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# Introduction

Analytical representation of traveltime equations is necessary for efficient velocity estimation. For isotropic media, traveltimes as a function of offset and common midpoint (CMP) are given by a simple analytical equation [the double-square-root equation (DSR)] in homogeneous media Yilmaz and Claerbout (1980). In vertically inhomogeneous media, the velocity in this analytical equation is replaced by its root-mean-squared (RMS) average velocity. The DSR equation is often used to implement efficient prestack time migration Bancroft and Geiger (1994); Karrenback and Gardner (1988); the kind that can be used for iterative velocity estimation.

However, velocities estimated based on the isotropic assumption have seldom provided us with the full story; seismic depth mis-ties with well logs, among other shortcomings, have resulted from such a medium restriction. Some of these shortcomings can be attributed to the presence of complicated lateral velocity variations, however, many can only be explained by the presence of anisotropy Banik (1984).

Since no analytical relation between the group velocity and ray angle exists, traveltimes in transversely isotropic media with a vertical symmetry axis (VTI media), even for the homogeneous case, are often calculated numerically Richards (1960). Such a limitation has restricted parameter estimation in VTI media, especially with regard to using prestack time migration. Equations for anisotropic media are better represented using plane waves, with phase velocities that can be described analytically as a function of propagation direction. Efficient treatment of plane waves, however, is only possible in the Fourier domain. The main drawback of this domain is the loss of the lateral position information, and so the Fourier domain cannot efficiently treat media with lateral inhomogeneity. With stationary phase approximations Alkhalifah (1997b), we can, however, obtain analytical representations of traveltime in the space-time domain from the well-known analytical equations in the Fourier domain.

In this paper, I derive approximate analytical equations that describe traveltime as a function of offset and midpoint in VTI media. Though an approximation, its accuracy far exceeds any previous representations, even for the case of horizontal reflections. This equation, Cheop's pyramid for VTI media, will be used to implement efficient space-time domain Kirchhoff time migration. For vertically inhomogeneous media, average equivalent velocities and anisotropic parameters are used in the analytical equation. The accuracy of these equations are demonstrated on synthetic data.

Next: Anisotropic media parameterization Up: Alkhalifah: Analytical traveltimes in Previous: Alkhalifah: Analytical traveltimes in
Stanford Exploration Project
7/5/1998