Point-spread function associated with underwater imaging through a wavy air—water interface: theory and laboratory tank experiment

Point-spread function associated with underwaterimaging through a wavy air-waterinterface: theory and laboratory tank experiment

William C. Brown and Arun K. Majumdar

The point-spread function needed for imaging underwater objects is theoretically derived and comparedwith experimental results. The theoretical development is based on the emergent-ray model, in whichthe Gram-Charlier series for the non-Gaussian probability-density function for emergent angles througha wavy water surface was assumed. To arrive at the point-spread model, we used a finite-elementmethodology with emergent-ray angular probability distributions as fundamental building functions.The model is in good agreement with the experiment for downwind conditions. A slight deviationbetween theory and experiment was observed for the crosswind case; this deviation may be caused by thepossible interaction of standing waves with the original air-ruffled capillary waves that were not takeninto account in the model.

1. Introduction

The imaging of an underwater object viewed througha wavy air-water interface is challenging and complex.Recentlyl 2 attempts have been made to understandthe imaging of objects and to measure their degrada-tion when they are imaged through a wavy air-watersurface, but a detailed study of the random point-spread functions through a random surface was notmade. Here we present new results for modelingand comparing with experimental results of the ran-dom point-spread function for a wavy air-waterinterface as a function of the air speeds that generatethe surface capillary waves.

Many authors have treated the problem of imagingthrough various distorting media. Ishimaru3 consid-ered propagation through scattering media. Good-man4 treated the general problem of imaging throughrandom phase screens, and in a recent study Karp etal.5 characterized many optical media, including thesea. Yura6 considered imaging in ocean water, butlittle has been published on the problem of imagingthrough water surfaces in general. Because a watersurface is extremely difficult to characterize, only astatistical description of ocean surfaces is possible.

The authors are with the Department of Electrical Engineering,University of Colorado at Denver, Denver, Colorado 80204.

Received 28 May 1991.0003-6935/92/367650-10$05.00/0.© 1992 Optical Society of America.

The long and short capillary surface waves on abody of water cause distortion of the image of sub-merged objects when the image is obtained fromabove the surface. Here we report the experimentalresults of measuring average point-spread functions(PSF's) for a laser source directed from below awind-ruffled water surface in a laboratory tank. Weused the Gram-Charlier model for wave-angle statis-tics, and a finite-element ray approximation of theincident wave fronts was used to compute the irradi-ance distribution in a viewing plane. The theoreti-cally developed PSF was compared with the measuredPSF for different air speeds for both downwind andcrosswind conditions.

2. Emergent-Ray Model

Several studies have shown that when the slopes ofwater surfaces are moderate, they can be character-ized by a Gaussian distribution. This characteriza-tion is true for directions parallel and perpendicularto winds that ruffle the surface. Light rays refractedfrom the water to the air will therefore be deflectedaccording to Gaussian statistics for small to moderateangles. For larger angles, however, purely Gaussianmodels do not fit the data well. A better model is theGram-Charlier model, which has been used by manyother researchers.8

In past reports the Gram-Charlier model wasmainly used to characterize the statistics of randomsurfaces. In our study we use the model to character-ize the angular distribution of rays refracted through

7650 APPLIED OPTICS / Vol. 31, No. 36 / 20 December 1992

the air-water interface. We investigate the behaviorof individual rays, which are approximated by narrowlaser beams that propagate in a small laboratory tankwith winds of variable speeds channeled above it.We studied the distribution of rays both downwindand crosswind. The detailed experimental setup andprocedure are described elsewhere.9 Essentially, air-ruffled water waves were produced in a 122 cm x 46cm x 52 cm tank by using a variable speed blower.A He-Ne laser with a beam diameter of 0.9 mm at the1/e2 point was used as a source. A one-dimensionallateral-effects sensor placed above the wavy watersurface was used to detect the deviation angle of thelaser propagating through and refracted by the ran-dom water surface. The calibrated sensor producedanalog signals that were converted to an 8-bit digitalformat sampled at 200 Hz. Interfacing of the sensorand signal conversion was done on an electronicboard that was internal to a 386-type computer,which was used for collecting, storing, and subse-quent processing of data.

