Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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2.4 The RTE for a horizontally homogeneous atmosphere 45
(1) Averaging the intensity I (τ,µ,ϕ) as defined by (2.70) yields
1
2π
2π
0
I (τ,µ,ϕ)dϕ =
1
2π
∞
m=0
(2 − δ
0m
)
2π
0
I
m
(τ,µ) cos mϕ dϕ
= I
m=0
(τ,µ) = I (τ,µ)
(2.77)
Only m = 0 contributes to the azimuthally averaged radiation field since the integral
of the cosine function over a complete cycle vanishes.
(2) Since the Planck function is isotropic the thermal emission term retains its form.
(3) Analogously to (1) the primary scattering term can be averaged according to the expan-
sion (2.68)
1
2π
2π
0
P(cos
0
)dϕ =
∞
l=0
p
l
P
l
(µ)P
l
(−µ
0
) = P(µ, −µ
0
) (2.78)
(4) For the multiple scattering term we proceed as in (2.77) and (2.78) obtaining
ω
0
8π
2
2π
0
1
−1
I (τ,µ
,ϕ
)
2π
0
P(cos )dϕ dµ
dϕ
=
ω
0
4π
2π
0
1
−1
I (τ,µ
,ϕ
)P(µ, µ
)dµ
dϕ
=
ω
0
2
1
−1
I (τ,µ
) P(µ, µ
)dµ
(2.79)
The azimuthally integrated form of the phase function P(µ, µ
)isgivenby
P(µ, µ
) =
1
2π
2π
0
P(cos )dϕ
=
1
2π
2π
0
P(µ, ϕ, µ
,ϕ
)dϕ =
∞
l=0
p
l
P
l
(µ)P
l
(µ
) (2.80)
Application of steps (1)–(4) results in the azimuthally integrated form of the RTE
µ
d
dτ
I (τ,µ) = I (τ, µ) −
ω
0
2
1
−1
P(µ, µ
)I (τ,µ
)dµ
−
ω
0
4π
S
0
exp
−
τ
µ
0
P(µ, −µ
0
) − (1 − ω
0
)B(τ )
(2.81)
Comparison of (2.81) with (2.76) reveals that for m = 0 both equations are
identical.
The separation of the ϕ-dependence of the radiance field results in an infinite
system of first-order ordinary differential equations for the Fourier modes I
m
(τ,µ).
46 The radiative transfer equation
It is convenient to introduce into (2.76) the following abbreviations
(a) R
m
(µ, µ
) =
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(µ
)
(b) J
m
(τ,µ) =
ω
0
2
1
−1
R
m
(µ, µ
)I
m
(τ,µ
)dµ
(2.82)
+
ω
0
4π
S
0
exp
−
τ
µ
0
R
m
(µ, −µ
0
) + (1 − ω
0
)B(τ )δ
0m
With these definitions the RTE reads in short-hand notation
µ
d
dτ
I
m
(τ,µ) = I
m
(τ,µ) − J
m
(τ,µ), m = 0, 1,...,
(2.83)
For practical applications the infinite series (2.70) must be truncated after a finite
number of terms, , as indicated above.
2.5 Splitting of the radiance field into upwelling and
downwelling radiation
It is common practice to distinguish between the upwelling (µ>0) and down-
welling (µ<0) radiation, see Figure 2.3. For the radiance field I
m
and the source
term J
m
the following quantities are introduced
2
I
m
+
(τ,µ) = I
m
(τ,µ > 0), I
m
−
(τ,µ) = I
m
(τ,µ < 0)
J
m
+
(τ,µ) = J
m
(τ,µ > 0), J
m
−
(τ,µ) = J
m
(τ,µ < 0)
(2.84)
Now the integral
1
−1
...dµ
in (2.82b) will be split into the two parts
0
−1
...dµ
and
1
0
...dµ
1
−1
R
m
(µ, µ
)I
m
(τ,µ
)dµ
=
0
−1
R
m
(µ, µ
)I
m
(τ,µ
)dµ
+
1
0
R
m
(µ, µ
)I
m
(τ,µ
)dµ
=
1
0
R
m
(µ, −µ
)I
m
(τ,−µ
)dµ
+
1
0
R
m
(µ, µ
)I
m
(τ,µ
)dµ
=
1
0
R
m
(µ, −µ
)I
m
−
(τ,µ
) + R
m
(µ, µ
)I
m
+
(τ,µ
)
dµ
(2.85)
With the above convention µ
and µ are always positive quantities. As a consequence
of this, the system (2.83) can be treated as two separate differential equations, one
governing the upwelling part and the other one the downwelling part of the radiation
2
Some authors define the zenith angle ϑ in the opposite way, that is µ>0 for downwelling radiation and µ<0
for upwelling radiation.
