Exact synthetic seismograms for an inhomogeneity in a layered elastic half-space

Summary. The propagation of a pulsed elastic wave in the following geometry is considered. An elastic half-space has a surface layer of a different material and the layer furthermore contains a bounded 3-D inhomogeneity. The exciting source is an explosion, modelled as an isotropic pressure point source with Gaussian behaviour in time.
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves.     

Simple analytical Green's functions for ray perturbations in layered media

The total Green's function for two-point boundary-value problems can be related to the propagator for initial-value problems. A very simple expression for the Green's function is obtained when the unperturbed medium may be described by material with a constant gradient in quadratic slowness. The derivation requires a correct understanding of assumptions made in the propagator solution. Expressions are also obtained for Green's function in multilayered media.     

Displacement, strain and stress fields due to shear and tensile dislocations in a viscoelastic half-space

Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.     

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.