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A new method is presented for computing the complete elastic response of a vertically heterogeneous half-space. The method utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series. The series representation is exact if summed to infinity and consequently eliminates the need to integrate a continuous Bessel transform numerically. In practice, a band-limited solution is obtained by truncating the series at large wavenumbers. The vertical and time dependence of the wave motion is obtained as the solution to a system of partial differential equations. These equations are solved numerically by a combination of finite element and finite difference methods which accommodate arbitrary vertical heterogeneities. By using a reciprocity relation, the wave motion is computed simultaneously for all source-observer combinations of interest so that the differential equations need only be solved once. A comparison is made, for layered media, between the solutions obtained by discrete wavenumber/finite element, wavenumber integration, axisymmetric finite element, and generalized rays.
Olson et al. (Tue,) studied this question.