Light pulse propagation along the pathatmosphere–rough-surface–sea water

Alexander G. LuchininInstitute of Applied Physics, Russian Academy of Science, 46, Ulyanov Street, Nizhny,

Novgorod, Russia, 603950 (luch@hydro.appl.sci‑nnov.ru)

Received 18 June 2010; revised 9 August 2010; accepted 10 August 2010;posted 11 August 2010 (Doc. ID 130365); published 17 September 2010

The influence of surface waves andmultiple scattering in water on the parameters of light pulses from anairborne source is studied. The contributions of variousmechanisms to variations in delay of pulse and itsvariance are estimated. It is shown that waves make the main contribution to these values at smalldepths. With strong wind, the allowance for waves is important for small receiving apertures in the wholepractically important depth range. For large receiving apertures or/and large widths of light beams in-cident on the surface, the determining factor is multiple scattering of light in water. © 2010 OpticalSociety of AmericaOCIS codes: 010.4450, 010.5620, 010.3310, 290.4210, 290.5880.

The study of nonstationary light fields in sea water isof considerable interest both for the theory of radia-tion transfer in scatteringmedia anddirectly for someapplied problems. Nevertheless, papers devoted tothis problem are extremely few. The majority of themdeal with simplified problems; in particular, they con-sider infinitelywide light beams. Exceptions areRefs.[1–8], in which analytic representations of the Greenfunction of the radiative transfer equation were ob-tained in approximations that are outside the limitsof the small-angle approximation and take into ac-count diffusion of photons along multipaths. In thispaper, an attempt is made to calculate the integralcharacteristics of a pulse signal with allowance forthe effects of longitudinal diffusion of photons andthe influence of surface waves. The base representa-tion is that of the Green function of the radiativetransfer equation in the simplified self-similar ap-proximation obtained by Dolin and Levin [4,9].

1. Basic Equations

First we specify the statement of the problem. Let asource located at the height H above average level ofthe sea surface radiate a short light pulse of power

approximated by the delta function:

P0ðtÞ ¼ W0δðtÞ; ð1Þ

where W0 is the initial energy of the pulse. Thespatial-angular distribution of the source radianceis described by the function

ISðr; nÞ ¼ W0DSðr; nÞδðtÞ; ð2Þ

where the vector r determines the position of thepoint in the horizontal plane, while the vector n isthe projection of the unit vector of the ray directiononto this plane. The function DSðr; nÞ satisfies the“small-angle” normalization condition:

ZZ ZZDSðr; nÞdrdn ¼ 1: ð3Þ

In Eq. (3) and below, the integration is performed ininfinite limits.

The sea surface is assigned by the random realiza-tion of elevations ξðrÞ, and its time dependence isinessential. Therefore, the radiance distribution inci-dent on the rough surface can be given as

Iðr; n; tÞ ¼ W0DSðrþ nH0; nÞδðt −H0=cÞ; ð4Þ0003-6935/10/275059-08$15.00/0© 2010 Optical Society of America

20 September 2010 / Vol. 49, No. 27 / APPLIED OPTICS 5059

whereH0 ¼ H − ξðrÞ, and c is the velocity of light. Thedeviation of rays from the initial direction after cross-ing the air–water interface is determined by thewater refraction index m and the surface slopeηðrÞ, which, in the linear approximation, coincideswith the surface elevation gradient ηðrÞ ¼ −∇ξðrÞ.For ray refraction at the interface, the vector n ofthe ray direction is transformed by the formula

n → mn − ðm − 1Þη: ð5Þ

The appropriate variation of the light field radianceis described as

Iðr; nÞ → m2TIðr;mn − ðm − 1ÞηÞ: ð6Þ

Thus, the radiance distribution under the air–waterinterface, with allowance for Eqs. (4)–(6), is yieldedby

Iðr; n; tÞ ¼ m2TW0DSðrþ ðmn − ðm − 1ÞηðrÞÞH0;mn

− ðm − 1ÞηðrÞÞδðt −H0=cÞ; ð7Þ

where T is the transmittance of the interface. Thisrelation can be interpreted as the boundary conditionfor the radiative transfer equation in the water layer.By using this equation and an analytic representa-tion of the Green function of this equation, one canobtain a complete description of the light fieldin water.

