Approximations for traveltime, slope, curvature, and geometric spreading of elastic waves in layered transversely isotropic media

Abstract

Each seismic body wave, including quasi-, P-, S-, and converted wave modes, carries useful subsurface information. For the processing, imaging, amplitude analysis, and forward modeling of each wave mode, we need approximate equations of the traveltime, slope (ray parameter), and curvature as a function of offset. Considering the large offset coverage of modern seismic acquisitions, we have developed new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis. Our approximation for traveltime is a modified version of the extended generalized moveout approximation that comprises six parameters. Our direct approximations for the ray parameter and curvature use new algebraically simple equations with three parameters. We define these parameters for each wave mode without ray tracing so that we have similar approximate equations for all wave modes that only change based on the parameter definitions. However, our approximations are unable to reproduce S-wave triplications that may occur in some strongly anisotropic models. Using our direct approximation of traveltime derivatives, we also obtain a new expression for the relative geometric spreading. We determine the high accuracy of our approximations using numerical tests on a set of randomly generated multilayer models. Using synthetic data, we show simple applications of our approximations for the normal moveout correction and relative geometric spreading compensation of different wave modes.