A Hybrid Collocation Method For Calculating Complete Theoretical Seismograms In Vertically Varying Media

Abstract

A numerical method is presented for calculating complete theoretical seismograms, under the assumption that the earth models have velocity, density and attenuation profiles which are arbitrary piece‐wise continuous functions of depth only. Solutions for the stress‐displacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. Our method for solving the resulting two‐point boundary value problems differs from that of other investigators in three ways. First, collocation is used in traditionally troublesome situations, e.g. for highly evanescent waves, at turning points, and in regions having large gradient in material properties. Second, in some situations (high frequencies and small gradients) P and S‐waves decouple and we use a different solution method for each wave type, instead of trying to force a single method to find all solutions. For example, above the P‐ and S‐waves turning points an approximate fundamental matrix may be used for each wave type. At the P‐wave turning point, the fundamental matrix may be used for the S‐wave components but collocation is used for the P‐wave. Between the P‐ and S‐wave turning points collocation is used for the evanescent P‐wave and the fundamental matrix is used for the S‐wave. At the S‐wave turning point and below, collocation is used for both. Third, the computational algorithm chooses the appropriate solution method and depth domain upon which it is employed based upon a specified error tolerance and the known inaccuracies of the various approximations employed. Once solutions of the boundary value problems are obtained, a Fourier—Bessel transform is then applied to get back into the space‐time domain. Copyright © 1986, Wiley Blackwell. All rights reserved