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tions. Academic Press, San Diego (CA), 1980.

Mathematical Simulation of the Radiative

Transfer in Statistically Inhomogeneous Clouds

Evgueni I. Kassianov

Paciﬁc Northwest National Laboratory, 902 Battelle Boulevard, P.O. Box 999,

Richland, WA

Evgueni.Kassianov@pnl.gov

Abstract. The solar radiation transport through broken clouds can be treated

as photon transport through stochastic media. The stochastic radiative transfer

equation and a new statistically inhomogeneous Markovian model are used to de-

rive analytical equations for the ensemble-averaged intensity. The computational

method for solving these equations is introduced, and their accuracy and robust-

ness are discussed. Validation tests show good predictive performance for the mean

radiative properties.

1 Introduction

Transport theory, also called radiative transfer (RT) theory, has been ex-

tensively developed over a century. The equation of RT (or linear transport

equation) applies successfully to numerous areas, such as atmospheric RT,

neutron diﬀusion and sound propagation [e.g., CZ67, Lio02]. The current

book provides some new and important applications of this equation. The

latter has parameters that are deterministic functions of space and time, thus

the equation is deterministic. Sometimes the lack of a complete description

of these parameters renders their statistical treatment: although one does

not know their values at each space point and time instant, one does know

the values they can assume and probabilities of these values. Thus, the RT

theory can be formulated in a statistical setting and the corresponding RT

equation can be viewed as stochastic [e.g., Pom91].

The stochastic RT equation is applicable for diﬀerent problems: for ex-

ample, the neutron transport in a boiling water reactor, where liquid water

and vapor are considered as stochastic mixture. Broken clouds are an other

example of such a potential application. Since the location, geometry, and

optical properties of each individual cloud are not known in the determin-

istic fashion, the cloud properties can be considered as random variables.

A radiation ﬁeld transformed by such a cloud ﬁeld also becomes random.

This fact dictates the necessity to use statistical methods to study the cloud-

radiation interaction. The ultimate goal of a statistical approach is to suggest

a relatively simple and practical useful treatment of the stochastic radiative

transfer problem and to establish the relationship between the statistical pa-

rameters of clouds and radiation.

142 E.I. Kassianov

Here we introduce the statistical treatment of solar radiation in multi-

layer broken clouds. This treatment is based on the stochastic RT equation

(Sect. 2) and a new statistically inhomogeneous Markovian model (Sect. 3).

The latter has been used to derive a closed system of equations for the mean

solar radiance (Sect. 4). The computational method for their solution is also

described in Sect. 4. The accuracy and robustness of the obtained equations

are discussed in Sect. 5.

2 Stochastic RT Equation

We will assume that broken clouds occupy a certain layer Λ: hb ≤ z ≤ ht.

Within Λ, the random monochromatic radiance (or intensity) I (r, ω)atthe

point r =(x, y, z) in direction ω =(a, b, c) satisﬁes the three-dimensional

(3D) stochastic transfer equation

ω ∇I (r, ω)+σ (r) κ (r) I (r, ω)

= ω

(r) σ (r) κ (r)



4π

g (r, ω, ω



) I (r, ω, ω



) dω



, (1)

with the boundary conditions

I(r

, ω)=I(x, y, hb, ω)=I

↑

, ω),c>0(2)

I(r

, ω)=I(x, y, ht, ω)=I

↓

, ω),c<0

where a, b, c are the projections of the unit vector ω on the axes X,Y ,Z

(Cartesian coordinate system), respectively, I

↑

, ω)andI

↓

, ω)arethe

radiances of the external sources at the boundary of layer Λ,z

= hb (the

cloud base altitude) and z

= ht (the cloud top altitude) σ(r) is the extinc-

tion coeﬃcient, ω

(r) is the single scattering albedo, and g(r, ω, ω



)isthe

scattering phase function. Here σ(r), ω

(r)andg(r, ω, ω



)arenonrandom

functions. In contrast to the deterministic transfer equation, the stochastic

one contains the random indicator ﬁeld κ(r), κ(r) = 1 inside clouds and

κ(r) = 0 outside the clouds. In other words, the broken cloud ﬁeld may

be considered a mixture of two atmospheric components (clouds and clear

air), with ﬂuctuating geometrical parameters. In the limiting case, where

randomness vanishes (κ(r) is a nonrandom function), (1) describes the 3D

deterministic radiative transfer. The statistics of the optical parameters of

broken clouds are determined completely by the statistical characteristics of

the cloud ﬁeld, κ(r).

