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In this section, we study the scattering of electromagnetic waves by lossy dielectric surfaces with large permittivity This has broad applications in natural media For wet soil, the relative permittivity can be as high as 17 + i20 at 14 GHz For ocean surfaces, the permittivity can be as high as 397 +i402 at 14 GHz For lossy dielectric rough surfaces with high permittivity, there can be rapid spatial variations of the dielectric medium Green's function and surface fields Also the wave can propagate from a point on the rough surface to another point on the rough surface through the lossy medium Based on physical reasons, the surface fields can have large spatial variations within short distances These large spatial variations can also be attributed to fine scale geometric irregularities on the surface or due to large permittivity of the lower medium.

From the point of view of spatial frequency, the high spatial frequency components can interact with the low spatial frequency components Thus a dense grid is needed to discretize the surface fields In 5, we have developed the physics-based two-grid (PBTG) method In PBTG, two grids are used: a dense grid and a sparse grid The sparse grid is that of the usual 8 to 10 points per wavelength The dense grid ranges from 16 or higher number of points per wavelength depending on the relative permittivity of the lossy dielectric medium The surface fields are calculated on the dense grid In the formulation of the surface integral equations, two Green's functions are used The free space Green's function and the Green's function of the lossy dielectric medium.

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Although the surface fields have large spatial variations, what need to be calculated are the convolution of the surface fields with the two Green's functions The PBTG is based on two observations: (1) the Green's function of the lossy dIelectric is attenuative (spatial limited), and (2) the Green's function of free-space is slowly varying on the dense grid (spatial frequency limited) The first observation results in a sparse matrix for the Green's function of the lossy.

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an average of NeA c times where A e is the acceptance rate. Step 5: Count the frequency of occurrence of different pair separations. For each configuration so reached, we will count, Cf](r), the number of particles of size Uj having their centers located in a spherical shell with inner radius r and outer radius r + dr, where r is measured from a tagged particle I of size ai during the simulation cycle of realization t. The subscripts i and j are indices of particle sizes, the superscript I is for labeling the particle, and the superscript t is the index of realization. These tabulations are carried out for a number of divisions of r between R ij and 5Rij , where R ij' = ai + aj. The thickness of the spherical shell dr will then be the smallest length over which the pair distribution functions are resolved. The above process is repeated for a large number of times to generate many different realizations and to record the frequency of occurrence of different pair separations. Thus, the average number of particles of size Uj contained within a spherical shell of thickness dr at a distance r from a particle of size ai, averaged over all particles of size Ui and total T realizations, can be expressed as

L L CfJ(r)

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