1i11Eih12

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). (K. ) = [

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-2TieTihRe(EihEie)

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-2~e~,Jm(E:hEie)

Ti~IEie12

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(7.4.7)

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The quantities Tih and Tie are practically real since K:~ z ; and for convenience, we shall just take their respective real parts when calculating (7.4.7). The second term in (7.4.5) is the source term for the incoherent wave and corresponds to a single scattering of the coherent wave into incoherent wave. The elements of the 4 x 4 phase matrix P (0, cP; 0'Y' cP'Y) are P u (0, cP; 0'Y' cP'Y) =

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~ [sin

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sin 0'Y

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+ cos

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cos 0'Y cos( cP - cP'Y)]2

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(7.4.8) (7.4.9)

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P12 (O,cP;0'Y,cP'Y) =

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~Cos20sin2(cP-cP'Y)

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H3 (0, cP; 0'Y' cP'Y) = -

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~ [sin

. cos sin( cP - cP'Y)

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P21 (0, cP; 0'Y' cP'Y) =

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~ cos2 0'Y sin 2(cP 3

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sin 0'Y

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+ cos

cos 0'Y cos( cP - cP'Y)] (7.4.10)

cP'Y)

(7.4.11)

(7.4.12)

P22 (B, cP; B'Y' 'Y) = "2 cos (cP - 'Y)

(B, ; B'Y' 'Y)

~ cos B'Y sin( -

'Y) cos( - 'Y)

(7.4.13)

4 DMRT for Active Remote Sensing

P31 (O, ;0"Y'cP"Y) = 3 cos0"Y sin( - "Y) . [sin 0 sin 0"Y + cos 0 cos 0"Y cos( cP - cP"Y)] P3dB, ;O-y,cP"Y) = -3cosOcos(cP-cP"Y)sin(cP-cP-y) P33 (0, cP; 0"Y' cP"Y) =

(7.4.14) (7.4.15)

~ [sin 0 sin 0"Y cos( cP ~ [sin OsinB"Y cos(cP -

cP"Y)

+ cosBcosB"Ycos2( - "Y)]

P44 (0, cP; 0"Y' cP"Y) = P 14 cP"Y)

+ cos

(7.4.16) (7.4.17) (7.4.18)

cos 0"Y]

= P24 = P34 = P41 = P42 = P43 = 0

The phase matrix is identical to the Rayleigh phase matrix of the conventional radiative transfer theory. This is because of the small-particle assumption and because the pair distribution functions are only correlated for the range of a few diameters. At z = 0 the boundary condition for the Stokes vector is, from (7.3.52)(7.3.57)

I. (z, 1f - 0, cP)

where R(B) =

= R (0) I. (z, 0, cP)

(7.4.19)

[IRh~e)I'

IR,+)1

",,(Rh(~)R:(e)) -lm(Rh~')R:('))]

1m (Rh(O)R;UJ)) Re (Rh(O)R; UJ))

(7.4.20)

where Re(O) and Rh(O) are given by (7.2.19)-(7.2.20), respectively, with K z = K cos 0 and k z = (k 2 - K 2 sin2 0)1/2. The dense media radiative transfer equation (7.4.5) is to be solved subject to the boundary condition (7.4.19). After the solution is calculated, the incoherent Stokes vector that is transmitted back into region 0, 'Io (0 0 , cP), with B = sin- 1 (K' sin O/k) according to Snell's law, is o

'Io(Bo, o) = T(B)'I(z = O,B, ) For 0 less than critical angle Oe (= sin- 1 (k/K')) we have

T (B) =

(7.4.21)

l-I R h(OW o o

o o o o

- cos 0 1m cosO

ccO:s~ Re( Th(O)T;(O))

cos 00 1m (Th (O)T: (0)) cos

0 (T (O)T* (0))

cco:sO Re ( Th (O)Te (0) ) *

(7.4.22)

7 DENSE MEDIA SCATTERING

Th(B) = 1 + Rh(B) and Te(f)) = 1 + Re(B). For () greater than the critical angle Be, T( B) = O. The dense media radiative transfer equation (7.4.5) with boundary conditions (7.4.19)-(7.4.20) resembles that of conventional radiative transfer equation of independent scattering except that the expressions for "'e and w in terms of physical parameters are different from those of the conventional theory because of correlated scattering and the fact that the effective propagation constant K has been taken into account. The procedure of numerical solutions of the dense media transfer equation follow the same procedure as that of the conventional ones as described in Volume 1.

General Relation between Active and Passive Remote Sensing with Temperature Distribution

Relations exist between the brightness temperatures of passive remote sensing and the bistatic coefficients of active remote sensing when the medium is of uniform temperature. The expressions were derived in Volume I, 3, Section 5. In this section, we derive a similar expression for nonuniform temperature distribution. An expression will be derived relating the thermal radiated field intensity of a medium with nonuniform temperature distribution to the electromagnetic fields in the problem of active remote sensing. Specifically, it is shown that the thermal radiated field intensity is equal to a constant times the integration of the product of the temperature distribution and the divergence of the Poynting's vector for the problem of active remote sensing. The derivation is based on the field equations of a medium of discrete scatterers and the fluctuation-dissipation theorem. The key quantity to utilize in the derivation is the dyadic Green's function of the random medium, which is common to both problems and is also reciprocal. In the following, active and passive remote sensing problems will be labeled as Problems A and B, respectively.

Problem A: