It is known that sea surface roughness significantlyimpedes the airborne observation of the sea bottom.Light refraction by a randomly rough sea surfacecauses random “refractive” distortions of subsurfaceimages. This refraction also adds spatial interferencein the form of the images of the waves themselvesand in fluctuations of the underwater radiance“haze” due to light focusing and defocusing by therandomly rough surface. The effects of light reflec-tion and refraction by a wavy surface are thoroughlyanalyzed in reference [1]. Numerous papers [2–9] aredevoted to the problem of imaging through a wavysurface with results presented in several books[10–13]. The author of the pioneering paper [2] for-mulated the concept of the modulation transfer func-tion (MTF) of the rough sea surface as a one-pointcharacteristic function of surface slopes. More de-tailed analysis of the imaging of underwater objectsaccounting for the multiple scattering and absorp-tion in water and the correlation between fluctua-
tions of water-entering and water-leaving radiationare given in [3–9]. Computer simulation algorithmsfor random realizations of surface-distorted imagingare presented in [13–15]. Reference [13] also de-scribes a technique for calculating the statistical mo-ments of an image required for its quality evaluation.
The results listed above can be used to predict ima-ging of the bottom under specified conditions and toincrease the quality of imagery by optimally choosingthe observation conditions, for example, the solaraltitude at the time of imaging and the flight direc-tion with respect to the Sun and the surface wind di-rections [13]. However, image quality can also beimproved in other ways. As shown in [16–18], refrac-tive distortions in bottom images can be eliminatedby appropriately processing the raw images employ-ing data on the spatial distribution of sea surfaceslopes present when the images were made. Thissea surface data may be retrieved either from sky-lit sea surface imagery or from the glitter pattern ob-tained when the surface is artificially illuminated.The possibility of using an artificially illuminatedglitter pattern to eliminate refractive distortionswas discussed in [19,20] and has been demonstratedin the laboratory [21].
A method to partially eliminate refractive distor-tions in a lidar imaging system is proposed and ana-lyzed in the present paper.
2. Model Description
We assume that a bottom image is formed by apulsed laser beam scanning the bottom while record-ing the signal power reflected from the bottom, whichdepends on the beam direction. The receiver consistsof an objective lens and a “single-element” photode-tector (e.g., photo-multiplayer), Fig. 1. The incidentbeam width is assumed to be small compared to thewavelengths of the wind waves contributing to thesurface slope variance. Under these conditions, dis-tortions in the bottom image are primarily due torandom variations in the angle that the laser beamis refracted by the sea surface. These distortions canbe removed if the true direction of the beam entrancein water is known. We suggest that the informationon the refraction angle of the laser beam can be re-trieved from the centroid of angular distribution ofthe water-backscattered signal radiance (or the cen-troid of irradiance distribution in the photodetec-tor plane).Changes in the propagation direction of the beam
lead to shifts of a light patch on the water layer(Fig. 1) from which the backscattered signal arrivesat a given instant of time. The receiver lens forms an“instantaneous” image of this patch. However, if asingle-element photodetector is employed, informa-tion on the position of the patch is lost. Therefore,to record beam shifts, it is necessary to use a multi-element sensor or additional photoreceivers withasymmetrical directivity diagrams, that is, withasymmetrical sensitivity of the photodetectors along
their surface capable of reacting to the angular var-iations of the backscattered water signal.
For example, this property is inherent in receiverswith a nonuniform linear sensitivity over their fieldof view. Such receivers can be fabricated by superim-posing a mask with a linearly nonuniform transpar-ence on the photocathode of a photomultiplier. Ifthe beam shifts toward the increasing transparency,the sensitivity of the photodetector increases and thebackscattered signal grows, while, if it shifts inthe opposite direction, the signal decreases. To recordthe beam shift in two orthogonal directions, oneneeds at least two additional photoreceivers withsensitivities varying in the appropriate x-y direc-tions. The imaging systems with three one-elementsensors and with a single multielement sensor areequally suitable for correction of the image refractivedistortion. However, the first one is more simplesince it makes possible the spatial image processingbefore converting the image into electrical signals. Inthis system, the patch centroid and refraction anglesof the laser beam are directly determined fromvalues of current on the exits of three sensors. Inthe system with one multielement sensor, the imagespatial moments should be computed in the electricalsection of the system, making it more complicated.
