Extended formulation for Rayleigh-wave computations at high frequencies in a spherical layered earth

Summary. The earlier formulation of dispersion function, amplitude response and surface ellipticity for Rayleigh waves in a spherical layered earth has been extended by the reduced-delta matrix to avoid computational instability at high frequencies. We derive a modified formulation, where the layer thickness is kept separated during matrix multiplications. The present formulations are valid for an unlimited frequency range. A numerical experiment with single precision (six decimal digits accuracy) on an IBM 360/44 computer is performed at frequencies from 0.4 to 12.5 Hz on a Shield structure down to a depth of 1090 km below which a complete sphere is assumed. The experiment shows that both reduced-delta matrix and modified formulations generate roots of the dispersion equation without any reduction of layers. However, to avoid unnecessary computation, a layer-reduction procedure has been derived from a modified formulation when the contribution of deep layers becomes insignificant on surface waves.     

Windowed f-k spectra of a three-dimensional wavefield excited by a point source in a two-dimensional multilayered elastic medium

When full 3-D modelling is too costly or cumbersome, computations of 3-D elastic wave propagation in laterally heterogeneous, multilayered 2-D geological structures may enhance considerably our ability to predict strong ground motion for seismological and engineering purposes. Towards this goal, we extend the method based on the combination of the thin-layer finite-element and boundary-element methods (TLFE-BEM) and calculate windowend f - k spectra of the 3-D wavefield. The windowed f - k spectra are spatially localized spectra from which the local properties of the wavefield can be extracted. The TLFE-BEM is particularly suited for calculating the complete wavefield where surface waves are dominant in multilayered media. The computations are performed in the frequency domain, providing the f - k spectra directly. From the results for the 3-D wavefield excited by a point source in a 2-D multilayered, sloped structure, it can be said that the phase velocity of the fundamental-mode Rayleigh wave in a laterally heterogeneous multilayered medium, estimated from the windowed f - k spectra, varies with the location of the point source. For the model calculated in this article, the phase velocity varies between the value for the flat layered structure of the thick-layer side and that for the structure just under the centre of the window. The exact subsurface structure just under the centre of an array in a laterally heterogeneous medium cannot be obtained if we use the f - k spectral analysis assuming a flat layered structure.