2.5 Splitting of the radiance field into up- and downwelling radiation 47
field
µ
d
dτ
I
m
+
(τ,µ) = I
m
+
(τ,µ) − J
m
+
(τ,µ)
µ
d
dτ
I
m
−
(τ,µ) =−I
m
−
(τ,µ) + J
m
−
(τ,µ)
m = 0, 1,...,, µ>0
(2.86)
In a similar manner the source terms of (2.82b) can be split to give
(a) J
m
+
(τ,µ) =
ω
0
2
1
0
I
m
+
(τ,µ
)R
m
(µ, µ
) + I
m
−
(τ,µ
)R
m
(µ, −µ
)
dµ
+
ω
0
4π
S
0
exp
−
τ
µ
0
R
m
(µ, −µ
0
) + (1 − ω
0
)B(τ )δ
0m
(b) J
m
−
(τ,µ) =
ω
0
2
1
0
I
m
+
(τ,µ
)R
m
(−µ, µ
) + I
m
−
(τ,µ
)R
m
(−µ, −µ
)
dµ
+
ω
0
4π
S
0
exp
−
τ
µ
0
R
m
(−µ, −µ
0
) + (1 − ω
0
)B(τ )δ
0m
(2.87)
with µ>0 and µ
> 0.
The above expressions demonstrate that (2.86) is a system of 2( + 1) coupled
integro-differential equations. The source terms J
m
+
and J
m
−
contain integrals over
µ
which may be approximated by means of quadrature formulas. Best suited for the
interval [−1, 1] are the Gauss–Legendre quadrature formulas, which are obtained
by neglecting in the expression
1
−1
f (µ)dµ =
r
i=1
w
i
f (µ
i
) + R
r
with R
r
=
2
2r+1
(r!)
4
(2r + 1)
[
(2r)!
]
3
f
(2r)
(ξ), − 1 <ξ <1
(2.88)
the error term R
r
(see, e.g. Abramowitz and Stegun, 1972). The w
i
are the weights
and the µ
i
are the zeros or roots of the Legendre polynomials P
r
(µ). The weights
are given by
w
i
=
2
(1 − µ
2
i
)
P
r
(µ
i
)
2
> 0,
r
i=1
w
i
= 2 (2.89)
where P
r
is the derivative of P
r
. The last expression follows immediately from
(2.88) by setting there f (µ) = 1.