Let a radiation receiver with an isotropic direc-tional diagram and an aperture of transparency func-tion DRðrÞ be located at the depth z. Then the powerof the light signal entering the photodetector of thereceiver is

Pðz; tÞ ¼ZZ ZZ

DRðrÞIðz; n; r; tÞdrdn

¼ZZ

DRðrÞEðz; r; tÞdr; ð8Þ

where Iðz; n; r; tÞ is the spatial-angular distribution ofradiance in the plane z at the instant t, and the cor-responding distribution of irradiance of the plane z is

Eðz; r; tÞ ¼ZZ

Iðz; n; r; tÞdn: ð9Þ

The aim of the present paper is to study the beha-vior of the first two temporal moments of the valuePðz; tÞ as a function of the depth z for various valuesof inherent optical properties and of a rough-surfacestate. Since the function Eðz; r; tÞ is random due tothe random character of surface waves and theappropriate boundary conditions, the temporal mo-ments of the pulse-response power are also random.Random realizations of the mean propagation timeand the variance (width) of the pulse response aredetermined by the known equations

tðzÞ ¼RtPðz; tÞdtRPðz; tÞdt ; ð10Þ

ΔtðzÞ2 ¼R ðt − tðzÞÞ2Pðz; tÞdtR

Pðz; tÞdt : ð11Þ

Another representation of these parameters, whichis more convenient for calculations, has the form

tðzÞ ¼ −imc∂ðlnPωðzÞÞ

dω

����ω¼0

; ð12Þ

ΔtðzÞ2 ¼ −m2

c2∂2ðlnPωðzÞÞ

∂ω2

����ω¼0

; ð13Þ

where PωðzÞ is the Fourier transform of the functionPðz; tÞ:

PωðzÞ ¼ 2π−1Z

Pðz; tÞ expð−iωtÞdt:

Now we specify the function PωðzÞ with allowancefor random boundary conditions and effects of multi-ple scattering and absorption in water. According toEq. (7), the boundary conditions for the radianceFourier transform can be written as

Iωðr; nÞ ¼ m2TW0DSðrþ ðmn − ðm − 1ÞηðrÞÞH0;mn

− ðm − 1ÞηðrÞÞ expðH0iω=cÞ: ð14Þ

An equation for random realization of irradiancedistribution at the plane z incident on the surfacecan be deduced as follows. The radiance distributionunder the interface can be considered as a set of ele-mentary point unidirectional sources. Each of thesesources has its own initial coordinates and propaga-tion direction assigned by Eq. (14). We find the radi-ance distributions at the fixed depth z for thesesources and make convolution of these distributionsof all the sources. The obtained distribution of radi-ance from the set of elementary sources is integratedwith respect to the angular variables. As a result, wehave the equation for random realization of the hor-izontal distribution of the irradiance density at thedepth z:

Eωðz; rÞ ¼W0Tm2

ð2πÞ3Z

…

ZexpðH 0iω=cÞFSðk1; k1H0

þ k2z0=mÞΦωðk2; z0Þ × expðik1r1 − ik2ðr1 − rÞ− ik2ηðr1Þqz0Þdk1dk2dr1; ð15Þ

where z0 ¼ zþ ξðrÞ þ zqηðrÞ2=2, q ¼ ðm − 1Þ=m, and

FSðk; pÞ ¼ ð2πÞ−4ZZ ZZ

DSðr; nÞ expð−ikr − ipnÞdrdn;

5060 APPLIED OPTICS / Vol. 49, No. 27 / 20 September 2010

while Φωðk; zÞ is the Fourier spectrum of the irradi-ance distribution in a cross section of an initiallyunidirectional light beam. More precisely, it is thespectral (over frequency ω) density of this spectrumat the depth z for vertical incidence of the beam on aflat surface. The change of variables z → z0 is causedby effective variation of the path length for each raydue to wave height at the place of incidence and itsdeviation from vertical. An analogous equation of therandom irradiance distribution for a light pulse(without allowance for diffusion of photon multi-paths) was obtained in Ref. [10].

In accordance with Eqs. (8) and (15), the spectraldensity of the signal power can be found as follows:

PωðzÞ ¼ ð2πÞ−1W0Tm2

Z…

ZexpðH0iω=cÞFRð−k2Þ

×FSðk1;k1H0 þk2z0=mÞΦωðk2;z0Þ×expðiðk1 −k2Þr1 − ik2ηðr1Þqz0Þdk1dk2dr1; ð16Þ

where FRðkÞ ¼ ð2πÞ−2 R RDRðrÞ expð−ikrÞdr. Equa-

tion (16) serves as a basis for calculation of the sta-tistical characteristics of the pulse signal, includingits temporal moments, averaged over an ensemble ofwave realizations.

