The P-SV inverse problem for a layered elastic medium: what data are required?

Abstract

Summary In an earlier paper we described a method whereby the elastic parameters of a horizontally layered medium could be determined from a matrix of P-SV reflection seismograms for obliquely incident plane waves. We suggested that in order to obtain the required matrix, it would be necessary to measure the response, at many locations along a line, due both to a compressional source and to a source generating shear waves. By slant stacking such data, we could simulate the plane wave response at some slowness, and hence obtain the plane wave reflection seismograms for the structure, enabling us to use the algorithm described by Clarke to retrieve the wave speeds, density and thickness for each layer. In this paper we consider the precise nature of the seismic sources required, and the data to be collected, in order to construct the P-SV reflection matrix. We show that it is possible to retrieve this information from three sets of data, each consisting of measurements, along a line, of the vertical or radial component of the response to a source possessing an appropriate degree of cylindrical symmetry. A certain amount of care is required, however, in choosing the three source/receiver combinations, since certain such pairs of data sets are not independent, and must be counted as a single set. As an example of three data sets which are independent, and hence sufficient, we show that measurements of the vertical and radial displacements caused by an explosive source, as well as the vertical component of the response to a vertical force, enable us to deduce the P-SV reflection seismograms, and hence to perform the structural inversion.