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SIMPLE RELATIONSHIPS BETWEEN RADIATIVE AND MICROPHYSICAL
CHARACTERISTICS OF CLOUDY MEDIA
A. A. Kokhanovsky
Institute of Physics
70 Skarina Avenue, Minsk 220072, Belarus
ABSTRACT
This paper is devoted to the derivation of simple
relationships between reflection and transmission matrices
of cloudy media and their microphysical characteristics
(average size of droplets, their concentration and refractive
index). Such relations are of importance for the
parametrization of radiative and polarization characteristics
of clouds. They can be used not only for the solution of
direct problems but for the development of semi-analytical
inversion algorithms as well.
1. INTRODUCTION
The influence of clouds on radiative fluxes in the earth
atmosphere has been studied by many authors both
experimentally and theoretically. The knowledge obtained
has been applied to the solution of inversion problem,
namely for the determination microstructure, thickness and
liquid water path of cloudy media.
However, cloudy media not only absorb and scatter
radiation. They also can be considered as major sources
(along with aerosols and gases) of polarized light in the
terrestrial atmosphere. They introduce a new property in
coming solar light, namely the preferential direction of
oscillation of an electic vector in a light beam.
The polatization arises mostly due to a single scattering
of light beam by a droplet. Multiple scattering of the beam
will lead to the randomisation both photons? directions and
polatization states. This will cause the decrease of light
polarization due to increase in the optical thickness of a
cloud. Another effect of multiple scattering is the
introduction of ellipticity in scattered light beams. It should
be pointed out that the solar light scattered by a spherical
droplet becomes partially linearly polarized. The ellipticity
of a singly scattered light is equal to zero.
Both polarizarion and radiative characteristics of
cloudy media can be studied with modern radiative transfer
codes. However, it is of importance to have approximate
formulae for optical chracteristics of cloudy media. This is
due to the fact that that simple solutions allow to make
rapid estimations of results expected. They also of
importance for the development of semi-analytical
inversion schemes.
2. TRANSMISSION AND REFLECTION
MATRICES
Asymptotic theory for thick layers can be used both for
studies of polarization signatures of light fields both inside
of clouds and on their boundaries. We are concern only with
the polarization characteristics of reflected light, which can
be used for a global coverage from instruments on board of
different satellites. Thus, we will limit ourselves by studies
of the reflection matrix ( )
y
Å
Å ,
,
? 0
R of a cloudy medium.
This matrix depends on the optical thickness of a cloud t ,
microstructure of a cloud, cosine 0
Å of the solar angle 0
J ,
cosine Å of the viewing angle J and the relative azimuth
0
j
j
y -
= , where 0
j and j are azimuths of the Sun and
observer respectively. The reflection matrix ( )
y
Å
Å ,
,
? 0
R can
be used to find the Stokes vector ( )
V
U
Q
I
S r
,
,
,
& of the
reflected light for arbitrary incident light beams. Namely, it
follows:
( ) ( ) ( )
j
Å
j
j
Å
Å
j
Å
Å
p
j
Å p
,
,
,
?
1
, 0
2
0
1
0
S
R
d
d
S r
&
& ? -
?
= (1)
where ( )
j
Å,
0
S &
represents the stokes vector of the incident
light beam. A similar expression could be written for the
transmission matrix.
One can consider the Sun a source of light incident from
only one direction 0
0 ,j
Å . Then it follows:
( ) ( ) ( ) F
S &
&
p
j
j
d
Å
Å
d
Å
j
Å 0
0
0
0
1
, -
+
-
= , (3)
where a vector F &
is normalized so that the first element of
F &
p is the net flux of solar beam per unit area of a cloud
layer. Eq. (1) becomes in this case:
( ) ( )F
R
S r
&
& 0
0 ,
,
?
, j
j
Å
Å
j
Å -
= , (4)

where only first element of the column vector
? ?
?
?
?
?
?
