JOSA COMMUNICATIONSCommunications are short papers. Appropriate material for this section includes reports of incidental research results,comments on papers previously published, and short descriptions of theoretical and experimental techniques.Communications are handled much the same as regular papers. Proofs are provided.

On the extinction of radiation by a homogeneousbut spatially correlated

random medium: comment

Anatoli Borovoi

Institute of Atmospheric Optics, Russian Academy of Sciences, Tomsk 634055, Russia

Received November 30, 2001; revised manuscript received March 19, 2002; accepted July 26, 2002

Some extinction laws for radiation transmitted through inhomogeneous random media were discussed by Ko-stinski [J. Opt. Soc. Am. A 18, 1929 (2001)] by means of a complicated use of concepts of statistical theory offluids. We show that these extinction laws are readily obtained in terms of classical probability theory. Thevalidity of exponential extinction laws for large observation distances (as compared with the size of inhomo-geneities of a medium) is proven and emphasized. It is shown that Kostinski’s results turn out to be appli-cable to small observation distances only, for which the concept of extinction law is hardly applicable. © 2002Optical Society of America

OCIS codes: 030.5620, 030.6600, 010.1290, 000.4930.

In a recent paper1 A. Kostinski considered several ex-tinction laws for radiation transmitted through randomscattering or absorbing media. According to the author,application of the concepts of the classical statisticaltheory of dense gases and liquids allowed him to obtaininsight into the extinction laws. As a result of sophisticalspeculations, he concluded that the well-known exponen-tial extinction law is valid for homogeneous scatteringmedia with statistically independent scatterers, whereasa number of power extinction laws appeared to be validfor spatially correlated scatterers. At the same time,there was no certainty regarding which of the power ex-tinction laws should be preferred.

The purpose of this communication is to attract atten-tion to the fact that the problem of extinction laws for ra-diation transmitted through random media is readilysolved without using the kinetic theory of dense gases.Following an earlier paper,2 here certain expressions forthe extinction laws are recalled along with their physicalmeanings and ranges of applicability, including the re-sults of Ref. 1 and their relevant interpretation.

As an introduction, it is expedient to outline this prob-lem through its relationships to other physical problems.In physics, propagation of various kinds of radiation inscattering or absorbing media is often described by the ra-diative transfer equation.3 This is an interdisciplinaryequation describing a number of phenomena in variousfields of physics. In particular, in optics this equation de-scribes propagation of light through interstellar dust,Earth’s atmospheric clouds, biological tissues, etc. Pas-sage of electrons, neutrons, and x-rays through matteralso obeys this equation. Even collisions of high-energyelementary particles inside heavy atomic nuclei followthis equation.4 Here, as a rule, radiation is treated as

1084-7529/2002/122517-04$15.00 ©

corpuscles, i.e., as a set of discrete photons, neutrons, etc.A scattering medium can be reasonably determined, bothby the properties of any scatterer and by scatterer concen-tration. Two examples of the scatterers are cloud waterdroplets for photons and atomic nuclei for neutrons.

The radiative transfer equation is based on the expo-nential law of radiation extinction also referred to as theBeer–Lambert law in optics. From the corpuscular pointof view, the exponential law means the probability thatone radiation particle will move along a straight-line seg-ment of length L without any collision with scatterers. Itis obvious that the collision probability is a product ofscatterer concentration c by extinction cross section s ofone scatterer. Assuming that the scatterers can appearon the particle path randomly, one obtains the exponen-tial extinction law,

T 5 exp~2t! 5 expF2sE0

L

c~l !dlG , (1)

which can also be derived from the trivial differentialequation for radiation intensity: dI/dl 5 2csI. In Eq.(1), T means the transmittance between two points r1 andr2 , L 5 ur2 2 r1u is the length of the straight-line seg-ment, and l corresponds to a point on the segment. Thequantity t is commonly referred to as the optical depth.For brevity, let us assume the scattering properties ofscatterers to be independent of the coordinates and thescatterer concentration c(r) to be a function of the three-dimensional coordinates.

Equation (1) describes only ‘‘direct,’’ or nonscattered,radiation, where the radiation particles did not undergoany collisions. For a given radiation source, this radia-tion is a simple convolution, with Eq. (1) being the convo-

2002 Optical Society of America

2518 J. Opt. Soc. Am. A/ Vol. 19, No. 12 /December 2002 Anatoli Borovoi

lution kernel. The total radiation including particle col-lisions of all orders is determined by a much morecomplicated radiative transfer equation. It is importantto note that under certain conditions, the direct and thescattered radiations can be differentiated experimentally.For instance, the different angular distributions of bothconstituents at a given point of space r can be used forthis purpose. Indeed, the angular distributions of par-ticles for direct radiation are often quite sharp in com-parison with those for scattered radiation.

