EFFICIENT COMPUTATION OF FREQUENCY-WAVENUMBER KERNELS FOR SYNTHETIC SEISMOGRAMS OF HORIZONTALLY LAYERED ELASTIC MEDIUM

Abstract

This study focuses on the efficient generation of synthetic seismograms for the class of linearlyelastic horizontally-layered half-space media excited by a kinematic shear dislocation point source. A convenient way to do so is to transform the governing wave equations from timespace (t-r) domain to first frequency-space (f-r) domain and then to frequency-wavenumber (fk) domain. These transformations make it convenient to obtain the analytical solution of Green’s functions (viz., the dynamic response of the aforementioned medium due to an impulsively applied concentrated load of unit magnitude at a given location within the medium) in the f-k domain. This solution is then inverted back to the frequency-space domain (f-r) and finally to the time-space domain (t-r). Whereas the transformation from f-r to t-r domain involves a routine Fourier inversion (e.g., using FFT), the transformation from f-k to f-r domain involves numerical evaluation of improper integrals with respect to the wavenumber k. Since the f-k kernels (that occur in the integrands of the improper integrals) can be evaluated independently for any given set of frequencies and wavenumbers, parallelizing the computations of these kernels can significantly reduce the computing time of the corresponding integrals (and hence, the synthetic seismograms). In this study, C programs based on the OpenMP (open source) framework are developed for computing the f-k kernels over multiple wavenumbers considering a single frequency. The developed programs are validated and reductions in the computing times (with respect to serial-based programs) are quantified. Attempt has also been made towards understanding the behavior of the f-k kernels with respect to wavenumbers, in the simple case of a homogeneous elastic half-space, in order to arrive at the smallest maximum wavenumbers and wavenumber step-sizes for which the improper integrals can be accurately and efficiently evaluated.