The total quantity of data collected for this studywas as follows. In the downwind direction, 18 runswere made at six different air speeds and nine differ-ent incidence angles, and nine additional runs weremade at an incidence angle of 0 (measured withrespect to the average surface normal) and nine otherwind speeds. In the crosswind direction, two sets ofruns were also made. The first set consisted of nineruns at 00 incidence and nine different wind speeds.The next consisted of 10 runs made at different windspeeds and five different incidence angles. Each runconsisted of 30,000 instantaneous samples of therefracted narrow laser-beam angle collected at 200Hz, representing 2.5 min of record time. A sum-mary of these runs and some of their statistics aregiven in Table 1.

Next we outline the development of a Gram-Charlier model to account for the data collected.The mean or expected emergent angle of a ray of lightrefracted through the water surfaces depends on boththe incidence angle and the wind speed above thesurface. For small surface waves the angle of refrac-tion is a function of the incidence angle, according toSnell's law. Figure 1 shows the passage of a raythrough the air-water interface with an angle ofincidence 0 and an angle of emergence E. Figure 2compares the expected emergent angles according toSnell's law with those measured experimentally (ob-tained from the probability-density functions). Forthe downwind case the results are close for the two airspeeds of 2.5 and 5 m/s, except at the 4 angle ofincidence run. For crosswind components the mea-sured results deviate from Snell's law. This pointwill be discussed further.

We derived an empirical relationship for the meanand standard deviation. The expected mean emer-gent angle of refracted rays as a function of downwindair speed is obtained by using Snell's law and aquadratic least-squares fit as follows:

pc = arcsin(nw sin 0jC) + 1.5378 - 1.0955v, (1)

ALD = arcsin(nw sin OID) + 0.5723 - 0.4748v, (2)

where !1C,D are mean emergent angles crosswind anddownwind, nw is the index of refraction of the waterat the He-Ne laser wavelength, 0 is the in-waterincidence angle in the downwind or crosswind direc-tion, and v is the downwind air speed measured at 10cm above the surface in units of meters per second.The data of Table 1 show that the measured standarddeviations of the emergent-angle distributions arestrongly dependent on air speed. A least-squares fitto the data yields the following empirical relationshipfor the standard deviations of the crosswind (c) anddownwind (D) components:

oc= 1.1663v - 1.1938,

CD = 0.6767v - 0.2546.

(3)

(4)

If the surface-slope statistics were truly Gaussianthen the relationships given by Eqs. (1)-(4) would besufficient to construct the model distribution. Amore accurate model can be obtained by using theGram-Charlier development, which requires addi-tional relationships. Slight deviations from Gauss-ian statistics can be modeled in terms of the Edge-worth or Gram-Charlier series. The Gram-Charliermodel is used here and is formulated in terms of abasic Gaussian probability density given by

1Eexp [ -e2] 1 (5)

The Gram-Charlier series is given by10

A B-(3) + 4+(4)P(E) = (E) - - ( + 4 E(61 [ (e-11)2

= - ; exp I IVF;uL 2o,2

+ 11 _3A ( - +LL3 - 42+ B 3 - 6p +(e)]' (7)

where A and B are constants and P(E) is the moreaccurate model of the refraction-angle probabilitydensity. Constants A and B depend on whether P(E)is a crosswind or a downwind model distribution.