Backscattered light passing through the sea sur-face is also refracted. Thus, the signal variations con-taining information on the propagation direction ofthe beam are observed against an interfering back-ground caused by the refraction of water-leavinglight through the sea surface. To diminish this inter-ference, the linear size of the receiver field of viewshould be large in comparison to the wavelengthsof the surface waves. Additionally, the backscatteredsignal should be received from rather deep waterlayers, where the effects of light focusing by windwaves are weak. To estimate the value of the refrac-tion angle of the laser beam and the error in itsdetermination by signals from nonsymmetrical di-rectivity diagrams, the equation for the power PðtÞof the lidar echo signal reflected by the water bodyis used [11]:
PðtÞ ¼ W0ΣΩ2πm2 ∭Vbbðz; rÞ
�Z∞
−∞Erðz; r; t − t0Þ
× Esðz; r; t0Þdt0�drdz; ð1Þ
where W0 is the laser pulse energy, Σ and Ω are thereceiving aperture area and solid angle, m is the re-fractive index of sea water, bb is the water backscat-tering coefficient at a depth z in a point withhorizontal coordinates r, Esðz; r; tÞ is the irradiancedistribution in the horizontal plane at a depth z atthe instant t induced by a radiation source of unit en-ergy, and Esðz; r; tÞ is the analogous virtual distribu-tion induced by an auxiliary source of unit energywith the same spatial–angular characteristics asthose of the receiver. Integration is conducted over allFig. 1. Schematic diagram of an imaging system.
of the space V. Equation (1) is a direct consequence ofthe optical reciprocity theorem assuming that back-scattering is isotropic.Let us assume that the depth of the water layer
from which the backscattered signal is received issuch that the longitudinal diffusion of the laser pulsemay be neglected. In this case, we calculate thefunctions Es;yðz; r; tÞ by applying the small-angle ap-proximation of the radiation transfer equation. Addi-tionally, to simplify calculations, assume that thesource radiates a short pulse approximated by thedelta function. It is convenient to write the equationsfor the functions as [22]
Es;rðz; r; tÞ ¼ Tδðt −H=c −mz=cÞ
×Z
…
ZFs;rðk1; k1H þ k2z=mÞΦðk2; zÞ
× exp½ik1 · rsf − ik2 · ðrsf − rÞ þ ik2
· ηðrsf Þqz�dk1dk2drsf ; ð2Þ
where T is the transmittance of the boundary,
Fs;rðk; pÞ ¼ ð2πÞ−4ZZ ZZ
Ds;rðr; nÞ
× expð−ik · r − ip · nÞdrdn ð3Þ
are the Fourier transforms of the initial spatial–angular distribution of the source radiation Dsðr; nÞand the receiver directivity diagram Drðr; nÞ, n is aprojection of the unit vector determining a directionof a ray on the horizontal plane, k and p are the spaceand angle wavenumber vectors, ηðrsf Þ is the surfaceslope (projection of the local normal to the surface atthe point on the horizontal plane), q ¼ ðm − 1Þ=m,Φðk; zÞ is the Fourier spectrum of the point-spreadfunction for a water layer of a thickness z, and His the lidar height above the average sea level.Now we assume that the source forms an infinitely
narrow vertical beam, that is,
Ds ¼ δðrÞδðnÞ; Fs ¼ ð2πÞ−4: ð4Þ
We assume three receivers coaxial with the emit-ter with directivity diagrams and correspondingFourier transforms of
where α is the rate of change of the sensitivity varia-tion, i.e., the gradient of the optical transparency
wedge value, and δ0 is the derivative of the delta func-tion. The relationship in Eq. (5a) describes the direc-tivity diagram of an isotropic point receiver. Therelationships in Eqs. (5b) and (5c) are the directivitydiagrams of the receivers with transparency wedgefilters placed over their sensitive surfaces. Thesewedges linearly vary the sensitivities inside the di-rectivity diagrams in either one of two orthogonal di-rections. The first terms in Eqs. (5b) and (5c)correspond to the photo receiver’s average sensitiv-ity. It is constant inside the directivity diagram, i.e.,P1. The second term describes the linear transpar-ency/sensitivity variation. Hence, the signal powerin these receivers may be represented appropriatelyas P2x;2y ¼ P1 þ ~P2x;2y. Therefore, the term P1 coin-cides with the average signal power in the receiver,whose characteristics are described by Eq. (5a).Hereafter, we assume that the water-backscatteringcoefficient does not depend upon horizontal coordi-nates. If the wave height through which the beampasses is neglected and the functions Es and Er areassumed to be uncorrelated, the power P1 averagedover an ensemble of surface realizations in the small-angle approximation is described by the expression
P1 ¼ W0ΣΩT2bbc
2πmðmH þ zÞ2 Φð0; zÞ; ð6Þ
where z ¼ ðct=2 −HÞ=m is the depth from which thebackscattered signal comes within time t after thesensing pulse had been emitted.
