Some useful approximations to generalized ray theory for regional distance seismograms

Summary . Seismograms recorded at regional distances (2°–12°) are quite complicated due to the waveguide nature of the crust. Generalized ray theory can be used to model the body waves in this distance range but a very large number of rays is required. Here I present a series of approximations to streamline generalized ray theory for the waveguide problem. If a layer over a half-space is used for the structure, then the de Hoop contour for a given ray is most strongly dependent on the fastest velocity of any leg of the ray. This results in analytic approximations to locate the contour. Each ray has two body wave arrivals (a headwave and a reflected arrival) so the displacement response of the ray need only be evaluated at a few points in time about the two arrival times and interpolated in between. A change in structure (increasing crustal thickness or Pn velocity) most strongly affects the relative timing of the headwave and the reflected arrival, so it is possible to 'stretch' or 'squeeze' the waveform of a representative model to simulate a whole suite of models.
Also discussed is the applicability of a single layer over a half-space structure for modelling the observed regional distance waveforms for shallow earthquakes. At periods greater than a few seconds crustal layering can be replaced by a single layer having the appropriate average velocities. Lateral variations in crustal thickness with scale lengths of less than about 100 km can also be modelled with a simple horizontal layer of appropriate average thickness.     

Three-dimensional geometrical ray theory and modelling of transmitted seismic energy of data from the Nevada Test Site

This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

2-Dreflectivity method and synthetic seismograms for irregularly layered structures -I. SH-wave generation

Summary. The reflectivity method for complete SH seismograms has been extended to two-dimensionally layered structures. The Aki-Larner technique is generalized to solve the integral equations for 2-D boundary conditions, and propagator matrices are enlarged to express a total SH wavefield. Synthetic seismograms in a soft basin are calculated for an incident plane-wave. They compare favourably with the results of the finite-element and finite-difference methods even in the later portion where asymptotic ray and beam theories break down. Synthetic seismograms due to a line force and a point dislocation are also presented.