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Transport Theory for Optical Oceanography 161

the eﬀects of any Dirac-delta-like boundary conditions and internal sources.

Analytically-based post processing is used to compute the radiance in direc-

tions other than at the speciﬁed collocation points. To date the method has

been used only for idealized atmospheric problems and no eﬀort has been

made to incorporate refractive eﬀects at an air-water interface.

4 Computing Challenges

Because of the relatively simple geometry for most optical oceanography

problems, computations to date largely have been performed with the one-

dimensional transfer equation, which makes these problems relatively simpler

than the complicated ones arising in other transfer applications. However,

three-dimensional optical oceanography problems arise if the:

• Incident sunlight is not uniform over the each pixel of the water surface

(e.g., caused by a partial cloud cover)

• Bottom is not parallel to the water surface

• Water constituents are not horizontally uniform within a layer

Such problems can be analyzed with Monte Carlo techniques [MS03], al-

though deterministic multi-dimensional techniques developed for solving

transport problems for other applications also can be used provided adjust-

ments are made to incorporate the eﬀects of the index of refraction at the

air-water interface and the statistical nature of surface waves.

To better simulate ocean optical radiances, a better characterization of

the water inherent optical properties is needed. The computational techniques

discussed here for solving forward problems, when combined with methods

for solving inverse problems, presently are being investigated for use in ob-

taining water properties [FSS98,FES01,LC02,LCA02]. The inate variability

of any case 2 ocean water, however, introduces uncertainties when attempt-

ing to extrapolate the results to other such waters. Depending on where the

measurements are taken, possible non-plane-parallel geometry (sea surface

waves and wave focusing) eﬀects also can complicate the analysis. Perhaps

for this reason the recent emphasis in funded optical oceanography research

has shifted from the development of more sophisticated computer programs

to more extensive experimental activities.

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Perturbation Technique in 3D Cloud Optics:

Theory and Results

Igor N. Polonsky

, Anthony B. Davis

and Michael A. Box

Los Alamos National Laboratory, Space and Remote Sensing Sciences Group

(ISR-2), Los Alamos, NM 87545, USA

polonsky@lanl.gov

School of Physics, Univ of New South Wales, Sydney, NSW, 2052 Australia

m.box@unsw.edu.au

1 Introduction

It is well known that generally to simulate accurately radiative transfer

through a realistic cloudy atmosphere one should use numerical approaches

such as Monte Carlo [12], or SHDOM [3]. However, it is usually required too

much time to make a simulation which is inconvenient when just we need

an answer on a simple question like how signiﬁcant the 3D eﬀects are for a

given problem. The perturbation method is what comes to mind ﬁrst if we

need to go further into modelling of the radiative transfer through cloud at-

mosphere starting from the simplest framework of one dimensional radiative

transfer [2, 8].

Recently, two perturbation approaches have been used. One is somewhat

orthodox [14] and based on assuming that some term in the radiative transfer

equation is small enough to be considered as a small parameter to construct a

perturbation series [4,5,9–11]. A diﬀerent type of perturbation approach may

be formulated on the basis of the joint consideration of both the direct and

corresponding adjoint problem [1,7,13,15,18]. This approach can be derived

using a variational principle [17] allowing one to obtain a required solution

practically without eﬀort.

The goal of this paper is to formulate a variational principle to derive the

perturbation approach speciﬁcally for the problem of cloud optics. We will

also demonstrate how it can be used to explain some eﬀects in cloud optics.

2 Deﬁnition of the Problem

Let us consider radiation propagation through a cloud which has a shape of

a slab and is illuminated by a steady, uniform, and collimated beam (e.g.,

sunlight). We introduce a Cartesian coordinate system with the origin on

the upper slab surface with the z-axis directed toward the inner normal.

The direction is deﬁned by the Euler polar, θ and azimuth φ angles. The

radiance I(r,vn)atthepointr in the direction n can be calculated using the

framework of the radiative transfer equation (RTE) [e.g, 6]

166 I.N. Polonsky et al.

n · I(r, n)+σ

(r)I(r, n)=

(r)

4π



4π

P (r, n, n



)I(r, n



)dn



+ S(r, n) . (1)

Here n and r are the vectors which deﬁne the direction and position, respec-

tively, I(r, n) is the radiance at the point r in the direction n, σ

(r)andσ

(r)

are the extinction and scattering coeﬃcients, respectively, and P (r, n, n



)is

the phase function normalized as

4π



4π

P (r, n, n



)dn



S(r, n) is the source function which in the case of the sunbeam has the form

S(r, n)=S

δ(n −n

)δ(z − ε) .

Here S

is the Sun ﬂux at the top of the cloud, and ε is inﬁnitesimal small and

has been introduced for convenience. Certainly, (1) has to be complimented

with the boundary conditions

I(r, n)|

z=0

=0,µ>0 , (2)

I(r, n)|

z=H



µ>0

A(r, n, n



z=H

I(r, n



z=H

|dn



,µ<0 . (3)

Here A(r, n, n



) describes the reﬂection properties of the underlying surface

if one exists.

