Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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4.1 The matrix operator method 85
combined layer these unknown quantities can be determined as functions of the
radiative quantities of the individual sublayers (0, 1) and (1, 2).
In a first step we will express the radiances I
m
±
(τ
1
)atlevelτ
1
in terms of the
radiances I
m
−
(τ
0
), I
m
+
(τ
2
) illuminating the combined layer (0, 2). Replacing I
m
−
(τ
1
)
in (4.2a) by (4.1b) and in (4.1b) the term I
m
+
(τ
1
) by (4.2a), we obtain two rather
complicated expressions
(a) I
m
+
(τ
1
) = r
m
(1, 2)[r
m
(1, 0)I
m
+
(τ
1
) + t
m
(0, 1)I
m
−
(τ
0
) + J
m
−,1
(0, 1) + J
m
−,2
(0, 1)]
+ t
m
(2, 1)I
m
+
(τ
2
) + J
m
+,1
(2, 1) + J
m
+,2
(2, 1)
(b) I
m
−
(τ
1
) = r
m
(1, 0)
t
m
(2, 1)I
m
+
(τ
2
) + r
m
(1, 2)I
m
−
(τ
1
)+J
m
+,1
(2, 1) + J
m
+,2
(2, 1)
+ t
m
(0, 1)I
m
−
(τ
0
) + J
m
−,1
(0, 1) + J
m
−,2
(0, 1) (4.4)
Solving for I
m
±
(τ
1
) yields
(a) I
m
+
(τ
1
) = [E − r
m
(1, 2)r
m
(1, 0)]
−1
[t
m
(2, 1)I
m
+
(τ
2
) + r
m
(1, 2)t
m
(0, 1)I
m
−
(τ
0
)
+ r
m
(1, 2)[J
m
−,1
(0, 1) + J
m
−,2
(0, 1)] + J
m
+,1
(2, 1) + J
m
+,2
(2, 1)]
(b) I
m
−
(τ
1
) = [E − r
m
(1, 0)r
m
(1, 2)]
−1
[r
m
(1, 0)t
m
(2, 1)I
m
+
(τ
2
) + t
m
(0, 1)I
m
−
(τ
0
)
+ r
m
(1, 0)[J
m
+,1
(2, 1) + J
m
+,2
(2, 1)] + J
m
−,1
(0, 1) + J
m
−,2
(0, 1)]
(4.5)
where E is the unit matrix.
It is important to realize that (4.5a,b) contain the optical properties of both
sublayers (0, 1) and (1, 2) only, but none of layer (0, 2). This fact can be used
to derive the so-called addition theorems for the optical properties of the layer
(0, 2). Substituting I
m
+
(τ
1
) from (4.5a) into (4.1a) and I
m
−
(τ
1
) from (4.5b) into (4.2b)
yields
(a) I
m
+
(τ
0
) = t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
t
m
(2, 1)I
m
+
(τ
2
)
+t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
r
m
(1, 2)t
m
(0, 1)I
m
−
(τ
0
)
+t
m
(1, 0)[E−r
m
(1, 2)r
m
(1, 0)]
−1
r
m
(1, 2)[J
m
−,1
(0, 1)+J
m
−,2
(0, 1)]
+t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
[J
m
+,1
(2, 1) + J
m
+,2
(2, 1)]
+r
m
(0, 1)I
m
−
(τ
0
) + J
m
+,1
(1, 0) + J
m
+,2
(1, 0)
(b) I
m
−
(τ
2
) = t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
r
m
(1, 0)t
m
(2, 1)I
m
+
(τ
2
)
+t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
t
m
(0, 1)I
m
−
(τ
0
)
+t
m
(1, 2)[E−r
m
(1, 0)r
m
(1, 2)]
−1
r
m
(1, 0)[J
m
+,1
(2, 1)+J
m
+,2
(2, 1)]
+t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
[J
m
−,1
(0, 1) + J
m
−,2
(0, 1)]
+r
m
(2, 1)I
m
+
(τ
2
) + J
m
−,1
(1, 2) + J
m
−,2
(1, 2)
(4.6)
86 Quasi-exact solution methods for the RTE
Comparison of these expressions with (4.3a,b) gives the required optical properties
for the combined layer (0, 2)
t
m
(0, 2) = t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
t
m
(0, 1)
t
m
(2, 0) = t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
t
m
(2, 1)
r
m
(0, 2) = r
m
