Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

The seismic body wave tomography method has been improved and extended to adapt to a general velocity structure with a number of complexly shaped seismic velocity discontinuities (SVDs) and with three-dimensional variations in the velocities in the modeling space. An efficient three dimensional ray tracing algorithm which iteratively uses the pseudobending technique and Snell's law is developed. The large and sparse system of observation equations is solved by using the LSQR algorithm. This method is applied to 18,679 arrival times from 470 shallow and intermediate-depth earthquakes in order to study P and S wave tomographic images beneath northeastern Japan. In addition to first P and S wave arrivals, clear later arrivals of SP waves converted at the Moho and PS and SP waves converted at the upper boundary of the subducted Pacific plate (UBPP) are also used in the inversion. The UBPP, Conrad and Moho are taken as three SVDs, and their depth distributions obtained by previous studies are used. High-resolution P and S wave tomographic images down to a depth of 200 km have been determined. Large velocity variations amounting to 6% for P wave and 10% for S wave are revealed in the crust and upper mantle. In the crust low-velocity (low-V) zones exist beneath active volcanoes. In the upper mantle the low-V zones dip toward the west from the volcanic front. A high-velocity (high- V) zone corresponding to the subducted Pacific plate is clearly delineated. Most earthquakes in the lower plane of the double-planed deep seismic zone are found to occur in relatively high-V areas. The obtained tomographic images are also found to explain other seismological observations well.

Tomographic imaging of P and S wave velocity structure beneath northeastern Japan

Previous work on the simultaneous inversion method has been improved and extended to incorporate iterative solution for earthquake locations and laterally heterogeneous structure. Approximate ray tracing and parameter separation are important elements of the improved method. Application of the method to P wave arrival time data recorded by stations of the U.S. Geological Survey Central California Network yields a three-dimensional model for the velocity structure of the upper crust in an area encompassing the rupture zone of the Coyote Lake earthquake of August 1979. Very strong correlations between the velocity model and the geology and gravity and magnetic anomalies are observed. Improved estimates of the locations of earthquakes in the study area are also determined. The relocation of explosions indicates epicentral accuracies of the order of a kilometer or better. Based on the revised hypocentral locations, it is concluded that the San Andreas fault is vertical in this area, with no actual offset between the epicenters and the fault trace. In contrast, the Calaveras has two (or more) active fault surfaces, one nearly vertical and another dipping 75° to the northeast.

Earthquake locations and three-dimensional crustal structure in the Coyote Lake Area, central California

On etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d'ordre arbitraire sur des grilles a une dimension d'espacement arbitraire

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Generation of finite difference formulas on arbitrarily spaced grids

A new approximate algorithm for two-point ray tracing is proposed and tested in a variety of laterally heterogeneous velocity models. An initial path estimate is perturbed using a geometric interpretation of the ray equations, and the travel time along the path is minimized in a piecewise fashion. This perturbation is iteratively performed until the travel time converges within a specified limit. Test results show that this algorithm successfully finds the correct travel time within typical observational error much faster than existing three-dimensional ray tracing programs. The method finds an accurate ray path in a fully three-dimensional form even where lateral variations in velocity are severe. Because our algorithm utilizes direct minimization of the travel time instead of solving the ray equations, a simple linear interpolation scheme can be employed to compute velocity as a function of position, providing an added computational advantage.

A fast algorithm for two-point seismic ray tracing

A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays.

A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology.

In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method

Tables of coefficients for high order accurate, compact approximations to the first ten derivatives on and at the midpoints of uniform nets are presented. The exact rational weights are generated and tested by means of symbolic manipulation implemented through MACSYMA. These weights are required in the application of deferred corrections to new methods for solving higher order two point boundary value problems.

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H. B. Keller

Papers

Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method

Symbolic generation of finite difference formulas