It is convenient to divide the interval [−1, 1] according to the zeros of even-
order Legendre polynomials, which is achieved by choosing r = 2s. For such even
divisions we have 2s different weights and zeros. By denoting them as (w
k
,µ
k
,
48 The radiative transfer equation
k =−s,...,−1, 1,...,s) the following relations hold
w
i
= w
−i
, µ
i
=−µ
−i
, i = 1,...,s (2.90)
Now we split the integral of (2.88) into two parts yielding
1
−1
f (µ)dµ =
0
−1
f (µ)dµ +
1
0
f (µ)dµ =
1
0
f (−µ)dµ +
1
0
f (µ)dµ
(2.91)
For the sum on the right-hand side of (2.88) we may write
s
i=−s
w
i
f (µ
i
) =
−1
i=−s
w
i
f (µ
i
) +
s
i=1
w
i
f (µ
i
)
=
s
i=1
w
i
f (−µ
i
) +
s
i=1
w
i
f (µ
i
) (2.92)
where use was made of (2.90). Hence, the notation
is equivalent to the summa-
tion from i =−s to i = s thereby omitting the term i = 0. By comparing (2.91) with
(2.92) we obtain the Gauss–Legendre quadrature formulas of order 2s in the form
1
0
f (−µ)dµ ≈
s
i=1
w
i
f (−µ
i
),
1
0
f (µ)dµ ≈
s
i=1
w
i
f (µ
i
)
(2.93)
For ease of notation (2.93) will henceforth be called the Gaussian quadrature
formulas. It is important to note that the µ
i
, i = 1,...,s are the positive zeros
of the Legendre polynomials P
2s
(µ). Furthermore, from (2.89) and (2.90) we see
that the weights fulfill the requirement
s
i=1
w
i
= 1 (2.94)
We conclude that a particular Gaussian quadrature formula is defined by the set
of quantities (w
i
,µ
i
, i = 1,...,s). The weights and the nodes may be found in
tabulated form (see e.g. Abramowitz and Stegun, 1972) or may be found using effi-
cient numerical algorithms (Press et al., 1992). Thus the nodes define the directions
of the radiances for which the system of equations (2.86) will be discretized.
In order to express the directional dependence of the radiance it is convenient
to introduce a compact matrix–vector notation. One defines two vectors for the
upwelling and downwelling radiation field by means of
I
m
+
(τ ) =
I
m
+
(τ,µ
1
)
.
.
.
I
m
+
(τ,µ
s
)
, I
m
−
(τ ) =
I
m
−
(τ,µ
1
)
.
.
.
I
m
−
(τ,µ
s
)
(2.95)
2.5 Splitting of the radiance field into up- and downwelling radiation 49
In a similar manner vectors for the sources of the up- and downwelling primary
scattered solar radiation and the thermal emission will be defined
J
m
+,1
(τ ) =
ω
0
4π
S
0
exp
−
τ
µ
0
R
m
(µ
1
, −µ
0
)
.
.
.
R
m
(µ
s
, −µ
0
)
, J
m
+,2
(τ ) = (1 −ω
0
)B(τ ) δ
0m
1
.
.
.
1
J
m
−,1
(τ ) =
ω
0
4π
S
0
exp
−
τ
µ
0
R
m
(−µ
1
, −µ
0
)
.
.
.
R
m
(−µ
s
, −µ
0
)
, J
m
−,2
(τ ) = J
m
+,2
(τ )
(2.96)
where all vectors contain s rows.
For the evaluation of the multiple scattering integral the phase function term
R
m
(µ, µ
) needs to be discretized. To give an example, let us first evaluate the mul-
tiple scattering term in J
m
+
for a particular direction µ
k
by employing the quadrature
formula (2.93)
ω
0
2
1
0
I
m
+
(τ,µ
)R
m
(µ
k
,µ
) + I
m
−
(τ,µ
)R
m
(µ
k
, −µ
)
dµ
≈
ω
0
2
s
i=1
w
i
I
m
+
(τ,µ
i
)R
m
(µ
k
,µ
i
) + w
i
I
m
−
(τ,µ
i
)R
m
(µ
k
, −µ
i
)
=
ω
0
2
s
i=1
s
r=1
p
m
kr,++
w
r
δ
ri
I
m
+
(τ,µ
i
) +
s
r=1
p
m
kr,+−
w
r
δ
ri
I
m
−
(τ,µ
i
)
=
ω
0
2
P
m
++
W I
m
+
(τ ) + P
m
+−
W I
m
−
(τ )
(2.97)
Here the following matrices have been introduced
P
m
++
=
p
m
kr,++
=
R
m
(µ
1
,µ
1
) ··· R
m
(µ
1
,µ
s
)
.
.
.
.
.
.
R
m
(µ
s
,µ
1
) ··· R
m
(µ
s
,µ
s
)
P
m
+−
=
p
m
kr,+−
=
R
m
(µ
1
, −µ
1
) ··· R
m
(µ
1
, −µ
s
)
.
.
.
.
.
.
R
m
(µ
s
, −µ
1
) ··· R
m
(µ
s
, −µ
s
)
W = (w
i
δ
ij
) =
w
1
0 ··· 0
0 w
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
00··· w
s
(2.98)
50 The radiative transfer equation
Since
P
m
l
(−µ) = (−1)
l+m
P
m
l
(µ)
P
m
l
(−µ)P
m
l
(−µ
) = (−1)
2(l+m)
P
m
l
(µ)P
m
l
(µ
) = P
m
l
(µ)P
m
l
(µ
)
(2.99)
with the help of (2.82a) we obtain the equalities
P
m
++
= P
m
−−
, P
m
+−
= P
m
−+
(2.100)
With the above definitions the RTE can now be written in the form
M
d
dτ
I
m
+
(τ ) = I
m
+
(τ ) − J
m
+
(τ )
M
d
dτ
I
m
−
(τ ) =−I
m
−
(τ ) + J
m
−
(τ ), m = 0, 1,...,
(2.101)
where the matrix M is defined by
M = (µ
i
δ
ij
) =
µ
1
0 ··· 0
0 µ
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
00··· µ
s
(2.102)
and the source vectors J
m
+
, J
m
−
are given in matrix notation as
J
m
+
(τ ) = J
m
+,1
(τ ) + J
m
+,2
(τ ) +
ω
0
2
P
m
++
W I
m
+
(τ ) + P
m
+−
W I
m
−
(τ )
J
m
−
(τ ) = J
m
−,1
(τ ) + J
m
−,2
(τ ) +
ω
0
2
P
m
+−
W I
m
+
(τ ) + P
m
++
W I
m
−
(τ )
(2.103)
Multiplying (2.101) from the left by M
−1
we find the matrix–vector form of the
RTE as
d
dτ
I
m
+
(τ )
I
m
−
(τ )
=
I
m
++
−I
m
+−
I
m
+−
−I
m
++
I
m
+
(τ )
I
m
−
(τ )
+
−Σ
m
+
(τ )
Σ
m
−
(τ )
(2.104)
where the following abbreviations have been introduced
I
m
++
= M
−1
E −
ω
0
2
P
m
++
W
I
m
+−
= M
−1
ω
0
2
P
m
+−
W
Σ
m
+
(τ ) = M
−1
J
m
+,1
(τ ) + J
m
+,2
(τ )
Σ
m
−
(τ ) = M
−1
J
m
−,1
(τ ) + J
m
−,2
(τ )
(2.105)
E is the s × s unit matrix. From the first two expressions of (2.105) it may be easily
seen that due to (2.100) I
m
++
= I
m
−−
and I
m
+−
= I
m
−+
.
2.6 Solution of RTE for a horizontally homogeneous atmosphere 51
The system (2.104) represents the discretized form of the RTE, discretized in
µ-space. It has been derived under the assumptions that
(i) the phase function P can be developed as an infinite series of Legendre polynomials;
and
(ii) the ϕ-dependence of both the radiance as well as the phase function can be separated
in product form from the dependency of the other variables.
If the solution of the Fourier modes I
m
+
, I
m
−
of the radiance is known, the full
dependence of the radiance function follows from (2.70)
3
I
+
(τ,ϕ) =
∞
m=0
(2 − δ
0m
)I
m
+
(τ ) cos mϕ
I
−
(τ,ϕ) =
∞
m=0
(2 − δ
0m
)I
m
−
(τ ) cos mϕ
(2.106)
The µ
i
-dependency of the radiance vectors I
+
, I
−
is expressed by (2.95).
