Generalized Dix equations for a layered transversely isotropic medium

Abstract

In standard Dix equations, normal-incidence traveltimes and rms or NMO velocities for PP reflections are used to estimate interval velocity and thickness for homogeneous isotropic horizontal layers. For a stack of homogeneous horizontal layers that are transversely isotropic with a vertical symmetry axis (VTI), the qPqP traveltimes (and the qPqSV traveltimes) depend on four parameters and the thickness of each layer. These can be estimated from the traveltimes of the qPqP and qPqSV reflected waves. The estimated qPqP traveltime parameters are P-wave normal-incidence traveltime, P-wave NMO velocity, and a heterogeneity factor entering in the shifted hyperbolic traveltime approximation or in a continued fraction traveltime approximation. The estimated qPqSV traveltime parameters are PS normal-incidence traveltime and PS NMO velocity combined with the qPqP traveltime parameters. These are used to compute S-wave normal-incidence traveltime and SV-wave NMO velocity. The estimated traveltime parameters are entered into explicit expressions for the four layer parameters and the thickness of each layer. These layer-recursive formulas, except for slightly different initial conditions, are also valid for ocean-bottom seismic data. For reflected SH-wave traveltimes, there are two elastic parameters and thicknesses per layer, and we can estimate three traveltime parameters. However, two of these traveltime parameters are functionally related, and it is impossible to estimate the three layer parameters from SH-wave traveltimes only.