The determination of the constants A and B is moredifficult. From the least-squares fit of the relation inEq. (7) to the emergent angular histogram, whosearea was normalized, the constants A and B in theseries were determined for each downwind and cross-wind direction. The constants were determined foreach of the 46 runs. The actual parameters are astrongly dependent function of ray incidence angles.Therefore, the shape of the angular ray distributionsalso depends heavily on the incidence angles. After

20 December 1992 / Vol. 31, No. 36 / APPLIED OPTICS 7651

(6)

Table 1. Run Summary and Statistics

Wind Speed Incidence EmergentRun Direction (m/s) Angle (deg) Means (deg) SD (deg) Skew

123456789

10111213141516171819202122232425262728293031323334353637383940414243444546

DownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindCrosswindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwindDownwind

4.92.52.54.94.92.42.45.05.02.42.45.05.02.55.02.42.73.71.02.02.63.53.73.94.14.75.45.033.163.165.095.093.003.005.195.192.781.02.02.62.73.53.73.94.15.4

313121.321.3

8.08.04.04.0

-4.0-4.0-8.0-8.0

-20.0-20.0-26.0-26.0

00000000000004.124.12

-4.12-4.1211.7511.75-9.05-9.05

000000000

40.6244.4728.9226.43

9.1811.20-7.56-6.32-6.83-5.12

-10.96-12.93-27.45-26.43-36.42-35.79

0.075-2.19-0.310-0.721

0.880-0.973-3.241

0.00465-4.439-5.211-4.236-2.571-1.754

1.5330.482

-6.278-3.82212.207.402

-11.59-8.944-0.324-0.0068-1.213

0.0751-1.113-2.187-1.671-1.501-2.132

2.681.571.0073.524.0580.8810.9431.434.240.6261.934.0463.831.504.021.9840.7283.650.6420.8050.7682.3804.5841.9904.9734.1904.9884.5154.4153.8144.3053.7631.5370.5444.1683.331.9160.7950.7750.7710.7282.2283.6483.8611.8372.702

0.0885-4.83

2.0068.313

-34.550.3233.383.36

-63.26-0.055-8.59

-25.20-32.24-3.52-5.47-2.39

0.155-56.88-0.60-1.52-0.116

-41.25-96.91-14.51-66.91-27.42-72.82-59.84-83.64-69.92-57.89-63.94-15.74-0.899

-39.209-34.35-20.28

0.4840.374

-0.08540.155

-23.01-56.885-64.42-40.73-45.264

obtaining these best-fit A and B constants, we deter-mined cubic least-squares polynomials to approxi-mate the best-fit set as a function of incidence angles.

The results of the cubic polynomial determinationare as follows. In the crosswind direction the Gram-Charlier constants are given by

AC = 8.4426 x 10-40lC3 + 4.8308 x 10-30IC2

- 4.3547 x 10- 20, - 1.8582,

BC = 2.745 x 10- 3 0'C3 - 1.7138 x 10- 3 0IC2

- 4.34755 X 10-20,,C + 0.392204.

In the downwind direction the constants are

AD = 8.77507 x 10- 50,D 3 + 8.9985 x 10 401,D2

+ 6.4387 x 10 201,D - 0.50923, (10)

BD = 6.8299 x 10 5OID3 - 1.31079 x 10- 3O,D2

- 2.629675 x 10-2 OD + 0.378067. (11)

(8) We used these results to compute the Gram-Charlier model by substituting Eqs. (1)-(4) and Eqs.(8)-(11) into Eq. (7) for both directions. As a result,

(9) we can accurately predict the angular distribution of


Random

I I WaterI o I Surface

1/ 1I I

Fig. 1. Ray passage through an air-water interface.

refracted light rays through a random water surfacefor arbitrary incidence angles and air speeds in boththe downwind and crosswind directions. This predic-tion will give two distinct angular ray densities; if weassume that they are independent in the two orthogo-nal directions, then the two-dimensional probabilitydensity is

P(0I,C, OI,D) = P(OI,C) P(OI,D), (12)

where P(0I,C) and P(OID) are the Gram-Charlierdensities in Eq. (7) after substituting the appropriate

DOWNWIND MEAN EMERGENT ANGLE vs INCIDENCE ANGLE

I

zUjw

Lu

cl

x

LU

E

0

zC4c)

Lu

0Lu

LuUa

50

30 .. .. . ....W

c~ ~ ~ ~ ~-305 a X

-0 20 10 0 ,,,, 10 20 30 4 ,,,,,,,,,,'.mS

20 .. . . . .. NCIENCE ANGLE degrees)

CROSS MEAN EEGNAG ..NC.DENCE ANL.E

-20 ............. ............/ ~ ~~~~~~~IndX Of Reckon Wail 133

430 -20 -10 0 10 20 30 41INCIDENCE ANGLE (degrees)

CROSS MEAN EMERGENT ANGLE ,s. INCIDENCE ANGLE

15

INCIDENCE ANGLE (degrees)

Fig. 2. Expected (according to Snell's law) and measured meanrefracted emergent angles for the downwind and crosswind direc-tion.

constants A and B for the crosswind and downwindcases, respectively.

We are now ready to compare the predicted emer-gent-angle distributions with the experimentally mea-sured distributions. Figures 3 and 4 show such acomparison for both the downwind and crosswindcomponents. In the downwind direction there is agood agreement for many of the runs. The applica-bility and versatility of the Gram-Charlier model areclearly shown in these figures. The distributions insome cases are skewed and even bimodal, yet themodel generally follows these variations. In thecrosswind direction the agreement is less strikingbecause of the limited transverse dimension of thetank. In this direction the standing wave patternsand their interactions with the small surface wavesmay make the model less valid. Figure 5 shows atwo-dimensional contour plot of the emergent-angledistribution that was captured on a 1.5-min exposureof Tri-X film and subsequently transferred by way ofa CCD camera and interfaced to a PC. The distribu-tion is asymmetrical and shows certain preferreddirections. This preference indicates that standingwave patterns significantly affect the one-dimen-sional densities so determined.

3. Finite-Element Formulation

For a point source located within the water, a solidcone of light will be emitted into the surface. Eachdiscrete ray will intercept the surface with an infini-tesimal facet over which the instantaneous slope isapproximately constant. A ray at each facet willeither be refracted or internally reflected, dependingon the instantaneous angle between the incidencevector and facet normal. If we imagine this set ofincident rays to be either reflected or refracted, theirradiance distribution above the surface will be, to afirst approximation, a superposition of the rays pene-trating the surface.

The irradiance distribution in a plane parallel tothe average water surface can then be obtainedapproximately by superpositing the positional proba-bility densities of individual rays. Because the finite-surface elements are random, the position of anyindividual ray will be constantly changing in time.Over times much greater than the characteristicperiod of the surface waves, the position of eachrefracted ray penetrating the surface at a given angleand position can be described in terms of a probabilitydensity in the parallel plane related to the emergent-angle densities derived in Section 2.

With a detector located above the surface, we cannow determine the shape and dispersion (width) ofthe positional probability densities in the plane. Ifnarrow raylike bundles are propagated upwardthrough the surface to an area detector, a time-averaged distribution is measured. The distributionin the plane of detection is then proportional to asummation of several ray-position probability densi-ties. As the number of rays increases, these distribu-tions become more and more continuous.


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!40 -35 -30 -25 -20 -15

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_ . ............. 1 ..-......... *-* ---- -. e--. --

.......

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run 18

0.2 ..

0. . .

-20 -1U U

WAVE ANGLE (degree) WAVE ANGLE (degree) WAVE ANGLE (degree)

Fig. 3. Angular ray probability densities for the downwind direction for various wind speeds and incident angles. The run numberscorrespond to the data as shown in Table 1. The dashed curves are the experimental model and the solid curves are from Gram-Charliermodels.


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WAVE ANGLE (degree)

run 33

run 36

WAVE ANGLE (degree)

50

WAVE ANGLE (degree)Fig. 4. Same as in Fig. 3 except for the crosswind direction.

