B, ; B', ')Iu ( e', ')

(3.5.14) Representing (3.5.13) and (3.5.14) by input and output relations and summing contributions from all directions of Id(e', ') and I u (e', ') to the (7f e, ) direction, we have

(7f -

B, )

D(){ {7r/2 de' sin e' {27r d '

()',

')Id ( e', ') + P( 7f - e, ; e', ')Iu ( B', ')]} (3.5.15)

Thus we have the following difference equation for input and output:

Id(e, , z - D) - Id(e, , z) = (Output - Input) in direction (7f - e, )

cos()

dB'sinB' {27r d '

[P(7f - e, ; 7f - e', ')Id(e', ', z) + P( 7f - e, ; e', ')Iu(e', ', z)] }

Based on (3.5.16), one can write a differential equation: - cose dz Id(e, , z)

(3.5.16)

7r /2 dB' sin B'

B', ')Id(()', ', z)

(3.5.17)

+ P(7f - e, ;e', ')Iu(e', ',z)]

be obtained similar to (3.5.16) and (3.5.17):

I ll (f),<p, z) - Iu(f), <p, z - D)

D = -f) {

I ll(f), <p, z)

+ j7f/2 df)' sin f)' j27f d<p'

[P(f), <p; f)', <p')Ill (f)', <p', z) cosf) dzIll(f),<p,z)

+ P(f), <p; 7f -

f)', <p')Id(f)', <p', z)]}

(3.5.18)

/'i,e I ll(f), <p, z)

r/ r + J df)' sin f)' Jo d<p' P(f), <p; f)', <p')Ill (fJ', <p', z)

2 27f [

+ P(f), <p; 7f -

f)', <p')Id(f)', <p', z)]

(3.5.19)

Equations (3.5.16) and (3.5.18) are the difference forms of the radiative transfer equations, while (3.5.17) and (3.5.19) are the differential forms.

(1) The differential form of radiative transfer equations in (3.5.17) and (3.5.19) can be misleading since it suggests an equation of intensity changes at a point in space. However, as we see from previous sections, it is the wave equation which governs wave propagation in space. The result of the difference equation in (3.5.16) is obtained by assuming l D and >. < D lmfp. The conditions >. < D and l Dare necessary because in such a limit of a layer D, averaging can be taken over random particle positions and over a spatial extent of several to many wavelengths. The condition D lmfp is ne ~ded because it means that one can represent input and output relations of a thin layer D by the first- and second-order scattering solutions. Thus (3.5.17) has to be interpreted in the sense that the distance has been calibrated with the distance scale D. (2) In microwave remote sensing problems, usually A, l lmfp, so that it is possible to find a distance scale D such that >., l < D lmfp with the appropriate radiative transfer equation. However, in problems of strong localization, >. :::; [mfp, the distance scale D such that >., [ < D lmfp does not exist. (3) We are able to derive simple expressions for the phase function because of the isotropic scattering of the point particle models. In problems of complicated particle shapes and correlation of particle positions, it is generally difficult to analytically determine the phas ~ function. In Volume II, Monte Carlo methods have been employed to determine solutions

of Maxwell's equations. However, to use Maxwell's equations to solve a problem of layer thickness d lmfp is a computationally formidable problem. The development in this section justifies the useful alternative that has been adopted in Volume II. Divide the problem into "thin layers" of thickness D with A, l < D lmfp. Use Maxwell's equations to calculate the input and output relations of layer of thickness D problem. Then use the cascading of layers approach to find the overall solution to include the multiple scattering for the large thickness d problem. By using D such that A, l, < D lmjp, we have carried out exte...nsive Monte Carlo simulation in Volume II to calculate the extinction coefficient and the phase matrix. (4) Based on (3.5.16), we can interpret the phase function as the averaged bistatic coefficients of a thin layer.