2. Statistically Averaged Beam Characteristics

It follows fromEq. (16) that the wave influence on thelight field and received signal power can be reducedto three factors. The first two factors result from var-iation of the path length of photons caused by eleva-tions and slopes of waves. Because of obviousinequalities

ξðrÞ=z ≪ 1; qηðrÞ2=2 ≪ 1; ð17Þthe allowance for these two factors can be made byseries expansion of Eq. (16) in terms of these para-meters, keeping terms of the second order of small-ness and further ensemble averaging over surfacewave realizations. The third factor of the wave influ-ence is associated with effective additional angularbroadening of the light field refracted at the interfaceand it cannot be, in general, considered small.

Now we pass over to calculation of average statisti-cal values hPωðzÞi, htðzÞi, and hΔtðzÞ2i (angle parenth-eses mean ensemble averaging over rough-surfacerealizations). Assuming that the average variationof the light field due to the first two factors is small,we can write

htðzÞi ¼ tðzÞ0 þ htðzÞ1i; ð18Þ

hΔtðzÞ2i ¼ ΔtðzÞ20 þ hΔtðzÞ21i; ð19Þ

where

t0ðzÞ ¼ −imc∂ðlnP0

ωðzÞÞ∂ω

����ω¼0

; ð20Þ

ΔtðzÞ20 ¼ −m2

c2∂2ðlnP0

ωðzÞÞ∂ω2

����ω¼0

; ð21Þ

ht1ðzÞi ¼ −imc

�1

P0ω

∂hΔPωi∂ω

����ω¼0

−hΔPωiðP0

ωÞ2∂P0

ω∂ω

����ω¼0

�;

ð22Þ

hΔtðzÞ21i¼−m2

c2

�1

P0ω

∂2hΔPωi∂ω2

����ω¼0

þ2hΔPωiðP0

ωÞ3�∂P0

ω∂ω

����ω¼0

�2

−2

ðP0ωÞ2

∂P0ω

∂ω

����ω¼0

∂hΔPωi∂ω

����ω¼0

−hΔPωiðP0

ωÞ2∂2P0

ω∂ω2

����ω¼0

�; ð23Þ

hP0ωðzÞi ¼ 2πW0Tm2 expðHiω=cÞ

Z…

ZFRð−kÞ

× FSðk; kðH þ z=mÞÞΦωðk; zÞΘð•Þdk; ð24Þ

Θð•Þ ¼ Θðσηxqzkx; σηyqzkyÞ: ð25Þ

hP0ωðzÞi is the average statistical spectral density of

signal power calculated byneglecting the first two fac-tors, while hΔPωi is its average variation, which takesthe factors into account.Θð•Þ is the single-point char-acteristic functionofwaveslopes; σ2ηx,σ2ηy are slopevar-iances in twomutuallyperpendicular directions.Foraflat surface (in the absence of waves), Θð•Þ≡ 1and hΔPωi≡ 0.

By setting Θð•Þ≡ 1 in Eq. (24) and substituting itinto Eqs. (20) and (21), we can calculate regular va-lues of delay tðzÞ0 and variance �ΔtðzÞ20 of a light pulsefor known characteristics of source, receiver, andwater layer. The characteristics are assigned by thefunction Φωðk; zÞ. An analogous operation can be ea-sily made with allowance for a finite value of slopevariance. In this case, rough surface assimilates toan equivalent phase screen, while the obtained dif-ference in the sought values enables one to estimatethe contribution of the surface to the variation of sig-nal power and its temporal characteristics related tothe third factor.