In a number of cases, a scattering medium can be se-verely inhomogeneous, such as clouds in Earth’s atmo-sphere. In addition to the spatial variability, the scat-terer concentration c(r) may also vary in time in acomplicated way. Therefore we need the stochastic ra-diative transfer equation, where all coefficients becomerandom values owing to the random function c(r). Notean analogy of the problem with the well-known problem oflight propagation in a turbulent atmosphere, where thestochastic wave equation has been extensivelyinvestigated.5 In both cases, only a few quantities aver-aged over a statistical ensemble of a random medium canbe calculated either analytically or numerically.

Though the solution of the stochastic radiative transferequation is quite complicated (see, e.g., Ref. 2), we canreasonably restrict ourselves to only the direct radiationin random media. In this case, we have to solve a sim-pler problem of statistical averaging of the transmittance[Eq. (1)]. In general, the statistically averaged transmit-tance will not decay exponentially with distance L. It isjust a new extinction law, which will be discussed in thiscommunication.

To make this overview complete, it is worthwhile tomention a stricter approach to the problem, which isbased on the theory of multiple scattering of waves.4,5

According this approach, the exponential law of Eq. (1) isobtained by a statistical averaging of interference pat-terns of multiple scattered waves. Within the wave-scattering theory, Eq. (1) corresponds to the so-called co-herent scattering.

If scatterers are statistically independent of each otherbut are interacting with an external field, such an en-semble will be characterized by a predetermined concen-tration c(r). Under certain restrictions, Eq. (1) for suchstatistical ensembles can be obtained by straightforwardanalytical calculations of the coherent scattering. Nowlet us assume this external field to be a random value.As a result, we get Eq. (1) with random concentrationc(r). In addition, some inhomogeneities can be causedby possible interaction among the scatterers. These, too,may be included in the random concentration c(r). Thisis also the case for the incoherent part of the radiation in-tensity. Thus we conclude that both the wave approachand the previous heuristic corpuscular approach give usthe same stochastic radiative transfer equation under cer-tain restrictions. A discussion of these restrictions is be-yond the scope of this communication.

Thus our final goal is to find an average of the trans-mittance T over a statistical ensemble of realizations ofthe function c(r). To begin with, it is instructive to pointout a simple inequality that is a consequence of the gen-eral Jensen inequality known in mathematics,

^exp~2t!& > exp~2^t&!, (2)

where the brackets ^...& mean the statistical average.In spite of mathematical triviality, Eq. (2) gives two

physically important conclusions: (A) Any scattering me-dium with a fluctuating concentration c(r) is, on average,more transparent along any ray than is the same mediumwith an average concentration ^c(r)&. (B) A layer havinga finite longitudinal size but infinite transversality exhib-its lowest longitudinal transmittance in the case of uni-form distribution of scatterers inside this layer. Any re-distribution of the scatterers in space forming transversalinhomogeneities will result in an increase in the longitu-dinal transmittance of the layer.

It is worthwhile to point out physical meanings of sta-tistical averages. In general, the average concentration^c& can depend on the coordinates g1(r) 5 ^c(r)&. So themedium can exhibit certain regular inhomogeneities. Inaddition, there are some random inhomogeneities thatare described by the correlation function g2(r1 , r2)5 ^c(r1)c(r2)& 2 ^c(r1)&^c(r2)&. A typical distance a5 ur2 2 r1u, where the function g2 is essentially nonzero,has a physical meaning of the size of the random inhomo-geneities. Correlation functions of higher orders gn de-scribe internal structure of the random inhomogeneities.Therefore the functions gn vanish at the same distance arelative to the difference variables, say, r2 2 r1 , r32 r1 , etc. A correlation function of concentration gn canbe readily related to the probability of locationsr1 ,r2 ,..., rn for n scatterers in the ensemble, as is wellknown in statistical mechanics. Within the statisticalmechanics, homogeneous media, where g1 5 const.,g2(ur2 2 r1u, etc., are common. In contrast, in the prob-lem, for example, of light scattering by clouds, regularmedium inhomogeneities g1(r) are natural. Let us em-phasize that any statistical averaging is usually appli-cable not only to a set of realizations of c(r) but also to afixed realization of the concentration c(r), owing to theergodic assumption.

