Computational aspects of synthetic seismograms for layered media

Abstract

A modified direct global matrix method for computation of synthetic seismograms in a layered viscoelastic media has been developed by combining information obtained from continuous and discrete formulations. The resulting code is stable for vertical seismic profiling as well as normal horizontal profiling and has proven efficient in the generation of full-scale seismograms. The wave field is first computed in the wave slowness domain. A proper integration back to the frequency domain is a critical part of a forward modeling code. By considering the discrete wavenumber formulation it is shown that numerical integration with equally spaced integration points corresponds to a fictitious cylindrical reflective surface with diameter inversely proportional to the sampling interval. It is demonstrated that for large ranges Filon integration is a natural extension. It is demonstrated that a trapezoidal Filon integration is more accurate than higher-order Filon schemes, which generate earlier spurious arrivals. In order to yield seismograms without spurious arrivals, the wavenumber or slowness sampling must be sufficiently dense. However, in some regions, the reflectivity function is quite regular. In these regions, linear interpolation is adequate and the number of sampling points is reduced by an adaptive scheme. Finally, in the slowness domain as opposed to the wave-number domain, the position of the peaks in the reflectivity function is nearly independent of frequency. This enables an efficient vectorization over frequency.