These values can be calculated by using the repre-sentation of the functionΦωðk; zÞ for stationary fieldsand substituting differentiation with respect to ω bydifferentiation with respect to the absorption coeffi-cient a. This can be done because, in the radiativetransfer equation for the spectral density of radiance,the absorption coefficient and the parameter iω=c areadditive. Note that this substitution into Eqs. (20)–(23) is incorrect, because the random length of the airpart of the propagation path is not taken into ac-count. We represent the additional delay of the pulsecaused by the first two factors as the sum

20 September 2010 / Vol. 49, No. 27 / APPLIED OPTICS 5061

ht1ðzÞi ¼ ht1ξðzÞi þ ht1ηðzÞi; ð26Þ

where the indices ξ, η indicate the mechanism ofadditional delay occurrence due to the influence ofelevations and slopes of the surface, respectively,on the effective length of the photon path. To simplifyrecording, we assume waves to be isotropic with theslope variance σ2η . Then, omitting the index ω hereand below for brevity and having made necessaryoperations, we can write

t0ðzÞ ¼ −mc∂ðlnP0Þ

∂a; ð27Þ

ht1ξðzÞi ¼mσ2ξ

2cP0ω¼0

�1

P0

∂P0

∂a∂2P0

∂z2−

∂3P0

∂z2∂a−2m

∂P0

∂z

�;

ð28Þ

ht1ðzÞiη ¼mzq2σ2η2cP0

�1

P0

∂P0

∂a∂P0

∂z−∂2P0

∂z∂a

�; ð29Þ

where P0 ¼ hP0ω¼0i.

Analogous operations made with respect to thesecond moment of pulse signal yield the equations

ΔtðzÞ20 ¼ m2

c2∂2ðlnP0Þ

∂a2 ; ð30Þ

hΔtðzÞ21ξi ¼m2σ2ξc22P0

�∂4P0

∂a2∂z2þ 2

ðP0Þ2�∂P0

∂a

�2 ∂2P0

∂z2

−2

P0

∂P0

∂a∂3P0

∂a∂z2−

1

P0

∂2P0

∂z2∂2P0

∂a2

−4

mP0

∂P0

∂a∂P0

∂zþ 4m

∂2P0

∂a∂zþ 2P0

m2

�; ð31Þ

hΔtðzÞ21ηi ¼m2q2σ2ηzc22P0

�∂3P0

∂a2∂zþ 2

ðP0Þ2�∂P0

∂a

�2 ∂P0

∂z

−1

P0

∂P0

∂z∂2P0

∂a2 −2

P0

∂2P0

∂a∂z∂P0

∂a

�: ð32Þ

3. Model of Source and Propagation Path of LightPulse

To carry out further calculations, the functions de-scribing light source, path characteristics, and recei-ver should be specified. For simplicity, we assumethat the initial spatial-angular distribution of radia-tion is described by the Gaussian function. Accordingto the normalization condition, the spectrum func-tion DSðr; nÞ has the form

FSðk; pÞ ¼ ð2πÞ−4 expð−α2k2=2 − β2p2=2Þ; ð33Þ

where α and β are the parameters characterizing theinitial aperture of the beam and its angular diver-gence, respectively. The receiver parameters areassigned in a similar way. When deriving basic equa-tions, it is assumed that the signal enters the recei-ver with an isotropic directional diagram. We alsoassume that the transmittance of its aperture is de-scribed by the Gaussian function and the aperturecenter is on the beam axis. Then

FRðkÞ ¼ ð2πÞ−2πR2 expð−R2k2=4Þ; ð34Þ

where R is the effective radius of the receivingaperture.

The following assumptions are made with respectto surface wave properties. We assume that wavesare isotropic andnormalwith the corresponding slopecharacteristic function:

Θðk; zÞ ¼ expð−ðqzσηkÞ2=4Þ; ð35Þ

where σ2η ¼ σ2ηx þ σ2ηy and, in accordance with the Coxand Munk data, are assigned by the equation σ2η ¼0:003þ 0:00516V , where V is the near-surface windvelocity in meters per second. The surface elevationvariance is determined using the wave descriptionby the Pierson–Moskowitz spectrum [11], modifiedso that the calculated slope variance corresponds tothe Cox and Munk data [12]. This modification con-sists inmultiplying the Pierson–Moskowitz spectrumby the “cutting-off” function expð−k2k−20 Þ, where k isthe wave number. The modified spectrum has theform

GξðkÞ ¼ β0=2πk4 exp�−0:74g2

k2V4 −k2

k20

�; ð36Þ

where β0 ¼ 0:008, g is the acceleration of gravity, andV is the wind speed. In this case, the parameter ν de-pends on the wind speed according to the equation

ν ¼ 10ð−4:13þ 1:23VÞβ−10 : ð37Þ

The energy carrying part of the wave spectrum doesnot essentially change.

