Elastic modeling by phase-shift cascade with scattering matrix control
Chanpen Silawongsawat, Gary F. Margrave
A new numerical technique for elastic modeling in a stratified medium is introduced and applied in the following procedure. A surface point source emits a pulsed cylindrical P (or S) wave in a stratified medium (x-z plane). Using Fourier analysis, the cylindrical wave is decomposed into a sum of harmonic plane waves and the numerical operations are defined in f-k domain. The technique iteratively uses phase shift extrapolation in depth to propagate elastic potentials and modified-for-potential Zoeppritz equations to compute reflection, transmission and conversion coefficients. All multiples and mode conversions are generated and may be turned off via scattering-matrix modification. The spectral result is obtained by cascading each plane wave (each f-k component) through a computational grid and, then, transformed to yield 2-D elastic seismograms in time-space domain. Options for free surfaceeffects and displacement conversions are made before the inverse Fourier transformation.
The program is built in the Matlab environment and illustrated with three simple models. The results from two single-interface models show the method produces head waves. The cylindrical PP reflection coefficients extracted from synthetic data are shifted and smoothed compared to the plane wave analytic from Zoeppritz equations. Multiples and converted modes from another model are very realistic though there are some algorithmic artifacts have not been completely solved yet. The method works and is fast and stable. It produces seismograms with reasonable agreement to theory and with some interesting characteristics which need to be investigated more. The program itself has great potential for many extensions and applications including 3-D synthetics, anisotropy, anelasticity and wave effects at any specific range of depth.