Ray tracing in 3-D complex isotropic media: An analysis of the problem

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.

[1]  H. B. Keller,et al.  Fast Seismic Ray Tracing , 1983 .

[2]  J. Virieux,et al.  Ray tracing for earthquake location in laterally heterogeneous media , 1988 .

[3]  Raul Madariaga,et al.  Seismic waveform modeling in heterogeneous media by ray perturbation theory , 1987 .

[4]  J. Lane,et al.  A generalized scan line algorithm for the computer display of parametrically defined surfaces , 1979 .

[5]  Klaus H. Jacob,et al.  Three‐dimensional seismic ray tracing in a laterally heterogeneous spherical Earth , 1970 .

[6]  Victor Pereyra,et al.  Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method , 1980 .

[7]  H. Inoue.  A least-squares smooth fitting for irregularly spaced data; finite-element approach using the cubic B-spline basis , 1986 .

[8]  Bjørn Ursin,et al.  Traveltime and wavefront curvature calculations in three-dimensional inhomogeneous layered media with curved interfaces , 1984 .

[9]  L. Johnson.  Some mathematical topics in seismology by Robert Burridge, Courant Institute of Mathematical Sciences, New York University, 317 pp, 1976, $9.75 , 1978 .

[10]  James D. Foley,et al.  Fundamentals of interactive computer graphics , 1982 .

[11]  V. Farra.  Amplitude computation in heterogeneous media by ray perturbation theory: a finite element approach , 1990 .

[12]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[13]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[14]  Jean Virieux,et al.  Ray perturbation theory for interfaces , 1989 .

[15]  Paul G. Richards,et al.  Quantitative Seismology: Theory and Methods , 1980 .

[16]  Victor Pereyra.  Two-point Ray Tracing In Complex 3-D Media , 1988 .

[17]  J. Virieux.  Fast and accurate ray tracing by Hamiltonian perturbation , 1991 .

[18]  R. K. Luneburg,et al.  Mathematical Theory of Optics , 1966 .

[19]  Chris Chapman,et al.  Ray theory and its extensions; WKBJ and Maslov seismograms , 1985 .

[20]  V. Červený,et al.  Computation of Geometric Spreading of Seismic Body Waves in Laterally Inhomogeneous Media with Curved Interfaces , 1974 .

[21]  Tom Lyche,et al.  Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics , 1980 .

[22]  T. Lyche,et al.  Making the Oslo algorithm more efficient , 1986 .

[23]  The Maslov Method and the Asymptotic Fourier Transform: Caustic Analysis. , 1980 .

[24]  Malcolm Sambridge,et al.  Boundary value ray tracing in a heterogeneous medium: a simple and versatile algorithm , 1990 .

[25]  G. Müller.  Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media , 1984 .

[26]  Stephen K. Chiu,et al.  Three-dimensional determination of structure and velocity by seismic tomography , 1986 .

[27]  Samuel D. Conte,et al.  Elementary Numerical Analysis: An Algorithmic Approach , 1975 .

[28]  V. Červený.  Ray tracing algorithms in three-dimensional laterally varying layered structures , 1987 .

[29]  Bryan D. Stone,et al.  Optimal interpolants for Runge-Kutta ray tracing in inhomogeneous media , 1990 .

[30]  Robert Burridge.  Some mathematical topics in seismology ; 1974 - 1975 , 1976 .

[31]  G. Deschamps,et al.  Ray techniques in electromagnetics , 1972 .