For a random realization of the term ~P2x, we have,from Eqs. (1)–(5) using the same small-angle approx-imations:
~P2x ¼W0ΣΩT2αðmHÞ2ð2πÞ4
bbc2πm
×Z
…
ZΦðk1; zÞΦðk2; zÞ exp½ir · ðk1 þ k2Þ − ik2
· rsf γ − ik1 · η0qz − ik2 · ηðrsf Þqz�rx;sfdrdrsfdk1dk2; ð7Þwhere η0 is the surface slope at the point of the laserbeam incidence on water, ηðrsf Þ is the surface slope atthe point rsf and γ ¼ 1þ z=mH.
Note that the assumption that the functions Esand Er are not correlated, mentioned above, isequivalent to the assumption that the vector η0and the surface slopes in the area forming the re-ceived signal, i.e., that area ”covered” by the receiv-ing directivity diagram, are not correlates, as well.Using this approximation one can easily obtain theexpression for the statistically averaged value ~P2x as
~P2x ¼W0ΣΩT2
ðmH þ zÞ3bbc2π Φð0; zÞqzη0x; ð8Þ
where η0x is the averaged estimate of the projection ofthe vector η0 on the axis x. It follows fromEqs. (6) and(8) that
An analogous expression can also be easily foundfor the second orthogonal y component of the vectorη0 if the corresponding variation in the sensitivity inthe orthogonal direction of the receiving directivitydiagram is assigned according to Eq. (5b). It followsthat a trivial combination of the measurements ob-tained by the three receivers with the characteristicsgiven by Eqs. (5) enables one to determine “an aver-age” of the required value. This information can beemployed to specify the direction of propagation ofthe laser beam in water and to appropriately correctthe characteristics of the images.
3. Model Statistics
The propagation direction estimated in the way de-scribed is obviously probabilistic in nature. There isthus a statistical error in determining the vector η0.To estimate this error, we calculate the variance of anestimate for the value η0 as
Δη0x2 ¼�H þ z=m
αqz
�2�~P2x þΔ~P2x
P1 þΔP1
−~P2x
P1
�2
; ð10Þ
where Δ~P2x and ΔP1 are random deviations of thesignal power from their average values. If ΔP1 ≪P1, then, from Eqs. (9) and (10), we obtain
Δη0x2 ¼ η0x2
0BBBB@
�Δ~P2x
�2
P22 þ
�ΔP1
�2
P12 −
2
�Δ~P2xΔP1
�
P1P2
1CCCCA:
ð11Þ
The random deviation from the average of the signalΔ~P2x containing information on the wave slope inthis linear approximation over the wave slopes ηðrsf Þis described by the expression
The analogous equation for the value ΔP1 has theform
ΔP1 ¼ iqzW0ΣΩT2αð2πÞ2m3H3
bbc2
ZZ ZZðη · kÞΦ2ðk; zÞ
× expðik · rsf γ − ik · η0qzÞrx;sfdrsfdk: ð13Þ
Substituting Eqs. (12) and (13) into Eq. (11) and aver-aging over the ensemble of surface slope realizations,we obtain the normalized variance of the η0;x as
δη20x ¼ 2ZZ
GξðγkÞ�−jkj
�k2x þ k2
�~Φ3 ∂ ~Φ
∂k
− k2k2x
�~Φ3 ∂2 ~Φ
∂k2þ ~Φ2
�∂ ~Φ∂k
�2��
dk; ð14Þ
where GξðkÞ is the spectrum of surface elevation ξ,and ~Φ ¼ Φðk; zÞ=Φð0; zÞ and is the normalized opti-cal transfer function of a water layer of a thicknessz. Note that in the small-angle approximationΦð0; zÞ ¼ expð−azÞ, where a is the water’s optical ab-sorption coefficient. Equation (14) requires someclarification. It has been previously shown [4] thatthe amplitude fluctuations of the backscattered sig-nal caused by the effect of double focusing of light bysurface waves are related to the wave spectrum asfollows:
ðΔP1Þ2P1
2 ¼ ðqzÞ2ZZ
ð ~Φðk; zÞÞ4k4GξðγkÞdk: ð15Þ