Construction of a physical model of κ(r) is a complex problem, the so-

lution of which can be based on results of a cloud-resolving model and/or

a large eddy simulation model. These models provide 3D cloud ﬁelds which

mimic real clouds, thus, the simulated cloud ﬁelds can be a good source for

cloud statistics [e.g., OK03; KAM03]. The construction of a physical model

Radiative Transfer in Statistically Inhomogeneous Clouds 143

has not yet been solved, and therefore researchers are forced to use some

mathematical models of κ(r). The latter should be selected from the follow-

ing circumstances: (i) the possibility for determining the input parameters of

the model from observations, and (ii) the possibility for deriving relatively

simple analytical equations, which links the ensemble-averaged intensity with

statistics of κ(r). Below we consider one such model.

3 Statistically Inhomogeneous Model

The broken clouds can be represented as a Markovian mixture of cloudy

and noncloudy segments [e.g., Tit90; MBP93]. In this case, random indica-

tor ﬁeld κ(r) is described completely by unconditional κ(r) = P {k(r)=1}

and conditional V (r

, r

)=P {κ(r

)=1|κ(r

)=1} probabilities of the cloud

presence. Here and in the following, the angular brackets will be used for en-

semble averages over κ(r) realizations. Recall that the conditional probability

of the event {κ(r

)=1}, given that the event {κ(r

)=1} has occurred, is

P {κ(r

)=1|κ(r

)=1} =

P {κ(r

)=1,κ(r

)=1}

P {κ(r

)=1}

, (3)

where P {κ(r

)=1,κ(r

)=1} is the joint probability that both events

{κ(r

)=1} and {κ(r

)=1} have occurred, and P {k(r

)=1} > 0. For the

statistically homogeneous ﬁelds, the unconditional probability κ(r) does

not depend on the vertical coordinate, and the conditional one V (r

, r

)

depends only on the distance between points r

and r

for a ﬁxed direction

ω =(r

− r

)/|r

− r

| [e.g., Tit90, ZT95; MBP93].

Real clouds have strong horizontal and vertical variability. To extend the

Markovian approach [e.g., Tit90; MBP93] to multi-layer (or statistically in-

homogeneous) broken clouds, one has to specify a statistical relationship

between cloud layers. It was demonstrated that (i) the overlap of clouds at

two levels tends to fall rapidly as their vertical separation is increased and

(ii) the degree of overlap as a function of level separation can be described by

a simple inverse exponential expression [e.g., HI00; OK03]. In our approach,

we assume that the statistical relationship between two adjusted layers is

described by inverse-exponential expression. We represent broken clouds as

a set of correlated cloud layers: each layer is homogeneous in vertical but

inhomogeneous in horizontal dimensions. Such a representation allows one to

use methods previously developed for a single layer of broken clouds.

To generalize the Markovian approach [e.g., Tit90; MBP93] to multi-

layer broken clouds, we suggested a statistically inhomogeneous model of

κ(r) [Kas03]. For this model, κ(r) can vary strongly with altitude and the

conditional probability is a function of the relative positions of points r

and

. As an illustration, some examples are given below:

if the point r

and r

belong to the same kth layer, then

144 E.I. Kassianov

, r

)=(1− p

)exp(−A

− r

|)+p

(4)

where p

= κ(r

) = κ(r

) is cloud fraction in the kth layer, and A

parameter; if the point r

and r

belong to diﬀerent adjacent layers, namely

kth and mth layers, then

k,m

, r

)=exp(−A

∗

− r

|) {V

, r

∗

) − p

} + p

, (5)

where r

∗

= r

+ ω(z

∗

− z

)/c, ω =(a, b, c)=(r

− r

)/|r

− r

For the upward direction (c>0),m = k +1,A

∗

= A

and z

∗

equals

the top altitude of the kth layer, z

∗

= z

k+1

; for the downward direction

(c<0), m = k −1, A

∗

= A

and z

∗

equals the base altitude of the kth layer

∗

= z

k−1

. Parameters A

and A

determine the statistical relationship

between kth layer and its upper (k +1) and lower (k − 1) adjacent layers,

respectively. Note, all these parameters A

, A

and A

depend on both

the 3D cloud structure and the positions of points r

and r

.Bychanging

the values of these parameters, one can describe diﬀerent combinations of

maximum and random cloud overlaps [Kas03]. This sketched ﬂexibility of

the inhomogeneous model is its appealing feature. Also, all input parameters

of this model can be estimated from observations [KAM03, KAK05].

4 Ensemble Averaged Radiance

The statistically inhomogeneous model of broken clouds and the stochas-

tic transfer equation were used to derive approximated equations for the

mean solar radiance [Kas03]. It was assumed that for each kth layer the

domain-averaged optical properties are constant (piecewise constant approx-

imation), e.g., the extinction coeﬃcient σ(r)=σ(z)=σ

, the single scat-

tering albedo ω

(r)=ω

(z)=ω

0,k

, and the scattering phase function

g(r, ω, ω



)=g(z,ω, ω



)=g

(ω, ω



). Also it was assumed that a parallel

unit ﬂux of solar radiation is incident on the upper boundary of a given

cloud ﬁeld in direction ω

⊕

. To get the absolute values, one should multiply

the calculated radiative properties by the spectral solar constant weighted

with cos(ξ

⊕

), where ξ

⊕

is the solar zenith angle. The equations were ob-

tained without considering the aerosol-molecular atmosphere and underlying

surface; the latter can be added quite easily.