The finite-element approach can be understood inthe following way. Let us assume that the pointsource emits a total radiant power PRAD into theupward solid hemisphere consisting of 2 sr. If afinite number N of rays is assumed to approximatethe continuous radiant intensity (W/sr), then eachray will transport an amount of power equal to

Pj = PRAD/N, (13)

assuming a uniform distribution of power over allrays.

The irradiance distribution measured by a detectoris averaged over the integration period. A detectorlocated in a viewing plane will add the energy of eachrefracted ray whenever it passes over the entrancepupil of the detector. The time an individual rayspends in the entrance pupil depends on the area ofthe pupil and the position probability density of theparticular ray refracted by the randomly changing

facet of the water surface. If the detector's integra-tion time is T and the differential area of an infinites-imal detector is dA, then the differential time a rayspends in the detector entrance pupil is

dt = Tfj(x, y)dA, (14)

where f(x y) is the position probability density (prob-ability/unit area) of the jth ray in the x-y plane,which is assumed to be above the water surface.

Combining Eqs. (13) and (14) and rememberingthat the detector is energy sensitive, we see that thedifferential energy contribution of the jth ray to asmall detector is then

dEj(x, y) = Pjdt = NRAD f/(x, y)dA. (15)

The net detected energy from this area dA of thedetector can be obtained by summing the contribu-


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50

0

z

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so

20

la 10 20 30 40

PIXEL NUMBER

0

. _

z

z a

50 60

Fig. 5. Two-dimensional contour plot of the endistribution at the wind-ruffled water surface. The cm/pixel, i.e., a total of 8.03 cm across (each side).

tions from the entire ensemble of rays:

N TP N

dE(x, y) = I dEj(x, y) = N E f(x,)j=1 T j=1

The average irradiance is then given by

dE(x, y) pED N

I(X A TdA =N 1f~'

Equation (17) represents a finite-elementtion for a distribution in a viewing plaiwater surface illuminated by a point scillustrated in Fig. 6. From a practical pothis equation will be used to approximatethe positional probability densities can be (as outlined in Section 4. Also note that tsion represents an incoherent summationthe ray intensities or positional probabilit

Itotal (X.

RandomSurface

PointSource

F do - d m o

Fig. 6. Finite-element summation.

aergent-scale is 0

')dA.

ipproxie aboturce;int of

can be added. We are not adding the amplitudesthat would otherwise cause interference. The ex-pected irradiance thus will be an incoherent sum,even if the original source is spatially coherent. Weassume that the random water surface will introducerandom transverse phase variations that effectivelydestroy the spatial coherence of any wave front.

4. Point-Spread Function in Random Medium

If a perfect, aberration-free optical system images anobject below the random water surface, the result willbe imperfect because of the random surface. Asmeared and distorted image results because of theimperfect transmission medium, which is the wavy

angle water surface. Because the detailed structure of the.1339 surface at all times cannot be known exactly, the

optical distortion must be treated as a random pro-cess.

Both Ishimaru3 and Goodman4 defined modulation-transfer functions and PSF's that are averages ofexpectations of irradiance of optical waves after trans-

(16) mission through random media. In addition to theassumption that the irradiance in the viewing plane isincoherent, we also assume that capillary wave-lengths on the random water surface are much longerthan the wavelength of the light used for imaging.

(17) The PSF is the incoherent irradiance distribution inthe image plane at a position (xi, yi) caused by a point

ima- source located in the object plane at a position (xo, yo),where

ve ait isdew,

S(xi, yi, X0, yo) = _7[( fu, fr)I (18)

the PSF if and the average PSF H of two spatial frequencies, f"determined and f, and the optical transfer function S are Fourierhis expres- transform pairs. When the linear-systems theoryL. Each of holds, S can be convolved with any arbitrary object-ty densities distribution function to obtain the expected output

image. Of course, the same result can be obtainedN by the usual manipulations in the frequency domain,= x.Y) by using the optical transfer function.=1I Although other approaches are possible, here the

PSF has been determined and modeled directly fromthe basic definition. Because we are primarily inter-ested in the degrading effects caused by the wavywater surface when imaging objects below the sur-face, a perfect aberration-free optical system will be

- i- (X.Y) assumed. The approach is to determine the averagePSF geometrically in terms of angular distributionsand the finite-element approximation discussed inSection 3.