This term is absent in Eq. (14) due to the fact that oneof the fluctuation components of the value ~P2 has ex-actly the same form and origin. Additionally, thisterm is fully correlated with the fluctuations of thevalue P1. The result is that, when determining therefracted angle, these fluctuations compensate foreach other. This implies that, in the absence of scat-tering, the variance of the estimate of the refractedangle using only one measurement is zero. In thiscase the MTF of a water layer does not depend onthe spatial frequency k because, in a nonscatteringmedium, the point-spread function, the spectrumof MTF, is a delta function. Note also that the fluctua-tion level described by Eq. (14) is anisotropic, i.e., itdepends on the orientation of the “wedge” directivitydiagram with respect to the surface’s anisotropicroughness. This seems to be quite natural. Note,however, that the effect of the fluctuation varianceanisotropy with wedge orientation is not fundamen-tal, since a decrease of the fluctuations in one of theorthogonal directions is accompanied by an increaseof fluctuations in the other direction. Of more impor-tance is the influence of the other parameters on thefluctuation level. These include the scattering coeffi-cient, the roughness intensity, and, obviously, thedepth of the layer from which the backscatteredsignal arrives. Hence, in making some specific esti-mates, we will assume the waves to be directionallyisotropic. Here the relative variances of fluctuationsin orthogonal projections of the vector ~η0 are equaland, according to Eq. (14), are described as
We make one more comment concerning the prop-erties of different terms in Eqs. (14) and (16). Waterscattering plays a dual role in the formation of fluc-tuation in our calculations. First, it gives rise to thefluctuations since, as mentioned above, in a nonscat-tering medium, fluctuations in the angle estimateare zero. The first term in Eqs. (14) and (16) is re-sponsible for the formation of these fluctuations atarbitrary sensed depths. The third term, which de-scribes the smoothing of the fluctuations due to scat-tering, has the opposite sign. The second term isproportional to the second derivative of the opticaltransfer function and has a more complicated char-acter. At small depths this term slightly increasesthe fluctuation level while, at larger depths, its con-tribution to the fluctuations changes sign and fluc-tuations decrease. This decrease is due to thenarrowing of the MTF with increasing depth.For calculations using Eq. (16) one should specify
the forms of the functions ~Φðk; zÞ describing the scat-tering properties of water and GξðkÞ describing thespatial spectrum of surface wave roughness. Accord-ing to [23], we assign the MTF of the water layer ~Φusing the relation
where μ is the phase function parameter, assumed tobe equal to 7 [24], and b is the water scattering coef-ficient. The spectrum of surface elevations is given bythe Pierson–Moskowitz formula [25] as
GξðkÞ ¼β0k4
exp�−0:74g2
k2V4w
�; ð18Þ
where k is a spatial frequency, Vw is the wind speedin m=s, g ¼ 9:81m=s2, and β0 ¼ 8:10−3 is an empiri-cal coefficient. To provide integrability and corre-spondence of the wave slope variance to the CoxandMunk observational data [26], the upper integra-tion limit in Eq. (16) is given as
k0 ¼ 16:3 expð0:63VwÞ=V2w: ð19Þ
In a strict sense, if the size of the illuminated spoton the surface is larger than the spatial scale definedby Eq. (19), the limits of integration should be chan-ged in accordance with the patch size. Though thisparameter has not been taken into account in state-ment of the problem [see Eq. (4)], the integration lim-it is evident.Calculations of possible errors in the wave slope
angle determination at different speeds of the near-surface wind are given as a function of the bottomdepth in Figs. 2 and 3. As is seen in these figures,the relative error has its maximum at some depth,which depends on the wind speed and the scatteringcoefficient.