÷ ÷
÷
÷
÷
ø
ö
=
4
3
2
1
F
F
F
F
F &
differs from zero.
One can see that the knowledge of matrices
( )
0
0 ,
,
? j
j
Å
Å -
R and ( )
0
0 ,
,
? j
j
Å
Å -
T is of primary
importance for understanding radiative and polarization
characteristics of light fluxes, which emerge from a cloud
body.
3. ASYMPTOTIC THEORY
The optical thickness of most of water clouds is much larger
then 1. Some mixed and crystalline clouds could be also
extended in the vertical direction for many hundred meters.
Thus, the asymptotic vector theory is quite applicable for
cloudy media. The reflection matrix ( )
j
Å
Å ,
,
?
0
R and
transmission matrix ( )
0
,
? Å
Å
T of a homogeneous plane-
parallel turbid layer of the large optical thickness t can be
written in the following form:
( ) ( ) ( )
0
0
0 ,
?
,
,
?
,
,
? Å
Å
j
Å
Å
j
Å
Å T
f
R
R -
= ? ,
(1)
( ) ( ) ( )
0
0
,
? Å
Å
Å
Å T
K
K
t
T &
&
= .
(2)
Here J
J
J
Å
J
Å and
,
cos
,
cos 0
0
0
=
= are incidence and
observation angles respectively, j is the relative azimuth of
incident and reflected or transmitted light beams and
( )
2
1
exp
f
k
m
t
-
-
= t
. (3)
Matrices ( )
j
Å
Å ,
,
?
0
R and ( )
0
,
? Å
Å
T can be used to calculate
the Stokes vectors of reflected and transmitted light beams
under arbitrary illumination of a scattering layer.
Note, that the vector ( )
0
Å
T
K & is transpose to the
vector ( )
0
Å
K & . Constants f, m and the vector ( )
Å
K & can be
found from following relations:
( )
t
k
s
f -
= exp (4)
( ) ( ) ( ) ( )
[ ]
? -
-
-
= 1
0
2 Å
Å
Å
Å
ÅÅ P
P
P
P
d
m T
T &
&
&
& ,
( ) ( ) ( ) ( ) ? ?
?
ú û
ù -
?
-
= ?
- x
x
Å
xx
Å
Å P
R
d
P
m
K &
&
& ,
?
2 1
0
1 ,
where
( ) ( )
s d K P
T
= ? -
2
0
1
ÅÅ Å Å
& &
,
and
( ) ( )
j
x
Å
j
p
x
Å p
,
,
?
2
1
,
? 2
0
?
? ?
= R
d
R
is the azimuthally averaged reflection matrix of the semi-
infinite medium with the same local optical characteristics
as those for a finite layer under study. Note, that the vector
( )
Å
K &
also occurs in the so-called Milne problem, i. e. the
problem of finding the angular dependence and the
polarization characteristics of light emerging from a semi-
infinite turbid layers with sources of radiation located deep
inside the medium. The constant k is the smallest discrete
eigenvalue, which is real and nondegenerate, obeying to the
following equation:
( ) ( ) ( ) ( )
1
2
0
1
1
- = ?
-
ku P u d Z u P
& &
w x x x
# , ,
which is called the vector characteristic equation of the
radiative transfer theory. Here u is the cosine of the
observation angle, w 0 is the single scattering albedo,
which is equal to the ratio of scattering and extinction
coefficients, ( )
x
,
? u
Z is the azimuth-averaged phase
matrix and ( )
x
P & is correspondent eigenvector.
Eqs. (1), (2) are very general and can be applied to optically
thick disperse media with arbitrary local optical
characteristics. However, their application to the solution of
practical problems is difficult due to the necessity to solve
integral equation (9) and perform integrations (5) - (8). The
reflection matrix ( )
j
x
Å ,
,
? ?
R obeys to the integral equation
as well.
The problem of finding matrices ( )
j
Å
Å ,
,
?