The geometrical method used is shown in Fig. 7.The light emitted by a point source in the object planepropagates through the water surface, through asimple lens, and then onto an image plane. Weassume that the lens is a thin, radially symmetricallens operating strictly in the paraxial mode. Toapproximate the average PSF by using the finite-element method, we need to determine the positionalprobability densities of rays falling on the image


ImagePlane I -do | d.

z

Point Random Lens ImageSource Surface Plane

Fig. 8. Ray trace through random surface and lens.Fig. 7. Point-spread function of a simple imaging system.

plane. These densities will be obtained by the trans-formation of variables as the angular deviation ofrays at the water surface map into positions in theimage plane.

The transformation of variables follows easily fromelementary ray-tracing techniques. Many research-ers, such as Yariv1 and O'Neill,12 provide elementaryray-tracing matrices that allow us to calculate thetransformation of angular deviation into the image-plane position. This calculation requires computa-tions of three matrices. Two of them are transfermatrices: one from the water surface to the lensplane and the other from the lens plane to the imageplane. The third is a lens matrix. If the three aremultiplied in the proper order, then the transforma-tion matrix, taking into account the random mediumand the imaging lens system, is given by

[r 1 (1 -do/f)

rJ -/f ] [e1(19)

In Eq. (19) f is the focal length of the lens and theother parameters are explained in Fig. 8. The ma-trix representation can be expanded into two separaterelationships:

r= (1 - do/f)h + fe, (20)

r'= -h/f. (21)The distance h is given by

h = do tan 0, (22)

which for small angles reduces to

h _ do. (23)

If Eq. (20) and relation (23) are combined then therequired transformation is

r = (1 - do/f )d00 + fE, (24)

with the inverse relation

e = (1/f)[(1 - do/f )doE - r (25)

Finally, the desired positional PSF in the image

plane can be determined as follows. If the individualangular probability densities that are already deter-mined are given by P(E), then the transformation ofprobability densities gives

, dElP(r) = PE[E(r)] -1

= - P E[E(r)].

We used Eq. (17) to obtain the average PSF

pN9(r) = N5_ E Pj[(r)].f *=

(26)

(27)

(28)

Using all the individual ray angular distributionmodels, we implemented Eq. (28) in computer soft-ware to simulate an actual measurement of the PSF.

5. Experimental Setup and Results

In a controlled laboratory environment, a comparisonof the measured PSF and that obtained by simulationbased on the proposed model was made. The detailsof the experimental arrangement are described else-where.9 A 189.28-L glass test tank was used. Awind flume through which variable air speeds weregenerated was mounted at the top of the tank on oneside.

The tank was lined with an optically opaque blackcloth to prevent reflections from the internal surfaceof the glass and to block stray outside light fromentering the experiment. A small pinhole located18.8 cm below the water surface was illuminated by aHe-Ne laser light with a wavelength of = 0.6328pm. An imaging lens with a focal length of 10.9 cmwas used. The lens was placed 42.5 cm above thesurface. The image plane was thus near the backfocal plane. Each of the average PSF's averagedover 2 min were recorded at four different selected airspeeds.

The two-dimensional images were first recorded onKodak Plus-X film and then transferred to a com-puter file by using a solid-state-array CCD camerasystem and a video-frame digitizer. The camerasystem was interfaced with a desktop computer to


f

point-sprcad unction

0.9 ,.., ... ,, ,, ;..,1,. 0.9 1. 9 rls air speed

0.6 ,: ',;,,,,.,

0.5 . .: , , _

0.3 . . .. .. .0.2 .. ' .._0.1 _ ......... . .

0 10 20 30 40 50 6C

PIXEL NUMBER

0.

0.

0.

