr , we get

U~h!5k2c2VU0

8ipz02z1

expF ikS h2

2z112z0D G

3E expF2k2c2

4 S h

z11

r

z0D2

r2

g2z01G

3expF ikS r2

z02 2z~r! D Gdr. ~7!

The radar signal~the intensity of the radiation recordeby a photodetector with coordinates given by vectorh! canbe written as

I ~h!5E U~h1!U* ~h1!expF2~h2h1!2

d2 Gdh1 ~8!

~the size of the photodetector is given by a Gaussian funcwith half-width d).

Neglecting diffraction at the objective by comparisowith the angular resolution:d/z1@l/c, after substituting Eq.~7! into Eq. ~8! and integrating overh1 , we get

148A. Z. Zurabyan and V. K. Kachurin

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I ~h!5k2c2U0

2V2

32p2z04z1

2 E expF ikS r122r2

2

z012z~r2!22z~r1! D

2r1

21r22

g2z02 2

k2c2

8

~r12r2!2

z02 G

3expF2z1

2

d2 S h

z11

r11r2

z0D 2Gdr1dr2 . ~9!

Let us introduce new variables of integrationr5 r1

1r2/2 andr25r12r2 and expand the exponential in E~9! in r2 . Restricting ourselves to the linear term and nglecting diffraction at the objective by comparison with tangular divergence of the probe beam (l/c!g), after inte-grating overr2 , we get

I ~h!5U0

2V2

4z12z0

2 E expF28z0

2

c2 S r

z02¹z~r! D 2

22r2

g2z02

2z1

2

d2 S h

z11

r

z0D 2Gdr. ~10!

As shown in Ref. 5, the limitation used here is valid fcarrying out further calculations when the following obvioinequality is satisfied:

kA^z2~r!&@1. ~11!

It follows from Eq. ~10! that, for large values ofz0 /cand z1 /d, the main contribution to the radar signal is frointegration in the neighborhood of pointr, at which the fol-lowing conditions are simultaneously satisfied:

r

z052

h

z1, ~12a!

r

z05¹z~r!. ~12b!

It is obvious that Eq.~12a! connects the coordinates ofpoint on the sea surface and its image. In this case, the raof a section of the surface imaged on the photodetector~thespatial resolution of the radar! is

dr5dz0 /z1 . ~13!

Equation~12b! describes the reflection of light from thsea surface to the detector objective.

Let us calculate the first two moments of the signalselements of the photodetector array~having in mind the sig-nals of the photodetectors with coordinatesh andh1 , h2).

In order to average Eq.~10! over the ensemble of implementations, we multiply the expression under the integrathe single-point distribution densityW1(¹z) of slopes of thesea surface and introduce integration overd¹z (dzx ,dzy).We should point out that functionW1 is independent ofrbecause of the spatial homogeneity of the surface. Whencondition ordinarily implemented in practice,

c

z0!A^~¹z!2& ~14!

149 J. Opt. Technol. 70 (3), March 2003

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is satisfied, after integrating over the components of vec¹z, we get

^I ~h!&5U0

2V2

4z02 E W1S r

z0DexpH 2

z12

d2 F r

z01

h

z1G2

22r2

g2z02J dr.

~15!

When the conditions

^~¹z!2&,g2!d2

z12 ~16!

are satisfied, we have for the mean value of the radar sig

^I ~h!&5A expF22h2

g2z12GW1S 2

h

z1D , ~17!

where

A5p2U0

2V2c2d2

32z02z1

2 .

To compute the correlation function of the signals of telements of the photodetector array having coordinatesh1

and h2 , we write the product of the corresponding radsignals

I ~h1!I ~h2!5U0

4V4

4z02z1

2 E expH 2z1

2

d2 F S r1

z01

h1

z1D 2

1S r2

z01

h2

z1D 2G2

2~r121r2

2!

g2z02 J

3expH 28z0

2

c2 F S ¹z~r1!2r1

z0D 2

1S ¹z~r2!2r2

z0D 2G J dr1dr2 . ~18!

In order to average over the ensemble of implementions, we multiply the expression under the integral by ttwo-point distribution density of the slopes of the sea surfaW2(¹z1 ,¹z2 ,r12r2) ~the dependence of the function othe difference of the coordinate vectors is caused by thetial homogeneity of the surface! and we integrate over thecomponents of vectors¹z1 , ¹z2 @introducing the notation¹z(r1)5¹z1 , ¹z(r2)5¹z2 , z15z(r1), z25z(r2)]. Thefurther computations of the second moment of the radarnals are analogous to those carried out earlier when calcing the first moment of the signal. As a result, we get

^I ~h1!I ~h2!&5A2 expS 22~h1

21h22!

g2z12 D

3W2S 2h1

z1;2

h2

z1;z0

z1~h12h2! D . ~19!

