A Generalized Reflection-transmission Coefficient Matrix and Discrete Wavenumber Method for Synthetic Seismograms

Expressions for displacements on the surface of a layered half-space due to point force are given in terms of generalized reflection and transmission coefficient matrices (Kennett, 1980) and the discrete wavenumber summation method (Bouchon, 1981). The Bouchon method with complex frequencies yields accurate near-field dynamic and static solutions. The algorithm is extended to include simultaneous evaluation of multiple sources at different depths. This feature is the same as in Olson's finite element discrete Fourier Bessel code (DWFE) (Olson, 1982). As numerical examples, we calculate some layered half-space problems. The results agree with synthetics generated with the Cagniard-de Hoop technique, P· SV modes, and DWFE codes. For a 10-layered crust upper mantle model with a bandwidth of 0 to 10 Hz, this technique requires one-tenth the time of the DWFE calculation. In the presence of velocity gradients, where finer layering is required, the DWFE code is more efficient. INTRODUCTION Economic near-field solutions of a point source in a layered half-space are important in the fields of seismology and earthquake engineering. Recently, many approaches have been proposed to evaluate the layered half-space response. For example, there are generalized ray theory (Heimberger, 1968; Heimberger and Harkrider, 1978), reflectivity method (Fuchs and Muller, 1971), reflection and transmission coefficients matrix method (Kennett, 1974, 198Q; Apsel, 1979; Kennett and Kerry, 1980), discrete wavenumber method (Bouchon, 1981), and discrete wavenumbers-finite element method (DWFE) (Olson, 1982), among others. In this paper, a generalized reflection-transmission matrix and discrete wavenumber method for near-field synthetic seismograms is proposed. This approach is based on Kennett's reflection and transmission matrix method for the wavenumber integrands (1974, 1981) and the discrete wavenumber summation method (Bouchon, 1981) for the wavenumber integration. The reflection-transmission matrix is an effective procedure to evaluate the wavenumber integrand. Phase-delayed reflection and transmission coefficients are used which are slightly different than Kennett's expressions (1980). The algorithm includes simultaneous evaluation of the Green's functions of multiple sources at different depths. INTEGRAND EXPRESSIONS The displacement integrands on the free surface for buried source problems given by Kennett and Kerry (1979) in equation (5.22) are The notation used is that of Kennett and Kerry (1979). Slightly different relations are used for the reflection and transmission coefficients, which except for differences 1685 1686 Z. X. YAO AND D. G. HARKRIDER in normalization, are given by their equation (4.26). The relations and additional definitions are found in the Appendix. R is the reflection coefficient matrix on the free surface, (Mu + MvR) is the receiver function matrix (Heimberger, 1974).