0.

0.

0.

0.

0.

0.

point-spread function

-0 10 20 30 40 50

zW.z

W

point-spread function

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IU u 30

PIXEL NUMBER

point-spread function.83

z"at-

S

60

PIXEL NUMBER

Fig. 9. Comparison of the point-spread model (dashed curve) withthe experimental model (solid curve) in the downwind direction forlow and high air speeds. The scale is 0.1339 cm/pixel, i.e., 8.03cmacross.

store the data for further processing. Figures 9 and10 show the one-dimensional slices for both down-wind and crosswind components and for high and lowair speeds as indicated.

The results show that for the downwind compo-nent the modeled and measured PSF's are in closeagreement. For the crosswind component there issome deviation between the results from the modeland those from the experiment. This deviation maybe caused by some edge and multiple-wave effects inthe wave test tank, which had a limited transversedimension. The possible interactions between thestanding waves generated in the transverse directionand the original air-ruffled capillary waves were nottaken into account in the model. With higher airspeeds the crosswind results deviate more from theproposed model.

6. Summary and Conclusions

A general theory based on the emergent-ray model forlight propagating through a wind-ruffled water sur-face was developed. We used a finite-element formu-lation and emergent-ray angular probability-distribu-tion functions to derive the PSF in a random watermedium. The PSF model was compared with mea-

40 50 60

PIXEL NUMBER

Fig. 10. Same as in Fig. 9 except for the crosswind direction.

sured data obtained in a wave-tank experiment per-formed in a controlled laboratory environment. Anexcellent agreement between theory and experimentwas obtained. The model PSF fits closer to themeasurements for a downwind component than forthe crosswind situation.

The deviation for crosswind and for higher airspeeds may be caused by some edge and multiple-wave effects in the wave test tank of limited trans-verse dimension. The possible generation of stand-ing waves within the tank in the transverse directionof the wind and their compiling effects on the originalair-ruffled capillary waves may cause the deviation ofthe model PSF from the experimental measurements.Our results will be useful in evaluating the quality ofthe image formed through a wavy air-water inter-face, e.g., in underwater imaging through the oceansurface. Finally, the PSF for a random mediumdeveloped from the model we described here and fromthe experimental results in a laboratory test tank canbe used to calculate the image of an underwater objectthrough a wavy air-water interface.

References1. P. F. Schippnick, "Imaging of a bottom object through a wavy

air-water interface," in Ocean Optics IX, M. A. Blizard, ed.,Proc. Soc. Photo-Opt. Instrum. Eng. 925, 371-382 (1988).

2. N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price,and R. Miller, "Measured degradation in image quality when


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imaging through a wavy air-water interface," in Ocean OpticsIX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383-390 (1988).

3. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, San Diego, Calif., 1978), Vols. 1 and 2.

4. J. Goodman, Statistical Optics (Wiley, New York, 1985).5. S. Karp, R. Garliardi, S. Moran, and L. Stotts, Optical Chan-

nels (Plenum, New York, 1988).6. H. Yura, "Imaging in clear ocean water," Appl. Opt. 12,

1061-1066 (1973).7. C. Cox and W. Munk, "Statistics of the sea surface derived

from Sun glitter," J. Mar. Res. 13, 198-227 (1954).

8. C. Cox and W. Munk, "Measurement of the roughness of thesea surface from photographs of the Sun glitter," J. Opt. Soc.Am. 44, 838 (1954).

9. W. Brown and A. Majumdar, "Laser probe for measuringstatistics of small random surface waves on water in alaboratory tank," submitted to Appl. Opt. (1991).

10. H. Cramer, Mathematical Methods of Statistics (Princeton U.Press, Princeton, N.J., 1946).

11. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, NewYork, 1985).

12. E. O'Neill, Introduction to Statistical Optics (Addison-Wesley,Reading, Mass., 1963).


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Point-spread function associated with underwater imaging through a wavy air—water interface: theory and laboratory tank experiment