Extending Shuey’s approximation using Taylor expansions for forward and inverse modelling

Abstract

As seismic imaging moves towards the imaging of more complex media, properly modelling elastic effects in the subsurface is becoming of increasing interest. In this context, elastic wave conversion, where acoustic, pressure (P-) waves are converted into elastic, shear (S-) waves, is of great importance. Accounting for these wave conversions, in the framework of forward and inverse modelling of elastic waves, is crucial to creating accurate images of the subsurface in complex media. The underlying mechanism of wave conversion is well understood and described by the Zoeppritz equations. However, as these equations are highly nonlinear, approximations are commonly used. The most well-known of these approximations is Shuey’s approximation. However, this approximation only holds for small angles and small contrasts, making it insufficient for realistic forward and inverse modelling scenarios, where angles and contrasts may be large. In this paper we present a novel set of approximations, based on Taylor expansions of the Zoeppritz equations, which we name the extended Shuey approximations. We examine the quality of these approximations to the Zoeppritz equations and compare them to existing approximations described in literature. We then apply these extended Shuey approximations to the elastic full-wavefield modelling algorithm for a simple, synthetic, 1.5-D example, where we show that we can accurately model the P- and S-wavefields in a forward modelling case. Finally, we apply our approximations to the elastic full-wavefield migration algorithm for a simple, synthetic, 1.5-D example, where we show that we can recover an accurate image in an inverse modelling case.