Decomposition and Inversion of Elastic Reflection Data; First-Order Angular Dependence and Applications

Abstract

The elastic wave displacement equation is transformed into pressure-stress coordinates, where the Born approximation of the Lippman-Schwinger equation in the Fourier-transform domain is employed to decompose the observed fields into their scattered components: P-P, P-S, S-P, and S-S. Triple Fourier transforms of the scattered elastic wave data are linear combinations of the double Fourier transforms of the relative changes in the medium properties. Angular-dependent reflection coefficients for each of the scattering modes are constructed, and an inversion algorithm is outlined. Inversion of the observed elastic wave fields is accomplished in a manner similar to the acoustic problem. Density, bulk modulus, and shear modulus variations in an elastic earth can be recovered by utilizing the angular-dependent information present in the observed wave fields. Examples illustrate these points. Transforming the elastic wave data back to displacement coordinates and assuming a compressional source, an analysis of recorded amplitudes yields some practical answers about converted-wave data. Significant amounts of P → S data should typically be generated by compressional sources, with significant contributions at smaller angles, However, signal-to-noise calculations suggest that more sweeps and more geophone channels at longer offsets will typically be necessary to get P-S sections of comparable quality to P-P sections.