0
R and
( )
0
,
? Å
Å
T in Eqs. (1), (2) can be simplified for a broad
class of so-called weakly absorbing media, which are
characterized by a small value of the probability of photon
absorption 0
1 w
b -
= . It was obtained as 0
?
b that
( )( )
0
1
1
3 w
-
-
= g
k , (10)

( )
g
k
m
-
=
1
3
8 , (11)
( )
g
k
s
-
-
=
1
3
4
1 a , (12)
( ) ( ) ( ) ( ) ( )
0
0
0
0
0
0
1
3
4
,
,
?
,
,
? Å
Å
j
Å
Å
j
Å
Å T
K
K
g
k
R
R &
&
-
-
= ?
? ,
where ( )
Å
0
K & and ( )
j
Å
Å ,
,
? 0
?
R are the escape vector and
the reflection matrix respectively of the nonabsorbing
medium with the same phase matrix as a finite layer under
study,
( ) q
q
q
q
p
d
p
g cos
sin
2
1
0
?
=
(14)
is the asymmetry parameter of the phase function ( )
q
p and
( )
?
= 1
0
2
3 Å
ÅÅ
a S
d , (15)
where ( ) ,
)
( 1
0 e
K
S T &
& Å
Å = 1
e & is the unit column vector. It
should be pointed out, that integral (15) is close to one for
all types of phase matrices 3
1- . Thus, will neglect small
differences a
-
=
D 1 in the discussion, which follows. Note
that the first moment of the function ( )
Å
S does not depend
on the phase matrix at all. This is due to the normalization
condition 2 :
( ) 1
2 1
0
=
? Å
ÅÅ S
d (16)
Thus, Eqs. (10) - (15) allow to reduce the complexity of
Eqs. (1), (2). Only functions ( )
Å
0
K & and ( )
j
Å
Å ,
,
?
0
0
?
R for a
nonabsorbing medium should be found numerically in this
more simple case. They depend on the phase matrix of a
random medim in question. However, they do not depend
on the single scattering albedo 0
w and the optical thickness
t . Another interesting feature of these functions is their
weak dependence on the microstructure parameters (size,
shape, and chemical composition of particles) of a medium
under study in the broad range of angular
parameters .
,
, 0 j
Å
Å This is due to the randomization both
polarization states and propagation directions of photons in
highly scattering semi-infinite layers irrespectively to the
local optical properties of a medium, where they propagate.
4. APPROXIMATE EQUATIONS FOR WEAKLY
ABSORBING MEDIA
Numerical calculations show that expansions (11) - (15) can
be applied only for very small values of
( )
4
0 10
~
1 -
-
= b
w
b . This limits the power of the
asymptotic theory for a real life problems in a great extent.
Different methods to overcome this problem have been
used. The most straight forward approach calls for the
derivation of higher order terms in expansions (10)-(13).
However, we will use here another method, which has
initially been applied in the scalar radiative transfer theory.
According to this alternative approach expansions (11),
(12) are substituted by exponential functions:
y
e
m 2
1 -
-
= , (17)
y
e
s -
= , (18)
where
( )
g
k
y
-
=
1
3
4
(19)
and we assumed that 1
=
a .
Expansions (17), (18) transform to exact asymptotic results
(11), (12) as 0
?
y , providing that 1
-
a . However, they
do allow to calculate radiative and polarization
characteristics for more wide class of absorbing and light-
scattering media (up to 1
.
0
05
.
0 ?
Ë
b , depending on the
characteristic in question).
Thus, Eqs. (1), (2) can be transformed to the following
forms:
( ) ( ) ( ) ,
exp
)
,
(
?
,
,
?
,
,
?
0
0
0
t
Å
Å
j
Å
Å
j
Å
Å k
y
T
R
R -
-
-
= ?