(0, 1) + t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
r
m
(1, 2)t
m
(0, 1)
r
m
(2, 0) = r
m
(2, 1) + t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
r
m
(1, 0)t
m
(2, 1)
(4.7)
and
J
m
−,1
(0, 2) = J
m
−,1
(1, 2) + t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
×
J
m
−,1
(0, 1) + r
m
(1, 0)J
m
+,1
(2, 1)
J
m
−,2
(0, 2) = J
m
−,2
(1, 2) + t
m
(1, 2)[E − r
m
(1, 0)r
m
(1, 2)]
−1
×
J
m
−,2
(0, 1) + r
m
(1, 0)J
m
+,2
(2, 1)
J
m
+,1
(2, 0) = J
m
+,1
(1, 0) + t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
×
J
m
+,1
(2, 1) + r
m
(1, 2)J
m
−,1
(0, 1)
J
m
+,2
(2, 0) = J
m
+,2
(1, 0) + t
m
(1, 0)[E − r
m
(1, 2)r
m
(1, 0)]
−1
×
J
m
+,2
(2, 1) + r
m
(1, 2)J
m
−,2
(0, 1)
(4.8)
It is clear that the above formulas can be generalized to state the addition formulas
of two arbitrary adjacent layers (i, i + 1), (i + 1, i + 2) by the substitutions 0 → i,
1 → i + 1, 2 → i + 2.
1
The physical interpretation of the inverse matrix A =
[
E − r
m
(1, 2)r
m
(1, 0)
]
−1
occurring in (4.6a) will be demonstrated by means of Figure 4.3. It should be
observed that the inverse matrix can be developed analogously to the scalar
expression
1
1 − x
= 1 + x + x
2
+ x
3
+ ..., |x| < 1 (4.9)
i.e. A can formally be developed as the infinite series
A =
∞
k=0
[r
m
(1, 2)r
m
(1, 0)]
k
(4.10)
The series of matrix products converges if the eigenvalues λ
k
of A fulfill the condi-
tion |λ
k
| < 1, for more details see Gantmacher (1986). Summing up the individual
contributions of the multiple reflections between the two sublayers in Figure 4.3
1
Note that t
m
(i, i + 1), r
m
(i, i + 1) represent (s × s) matrices for the transmission and reflection of downward
radiation incident at level i.
4.1 The matrix operator method 87
I
m
+
()
t
m
r
m
r
m
r
m
t
m
t
m
t
m
r
m
r
m
r
m
r
m
r
m
t
m
(2, 1)I
m
+
()
(1
, 0) (2, 1)I
m
+
()
(1
, 2) (1, 0) (2, 1)I
m
+
()
(1
, 0) (1, 2) (1, 0) (2, 1)I
m
+
()
[
(1
, 2) (1, 0)]
2
(2, 1)I
m
+
()
k =0 k =1 k =2
τ
2
τ
2
τ
2
τ
2
τ
2
τ
0
τ
2
τ
1
τ
2
Fig. 4.3 Physical interpretation of the matrix A =
[
E − r
m
(1, 2)r
m
(1, 0)
]
−1
as an
infinite sum of successive reflections between layer (0, 1) and layer (1, 2).
is equivalent to the expansion terms given in (4.10) pre-multiplying the expression
t
m
(2, 1)I
m
+
(τ
2
) in (4.6a).
4.1.2 The optical properties of a homogeneous elementary layer
So far we have not determined the optical properties of a homogeneous elementary
layer. This layer is required to be sufficiently thin so that the photons entering this
layer will experience primary scattering only. In admitting only primary scattering,
the optical transmission, reflection and emission properties of the layer can be
directly extracted from the RTE (2.104). For τ = τ
1
− τ
0
1, the left-hand side
of (2.104) can be written in finite difference form as
d
dτ
I
m
+
(τ )
I
m
−
(τ )
≈
1
τ
I
m
+
(τ
1
) − I
m
+
(τ
0
)
I
m
−
(τ
1
) − I
m
−
(τ
0
)
(4.11)
with τ
0
<τ
1
and τ = (τ
0
+ τ
1
)/2.