3 Variational Principe

to Derive the Radiative Transfer Equation

We may assume that the most measurements in cloud optics can be describe

by integration of the radiance with some receiver function R(r, n)

P =



R(r, n)I(r, n)dndr (4)

where Ξ denotes the region of interest in the position-direction space. Ac-

cording [17] we may introduce a functional

J =



I(r, n)LI(r, n)dndr −





R(r, n)I(r, n)+

I(r, n)S(r, n)



dndr



z=0,µ>0

I(r, n)I(r, n)dndr

−



z=H,µ<0

I(r, n)

⎡

⎣

I(r, n) −



µ>0

A(r, n, n



)I(r, n



)dn



⎤

⎦

dndr . (5)

Perturbation Technique in 3D Cloud Optics: Theory and Results 167

Here to simplify notation the operator

L = n ·  + σ

(r) −

(r)

4π



4π



P (r, n, n



)⊗ , (6)

has been introduced. Note that the notation ⊗ is used to indicate that the

ﬁnal term is an integral operator, not merely a deﬁnite integral. This func-

tional has a clear property that the variation of it over

I(r, n) results in RTE

(1) with boundary condition (2). The variation of the functional over I(r, n)

results in the equation for

I(r, n)

−n ·

I(r, n)+σ

(r)

I(r, n)=

(r)

4π



4π

P (r, n , n



)

I(r, n



)dn



+ R(r, n)(7)

with boundary conditions

I(r, n)|

z=0

=0,µ<0, (8)

I(r, n)|

z=H



µ<0

A(r, n



, n)|

z=H

I(r, n



z=H

|dn



,µ>0 . (9)

This equation constitutes the adjoint formulation of our problem. The ad-

joint solution has a strict physical meaning. As can be noticed if we reverse

direction in (7–8) we obtain a RTE which is similar to (1–2). This means that

the adjoint formulation of a given problem tracks photon backward starting

from the receiver and ending up at the source.

4 Perturbation

If we substitute either (1–2) or (7–8) into (5) shows that at the minimum the

functional value is the same as our measured signal (4)

min

= −



R(r, n)I(r, n)dndr = −



S(r, n)

I(r, n)dndr = −P.

This fact provides us with possibility to estimate how small variations of the

problem parameters aﬀect the signal. Expanding our functional into series

over small variations of all functions in the neighborhood of its minimum,

keeping only the ﬁrst order terms, and taking into account (1–2, 7–8), we

obtain that

168 I.N. Polonsky et al.

∆J = −∆P =



I(r, n)∆LI(r, n)dndr

−





∆R(r, n)I(r, n)+

I(r, n)∆S(r, n)



dndr



z=H,µ<0

I(r, n)



µ>0

∆A(r, n, n



)I(r, n



)dn



dndr . (10)

where

∆L = ∆σ

(r) −

∆σ

(r)

4π



4π



P (r, n, n



) ⊗−

(r)

4π



4π



∆P (r, n, n



) ⊗ .

(11)

Equation (10) is the major formula of the perturbation approach. It says that

if both the direct and the adjoint solutions of the same problem are known

the eﬀect of the variation can be simply estimated by integration. To reveal

the physical sense of (10) let us consider a special point perturbation of L,

∆L = ∆σ

(r)δ(r −r

)

We have that the eﬀect has the form

∆J =



4π

I(r

, n)∆σ

(r)I(r

, n)dn .

Recalling the physical sense of the adjoint solution, this formula shows that

the eﬀect generated by this perturbation is the same as an eﬀective source

∆σ

(r)I(r

, n)δ(r − r

)

has been added at point r

. Thus, in the most general case we can treat vari-

ations of the media parameters as addition some distribution of the eﬀective

sources.

5 A Toy Example

As an example let us consider how addition of the periodic in x extinction

coeﬃcient aﬀect the reﬂected radiance. This problem in more details was

considered in [9, 16]. The formulation of the problem assumes that our base

problem parameters are constant over the medium and

∆L = ε

sin(k

The variation of the reﬂected radiance has the form

Perturbation Technique in 3D Cloud Optics: Theory and Results 169

0 1 2 3 4 5 6 7 8 9 10

−180

−160

−140

−120

−100

−80

−60

−40

−20

/σ

φ (k),[deg]

1.0

0.9999

0.99

0.95

0.9

0.7

0.2

0 0.5 1 1.5 2 2.5 3 3.5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

/σ

ρ,[a.u]

1.0

0.999

0.99

0.95

0.9

0.7

0.5

0.2

Fig. 1. Left panel. The phase shift, φ(k

) as a function of k

/σ

. Right panel.

Amplitude coeﬃcient, φ(k

), as a function of k

/σ

. Isotropic phase function. The

ﬁgures at the curves show the solar zenith cosine µ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−2

−1.5

−1

−0.5

0.5

1.5

x 10

δE,[a.u]

Fig. 2. The upwelling zenith radiance proﬁle as a function of k

x simulated using

the SHDOM. Isotropic phase function. Wavenumber, k

,is1.0σ

(dotted line), 2.0σ

(solid line), and 3.0σ

(dashed line)

170 I.N. Polonsky et al.

∆J(x)=ερ(k

)sin[k

x + φ(k

)] .

where [16] determined analytical expression for ρ(k

)andφ(k

Figure1showshowparametersρ(k

)andφ(k

) depend on k

. For the

simulation a slab of optical thickness 10.0 has been chosen. For the sake

of simplicity the medium is assumed to be non absorbing with an isotropic

phase function. The ﬁgure shows that for the case of normal illumination at

/σ

≈ 2.0 the amplitude coeﬃcient, ρ, is close to zero which means that for

this kind of the extinction coeﬃcient perturbation the reﬂected radiance does

not change and the radiance measurement cannot reveal periodicity of the

extinction coeﬃcient proﬁle. We tested this prediction performing the cor-

responding simulation employing SHDOM [3]. The results of the simulation

are shown on Fig. 2 which clearly shows that there is quite a weak variation

of the reﬂected radiance in the case of k

/σ

≈ 2.0whichshapedoesnot

resemble the sin-wave shape of the extinction coeﬃcient variation.

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