4. Advantages and Limitations of the Method
It should be noted that measurements obtained fromsmall depths are not practically applicable for deter-mining the local sea slope angle and correcting thedistortions in imaging systems. This is due to contri-butions of the surface reflected signal and the influ-ence of finite wave heights. However, measurementsfrom large depths may be too weak, with a large errorcaused by their own noise. We do not estimate thisnoise here because its level depends greatly on theequipment employed, e.g., the source power, thequantum efficiency of the photodetector, etc. Whenthe equipment is specified, simple techniques to cal-culate noise are well known. Methods of noise reduc-tion are also clear. Therefore, an estimate of the localsurface slope at the laser beam entrance point inwater with a variation coefficient of from 0.1 to 0.2may prove to be quite realistic. Such an estimatemay serve as a basis for finding the limits of imagingsystems, allowing for corrections. This means that, inthe cases where the resolution of an imaging systemis mainly determined by the random refraction onthe surface with the scattering in the water layerplaying only a minor role, the width of the averagepoint-spread function can be decreased by 2 ordersof magnitude. Accordingly, the MTF of the imagingsystem with such a correction is much wider thanthe MTF determined by the surface slope varianceση2 alone, which has the form [2]
Θ ¼ expð−σ2ηq2z2bk2=2Þ; ð20Þ
where k is the spatial frequency and zb is the bottomor object depth. When we use our system to retrievethe correct (not a randomly shifted) target pointposition of the laser beam in the bottom or objectplane with a variance of estimated slope angle of�δη20, we may write for the corrected surface MTFthe following:
Fig. 2. Variance of the refraction angle estimate as a function ofdepth. Numbers near the curves show the near-water wind speed.The water-scattering coefficient is b ¼ 0:1=m.
Here, Φðk; zbÞ is the MTF of the water layer of thedepth zb. Note that the finite width of the light beamcan be taken into account by multiplying Eq. (20) bythe function Fsðk; kH þ kzb=mÞ. The gain in the reso-lution of the system due to the applied correctiontechnique is illustrated by the MTF curves in Figs. 4and 5. As indicated above, these estimates relate tothe case when the size of the illuminated patch on thesurface is less than the value of 2π=k0 determinedfrom Eq. (19). For the greater patch size, Eq. (22)should be transformed to the equation
Θ0 ¼ exp�−ðσ21;ηδη20 þ σ22;nÞq2z2bk2=2
�; ð23Þ
where σ21;η and σ22;η are the wave slope variances ap-propriated to the low-frequency spectral region (withthe scales, greater than the illuminated patch) and tothe high-frequency one. The long waves deflect thesensing beam as a whole and this effect can be cor-rected by use of the algorithm described above.The short waves lead to irreversible change of thebeam angular structure. Therefore, a gain in the sys-tem resolution due to using the proposed algorithmin this case is less than for a narrow light beam.Table 1 demonstrates how much the beam size af-fects the resolution increase. The table gives esti-mates of the width of MTF and MTF0 at levels 0.1and 0.01 for different wind speeds and two diametersof illuminating patch, d1 ≪ 2π=k0 and d2 ¼ 1m. Thewater scattering coefficient equals 0:11=m; the bot-tom depth is 20m. One can see that a rather large
increase of resolution can be obtained even for thewide beam.
As our analysis has shown, the method proposedfor determining the refraction angle of the light beamcan significantly upgrade imaging by airborne obser-vation systems. Though it requires complications inthe receiving equipment and the signal processingsystems, it seems that such complications are justi-fied due to the large improvement in the quality ofthe resulting images.
