Combining Volume Scattering Simulations with Radiative Transfer in .NET

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Combining Volume Scattering Simulations with Radiative Transfer
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Figure 7.1.1 An elemental volume element V with arbitrary cross section area A and length ds for a random medium containing discrete scatterers. The elemental volume element contains many particles.
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in clusters. We also describe the iterative approach for solving the multiple scattering equations. Forward scattering amplitude in conjunction with Foldy's approximation has been used to calculate the extinction rate. It is shown that this is only applicable if the N-particle forward scattering amplitude is calculated to the second order. It is also shown that the coherent wave approaches a delta function in the forward direction in the limit of a large volume and large number of scatterers (large N). Thus, the coherent field intensity has to be subtracted from the total scattered intensity to give (a) the phase matrix which describes the scattering of energy away from the forward direction and (b) the extinction coefficient which describes the wave decay in the forward direction. We perform Monte Carlo simulations by solving the wave equations and averaging over many realizations.
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Combining Simulations of Collective Volume Scattering Effects with Radiative Transfer Theory
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In this section we redefine the scattering and extinction parameters of radiative transfer theory to include the collective scattering behavior of particles. This distinguishes the definition from the conventional definition of averaged single-particle scattering behavior. In radiative transfer theory we consider a specific intensity incident upon an elemental volume with cross-sectional area A and length ds (Fig. 7.1.1). The cross-sectional shape is arbitrary. It is an elemental test volume in the development of the radiative transfer equation. It is not small in the sense of the wave equation. The elemental volume V = Ads needs to obey the
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7 VOLUME SCATTERING SIMULATIONS
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following criteria: (i) V ~ >.3 or V~>.3. In other words, V = NV>.3, where N v is a number of moderate size. (ii) There is a large number of particles, N, contained in V. The particles are randomly positioned. Criteria (i) and (ii) are necessary to ensure that the coherent multiple scattering in volume V will include random phase fluctuations so that the energy transfer in and out of volume V can be averaged. Furthermore, if .the particles are of different sizes, shapes, and permittivities in volume V, averaging of particle scattering behavior will also be included in the radiative energy transfer within elemental test volume V. Criteria (i) and (ii) establish a lower limit of volume V. Let E i be the incident field in the direction k = (Oi, <lJi), and let E s be i the scattered field from volume V of N particles.
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E i = exp(iki . r) (7.1.1)
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The scattered field can be decomposed into coherent field (E s ) and incoherent field Es . (7.1.2) The scattered field E s is a statistical field that contains the coherent multiple scattering within volume V. From one elemental volume V to most other elemental volumes, the far field of E s can be taken. We have
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(7.1.3)
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where k s = (e", cP,,) is the scattered direction. In (7.1.3), F(k", ki ) represents the N-particle collective scattering amplitude. It can be decomposed into coherent component (F(k s , ki )) and incoherent component F(k s , ki ). (7.1.4) The optical theorem can be applied to N scatterers, which can be viewed as a single object called N-scatterer. Thus, for the case of nonabsorptive scatterers we have 41f , , (7.1.5) kIm { F(k i1 ki)} = dn s [F(k s1 ki)1 2
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The phase function P(k s ' ki) and the scattering coefficient f'i,s are next defined to take into account the collective scattering behavior of N particles.
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