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(8.1.103)
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(8.1.104)
1.5 Second-Order Scattering from Isotropic Point Scatterers
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Figure 8.1.12 Wave incident on a half-space of isotropic point.
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The iterative solution will be carried to second order. For fL > 0 the zeroth-order solutions are
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1(0) (T,
(8.1.105a)
fL, ) = 0
(8.1.105b)
The first-order solutions are
(1) _
eT//-Li
(T,fL, ) - 471'fL
1/fL+ 1/fLi
(eT//-L i eT//-L )
(8.1.106)
(8.1.107)
fL, ) - 471"fL
1/fLi -l/fL
The second-order iterative solution can be calculated readily by substituting (8.1.106) and (8.1.107) into (8.1.103) and (8.1.104). The solution of the outgoing specific intensity at T = 0 is, to second order,
= O,fL, ) = 1(1)(T = O,fL, ) +1(2\T = O,fL, ) = 471"fL
1I fL
+ 1I fLi
( w)
471"
271" _
dil' _'_ fL'
+ ~ + -, /-L, /-L 8 1 108 (-1.. + -1..) (-1.. + 1) (-1.. + 1) ( .. ) /-Li /-L' /-Li /-L /-L' /-L
The first-order solution is proportional to the albedo w, and the second2 order solution is proportional to w In general, the nth-order solution is n . Thus, the iterative solution converges quickly for small proportional to w albedo when scattering is dominated by absorption. We also note that there is an integration over angles for the second-order solution because in the
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
scattered direction
Figure 8.1.13 Double scattering processes involving two particles.
double scattering involving two particles, the direction of propagation from the first particle to the second particle can be arbitrary (Fig. 8.1.13). Thus, the general nth-order solution will involve an (n - 1)th-fold integration.
Discrete Ordinate-Eigenanalysis Method
Radiative Transfer Solution for Laminar Structures
For media with laminar structure, scattering only couples in two directions, the upward-going intensity I((), z) and its specular downward-going counterpart I(1f-(), z) at the same angle (). The two polarizations are uncoupled. The result is a set of coupled equations for the upward and downward intensities that can be solved analytically. Let I u denote upward-going specific intensity, and let Id denote downward-going specific intensity. Denoting the scattering region as region 1, we have, for 0 < () < 1f /2, d 1 cos () dz I u = -/'i,e1u + /'i,aCT(z) + 'i/'i,s(Pj I u + Pb Id) (8.2.1)
d 1 cosO dz Id = /'i,e1d - /'i,aCT(z) - 'i/'i,s(Pj Id + Pb I u )
(8.2.2)
where the subscript u denotes upward; d, downward; j, forward; and b, backward. Extinction is the sum of absorption and scattering /'i,e = /'i,a + /'i,s For horizontal polarization we have
/'i,s
8kl~lz 1 + 2kl~l; cos 2 0 = cos () 1 + 4k'2 [2 cos 2 ()
(8.2.3)
(8.2.4)
Pb = 1 + 2kl21Z2 COS2 0
1m z
2.1 Radiative Transfer Solution for Laminar Structures
:;--;'adiometer
Region 0 Region 1 I I
Figure 8.2.1 Thermal emission of half-space of laminar structure.
where kim denotes the real part of kIm. For vertical polarization we have
/'i,s
6k' l [ cos 2() ] 1m Z 1 + 2 2 2 2 cos () 1 + 4kimlz cos ()
(8.2.5)
2 cos 2() 2 2 2 1 + 4kimlz cos () + cos 2() For both cases of polarization we have
( 6) 8.2.
PI = 2 - Pb
(8.2.7)
We illustrate the solution for the case of thermal emission of a half-space laminar medium (Fig. 8.2.1). The following nonuniform temperature profile is assumed
T(z) = To
+ Th e'Y z
(8.2.8)
A typical subsurface temperature profile of the Antarctica is shown in Fig. 8.2.2. The temperature profile in the Amundsen-Scott Station in Antarctica [Lettau, 1971] is fitted with the exponentials as follows: For December 31 (summer) we have T l (z) = 222 + 34eo. 81z for August 31 (winter), we have
T2 (z) = 222 - 10eO. 37z
and for April 1 (autumn), we have T3(Z) = 222 + 81 eO. 51z
88eo. 66z
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
260,-----.,----,.--,------,--,----,---,-----.,----,.---, 255
g240
~23O
215 I
210'----'------'---'-----'----'----'---'-----'------'---' o 2 4 7 10 3 5 6 8 9
Depth (m)
Antarctica.
Subsurface temperature distributions at the Amundsen-Scott Station in
~ 220
T 3 (z)
a. E
CJ) CJ)
T2 (z)
Frequency (GHz)
Figure 8.2.3 Brightness temperature (without scattering) as a function of frequency for
the three temperature distributions shown in Fig. 8.2.2: (~m
1.8(0' (~m
0.00054 0'
2.1 Radiative Transfer Solution for Laminar Structures
240.------.-----.--,------r-----,--,---,----,-------,
220~--~======----=-=-=--=:-::=-
.5 = 0.002
180 160
--==
TE .5 = 0.002
.E OJ
.;;:
3l c::
140 120 100
Observation angle (deg)
Figure 8.2.4 Brightness temperature as a function of viewing angle for TE and TM waves at 20 GHz, E~m = 1.8E o , E~m = 0.00054E o , lz = 2 mm, and T = Tl (z).
where z is in meters. The equations for the specific intensities are two coupled first-order differential equations that can be solved exactly. We find the following solution for the brightness temperature for the temperature distribution of (8.2.8).
TB(()o) = (
+ I\;a
21\;)(1
- rOl a - I\;a
{To +
+ I cos ()
(8.2.9)
where rOl is the Fresnel reflectivity at the interface z = 0, and
a = [l\;a(l\;a
+ I\;sPb)] 1/2
(8.2.10)
In the absence of scattering (0 = 0), the brightness temperatures for the three temperature distributions are illustrated in Fig. 8.2.3. By comparing Figs. 8.2.2 and 8.2.3, we note that high-frequency emission originates primarily from the surface while subsurface emission dominates the low-frequency brightness temperatures. In Fig. 8.2.4, the brightness temperature as a function of viewing angle is plotted for the case of <5 = 0 (no scattering) and 0 = 0.002, where 0 is the variance of permittivity fluctuations. We note that the presence of scattering induces darkening, because scattering hinders the emission from reaching the radiometer.