Multilevel expansion of the sparse-matrix canonical grid method for two-dimensional random rough surfaces

Wave scattering from two-dimensional (2-D) random rough surfaces up to several thousand square wavelengths has been previously analyzed using the sparse-matrix canonical grid (SMCG) method. The success of the SMCG method highly depends on the roughness of the random surface for a given surface area. We present a multilevel expansion algorithm to overcome this limitation. The proposed algorithm entails the use of a three-dimensional (3-D) canonical grid. This grid is generated by a uniform discretization of the vertical displacement along the height (z-axis) of the rough surface in addition to the uniform sampling of the rough surface along the x-y plane. The Green's function is expanded about the 3-D canonical grid for the far interactions. The trade-off in computer memory requirements and CPU time between the neighborhood distance and the number of discretization levels along the x-axis are discussed for both perfectly electric conducting (PEC) and lossy dielectric random rough surfaces. Ocean surfaces of the Durden-Vesecky (1985) spectrum with various bandlimits are also studied.