Equations (4.1a,b) can be used to express the difference of the upwelling and
downwelling radiances at levels τ
1
and τ
0
so that we obtain
1
τ
I
m
+
(τ
1
) − I
m
+
(τ
0
)
I
m
−
(τ
1
) − I
m
−
(τ
0
)
=
1
τ
E − t
m
(1, 0) −r
m
(0, 1)
r
m
(1, 0) t
m
(0, 1) − E
I
m
+
(τ
1
)
I
m
−
(τ
0
)
+
1
τ
−J
m
+,1
(1, 0) − J
m
+,2
(1, 0)
J
m
−,1
(0, 1) + J
m
−,2
(0, 1)
(4.12)
88 Quasi-exact solution methods for the RTE
By comparing (4.11) and (4.12) with the RTE (2.104) we find explicit expressions
for the optical matrices t
m
(0, 1), t
m
(1, 0), r
m
(1, 0), r
m
(0, 1), and for the source
vectors J
m
+,1
(0, 1), J
m
−,1
(0, 1), J
m
+,2
(1, 0), and J
m
−,2
(0, 1)
t
m
(0, 1) = t
m
(1, 0) = E − I
m
++
τ = F +
ω
0
2
τ M
−1
P
m
++
W
r
m
(0, 1) = r
m
(1, 0) = I
m
+−
τ =
ω
0
2
τ M
−1
P
m
+−
W
J
m
+,1
(1, 0) =
ω
0
4π
τ S(τ
0
)exp
−
τ
µ
0
M
−1
R
m
(µ
1
, −µ
0
)
.
.
.
R
m
(µ
s
, −µ
0
)
J
m
−,1
(0, 1) =
ω
0
4π
τ S(τ
0
)exp
−
τ
µ
0
M
−1
R
m
(−µ
1
, −µ
0
)
.
.
.
R
m
(−µ
s
, −µ
0
)
J
m
+,2
(1, 0) = J
m
−,2
(0, 1) = τ (1 − ω
0
)B(τ )δ
0m
M
−1
1
.
.
.
1
(4.13)
where we have used the definitions (2.96) and (2.105). Furthermore, the
abbreviation
F = E − M
−1
τ =
1 −
τ
µ
i
δ
ij
≈
exp
−
τ
µ
i
δ
ij
(4.14)
has been introduced. It is important to recall that in the homogeneous elementary
layer with optical depth τ only single scattering is assumed to take place. Numer-
ical experimentation shows that a choice of τ
2
−15
is sufficiently small so that
the single scattering approximation holds.
4.1.3 The doubling algorithm
The addition theorems for the optical properties (4.7) and (4.8) can be used to build
up the optical properties of an arbitrarily thick homogeneous layer. This is done in
the following way.
We start with a homogeneous elementary layer (0, 1). Assuming that this layer
is small enough so that only single scattering takes place, the optical properties of
this layer are given by (4.13). Now we construct the layer (0, 2) by adding to (0, 1)
the layer (1, 2) having the same optical properties as (0, 1). Since the layers are
homogeneous and their optical properties are identical, the addition theorems (4.7)
4.1 The matrix operator method 89
and (4.8) reduce to
t
m
(0, 2) = t
m
(2, 0) = t
m
(0, 1)[E − r
m
(0, 1)r
m
(0, 1)]
−1
t
m
(0, 1)
r
m
(0, 2) = r
m
(2, 0) = r
m
(0, 1) + t
m
(0, 1)[E−r
m
(0, 1)r
m
(0, 1)]
−1
r
m
(0, 1)t
m
(0, 1)
with (4.15)
t
m
(0, 1) = t
m
(1, 0) = t
m
(1, 2) = t
m
(2, 1)
r
m
(0, 1) = r
m
(1, 0) = r
m
(1, 2) = r
m
(2, 1)
and
J
m
−,1
(0, 2) = J
m
−,1
(1, 2) + t
m
(0, 1)[E − r
m
(0, 1)r
m
(0, 1)]
−1
×
J
m
−,1
(0, 1) + r
m
(0, 1)J
m
+,1
(2, 1)
J
m
−,2
(0, 2) = J
m
−,2
(0, 1) + t
m
(0, 1)[E − r
m
(0, 1)r
m
(0, 1)]
−1
×
J
m
−,2
(0, 1) + r
m
(0, 1)J
m
+,2
(1, 0)
J
m
+,1
(2, 0) = J
m
+,1
(1, 0) + t
m
(0, 1)[E − r
m
(0, 1)r
m
(0, 1)]
−1
×
J
m
+,1
(2, 1) + r
m
(0, 1)J
m
−,1
(0, 1)
J
m
+,2
(2, 0) = J
m
+,2
(1, 0) + t
m
(0, 1)[E − r
m
(0, 1)r
m
(0, 1)]
−1
×
J
m
+,2
(1, 0) + r
m
(0, 1)J
m
−,2
(0, 1)
with
J
m
+,1
(2, 1) = J
m
+,1
(1, 0) exp
−
τ
1
− τ
0
µ
0
J
m
−,1
(1, 2) = J
m
−,1
(0, 1) exp
−
τ
1
− τ
0
µ
0
J
m
+,2
(2, 1) = J
m
+,2
(1, 0)
J
m
−,2
(1, 2) = J
m
−,2
(0, 1)
(4.16)
Sometimes these addition theorems are called the star-product algorithm.No
distinction has been made between the upward and downward transmission and
reflection matrices as well as for the thermal source vectors. Only the expressions
J
m
+,1
(2, 1) and J
m
−,1
(1, 2) involving primary scattering require a special distinction.