Averaging of Eq. (1) over a statistical ensemble of c(r)is expressed simply by the probability theory. Let us usea well-known equation defining cumulants and cumulant(correlation) functions; then we directly obtain

^T& 5 ^exp~2t!& 5 exp (n51

`

~21 !nkn /n!

5 exp (n51

`

~2s!nE gndl1dl2 ...dln /n!, (3)

where kn are the cumulants of the optical depth; in par-ticular, k1 5 ^t& is the average optical depth, k2 5 ^t 2&2 ^t&2 is its variance, etc., and gn(r1 , r2 ,..., rn) are thespatial correlation functions for c(r).

In the case of statistically independent scatterers, allthe correlation functions gn are equal to zero except thefirst function g1(r1) 5 ^c(r1)&. Here we arrive at theconventional exponential extinction law:

^T& 5 exp~2^t&! 5 expF2sE0

L

^c~l !&dlG . (4)

Anatoli Borovoi Vol. 19, No. 12 /December 2002/ J. Opt. Soc. Am. A 2519

Taking into account Eq. (2), we see that Eq. (3) describesquantitatively the increase in transmittance due to ran-dom inhomogeneities. Now, it is reasonable to consider aviolation of the exponential law of Eq. (4) by the randominhomogeneities in two cases of either small a ! L orlarge a @ L inhomogeneities because they yield physi-cally different results.

It is obvious that for small random inhomogeneities, in-tegration over the difference variables l2 2 l1 , l32 l1 ,... in the last integral of Eq. (3) will result in cer-tain constants bn owing to the vanishing of the correla-tion functions beyond the distance a. Denoting the re-siduary local variable bn(l1), we obtain an exponentialextinction law again:

^T& 5 expF2sE0

L

c̃~l !dlG , (5)

where the local ‘‘effective’’ concentration is represented bythe series

c̃~l ! 5 ^c~l !& 1 (n52

`

~2s!n21bn~l !/n!. (6)

According to Eq. (2), the local effective concentration isless than the average one c̃(r) < ^c(r)& that provides anincreased transmittance owing to random inhomogene-ities.

In general, the local effective concentration of Eq. (6)can be found only by numerical calculations. Further-more, the correlation functions of higher orders gn are of-ten very poorly investigated. Nevertheless, there are afew situations in which the averaged transmittance canbe obtained explicitly. First, random media often happento be Gaussian where the correlation functions gn arezero for n > 3. Here, the sum of Eq. (6) is reduced toonly the first term, and only one coefficient b2 has to becalculated. Second, a number of theoretical models ofrandom media allow one to solve the problem analytically.Naturally, an analytical expression for the average trans-mission can also be interpreted as the sum of the series ofEq. (6). As an example, consider a model of scatteringrandom media consisting of a set of discrete inhomogene-ities

c~r! 5 (j51

N

cj~r!, (7)

where the jth inhomogeneity is a domain filled with theformer scatterers of extinction cross section s and scat-terer concentration cj(r). Each inhomogeneity is deter-mined by the location of its center and by its internalstructure, such as size, shape, and orientation. In gen-eral, the inhomogeneities are allowed to overlap eachother. For a certain volume V, the number of inhomoge-neities N is a random value. The value ^C& 5 ^N&/V isthe average concentration of the inhomogeneities that istaken as a constant, for simplicity. For a given inhomo-geneity, the integral

sj 5 EE H expF2sE2`

`

cj~x, y, z !dzG 2 1J dxdy (8)

has the physical meaning of an extinction cross section forthis inhomogeneity, where the z axis passes through bothof the points r1 and r2 of Eq. (1).

In the case of an ensemble of independent discrete in-homogeneities of Eq. (7), where the number N obeys thePoisson distribution p(N) 5 @exp(2^N&)#^N&N/N!, the av-erage transmittance of the medium is readily calculatedanalytically. The simplest way of calculation is to substi-tute Eq. (7) into Eq. (1) and then to average the productobtained by using the statistical independence of cofac-tors. For simplicity, let us assume that the internal pa-rameters of the inhomogeneities are independent of theirlocations and that ^C& 5 const. As a result, it yields thewell-known trivial exponential law

^T& 5 exp^2s^C&L&, (9)

where the former scatterer parameters are replaced onlyby the similar inhomogeneity parameters. In particular,s stands for the extinction cross section of Eq. (8) that isaveraged over the statistical ensemble. Note that thecondition of small inhomogeneities a ! L plays an essen-tial role in derivation of Eq. (9), which provides the valid-ity of the infinite limits in the internal integral of Eq. (8).Otherwise, the concept of the extinction cross sectioncould not be applied.