(20)
( ) ( ) ( )
0
0
0
0
,
? Å
Å
Å
Å T
K
K
t
T &
&
= ,
(21)
where we accounted for Eqs. (17), (18) and the approximate
equality
( ) ( ) ( ) ( ) ( )
0
0
2
0 1 Å
Å
Å
Å T
y
T K
K
e
K
K
m &
&
&
& -
-
= .
(22)
The value of
t
t k
y
k
y
e
e
e
t -
-
-
-
-
= 2
2
1
(23)
is the global transmittance, defined as 1
( )
0
11
0
1
0
1
0
0 ,
4 Å
Å
Å
Å
Å
Å T
d
d
t ? ?
= .

(24)
Eq. (24) follows from Eq. (21) accounting for the integral
relation 1
( ) 1
2
1
0
01
=
? Å
Å
Å
Å d
K
d , (25)
where ( )
Å
01
K is the first component of the escape vector
( )
Å
0
K & .
Note, that the global transmittance (24) could be
transformed to the following simple form 6 :
( )
y
x
sh
shy
t
+
= , (26)
where
t
k
x = . (27)
Eqs. (20), (21) are much simpler than initial asymptotic
formulae (1), (2). They allow to calculate the reflection and
transmission matrices of thick weakly absorbing disperse
media by simple means , if the solution of the problem for a
semi-infinite medium with the same phase matrix as for a
finite layer under consideration is available. This reduction
of a general problem to the case of a semi-infinite medium is
of importance for the radiative transfer theory.
It follows from results presented that the reflection and
transmission matrices of cloudy media depend mostly on the
asymmetry parameter, extinction and absorption coefficients
of a cloudy media. The phase function is of a secondary
importance for most observation geometries. These local
optical characteristics can be found with following equations,
which are valid up to 2.2 micrometers:
,
)
(
1
.
1
1
2
3
3
/
2
2 ? ?
?
?
?
÷ ÷
÷
ø
ö
+
=
ef
ef
ext
a
k
a
c
r
s
,
15
.
0
)
(
2
1
88
.
0 3
/
2 ef
ef
a
ka
g a
-
+
=
( )( )
))
/
8
exp(
1
(
34
.
0
1
1
25
.
1 ef
ef
abs a
a l
a
a
s -
-
+
-
= ,
where the effective radius is equal to the ratio of third and
second moments of the particle size distribution,
l
p /
2
=
k , l
pc
a /
4
= , c is the liquid water content, r
is the density of water, c is the imaginary part of the
refractive index of particles and l is the wavelength. Our
calculations show that the accuracy of Eq. (A2.1) is better
than 5% for water droplets with the effective radius in the
range 4-16 micrometers at l < 2.2 m
Å . It is up to 10% in
the same spectral range for values of the absorption
coefficient and co-asymmetry parameter 1-g. It is smaller
than 5% for both 1-g and abs
s at l < 1.2 m
Å . The
dependence of ( )
l
y , obtained with Eqs. (A2.1)-(A2.3), ,
and Mie calculations for spherical polydispersions of water
droplets with the gamma particle size distribution is
presented in Fig. A2.1. One can see that the accuracy is
high. The error is smaller than 7% up to m
Å
l 8
.
1
= . The
error of geometrical optics equation increases for clouds for
small ( )
m
a ef Å
4
= and large ( )
m
a ef Å
16
= droplets.
However, even in this case it is rather high(see Fig. A2.1).
ACKNOWLEDGEMENTS
This paper was prepared with the support from the INTAS
Project N 652
REFERENCES
Reflection and Transmission of Polarized Light by Planetary
Atmospheres, PhD thesis, Utrecht.
Zege E. P., Ivanov A. P. and I. L. Katsev, 1991, Image
Transfer in a Scattering Medium, Springer-Verlag, Berlin.
Kokhanovsky A. A., 1999, Light Scattering Media Optics,
Wiley-Praxis, Chichester.
Umov N., 1905, Phyz. Z., 6, 674.
Hapke B., 1993, Theory of Reflectance and Emittance
Spectroscopy