GEOPHYSICS, VOL. 56, NO. 12 (DECEMBER 1991); P. 2057-2069, 14 FIGS. Ray tracing in 3·0 complex isotropic media: An analysis of the problem Jean Virieux* and Veronlque Farra* ABSTRACT Procedures for accurate ray tracing in complex three-dimensional media with interfaces are proposed. The ray tracing equations and the associated paraxial linear equations are solved either by a numerical solver or by an analytical perturbation approach. Interfaces are described with an explicit representa- tion or an implicit representation using B-spline inter- polation. For the implicit representation, we exploit two important properties of B-splines, the convex hull and subdivision properties, in order to determine the intersection of the ray with the interface. At the free surface where the recording system is located, a sampling strategy is proposed: limits of branches at caustics, shadow zones, and medium boundaries are detected for a fixed azimuth while the take-off angle is automatically adjusted in order to have a roughly homogeneous spacing between end points of the rays. The same strategy is also possible for a fixed take-off angle. The assumed continuity of the traveltime surface between two adjacent azimuths enables one to obtain the initial condition of a ray arriving at any station located on the portion of surface delimited by these two azimuths. This procedure al- lows for the classification of rays arriving at a given station as we show on different synthetic examples. INTRODUCTION Tracing rays in a three-dimensional medium is a formida- ble problem which has many applications in seismology. Introducing complicated interfaces considerably increases the difficulty. In order to reduce the problem to a tractable one, different simplifications are usually made in the param- eterization of either the velocity or the interfaces. Chiu et al. (1986) or Haas et al. (1987),among others, assumed constant velocities between interfaces. These interfaces are simply described by explicit sinusoidal or polynomial functions of the horizontal coordinates and do not allow for complex shapes. These different simplifications result in fast ray- tracing methods, but their impact on the tomographic image, for example, are difficultto analyze. Several research groups (Cerveny and Psencik, 1983; Cerveny, 1987; Pereyra et al., 1980; Pereyra, 1988; Gjoystdal et al., 1984; Hanyga, 1988) have attempted to go one-step further and solve with good accuracy the ray-tracing problem in three-dimensional media with interfaces. In this article, we analyze the classes of difficulties met in ray tracing and we investigate methods to solve them in an efficient way. The first step is to solve the ray-tracing equations. Analytical perturbation techniques proposed by Farra (1990) or Virieux (1991) are emphasized as an efficient and accurate alternative to numerical solvers. We must also check intersections with interfaces and compute them when they occur. The intersection problem is strongly related to the geometrical description of interfaces and the assumed complex shape of interfaces leads us to introduce implicit representations. Finally, we investigate a first attempt to find rays arriving at a station for different branches of the traveltime surface. Before addressing these problems, let us summarize briefly the ray theory and the associated first-order linear technique called the paraxial approximation. RAY-TRACING THEORY AND THE PARAXIAL APPROXIMATION Tracing rays inside a medium is a powerful tool for extracting information, because the computed quantities (traveltime, slowness vector, polarization vectors, and am- plitudes) are related to simple quantities in a seismogram and are perfectly associated with different features of the me- dium. In this section, we only introduce the notations we use in this paper. In a three-dimensional medium, the rays can be found by solving the eikonal equation, (\I T) 2 = U 2 = C -2 , where c is the wave speed and u is the corresponding slowness. The Manuscript received by the Editor September II, 1990; revised manuscript received June 5, 1991. *Laboratoire de Sismologie, Institut de Physique du Globe de Paris, 4 Place Jussieu, F.75252 PARIS cedex 05. © 1991 Society of Exploration Geophysicists. All rights reserved. 2057 Downloaded 07 Nov 2010 to 86.211.8.78. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/