The optical properties of layer (0, 4) are obtained by adding to (0, 2) the layer
(2, 4) having the same optical properties as layer (0, 2). Layer (0, 8) is obtained by
adding layers (0, 4) and (4, 8) where again layer (4, 8) has the same optical proper-
ties as (0, 4). This process is continued until the final thickness of the homogeneous
layer is reached.
Figure 4.4 illustrates the adding and doubling procedure. One may easily see that
after l doubling steps one arrives at a homogeneous layer with total optical depth
2
l
(0, 1), that is after 15 doubling steps a layer with optical depth 1 is generated
if the thickness of the starting layer is (0, 1) = τ
1
− τ
0
= 2
−15
. In the case of a
90 Quasi-exact solution methods for the RTE
(0, 1)
(0
, 2) = 2(0, 1)
(0
, 4) = 4(0, 1)
(0
, 8) = 8(0, 1)
Fig. 4.4 The doubling algorithm for homogeneous layers.
homogeneous cloud with a typical optical thickness of 32, only five additional
doubling steps are required. Since the final layer is required to be homogeneous it
must also be isothermal in order to specify the thermal emission.
4.1.4 Inhomogeneous atmospheres
Each vertically inhomogeneous atmosphere can be well approximated by N sub-
layers of varying optical properties and thermal structure. Each of these, however, is
homogeneous and isothermal. Let τ
N
denote the total optical depth of such an atmo-
sphere. Reflection of radiation at the ground can be treated by means of an additional
fictitious layer (N, N + 1) to which the following special properties are assigned
t
m
(N, N + 1) = t
m
(N + 1, N ) = 0
r
m
(N, N + 1) = r
m
g
r
m
(N + 1, N ) = 0
J
m
−,1
(N, N + 1) = J
m
−,2
(N, N + 1) = 0
J
m
+,1
(N + 1, N ) =
A
g
µ
0
π
S
0
exp
−
τ
N
µ
0
δ
0m
1
.
.
.
1
J
m
+,2
(N + 1, N ) = (1 − A
g
)B
g
δ
0m
1
.
.
.
1
(4.17)
4.1 The matrix operator method 91
Here B
g
and A
g
are the thermal emission of the ground and the albedo of the
isotropically reflecting ground, respectively. The assumption of isotropic ground
reflection leads to the following form of the (s × s) reflection matrix r
m
g
r
m
g
= 2 A
g
µ
1
w
1
... µ
s
w
s
.
.
.
.
.
.
.
.
.
µ
1
w
1
... µ
s
w
s
δ
0m
(4.18)
Note that for isotropy the reflection matrix r
m
g
is independent of the azimuthal
expansion index m. The derivation of this particular form for r
m
g
will be given in
the Appendix to this chapter.
In addition to the reflection at the ground we need to specify the boundary
conditions for the diffuse radiation at the top of the atmosphere and at the ground.
Usually, external diffuse illumination will be ignored, that is
I
m
−
(τ = 0) = 0, I
m
+
(τ
N +1
) = 0, I
m
−
(τ
N +1
) = 0 (4.19)
In the following we will summarize the main steps of the MOM. The algorithm
proceeds in the following way.
(1) Calculation of the optical properties.
(i) First calculate the optical properties of the elementary layers by means of (4.13).
(ii) Utilizing the adding and doubling formulas (4.15) and (4.16), the optical properties
of all homogeneous sublayers (i, i + 1), i = 0,...,N are calculated.
(iii) The optical properties of the combined layer (0, 2) are calculated by means of (4.7)
and (4.8).