The equation for the mean solar radiance has the form

I(z, ω) =

|c|



(ξ)φ(z, ξ)dξ



4π

g(ξ,ω, ω



)f(ξ,ω



)dω



+ j(z,ω)δ(ω −ω

⊕

)(6)

where E

=(hb, z)ifc>0, and E

=(z,ht)ifc<0, ht and hb are the top

height and the base height of cloud ﬁeld, respectively; j(z,ω) is the mean

Radiative Transfer in Statistically Inhomogeneous Clouds 145

direct (unscattered) radiance, and f(z,ω)=σ(r)κ(r)I(r, ω) is the mean

collision density, δ(.) is Dirac’s delta function.

An integral equation for the mean collision density can be written as

f(x)=



k(x, x



)f(x



)dx



+Ψ(x)(7)

k(x, x



(z)g(z, ω, ω



)η(r, r



)

2π|r − r





r − r



|r − r



− ω



(8)

Ψ(x)=σ(z)p(z)v(z,ω)δ(ω −ω

⊕

)(9)

where X is the phase space of coordinates and directions, x =(r, ω). All

functions φ(r, r



), η(r, r



)andv(z, ω)=κ(r)j(r, ω)/p(z) are deﬁned by the

recurrent expressions [Kas03].

The closed system of (6)–(7) can be solved by using any appropriate nu-

merical methods or analytic techniques (e.g., the spherical harmonic method).

Also, the Monte Carlo method can be applied. For kernel (8) in the space

we have K≤ω

≤ 1, while for a bounded medium K

 < 1, which

ensures convergence of the von Neumann series of (7)

f(x)=

∞



n=0

Ψ , (10)

where the operator K

is deﬁned by the formula

Ψ](x)=



···



Ψ(x

)k(x

, x

) ···k(x

n−1

, x

)dx

···dx

n−1

. (11)

The Monte Carlo tracking of source photons that have not undergone

a collision provides an estimation of Ψ

(the initial collision density), the

continued tracking of photons that have undergone one collision provides an

estimation of K

, and so on. Equation (10) signiﬁes the existence, unique-

ness, and positiveness of the solution of (7), and the possibility for using the

Monte Carlo method to estimate the linear functionals J



f(x)h(x)dx.

According to (6), I(z, ω) is such a functional. The latter can be estimated

from the general theory of solution of integral equations by the Monte Carlo

method [e.g., MMN80]. To calculate radiative ﬂuxes, another Monte Carlo

method can be used. In this simulation technique, the photon trajectory

modeling is made in correspondence with the initial Ψ(x) and transitional

k(x, x



)/ω

densities of integral equation (7), while the radiative properties

were estimated in accordance with their physical contents [e.g., MMN80].

For instance, the upward ﬂux at the altitude z = z

is estimated from the

mean number of photon crossings of the plane z = z

in directions satis-

fying the inequality c>0. We used the second Monte Carlo algorithm for

solving the obtained equations (6), (7) and for evaluating their accuracy and

robustness [KAM03, KAK05].

146 E.I. Kassianov

5 Validation

As was outlined in the previous sections, the statistically inhomogeneous

model has relatively few input parameters, which describe only the bulk geo-

metrical statistics of the 3D broken cloud ﬁeld. Thus the question arises: how

accurately can equations derived on the basis of the statistically inhomoge-

neous model represent the mean radiative properties (e.g., mean ﬂuxes, mean

absorption) of the 3D broken cloud ﬁelds? We started the validation with the

direct radiation and demonstrated that the general analytical equations for

the direct radiance (derived by us) are equivalent to those obtained by others

for the three limiting cases: broken ﬁeld with maximum overlap, broken ﬁeld

with random overlap, and overcast vertically inhomogeneous ﬁeld [KAM03].

To evaluate the accuracy and robustness of equations for the diﬀuse ra-

diation, the following steps were taken [KAM03,KAK05]. First, we obtained

the mean radiative properties exactly by applying 3D radiative transfer cal-

culations directly to a given full 3D cloud structure. The calculated radiative

properties were considered as a reference. Note that an exhaustive survey of

the large number of methods for solving the 1D deterministic transfer equa-

tion is given in several monographs and textbooks [e.g., Lio02]. For 3D RT,

several approaches were suggested [e.g., Cah05 and bibliography therein].

Second, for a given 3D cloud ﬁeld, we calculated the bulk cloud statistics,

which were served as input data for the statistically inhomogeneous model.