It follows from Eqs.~17! and~19! that, by measuring thefirst two moments of the radar signal, it is possible to recostruct the one- and two-point distribution density of tslopes of the surface of interest.

149A. Z. Zurabyan and V. K. Kachurin

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Let us consider the case in which the divergence anglthe probe beam is much less than the rms slope of theface:

g!A^~¹z!2&. ~20!

Then, for all values ofh in the image frame of the illu-minated section of the surface, we have

^I ~h!&5^I ~0!&5AW1~0!, ~21!

^I ~h1!I ~h2!&5A2W2S 0;02z0

z1~h12h2! D . ~22!

It is obvious that this character of the dependence ofdistribution densities of the surface slopes onh makes itpossible to do the processing in two stages: The firstmoments of the signal are first determined for each framethe surface image~by integrating within the frame overh!,and the results are then averaged over the resulting pictof the surface~the number of frames must ensure a givmeasurement accuracy!. It is easy to see that, for the givedevice, the gain in the implementation volume by compason with a single-channel radar that scans the surface wnarrow beam along the line of motion of the vehicle and hthe same spatial resolution is equal to the ratio of the diaeters of the illuminated sections of the water surface. Ifspatial resolution implemented in practice~for a single-channel radar, it equals the diameter of the scanning ray! is 5mm and the size of the illuminated section is 1.5 m in ordof magnitude, this gain is a factor of 300.

It is clear that the method under consideration doesrequire the vehicle to be manoeuvred to find out how fution W2 depends on the spatial variables~in particular, it ispossible to operate from a stationary vehicle!.

Because an image frame of the surface is obtainedtually instantaneously, there is no need to satisfy the hypesis of ‘‘freezing’’ the surface.3

Equations~17! and~19! can be generalized to the casecomputing thenth moment of the radar signals~photodetec-tors with coordinatesh1 ,¯ ,hn):

150 J. Opt. Technol. 70 (3), March 2003

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^I ~h2!...I ~hn!&5An expS 22~h1

21...1hn2!

g2z12 D

3WnS 2h1

z1,...,2

hm

z1;

2z1

z1h11

,...,2z0

z1h1nD . ~23!

CONCLUSIONS

1. Relationships have been obtained that make it psible to develop a new method of reconstructing the one-two-point distribution densities of the slopes of a sea surffrom the results of a statistical analysis of its image framby means of multichannel optical radar operating in thetive mode. An estimate has been made of the spatial restion of this method.

2. The main advantage of the method is the high msurement efficiency. It is more than two orders of magnitugreater than that of the version using single-channel rad

1A. Z. Zurabyan, A. S. Tibilov, and V. A. Yakovlev, ‘‘Determining thestatistical characteristics of a random surface by optical radar,’’ Opt. Sktrosk.57, 1066~1984! @Opt. Spectrosc.~USSR! 57, 649 ~1984!#.

2A. Z. Zurabyan and A. S. Tibilov, ‘‘Determining the statistical characteistics of slopes of a sea surface by means of optical radar,’’ Izv. AkNauk SSSR Fiz. Atm. Okeana23, No. 3, 194~1987!.

3A. Z. Zurabyan, V. K. Kachurin, A. S. Tibilov, and V. A. Yakovlev, ‘‘Onthe theory for determining the spatial characteristics of statistically unesurfaces from optical measurements,’’ Opt. Spektrosk.65, 117 ~1988!@Opt. Spectrosc.~USSR! 65, 68 ~1988!#.

4I. V. Aleshin, A. G. Zhurenkov, A. Z. Zurabyan and V. A. Yakovlev, ‘‘Calculating the characteristics of a marine medium from the results of optmeasurements,’’ Opt. Zh. No. 8, 82~1997! @J. Opt. Technol.64, 769~1997!#.

5F. G. Bass and I. M. Fuks,Wave Scattering from Statistically Rough Sufaces~Pergamon Press, Oxford, 1977; Nauka, Moscow, 1972!.

6A. Papoulis, Systems and Transforms with Applications in Opti~McGraw-Hill, New York, 1968; Mir, Moscow, 1971!.

150A. Z. Zurabyan and V. K. Kachurin