Broad-band synthetic seismograms for a spherical inhomogeneity in a many-layered elastic half-space

Abstract

The propagation of elastic waves in a many-layered elastic half-space is considered. One of the layers contains a bounded inhomogeneity which in the numerical applications is taken as a sphere. The half-space is excited by an explosion, modelled as a sudden isotropic point source. First the time-harmonic problem is solved using the null field approach (the T-matrix method). Starting from surface integral representations containing the free space Green's dyadic and field expansions in plane and spherical vector wave functions, the null field approach leads to a set of algebraic equations whose solution can be given in a form that naturally has a multiple scattering interpretation. The null field approach thus has a building-block structure, i.e. the transition matrix of the inhomogeneity and the reflection and transmission coefficients of the interfaces are used as parts in the total solution. It is noted that in the absence of the inhomogeneity the null field approach yields a more decoupled set of equations than the classical methods. The main practical limitation of the null field approach, which is, in principle, an exact method, is that only low and intermediate frequencies can be treated. In the numerical applications the highest frequency corresponds to a sphere diameter of about 12 S wavelengths. The transformations from the frequency domain to the time domain is performed with an FFT algorithm and to limit the bandwidth appropriate filters are placed at the receivers. The numerical examples show synthetic seismograms consisting of data from 10 observation points at increasing distances from the source. The layers have been chosen relatively thick so that the reflections from the different interfaces can be separated in time. As long as the reflections from the spherical inhomogeneity are also separated in time from other reflections they can mostly be well recognized. As the distance between the source and inhomogeneity increases, it becomes progressively more difficult to see any influence of the inhomogeneity in the seismograms. (Less)