Difficulties with computing anelastic plane-wave reflection and transmission coefficients

Abstract

SUMMARY Consider a plane homogeneous wave incident upon an interface between two anelastic halfspaces. Computing the plane-wave displacement reflection and transmission coefficients requires determining the proper signs of the vertical slownesses of all the reflected and transmitted waves. In certain cases, this is not straightforward. Previous work has shown that choosing the signs by applying the elastic radiation condition results in certain vertical slownesses, and hence coefficients, varying discontinuously and unphysically with the angle of incidence, and that choosing the signs so that the vertical slownesses vary continuously can also produce errors. We suggest three approaches for treating these cases. In the first approach, the signs are chosen so that the vertical slownesses vary continuously up to a certain angle of incidence (close to the elastic critical angle), with the elastic radiation condition being applied beyond that. We show that this is actually just an extension of the elastic radiation condition to complex-valued squared slownesses. In the spherical wave case in which a point source lies in the upper half-space, the approach also agrees with the results obtained by applying the saddle point method to approximate the integral for the reflected wavefield. This approximation is just geometrical ray theory. The first approach also produces coefficients with unphysical aspects, but these are confined to the critical zone, where geometrical ray theory is not valid anyway. Outside of this zone, the coefficients compare quite well with the elastic ones. In the second approach, real values of the horizontal slowness are used to compute the coefficients. This results in coefficients which are continuous and have no unphysical aspects, but do not compare as well with the elastic ones. The third approach involves using other paths in the complex plane (of horizontal slowness) to evaluate the coefficients. We show two examples. The first example is an approximation which involves replacing certain vertical slownesses with their complex conjugates before the coefficients are computed. This is equivalent to evaluating the horizontal slowness along a curve just above the real axis. The second example involves using a path which yields coefficients that are almost identical to the elastic ones. The third approach also results in continuous coefficients with no unphysical aspects, which compare well with the elastic ones. The errors mentioned above suggest to us that the problem of how to correctly compute plane-wave anelastic reflection and transmission coefficients, even in the apparently simple case of a homogeneous incident plane wave, still requires further study.