2.6 The solution of the radiative transfer equation for a
horizontally homogeneous atmosphere
The energy budget of the atmosphere is determined by the solar insolation and the
emission of infrared radiation by the Earth and the atmosphere. Thus scattering,
absorption and emission of radiation influence the radiative exchange between the
individual atmospheric layers. These processes depend on both the wavelength of
the radiation as well as on the different radiatively active atmospheric species. As
already mentioned in Chapter 1, the solar radiation spectrum ranges from roughly
0.2–3.5
µm while the thermal radiation spectrum of the Earth and the atmosphere
covers the region of about 3.5–100
µm. Hence, for all practical purposes both
spectral regions may be treated separately. As a consequence of this, in the solar
spectral region one is justified to neglect the Planck function B(τ ), while in the
infrared spectral region S
0
may be omitted. This is most easily achieved by setting
in the source vectors J
m
±
for the short-wave region J
m
±,2
(τ ) = 0 and for the long-wave
region J
m
±,1
(τ ) = 0, see (2.96) and (2.103).
In later chapters we will show that in the infrared spectral region scattering pro-
cesses are important only in the so-called atmospheric window region extending
from about 8 to 12.5
µm. In this spectral region the clear atmosphere is almost
transparent to infrared radiation while scattering and absorption by aerosol and
cloud particles must be accounted for. The atmospheric window is of particular
3
Recall that the coordinate system has been rotated so that ϕ
0
= 0.
52 The radiative transfer equation
importance for the energy budget of the Earth since in this region the largest frac-
tion of thermal radiation is emitted to space. Outside the window region scattering,
in general, plays almost no role in the troposphere. Unless in the upper troposphere
layers are extremely dry, outside the CO
2
absorption spectrum it is sufficient to
include H
2
O absorption in the radiative transfer equation. Since even in the atmo-
spheric window scattering is relatively weak, very often scattering processes are
completely neglected in the entire infrared spectral region. In this case the source
function J
±
of the RTE reduces to J
m
±,2
with ω
0
= 0.
Neglecting all scattering processes yields a relatively simple form of the RTE
which has a particularly simple solution. This motivates us to present the solution
of the RTE for two different cases. In the first case, scattering and absorption
processes are included while in the second case only absorption is accounted for.
According to the above discussion, the first case applies to the short-wave region
and to the atmospheric window, whereas the second case may be applied to the
infrared spectral region excepting the atmospheric window.
2.6.1 The scattering atmosphere
Inspection of the system (2.101) shows that the RTE is an inhomogeneous first-order
differential equation of the form
dy
dx
+ P(x)y(x) + Q(x) = 0, y(x = x
0
) = y
0
(2.107)
with the general solution
y(x) = y
0
exp
−
x
x
0
P(t)dt
−
x
x
0
Q(x
)exp
−
x
x
P(t)dt
dx
(2.108)
Let τ
g
denote the total optical depth of the medium. By comparing (2.101) with
(2.107) we may substitute:
(a) Upwelling radiation
x = τ , x
0
= τ
g
, x
= τ
y(x) = I
m
+
(τ ), P(x) =−M
−1
, Q(x) = M
−1
J
m
+
(τ )
(b) Downwelling radiation (2.109)
x = τ , x
0
= 0, x
= τ
y(x) = I
m
−
(τ ), P(x) = M
−1
, Q(x) =−M
−1
J
m
−
(τ )
2.6 Solution of RTE for a horizontally homogeneous atmosphere 53
According to (2.108) the formal solution for the upwelling and downwelling
radiation is then given by
I
m
+
(τ ) = exp
−M
−1
(τ
g
− τ )
I
m
+
(τ
g
) +
τ
g
τ
exp
−M
−1
(τ
− τ )
M
−1
J
m
+
(τ
)dτ
I
m
−
(τ ) = exp
−M
−1
τ
I
m
−
(0) +
τ
0
exp
−M
−1
(τ − τ
)
M
−1
J
m
−
(τ
)dτ
(2.110)
These solutions are called formal solutions of the RTE because they do not represent
explicit expressions for I
m
+
and I
m
−
. This is due to the fact that the source terms J
m
+
and J
m
−
implicitly contain the radiance field I
m
+
and I
m
−
as follows from (2.103).
