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matrix with successive iterates and could therefore present a major computational bottleneck. To speed up such calculations, the concept of canonical grid (CG) is introduced. The essential nature of CG is that it is translationally invariant. In rough surface scattering problems, the CG is usually taken to be the mean flat surface. By translating the unknowns to the CG, the weak interactions can be performed simultaneously for all unknowns using FFT. This reduces memory requirements from O(N 2 ) to O(N) and operation counts from O(N 2 ) to O(Nlog N). We also introduce the physics-based two-grid (PBTG) method for dealing with lossy dielectric surfaces. In this method, a dense grid suitable for the lower half-space and a coarse grid suitable for the upper half-space are chosen. By taking advantage of the attenuative nature of the Green's function in the lower half-space and the slowly varying nature of the Green's function in the upper half-space with respect to the dense grid, one can achieve the accuracy of a single dense grid with the computational efficiency of a single coarse grid. Other fast methods discussed and illustrated in 5 include the steepest descent fast multipoles method (SDFMM) and the method of ordered multiple interactions (MOMI). In contrast to rough surface scattering, volume scattering involving dense distributions of discrete scatterers is often a full-fledged 3-D scattering problem. The additional degree of freedom makes direct simulations of scattering coefficients rather difficult. Radiative transfer theory is commonly used for such problems, but the conventional approach fails to take into account of coherent multiple interactions between the scatterers. A better approach is to perform the scattering simulations on a test volume that contains a large number of scatterers but forms only a small part of the whole system. Coherent interactions are captured through the simulated extinction coefficients and phase functions, which can then be used in the dense medium radiative transfer equation (rigorously derived in Volume III) to solve the large-scale problem. These concepts are discussed in 7, where idealized randomly distributed point scatterers are used to illustrate the methods. The multiple scattering problem is formulated using the Foldy-Lax self-consistent equations. In a dense medium, the correlation of scatterer positions could significantly affect the scattering results. The pair-distribution function quantifies the two-particle correlation property of the scatterers. In 8, we introduce the Percus-Yevick equation for the pair-distribution function and give closed-form solutions for hard and sticky spheres. For Monte Carlo simulations, statistical realizations of scatterer configurations are needed. Two
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methods are commonly employed to generate the particle positions: sequential addition and Metropolis shuffling, the latter method being more efficient when the particles are very closely packed. We show simulation results of the pair distribution functions for hard spheres and spheroids as well as sticky spheres. The simulated pair distribution functions are found to compare well with the Percus-Yevick pair distribution functions. Before dealing with 3-D dense media scattering, it is instructional to first study, in 9, the simpler problem of 2-D dense media scattering, where the volume scatterers are chosen to be infinitely long cylinders. We describe analytical pair distribution function and Monte Carlo simulations of particle positions in the 2-D case. The Foldy-Lax multiple scattering equations are then used to simulate extinction coefficients for densely packed hard and sticky cylinders. Finally, the SMCG method used in rough surface scattering is generalized to the volume scattering simulations. In 10, we perform 3-D dense media scattering calculations with dielectric spheres and spheroids. The volume integral equation approach as well as the T-matrix approach based on the Foldy-Lax equations are described in details. Simulation results for the extinction coefficients and phase matrices are shown and compared with analytical approximations. In 11, we describe the novel correlation phenomenon in random media scattering known as the memory effect, which manifests itself in wave scattering through the angular correlation function (ACF). ACF has been discussed in 6 of Volume I in the context of single scattering by point scatterers. Here, we provide a general derivation of the memory effect based on the statistical translational invariance of the random medium. The special property of ACF for random medium makes it a good candidate for the detection of a target embedded in random clutter. We explore such ideas by studying targets buried under rough surface and volume scatterers. The subject of multiple scattering by finite cylinders has important applications in the remote sensing of vegetation as well as signal coupling among multiple vias in high frequency circuits. In 12, we consider scattering by vertical cylinders in the presence of reflective boundaries, which introduce additional complications. We discuss Monte Carlo simulations of these systems as well as simple analytical results that take into account of first and second order scattering. In 13, more realistic modeling of vegetation structures through stochastic Lindenmayer systems are presented. We compare scattering results from such systems obtained using the methods of DDA, the coherent addition approximation, and independent scattering.
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