Next, we estimated the mean radiative properties by applying the approxi-

mated equations (6), (7). The obtained radiative properties were considered

as approximations of true ones. Finally, we compared the mean radiative

properties obtained by using the independent exact method (references) with

ones obtained for the statistically inhomogeneous model (approximations).

The accuracy of the obtained equations was evaluated by comparing the

ensemble-averaged radiative properties that were calculated for 3D cloud

Markovian ﬁelds. These ﬁelds were provided by the stochastic Boolean model

[KAM03]. The robustness of these equations was estimated by comparing

the domain-averaged radiative properties obtained for diﬀerent 2D/3D cloud

ﬁelds: the 3D cloud ﬁelds were provided by large-eddy simulation model and

satellite and surface retrievals [KAM03], whereas 2D cloud ﬁelds were ob-

tained from radar observations [KAK05]. Figure 1 shows an example of such

cloud ﬁelds. It was shown that the approximated equations could provide

reasonable accuracy (∼15%) for both the ensemble-averaged and domain-

averaged radiative properties. Also, it was demonstrated that the angular

distribution histograms and the photon path length distributions of the mean

albedo and transmittance, which have been obtained by exact and approxi-

mated methods, agree qualitatively and quantitatively.

Radiative Transfer in Statistically Inhomogeneous Clouds 147

∆x = 200 m

∆z = 150 m

x, km

24.0 25.5 24.0 25.5 24.0 25.5 24.0 25.5

∆x

= 10 m

∆z = 150 m

x, km

∆x = 200 m

∆z

= 30 m

x, km

24 -- 30

18 -- 24

13 -- 18

6.8 -- 13

1.0 -- 6.8

∆x

= 10 m

∆z

= 30 m

x, km

z, km

0.9

1.8

(d)(c)(b)(a)

Fig. 1. Cross section (the horizontal and vertical dimensions) of the extinction

coeﬃcient at (a) high resolution and (b,c,d) degraded resolution in (b)thex-

direction, (c) the z-direction, and (d) both x- and z-directions. These 2D cloud

(x- and z-directions) ﬁelds were obtained from ground-based cloud radar observa-

tions [KAK05]

6 Summary

The deterministic RT theory has a long history, during which the simplest

1D radiation models have evolved to multi-dimensional ones. However, many

physical phenomena can be treated in a statistical fashion. For example,

the RT transfer through broken clouds can be considered as photon ﬂow

through the stochastic mixture of clouds and clear atmosphere. The ensemble-

averaged radiative properties (e.g., mean radiance and ﬂuxes) can be obtained

after numerical or analytical averaging of the stochastic RT equation over the

ensemble of realizations of clouds. Since the numerical averaging requires sig-

niﬁcant computer time, it is advisable to derive approximate solutions of the

problem. Such solutions may be obtained by using the analytical averaging of

the stochastic RT equation and an appropriate statistical cloud model. The

number of such models is quite limited.

A statistically homogeneous Markovian model was successfully used to

derive ensemble-averaging solutions analytically [e.g., Pom91; Tit90]. For the

statistically homogeneous ﬁelds, the unconditional probability of the cloud

presence does not depend on the vertical coordinate, and the conditional

probability depends only on the distance between two separated space points.

By using the assumption that broken clouds can be represented as a bi-

nary mixture with Markovian statistics, mathematically rigorous methods

148 E.I. Kassianov

for radiative calculations have been developed. However, all of these methods

are limited for a single layer of broken clouds.

We extended the Markovian approach to multi-layer broken clouds [Kas03,

KAM03]. In our approach, we assume that the statistical relationship between

two adjusted layers is described by inverse exponential expression. Such as-

sumption has some observation background [e.g., HI00; OK03]. To describe

such a statistical relationship, the statistically inhomogeneous Markovian

model was suggested [Kas03]. For statistically inhomogeneous ﬁelds, the un-

conditional probability of the cloud presence can vary strongly with altitude,

and the conditional probability is a function of the relative positions of two

separated space points.

The new statistically inhomogeneous Markovian model and the stochastic

RT equation were used to derive analytically the ensemble-averaged solutions

for the direct and diﬀuse intensity of solar radiation. It was demonstrated

that in the limiting cases the equations for the unscattered radiation yield

mathematically correct results. We applied the Monte Carlo method to solve

equations for the diﬀuse radiation. The accuracy and robustness of these

equations has been estimated using a numerical simulation method and dif-

ferent cloud ﬁelds. Validation tests show good predictive performance for the

mean radiative properties [KAM03, KAK05].

Acknowledgments

This work was supported by the U.S. Department of Energy’s Oﬃce of Bi-

ological and Environmental Research as part of the Atmospheric Radiation

Measurement (ARM) Program. It is my pleasure to express gratitude to

Professor Georgii Titov for having introduced me to the exciting ﬁeld of sto-

chastic RT and for his continued support through many years.

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