Some remarks concerning the so-called exponential matrices appearing in
(2.110) may be helpful. A few important rules for these matrices are stated
below.
exp
(
A
)
=
∞
k=0
1
k!
A
k
,exp
(
O
)
= E
exp
(
A
)
exp
(
B
)
= exp
(
A + B
)
,exp
(
A
)
exp
(
B
)
= exp
(
B
)
exp
(
A
)
(2.111)
Thus exponential matrices do not commute. If A is a diagonal (s × s) matrix, i.e.
A = (λ
i
δ
ij
), then
exp
(
A
)
=
exp(λ
i
δ
ij
)
=
exp(λ
1
)0... 0
0 exp(λ
2
) ... 0
.
.
.
.
.
.
.
.
.
.
.
.
00... exp(λ
s
)
(2.112)
2.6.2 The nonscattering atmosphere
Now we will solve the radiative transfer equation in an atmosphere where only
absorption takes place. In this case the scattering coefficient and, thus, the single
scattering albedo ω
0
vanishes so that (2.53) reduces to
µ
d
dτ
I (τ,µ) = I (τ, µ) − B(τ ) (2.113)
For a given frequency interval the Planck function B depends on temperature only.
The functional notation B(τ ) simply implies that B has to be evaluated using the
temperature existing at τ . Due to the fact that the interior long-wave sources as
well as the radiation field at the boundaries of the medium are isotropic, the long-
wave radiance field is axially symmetric with respect to the z-axis. Therefore,
the radiative transfer equation does not depend on the azimuth angle ϕ so that, in
54 The radiative transfer equation
contrast to the solar spectral region, a Fourier cosine expansion of the radiance is not
necessary.
Splitting the radiation field into upwelling and downwelling radiation yields
µ
d
dτ
I
+
(τ,µ) = I
+
(τ,µ) − B(τ )
µ
d
dτ
I
−
(τ,µ) =−I
−
(τ,µ) + B(τ )
(2.114)
From (2.114) it is seen that, in contrast to (2.86), the system of ordinary differential
equations for I
+
and I
−
is decoupled. The reason for this is that the multiple
scattering term, which couples the upwelling and downwelling radiation field, has
been neglected. Hence it is possible to obtain an analytical solution of (2.114).
This solution requires proper boundary conditions at the top and the bottom of the
atmosphere.
By comparing (2.114) with (2.107) we obtain the following correspondences
(a) Upwelling radiation
x = τ , x
0
= τ
g
, x
= τ
y(x) = I
+
(τ ), P(x) =−
1
µ
, Q(x) =
1
µ
B(τ )
(a) Downwelling radiation (2.115)
x = τ , x
0
= 0, x
= τ
y(x) = I
−
(τ ), P(x) =
1
µ
, Q(x) =−
1
µ
B(τ )
The boundary conditions at the lower and upper boundary of the atmosphere are
given by
I
+
(τ
g
,µ) = B
g
, I
−
(0,µ) =
B(0) if a black body is present at τ = 0
0 else
(2.116)
B
g
is the Planckian black body radiation of the ground. Substituting (2.115) into
the general solution (2.108) we obtain for the up- and downwelling radiances at the
reference level τ :
I
+
(τ,µ) = B
g
exp
−
τ
g
− τ
µ
+
1
µ
τ
g
τ
B(τ
)exp
−
τ
− τ
µ
dτ
I
−
(τ,µ) = I
−
(0,µ)exp
−
τ
µ
+
1
µ
τ
0
B(τ
)exp
−
τ − τ
µ
dτ
(2.117)
These equations can be interpreted as follows: the first term on the right-hand side
of each equation describes that part of the radiation which is emitted at the boundary