Finally we make several comments:
1. It should be kept in mind that the estimatesand conclusions made above are based on approxi-mately linear wave slopes. However, as shown in[27], the linear approximation yields an underesti-mate of the variance of lidar signal fluctuations forsmall sensed depths, while supplying a satisfactoryaccuracy for depths larger than 5–10m. At thesesame depths, one can neglect the correlation betweenthe functions Es and Er. Because of this the signalmodel for large depths and corresponding determina-tion of the beam refraction angle given above givesan accurate and realistic estimate.
Fig. 3. Same as in Fig. 2. The water-scattering coefficient isb ¼ 0:3=m.
Fig. 4. Logarithms of MTF, MTF0, and Φ as functions of the spa-tial frequency. This calculation is carried out for the followingparameter values: a water-scattering coefficient of b ¼ 0:1=m, avariance of surface wave slopes of σ2η ¼ 0:0542 according to thedata by Cox and Munk for a near-water wind speed V ¼ 10m=s,and a bottom depth zb ¼ 20m.
Table 1. Width of MTF and MTF0 at Levels 0.1 and 0.01
2. The equations and estimates given above arevalid when the lidar receiving angle 2ϑr satisfiesthe condition
2ϑr >2z
ffiffiffiffiffibz
p
μðH þ z=mÞ ; ð24Þ
which means that the light patch observed on thedepth z falls completely in the sensor’s viewing field.For example, if z < 20m, b < 1m−1, H > 100m, theconsidered method of the control over the laser beamrefraction angle can be realized by use of sensorswith receiving angle 2ϑr < 15°.3. We supposed that backscattering in water is
isotropic. Taking into account the backscatteringanisotropy leads to some change of the irradiancedistribution in the sensor focal plane, but will notchange the position of this distribution centroidand its determination accuracy by the consideredmethod.4. The estimates that we have made are with re-
ference to signal measurements from given depths.An increase in the received signal duration due to in-creasing the temporal gate of the receiving systemmay lead to a decrease in the variance of the angleestimate. However, one should remember that asmall increase of the gate duration is inefficient be-cause of the high correlation of the signals obtainedfrom closely spaced depths. At the same time, a sig-nificant increase in the gate duration results in com-plications of the average angle estimate. In this case,a relationship analogous to Eq. (9) has the form
where Δz is the depth “extent” of the gate. A prioriknowledge of the water-absorption coefficient isneeded.
5. The relative signal fluctuations in the sensorswere estimated on the assumption that the sensor’snoise (in particular, shot noise) is small as comparedwith the multiplicative signal fluctuations inducedby the focusing effect. The sensor noise may be takeninto account when applied to the method under con-sideration by replacement done in Eqs. (21) and (23):
δη20 → δη20;Σ ¼ δη20 δη20;shδη20 þ δη20;sh
: ð26Þ
Here, δη20;Σ is the relative variance of fluctuationswith consideration for the sensor noise and fluctua-tions induced by surface waves, δη20 is determinedfrom Eq. (16), and δη20;sh is the relative variance of themeasured angle fluctuations due to shot noise (in theabsence of focusing). It is evident that Eq. (25) isvalid on the assumption that the sensor noise andthe wave-induced fluctuations are uncorrelated.
6. As mentioned above, the estimates and calcu-lations presented have been carried out assumingthat wave elevations are neglected. In this approxi-mation, the waves play the role of a phase screen,leading only to random deviations of light rays re-fracted at the interface. It is useful to specify this ap-proximation, while taking into account the randomdispersion of photon paths. This estimate has beenmade in reference to bathymetric lidars’ characteris-tics [28]. However, in the problem discussed herein,special consideration is required.
7. The technique proposed for determining therefraction angle may not only be employed in ima-ging systems, but also used to upgrade bathymetricinformation in airborne oceanologic lidars. Addition-ally, this techniquemay be used to solve various ocea-nographic problems related to the study of thesurface wave characteristics.
We thank Gary Gilbert and Iosif Levin for manyuseful comments and generous editing of the manu-script. We thank our reviewers for many usefulcomments. The work is supported by the RussianFoundation for Basic Research (project 08-05-00252)and Office of Naval Research (ONR) (grantN000140610741) through CRDF (project RUG1-1619-NN-06).
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