An equation for the irradiance distribution in thecross section of an initially narrow beam in the re-fined self-similar approximation of the radiativetransfer equation is obtained in Ref. [9]. The appro-priate expression for its spatial spectrum is

Φðk; zÞ ¼ ð1þ μσ1=k lnðkz=μþ ð1þ ðkz=μÞ2Þ1=2ÞÞ× expð−az − bzÞ þQðzÞ× expð−k2=SðzÞ − a1zÞ; ð38Þ

where μ is the parameter of the sharp part of the scat-tering phase function assigned by the function

XðγÞ ¼ 2μγ−1 expð−μγÞ: ð39Þ

5062 APPLIED OPTICS / Vol. 49, No. 27 / 20 September 2010

The first two terms in Eq. (38) correspond to non-scattered and single scattered radiation components;the third term in the form of the Gaussian functiondescribes the diffusion component formed by multi-ple scattering. According to Ref. [9], the parametersof this function are

QðzÞ ¼ ðcoshðνÞÞ−1 − ð1þ b1zÞ expð−b1zÞ; ð40Þ

SðzÞ ¼ zðν − tanhðνÞÞððcoshðνÞÞ−1 − expð−b1zÞÞ2aνððcosh νÞ−1 − ð1þ b1zÞ expð−b1zÞÞ

; ð41Þ

where ν ¼ ðb1aÞ1=2z=μ, b1 ¼ bð1 − φ45Þ, and a1 ¼aþ bφ45 are “small-angle” scattering and absorptioncoefficients; a and b are absorption and scatteringcoefficients; and φ45 is a fraction of light scatteredin the angular range 45° ≤ γ ≤ 180° [9]. We assumethat absorption and scattering coefficients and scat-tering phase function parameters are related bythe relations [13] a ¼ 0:059bþ 0:051, μ ¼ ð2bÞ1=2ð0:0356bþ 0:0007Þ−1=2, while the parameter φ45 isexpressed through the absorption coefficient by therelation [14] φ45 ¼ −0:072þ 6:71μ−2.

It should be stressed that the choice of the inherentoptical properties (IOP) model, as well as of the wavespectrum model, is not principal; hence, appropriatecalculations can be carried out for other models.

4. Some Calculation Results

The given equations enable one to obtain the neces-sary quantitative estimates of the considered effects.Since the number of significant parameters in theproblem is rather large, we confine ourselves to aminimal set of combinations to evaluate the influ-ence of different factors on temporal characteristicsof the pulse signal. In particular, in calculations (withonly a few exceptions), we use a single wind speedvalue V ¼ 15 m=s, assuming that, at weaker winds,the surface influence decreases. In addition, we con-fine ourselves to two values of the water scatteringcoefficient, b ¼ 0:1 m−1 and b ¼ 0:3 m−1. We also as-sume that the source is located at a height of 400 mabove the average surface level and that its angulardivergence is β ¼ 0:001 rad. Before calculating thetemporal parameters of pulse, we will show thatthe contribution of the angular spread of rays to sta-tistically averaged signal power cannot be, in gener-al, considered small. To estimate this effect, weintroduce the ratio of received signal power at flatinterface to its average value on rough surfaceQ ¼ ~P0=P0, where ~P0 is the signal power calculatedby Eq. (24) for Θ ¼ 1 and ω ¼ 0. Dependence of thisvalue on depth for various values of effective radiusof the receiving aperture (the entrance pupil of theoptical receiving system) is shown in Fig. 1. Thecurves in this figure demonstrate that, in some depthrange, the contribution of surface to power variation(its decrease) is large, especially for small apertures.

This is due to the decrease of part of nonscatteredand single scattered radiation components as a re-sult of initial angular beam spread. In fact, the valueof this effect also depends on water transparency. Inparticular, for more turbid water, the maximum ofthe value Q becomes smaller and moves towardsmaller depths.

