On the adequacy of plane-wave reflection/transmission coefficients in the analysis of seismic body waves

In order to estimate the effect (on body waves) of discontinuities within the Earth, it is common practice to use the theory for plane waves incident upon the plane boundary between two homogeneous half-spaces. The resulting reflection/P~ V conversion/transmission coefficients are shown here to he inaccurate for many problems of current interest. Corrected coefficients are needed, in particular, for cases where the discontinuity (upon which boundary conditions are to be applied) is near a turning point of the P- or S-wave rays, or if one of these rays intersects the discontinuity at a near-grazing angle. Adequate corrections, based upon the Langer approximation to a full wave theory, are shown to be easily derived in practice. The method is first to write out the plane-wave coefficients as a rational polynomial, in sines and cosines of the angles of incidence upon the boundary, and second to introduce a multiplicative factor for each cosine. The new factors depart from unity only when the associated cosine tends to zero; i.e., when a turning point is approached. They incorporate all the corrections required for curvature of the boundary, frequency dependence, and Earth structure (velocity gradients) near the boundary.