Diffusion Approximation in .NET

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4 Diffusion Approximation
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when s points into the scattering medium. This is because diffuse intensity comes from scattering and must originate from inside the scattering medium. However, we cannot impose (8.4.37) under the diffusion approximation because the diffusion approximation has an intensity that is close to independent of the direction of S, and the diffusion approximation is an approximation on the diffuse equation. A common approximate boundary condition is as follows. Let n be the normal on the boundary that points inward into the scattering medium, the approximate boundary condition is
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F = Fnn + Ftt
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(8.4.38)
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where the integration is over 271" solid angle that includes those > O. We use (8.4.15) and (8.4.38) and so
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s such that
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(8.4.39)
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where t is tangential to the boundary. We then have
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Note that
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r d8(8' n) {U + ~Fnn. 8 + ~Ftt. 8} 471" 471"
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(8.4.40)
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Thus the approximate boundary condition of (8.4.38) is Fn 7I"U + 2 = 0 or
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dS(8' n)U = U
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d8(S'
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n)(n s) =
7r/2
dO sinO
127r
dcjJcosO = 7I"U
(8.4.41)
dO sinO
dO sin 0
dcjJ cos O(sin 0 cos cjJ)
dcjJcos 2 0
(8.4.42) (8.4.43)
7r/2
127r
271"
~U(-) + F(r) . n = 0 2 r 471"
() 8.4.44
for r on the boundary. Note that F(r) can be obtained from U(r) by using (8.4.33). Thus, the diffusion approximation consists of solving (8.4.34) subject to the boundary condition of (8.4.44) and also using (8.4.33) for
F(r).
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
REFERENCES AND ADDITIONAL READINGS
Bellman, R. and G. M. Wing (1975), An Introduction to Invariant Imbedding, John Wiley & Sons, New York. Chandrasekhar, S. (1960), Radiative Transfer, Dover, New York. Deirmendjian, D. (1969), Electromagnetic Scattering on Spherical Polydispersions, Elsevier, New York. EL-Rayes, M. A. and F. T. Ulaby (1987), Microwave dielectric spectrum of vegetation, Part I: Experimental observations, IEEE Trans. Geosci. Remote Sens., 25, 541-549, 1987. Fung, A. K. (1994), Microwave Scattering and Emission Models and Their Applications, Artech House, Norwood, Massachusetts. Glisson, A. W. and D. R. Wilton (1980), Simple and efficient numerical techniques for treating bodies of revolution, Technical Report 105, Engineering Experiment Station, University of Mississippi, University, Mississippi. Harrington, R. F. (1968), Field Computation by Moment Method, Macmillan, New York. Ishimaru, A. (1978), Wave Propagation and Scattering in Random Media, 1 and 2, Academic Press, New York. Jin, Y.-Q. (1994), Electromagnetic Scattering Modelling for Quantitative Remote Sensing, World Scientific, London. Joseph, J. (1990), Application of integral equation and finite difference method to electromagnetic scattering by two dimensional and body of revolution geometries, Ph.D. thesis, Department of Electrical and Computer Engineering, University of Illinois at UrbanaChampaign, Urbana, Illinois. Lane, J. and J. Saxton (1952), Dielectric dispersion in pure liquids at very high radio frequencies, Proc. Roy. Soc., A213, 400-408. Laws, J. O. and D. A. Parsons (1943), The relationship of raindrop size to intensity, Trans. Am. Geophys. Union, 24, 452-460. Lettau, H. (1971), Antarctic atmosphere as a test tube for meteorological theories, in Research in the Antarctic, 443-375, Am. Assn. for the Adv. of Sci., Washington, DC. Karam, M., and A. K. Fung (1988), Electromagnetic scattering from a layer of finite length, randomly oriented dielectric circular cylinders over a rough interface with application to vegetation, Int. J. Remote Sensing, 9, 1109-1134. Marshall, J. S. and J. W. M. Palmer (1948), The distribution of rain drops with size, J. Meteorol., 5, 165-166. Newton, R. W. and J. W. Rouse, Jr. (1980), Microwave radiometer measurements of soil moisture content, IEEE Trans. Antennas Propagat., 28, 680-686, 1980. Saxton, J. A. and J. A. Lane (1952), Electrical properties of sea water, Wireless Engineer, 29, 269-275. Sarabandi, K. (1992), Derivation of phase statistics of distributed targets from the Mueller matrix, Radio Sci., 27(5), 553-560. Tsang, L. and K. H. Ding (1991), Polarimetric signatures of a layer of random nonspherical discrete scatterers overlying a homogeneous half space based on first and second order vector radiative transfer theory, IEEE Trans. Geosci. Remote Sens., 29(2), 242-252. Tsang, L. and J. A. Kong (1977), Thermal microwave emission from a random homogeneous layer over a homogeneous medium using the method of invariant imbedding, Radio Sci., 12, 185-194.