Alexander A. Kokhanovsky Vol. 15, No. 11 /November 1998 /J. Opt. Soc. Am. A 2877

Reflection of light from semi-infinite turbid media

Alexander A. Kokhanovsky

Institute of Particle Technology and Environmental Engineering, Technical University of Clausthal, Leibnizstrasse19, D-38678 Clausthal-Zellerfeld, Germany, and Stepanov Institute of Physics, National Academy of Sciences

of Belarus, 68 F. Scarina Avenue, 220072 Minsk, Belarus

Received December 2, 1997; revised manuscript received July 10, 1998; accepted July 13, 1998

The simple equation for the spherical albedo of a semi-infinite turbid medium is obtained. The accuracy of theapproximation was studied with the numerical solution of the radiative transfer equation. The error issmaller than 10% for water clouds in the visible and the near infrared. © 1998 Optical Society of America[S0740-3232(98)00111-2]

OCIS codes: 290.7050, 120.5700, 030.5620.

Studies of light reflection from semi-infinite turbid mediaare important for many applications, including medicalimaging and cloud, snow, and foam optics. The sphericalalbedo of such media R can be obtained by integration ofthe reflection function r(m, m0 , c) (Ref. 1):

R 52p E

0

1

mdmE0

1

m0dm0E0

2p

r~m, m0 , c!dc, (1)

where m 5 cos n ; m0 5 cos n0 ; n and n0 are the observa-tion and the incidence angles, respectively; and c is theazimuth. The function r(m, m0 , c) is the solution of theradiative transfer equation.1 This solution depends onthe single scattering albedo v and the phase functionp(u), where u is the scattering angle. The value of v rep-resents the probability that a photon injected into a tur-bid medium will be scattered after interaction with a par-ticle. The function p(u) is the probability that a photonwill scatter in the direction specified by the scatteringangle u. According to the similarity principle,1 turbidmedia that have different phase functions p(u) but thesame value of the asymmetry parameter

g 512 E

0

p

p~u!cos u sin udu (2)

and the single scattering albedo v have approximatelythe same radiative characteristics. Thus in the firstcoarse approximation the value of R depends only on twoparameters, namely, v and g.

An approximate formula for the spherical albedo ofsemi-infinite turbid media was obtained by Perelmanet al.2 within the framework of the path integral formal-ism:

R 5 expF 2aS ln1v

1 2 gD G 1/2

, (3)

where, according to Perelman et al.,2 the value of a is afunction of g.

The purpose of this paper is to show that the value of ashould be constant and to compare Eq. (3) with exact ra-diative transfer calculations.

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The following exact asymptotical relation for thespherical albedo of a semi-infinite medium can beobtained from the radiative transfer theory as v → 1(Ref. 1):

R 5 1 2 4F 1 2 v

3~1 2 g !G1/2

. (4)

This relation is valid for any phase functions and valuesof g. It is possible to find the value of a in Eq. (3) by com-paring Eqs. (4) and (3) as v → 1. Indeed, it follows fromEq. (3) as v → 1 that

R 5 1 2 aS 1 2 v

1 2 g D 1/2

, (5)

where the asymptotical formula

limv→1

F lnS 1v D G 5 1 2 v (6)

is used. Thus one can see that the value of a does notdepend on g and that

a 54

A3. (7)

The value of a in the paper by Perelman et al.2 was ap-proximately 3.1 at g 5 0.908 and approximately 2.9 at g5 0.757. This value is nearly 30% larger than it shouldbe according to Eq. (7). It follows from Eqs. (3) and (7)that

R 5 expH 24F ln1v

3~1 2 g !G 1/2J . (8)

One can see that

limv→0

R~v! 5 0, limv→1

R~v! 5 1, (9)

as expected. The dependence R(v), calculated with Eq.(8) at g 5 0.85, is represented in Fig. 1. It is interestingthat at v . 0.9 (R . 0.2) a somewhat simpler formulafor calculation of the spherical albedo can be applied [seeEq. (6) and Fig. 1]:

1998 Optical Society of America

2878 J. Opt. Soc. Am. A/Vol. 15, No. 11 /November 1998 Alexander A. Kokhanovsky

R 5 expH 24F 1 2 v

3~1 2 g !G1/2J . (10)

Equation (10) was obtained by Rozenberg3 for a specialcase of weakly absorbing media. A comparison of the re-sults of the spherical-albedo calculation obtained with Eq.(8) and with a numerical solution of the radiative transferequation1 for semi-infinite water clouds with the gammaparticle size distribution

f~a ! 5 Ba6 exp~21.5a !, (11)

where B is the normalization constant @*0`f(a)da 5 1#, is

presented in Fig. 2. One can see that the accuracy of theapproximate Eq. (8) is high.

In conclusion, a simple approximation (8) was proposedto calculate the spherical albedo of turbid semi-infinitemedia. This approximation can be used to avoid complexnumerical calculations of the total reflectance R and ab-

Fig. 1. Dependence of the spherical albedo of a semi-infinite me-dium on the single scattering albedo at g 5 0.85, calculationedwith Eq. (8) (solid curve) and with Eq. (10) (dashed curve).

sorption A 5 1 2 R of radiation by semi-infinite thickturbid media.

ACKNOWLEDGMENTThis research was supported by the Alexander von Hum-boldt Foundation.

REFERENCES1. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer

through a Scattering Medium (Springer-Verlag, Berlin,1991).

2. L. T. Perelman, J. Wu, I. Itzkan, and M. S. Feld, ‘‘Photonmigration in turbid media using path integrals,’’ Phys. Rev.Lett. 72, 1341–1343 (1994).

3. G. V. Rozenberg, ‘‘Optical characteristics of thick weaklyabsorbing scattering layers,’’ Dokl. AN SSSR 145, 775–777(1962).

Fig. 2. Dependence of the spherical albedo of a semi-infinite wa-ter cloud with particle size distribution (11) on the wavelength,calculated with Eq. (8) (solid curve) and with the numerical so-lution of the radiative transfer equation (symbols).