Analysis of reflection traveltimes in 3-D transversely isotropic heterogeneous media

Abstract

A two‐point ray‐tracing technique for rays reflected from irregular, but smooth, interfaces in 3-D transversely isotropic heterogeneous media is developed. The method is based on Chebyshev parameterization of curved segments of the reflected rays, of the reflectors, and of the velocity and anisotropy distributions in the model. Chebyshev approximation also can describe the reflection traveltime surfaces to compress traveltime data by replacing them with coefficients of the corresponding Chebyshev series. The advantage of the proposed parameterization is that it gives traveltime as an explicit function of the model parameters. This explicitly provides the Frechét derivatives of the traveltime with respect to the model parameters. The Frechét derivatives are used in two ways. First, a two‐term Taylor series is constructed to relate variations in the model parameters to the corresponding perturbations in the traveltimes. This makes it possible, based on the results of a single ray tracing in a relatively simple model, to predict traveltimes for a range of more complicated models, without any additional ray tracing. Second, singular‐value decomposition of the Frechét matrix determines the influence of various model parameters on common‐source and common‐midpoint traveltimes. The singular‐value analysis shows that common‐source traveltimes depend mainly on the reflector position and shape. The common‐midpoint traveltimes also contain additional information about lateral velocity heterogeneity and anisotropy. However, both of these parameters affect the traveltimes in similar ways and so usually cannot be determined separately.