(iv) The combined layer (0, 3) is obtained by again applying the addition theorems
(4.7) and (4.8) to the sublayers (0, 2) and (2, 3).
(v) This procedure is continued until the optical properties of all required combinations
of sublayers (i, j), i = 0,...,N , j = 0,...,N are determined.
(vi) Utilizing the special properties of the fictitious layer (N, N + 1) as listed in (4.17),
the optical properties of the total layer (0, N + 1) are obtained by replacing in (4.7)
and (4.8) 1 → N and 2 → N + 1. This yields
t
m
(N + 1, 0) =0, t
m
(0, N + 1) = 0
r
m
(0, N + 1) =r
m
(0, N ) + t
m
(N , 0)
E − r
m
g
r
m
(N , 0)
−1
r
m
g
t
m
(0, N )
r
m
(N + 1, 0) =0
J
m
−,1
(0, N + 1) =0, J
m
−,2
(0, N + 1) = 0
J
m
+,1
(N + 1, 0) =J
m
+,1
(N , 0) + t
m
(N , 0)
E − r
m
g
r
m
(N , 0)
−1
×
J
m
+,1
(N + 1, N ) + r
m
g
J
m
−,1
(0, N )
J
m
+,2
(N + 1, 0) =J
m
+,2
(N , 0) + t
m
(N , 0)
E − r
m
g
r
m
(N , 0)
−1
×
J
m
+,2
(N + 1, N ) + r
m
g
J
m
−,2
(0, N )
(4.20)
92 Quasi-exact solution methods for the RTE
(2) Calculation of the radiances.
Replacing in (4.1b) and (4.2a) 1 → i we may formally write
I
m
+
(τ
i
) = α
+
(τ
i
)I
m
+
(τ
i+1
) + β
+
(τ
i
)I
m
−
(τ
i
) + γ
+
(τ
i
)
I
m
−
(τ
i
) = α
−
(τ
i
)I
m
+
(τ
i
) + β
−
(τ
i
)I
m
−
(τ
i−1
) + γ
−
(τ
i
)
(4.21)
The terms α
±
(τ
i
),β
±
(τ
i
),γ
±
(τ
i
) are functions of the optical properties of the medium
which have been determined in part (1) of the procedure, i.e. they are known quantities.
Writing (4.21) for i = 0,...,N + 1 we obtain a system of 2(N + 2) equations for
the radiances I
±
(τ
i
), i = 0,...,N + 1. Utilizing the boundary conditions (4.19) this
system may be solved by means of standard methods of linear algebra.
Finally, we list various properties of the MOM.
(1) MOM can be applied to layers with arbitrary large optical thickness.
(2) MOM offers high accuracy so that it can be used to compute benchmark results for
testing simpler techniques.
(3) MOM requires a significant amount of computer time due to the numerous inversions
and matrix-vector multiplications. This is particularly true if the radiance field needs
to be calculated at all interior levels.
(4) MOM can be extended by known methods to account for polarization effects.
The mathematical development as described above in large parts follows the
work of Plass et al. (1973). Numerous other authors have also contributed to the
development of the matrix operator method, for details see Lenoble (1985).
4.2 The successive order of scattering method
As already mentioned at the beginning of this chapter, the solutions (2.110) of the
radiative transfer equation are only formal because the source functions J
±
them-
selves are functions of the radiances I
±
. However, this fact might be a motivation
to construct an iterative solution of the RTE. For this iterative approach we first
define the following vectors
(a) Y
m
+
(τ ) =exp
− M
−1
(τ
N
− τ )
I
m
+
(τ
N
)
+
τ
N
τ
exp[−M
−1
(τ
− τ )]M
−1
J
m
+,1
(τ
) + J
m
+,2
(τ
)
dτ
(b) Y
m
−
(τ ) =exp(−M
−1
τ )I
m
−
(0)
+
τ
0
exp[−M
−1
(τ − τ
)]M
−1
J
m
−,1
(τ
) + J
m
−,2
(τ
)
dτ
(4.22)
The source vectors J
±,1
, J
±,2
are given by (2.96). From these expressions it is
seen that Y
m
+
and Y
m
−
contain only those quantities that are fixed with respect to
4.2 The successive order of scattering method 93
the iteration process, namely the boundary conditions, the primary solar scattering
term and the thermal emission.
For the remaining contributions to the formal solution (2.110) we define
(a) Q
m
+
[τ,I
m
+
(τ ), I
m
−
(τ )] =
1
2
τ
N
τ
ω
0
exp[−M
−1
(τ
− τ )]M
−1
× [P
m
++
(τ
)W I
m
+
(τ
) + P
m
+−
(τ
)W I
m
−
(τ
)]dτ
(b) Q
m
−
[τ,I
m
+
(τ ), I
m
−
(τ )] =
1
2
τ
0
ω
0
exp[−M
−1
(τ − τ
)]M
−1
× [P
m
+−
(τ
)W I
m
+
(τ
) + P
m
++
(τ
)W I
m
−
(τ
)]dτ
(4.23)
where use was made of (2.103). Utilizing these equations the formal solution of the
RTE can now be written as
I
m
+
(τ ) = Y
m
+
(τ ) + Q
m
+
τ,I
m
+
(τ ), I
m
−
(τ )
I
m
−
(τ ) = Y
m
−
(τ ) + Q
m
−
τ,I
m
+
(τ ), I
m
−
(τ )
(4.24)
For these equations we can set up an iteration process by defining
I
m
+
(τ )
n
= I
m
+
(τ )
n=0
+ Q
m
+
τ,I
m
+
(τ )
n−1
, I
m
−
(τ )
n−1
I
m
−
(τ )
n
= I
m
−
(τ )
n=0
+ Q
m
−
τ,I
m
+
(τ )
n−1
, I
m
−
(τ )
n−1
, n ≥ 1
(4.25)
where n is the iteration step. For the starting value n = 0wehave
I
m
+
(τ )
n=0
= Y
m
+
(τ ), I
m
−
(τ )
n=0
= Y
m
−
(τ )
(4.26)
Equations (4.25) and (4.26) are known as the successive order of scattering (SOS)
method of radiative transfer.
Equation (4.25) is based on the following method of solution. Consider the
integral equation
y(x) = g(x) +
x
x
0
f
[
t, y(t)
]
dt (4.27)
A solution of this integral equation can be found by successive iteration. Let n
represent the iteration step then
y
(n)
= y
(n=0)
(x) +
x
x
0
f [t, y
(n−1)
(t)] dt, n ≥ 1
with y
(n=0)
(x) = g(x)
(4.28)
The boundary conditions in (4.25) can be treated in the following manner.
94 Quasi-exact solution methods for the RTE
(1) At the upper boundary of the medium we assume that no diffuse radiation enters
I
m
−
(τ = 0) = 0 (4.29)
(2) At the lower boundary we again adopt an isotropically reflecting ground with albedo
A
g
. The diffusely reflected radiation can then be expressed as
I
m
+
(τ = τ
N
) = r
m
g
I
m
−
(τ
N
) +
A
g
π
δ
0m
µ
0
S
0
exp
−
τ
µ
0
1
.
.
.
1
+ B
g
(1 − A
g
)δ
0m
1
.
.
.
1
(4.30)
where r
m
g
is the reflection matrix of the ground, see (4.18).
The individual contributions in (4.30) can be interpreted in the following way.
(1) The first term on the right represents the isotropic reflection of the diffuse light I
m
−
(τ
N
)
which is incident at the ground.
(2) The second term stands for the isotropic reflection of the direct sunlight.
(3) The third term accounts for the black body emission B
g
of the ground with temperature
T
g
.
The azimuthally dependent radiance field can be reconstructed with the help of
(2.106). For the flux densities the expansion term m = 0 is required only.
In the successive order of scattering (SOS) method the first and the n-th iteration
step can be interpreted as follows.
(1) For n = 1 in (4.25) the radiance vectors I
m
±
(τ )
n=0
include:
(i) the influence of the boundary conditions;
(ii) primary scattering of the direct sunlight;
(iii) the isotropic thermal emission of each layer.
(2) The source vectors Q
m
±
(τ,I
m
±
(τ )
1
) include:
(i) secondary scattered sunlight;
(ii) primary scattered thermal radiation.
(3) In going from iteration step n to step n + 1 one more scattering process is simulated.
The iteration is continued until convergence is achieved, i.e. the changes in the
radiance vectors remain below a certain tolerance.
The convergence properties of the SOS method depend primarily on:
(1) the total optical depth of the medium;
(2) the total number of terms considered in the expansion of the phase function;
(3) the inhomogeneity of the optical parameters within the medium;
(4) the total number s of discrete directions for representing the vectors I
m
±
.
The SOS method has the following two advantages over the MOM.