Status of the Rayleigh scattering method in three-dimensional electromagnetic modeling

Abstract

The Rayleigh scattering method offers distinct advantages in geophysical modeling over differencing or integral formulations but an inherent error in the Rayleigh hypothesis has severely limited its wider application. The method can be numerically implemented to be very fast and topographic and subsurface geometries can be described simply by digitized surface and interface values. This contrasts with gridding throughout a volume as used in finite difference and finite element methods. However, the error in the Rayleigh method prevents the modeling of steep topography and very high subsurface slopes such as the vertical sides of prismatic bodies. The inherent Rayleigh error, or famous Rayleigh ansatz (approximation), exists because the formulation assumes that scattering above a rough surface is described by a superposition of upward-scattered plane waves or plane wavelets only. This does not allow multiple scattering with downward components that should occur in topographic depressions. Manifestations of this omission are non-convergence behavior, numerical instabilities, and incorrect results. Valid results are obtained for plane wave EM sources normally incidence on buried 2-D and 3-D structures having interface slopes as high as 50-60°. In the far-field, above the topography, the surface slope limit can be extended to over 74° by adapted-regularization. In the near-field, on 2-D topography, surface slopes of only 26° are properly modeled for the TM (transverse magnetic) mode. Since the physics of 3-D scattering can be dominated by TM-like (galvanic) effects, near-field Rayleigh EM scattering from 3-D topography is probably limited to slopes of about this value unless regularization schemes are used. Regularization and the identification and reduction of specific numerical instabilities are considered the keys to extending the Rayleigh method to general 3-D EM modeling. New ways of extending the validity of the Rayleigh scattering method justify more interest in the geophysical EM community especially in 3-D subsurface and airborne (i.e., far-field) modeling.