The examples described above prove that the exponen-tial extinction law is true in the case of small (a ! L) in-dependent scattering or absorbing obstacles. The natureof the obstacles can be different. For example, these ob-stacles for light may be either separate cloud droplets,blobs of droplets, or separate cumulus clouds. This con-clusion is also in conformity with the mathematical pointof view. In mathematics, the exponential extinction lawis associated with the Poisson random process, wherepoint obstacles appear randomly along a ray.

Now consider the opposite case of large random inho-mogeneities a @ L. This case is mathematically simpler.Indeed, in the initial Eq. (1), we can ignore a variation ofthe function c(r) along the observation distance L. Thusthe function c(r) can be treated as a random number cwith a probability distribution p(c) corresponding to theone-point distribution of the random field c(r) in the vi-cinity of the observation path. As a result, the expres-sion for average transmittance is exactly reduced to theLaplace transform of the probability density p(c):

^T& 5 E0

`

p~c !exp~2csL !dc 5 Lp~c !~ sL !. (10)

Thus practically any decreasing analytical function of dis-tance L can serve as an extinction law for the case of largerandom inhomogeneities.

Meanwhile, the physical interpretations of either Eq.(5) or Eq. (10) are a little different. It is easy to imaginean increasing distance L in an inhomogeneous mediumwith small inhomogeneities for both a given realization ofthe medium c(r) and for a statistical ensemble of theserealizations. For these inhomogeneities, an exponentialdecrease of radiation with distance L is common and ob-vious. Let us now consider a given medium realizationc(r) with large inhomogeneities. Here, we are not per-mitted to increase the distance L because the observation

2520 J. Opt. Soc. Am. A/ Vol. 19, No. 12 /December 2002 Anatoli Borovoi

path must be inserted into one inhomogeneity. Thus Eq.(10) describes the situations when a medium is fluctuat-ing as a single whole without changing the concentrationc(r) along the observation distance. For example, if asource and a receiver of light are mounted on an airfoil ofan aircraft at a distance, say, of 0.5 m from each other,and the aircraft is crossing inhomogeneous cloudiness,Eq. (10) just describes an average detector signal. Butthis signal can hardly be interpreted as an extinction lawin the cloudiness.

The results described above allow us to give a physicalinterpretation and to assess a range of validity for the re-sults obtained in Ref. 1. We believe that the author ofRef. 1 eventually ignored the spatial correlations along anobservation path in the last section containing his origi-nal results, even though the whole preceding section wasdevoted to a discussion of the spatial correlations in ran-dom media. This makes the results of Ref. 1 applicableto the case of large medium inhomogeneities a @ L only.In particular, his Eq. (11) is, in our notation, the followingprobability distribution for the whole concentration calong an observation path:

p~c ! 5 ^c&21 exp~2c/^c&!. (11)

Substitution of Eq. (11) into Eq. (10) immediately yieldshis power-extinction law [his Eq. (15)]:

^T& 5 1/~1 1 ^c&sL !. (12)

But in the case of large inhomogeneities, as noted above,one cannot interpret any function of distance L includingthe power function of Eq. (12) as an extinction law forrandom media.

Thus the common exponential extinction law proves tobe valid for random scattering media in the case of smallinhomogeneities a ! L. In this case, it is possible eitherto obtain an effective scatterer concentration that is lessthan the average scatterer concentration or to determinecertain discrete inhomogeneities of concentration as inEq. (9). In the intermediate case of a ' L, the exponen-tial law will naturally be violated, according to Eq. (3).And, finally, for large inhomogeneities a @ L, practicallyany decreasing function can serve as the average trans-mittance ^T&, but this function can hardly be interpretedas an extinction law.

Anatoli G. Borovoi can be reached by e-mail atborovoi@iao.ru.

REFERENCES1. A. B. Kostinski, ‘‘On the extinction of radiation by a homo-

geneous but spatially correlated random medium,’’ J. Opt.Soc. Am. A 18, 1929–1933 (2001).

2. A. G. Borovoi, ‘‘Radiative transfer through inhomogeneousmedia,’’ Dokl. Akad. Nauk SSSR 276, 1374–1378 (1984); inRussian.

3. S. Chandrasekar, Radiative Transfer (Oxford U. Press, Lon-don, 1950).

4. M. L. Goldberger and K. M. Watson, Collision Theory(Wiley, New York, 1964).

5. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978), Vols. 1 and 2.