Let us estimate the temporal characteristics. Forthe sake of clarity, delay and broadening of the pulseis reduced to the variation of the effective path lengthand the pulse broadening along it. For this purpose,we introduce new designations for the sought para-meters in Eqs. (27)–(32):

l ¼ h�tic=m; l0 ¼ t0ðzÞc=m; l1ξ ¼ ht1ξic=m;

l1η ¼ ht1ηic=m; Δl2 ¼ hΔt2iðc=mÞ2;Δl20 ¼ Δt20ðc=mÞ2; Δl21ξ ¼ hΔt21ξiðc=mÞ2;Δl21η ¼ hΔt21ηiðc=mÞ2: ð42Þ

In the calculations, we assume that the water refrac-tion index is m ¼ 1:33. First we calculate the addi-tional delay and the pulse broadening caused bywave contributions to the spread of optical paths.The value l1ξ versus depth for various effective radiiof receiving aperture is shown in Fig. 2. Note severalpeculiarities of the obtained result. The first pecu-liarity is due to the sign of the calculated value.The initial value of the additive to the total pulse de-lay can be estimated without complex calculations.In fact, photons transmitted through wave crestsmake a smaller contribution to the total value ofthe signal. The same photons enter the receiver laterthan photons transmitted through the wave troughsbecause of differential velocity in air and in water. Asa result, the average “initial” advance of the pulsecompared to a flat interface can be estimated by asimple formula:

Fig. 1. Parameter Q as a function of depth. Digits close to thecurves here and in other figures indicate the effective radius ofthe area of the entrance pupil of the receiving aperture R in me-ters. The water scattering coefficient b ¼ 0:1 m−1.

20 September 2010 / Vol. 49, No. 27 / APPLIED OPTICS 5063

l1ξða1z ≪ 1Þ ¼ −qa1σ2ξ : ð43Þ

The horizontal line in Fig. 2 corresponds to this va-lue. This estimate is, obviously, valid only for amedium without scattering. Scattering leads to a re-latively rapid decrease of the average delay of thepulse, as depth increases. This effect is, probably, re-lated to growth of a “fast” photon contribution due tosingle scattering in water and extending close to thebeam axis. For further growth of depth, the delay be-gins to increase due to multiple scattering andasymptotically tends to the value described by theformula

l1ξða1z ≫ 1Þ ¼ −σ2ξ ðqþ ð2μÞ−1ðb1=a1Þ1=2Þ× ða1 þ ðb1a1Þ1=2=μÞ: ð44Þ

An estimate of Eq. (44) can be obtained easily byusing asymptotic representation of the functionΦðk; zÞ and signal power at large depths:

P0ða1z ≫ 1Þ∼ expð−ða1 þ ðb1a1Þ1=2=μÞzÞ: ð45Þ

Actually, this means that, even in deep water, the in-fluence of surface elevations on effective pulse delayremains. As for the spectral representation of thefield, this effect is equivalent to variation of the in-itial phase formed at field depth (or, in other words,of the phase of the excitation coefficient in a deep-water regime). Naturally, the values of the describedeffects strongly depend on IOP. In particular, theasymptotic structure of the light field in more turbidwater begins at smaller depths. However, it will beshown below that the relative contribution of the con-sidered mechanism to the total delay is larger forclear water. Now we estimate the influence of photondiffusion along traveled multipaths, caused by thedeviation of rays refracted on the surface. Direct cal-culation of the value l1η shows that, in a wide depthrange, it can be calculated by the formula l1η ¼σ2ηq2z=2. Slight deviation from linear dependence

Fig. 3. Total variation of delay due to waves. The water scatteringcoefficient is b ¼ 0:1 m−1.

Fig. 4. Total pulse delay (with allowance for the medium). Thewater scattering coefficient is b ¼ 0:1 m−1. Dashed curves arecalculated for a flat air–water interface.

Fig. 2. Additional delay caused by surface elevations. The waterscattering coefficient is b ¼ 0:1 m−1.

Fig. 5. The same as in Fig. 4. The water scattering coefficient isb ¼ 0:3 m−1.

5064 APPLIED OPTICS / Vol. 49, No. 27 / 20 September 2010

on depth begins at depths larger than 100 m, whereadditions caused by multiple scattering begin toaccumulate.

As for the influence of the initial spread of the an-gular beam structure on pulse delay, calculationsshow that it manifests itself at comparatively largedepths, at which the delay of pulse increases dueto a decrease of a fraction of nonscattered and singlescattered radiation. This effect operates in a finitedepth interval and is relatively strong for clear water.The total delay variation as a result of all three me-chanisms of wave influence is shown in Fig. 3. It isshown in Figs. 4 and 5 how important it is to takeinto account the wave influence on signal delay incomplete consideration of all the mechanisms lead-ing to photon diffusion along travel paths. It is seenin these figures that waves affect the delay of pulsemost strongly at small receiving apertures and hightransparency of water.

Now we consider the wave influence on pulselength increase. The initial variation of pulse var-iance does not depend on water properties and is de-

termined by the formula Δl21ξða1z ≪ 1Þ ¼ q2σ2ξ . Asdepth grows, the addition to variance caused bywaves varies in a complicated way due to the jointaction of various mechanisms. Total variation ofthe addition to variance caused by wave influenceis shown in Fig. 6. At some depths, the contributionof the initial angular spread of rays on the surfaceleads to a decrease of this addition and even to itsnegative value for high transparency of water. Thiseffect is, evidently, caused by a decrease of the con-tribution of nonscattered and single scattered radia-tion components, which results in some shortening ofthe pulse precisely due to some “consumption” of theleading edge of the pulse. The total value of the var-iance, with allowance for waves and photon diffusionalong travel paths as a result of multiple scattering,is shown in Figs. 7 and 8. The results given aboverefer to well-developed waves and were calculatedfor the wind speed 15 m=s. It is natural that, atweaker winds, the surface wave influence is weaker.Its value can be determined by the curves in Fig. 9,

Fig. 6. Additive to pulse variance caused by waves. The waterscattering coefficient is b ¼ 0:1 m−1.

Fig. 7. Root-mean-square pulse lengthffiffiffiffiffiffiffiffiΔl2

pas a function of

depth. Dashed curves are calculated for a flat interface. The waterscattering coefficient is b ¼ 0:1 m−1.

Fig. 8. Same as in Fig. 7. The water scattering coefficient isb ¼ 0:3 m−1.

Fig. 9. Delay of pulse and its effective width as a function ofdepth. V ¼ 5 m=s, b ¼ 0:1 m−1, R ¼ 0:1 m.

20 September 2010 / Vol. 49, No. 27 / APPLIED OPTICS 5065

which display the dependence of the values l, l0, Δl,and Δl0 on depth for the near-water wind speedV ¼ 5 m=s.

5. Conclusions

The performed studies enable one to draw some con-clusions on the influence of waves on the pulse signalpropagating from airborne source in water.

1. At the initial depth interval, waves play thedetermining role in the formation of the first twopulse moments, i.e., its delay and broadening. In thiscase, the wave effects manifest themselves moststrongly in clear water.

2. At small depths, an important role is played bythe contribution of finite wave height. At meandepths and for small radii of receiving apertures,the angular spread of rays caused by refraction onsurface waves is significant.

3. At large depths and for large area of thereceiving apertures, the wave effect is practicallyinsignificant.

4. The considered phenomena are invariant withrespect to the value R2

Σ ¼ R2 þ α2 þ β2ðH þ z=mÞ2.Therefore, the given calculation results can be usedfor estimation of the phenomena for other combina-tions of initial parameters of beam and receiver. Inparticular, equal pulse delay and variance areformed for wide beams and small receiving aper-tures, and for narrow beams and large receivingapertures.

5. The results given above in Section 4refer to thecase where the receiver is on the beam axis. It may beexpected that, when the receiver is displaced, thevalues of the considered effects vary significantly, be-cause the ratio of contributions of various radiationcomponents varies. According to preliminary calcula-tions, this can be clearly observed for small values ofthe parameter R2

Σ.6. The technique developed in this paper enables

one to estimate only statistically averaged values ofdelay and pulse variance. Meanwhile, estimation ofthe second statistical moments, in particular, var-iance of the mean time of the pulse response, i.e.,the value hðhtðzÞi − tðzÞÞ2i, made similarly to that ofRef. [7] for the echo signal of pulse lidar, may be im-portant for practical applications. However, this re-quires a special study falling outside the limits ofthis paper.

Appendix A: Definition of Basic Variables

r is coordinate of point in the horizontal plane;z is depth measured from average surface level;H is height of source location above average sur-

face level;n is projection of the unit vector determining ray

direction onto the horizontal plane;k is vector conjugate to the Fourier vector r (has the

spatial frequency dimensionality);

p is vector conjugate to the Fourier vector n (hasthe angular frequency dimensionality);

ω is cyclic frequency conjugate to the Fouriertime t;

ξðrÞ is elevation of surface above its average level;and

ηðrÞ is surface slope, i.e., projection of local normalto the surface onto the horizontal plane.

The author is thankful to L. S. Dolin and I. M.Levin for valuable discussions and advice. The workis supported by the Russian Foundation of BasicResearch (RFBR) (project 08-05-00252).

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