Asymptotic Ray Theory Research Articles

Dynamic properties of reflected and head waves near the critical point

The classical problem of the incidence of spherical waves on a plane boundary has been reformulated from the computational point of view by providing a high frequency approximation to the exact solution applicable to any seismic body wave, regardless of the number of conversions or reflections from the bottoming interface. In our final expressions the ray amplitude of the interference reflected-head wave is cast in terms of a Weber function, the numerical values of which can be conveniently stored on a computer disk file and retrieved via direct access during an actual run. Our formulation also accounts for the increase of energy carried by multiple head waves arising during multiple reflections of the reflected wave from the bottoming interface. In this form our high frequency expression for the ray amplitude of the interference reflected-head wave can represent a complementary technique to asymptotic ray theory in the vicinity of critical regions where the latter cannot be used. Since numerical tests indicate that our method produces results very close to those obtained by the numerical integration of the exact solution, its combination with asymptotic ray theory yields a powerful technique for the speedy computation of synthetic seismograms for plane homogeneous layers.

Synthetic seismic sections of typical petroleum traps

Interpretation of seismic reflections remains a key problem in seismic exploration because reflections have complex behavior, especially near geologic structures. One method to gain an understanding of this complex behavior is to study synthetic seismic sections of models of typical petroleum traps as computed by zero‐offset ray tracing for primary P‐waves only. These synthetic sections have features of significant interpretative value to the practicing geophysicist, such as variations in reflection amplitudes and complexities in reflection‐time geometries. Asymptotic ray theory was applied to calculate reflection amplitudes, accounting for mode conversion and three‐dimensional geometric divergence of ray tubes in the presence of curvilinear interfaces. This suite of synthetic seismic sections illustrates the difficulties in making correct seismic interpretations of geologic structures and suggests three conclusions: (1) The customary assumption that seismic sections are simple images of geologic cross‐sections is an oversimplification and can lead to erroneous interpretations. (2) Variations in overlying strata produce marked disturbances in reflection amplitude and traveltime. (3) Critical reflections that provide key structural information are often difficult to recognize and are apt to be ignored, or misidentified, especially if trace processing has been cursory. A fundamental principle of interpretation is underscored by this study: CDP stacked seismic sections must be interpreted so as to be consistent with structural depth models because stacked sections are not simple images of geologic structures.

Weber function computation in the interference reflected‐head wave amplitude

In the critical region, the complex amplitude of the interfering system of a head wave and its associated reflected wave can be described by a single formula for all eight types of interference waves. The formula consists of transmission, reflection and head wave coefficients, geometrical parameters of the layers and an appropriate Weber function. The real and imaginary parts of the Weber function can be shown to be the solutions of a system of first order differential equations. The system may be solved numerically with the help of existing subroutines in the computer software libraries. Upon determining the Weber function, one may construct amplitude‐distance curves for the interference reflected‐head waves of any desired type. Some properties of the amplitude‐distance characteristics can be directly inferred from the Weber function. The very good matching between this method and asymptotic ray theory outside the critical region suggests that a combination of the two would provide an efficient technique for ray amplitude calculation.

Reflection and transmission coefficients for transversely isotropic media

abstract It has become necessary in seismology to consider more complicated models of the Earth's structure in order to obtain synthetic seismograms that are more consistent with actual field data. Gassmann (1964) and Postma (1955) have presented results dealing with travel-time methods in anisotropic media—in particular, transversely isotropic media. Kinematic properties alone, however, are not enough to conclusively interpret seismic records. Consequently, dynamic properties must be considered producing a need for synthetic seismograms. One of the most efficient methods for obtaining synthetic seismograms is through the use of asymptotic ray theory (Hron and Kanasewich, 1971; Hron, 1973; Hron, Kanasewich and Alpaslan, 1974). A necessary step in the implementation for layered media displaying transverse isotropy is the computation of reflection and transmission coefficients at the interface between two such layers. Reflection coefficients for a free interface and the corresponding surface conversion coefficients must be computed, as well. Theoretical formulas for reflection, transmission, and surface conversion coefficients corresponding to the zero-order approximation of asymptotic theory are presented for the above-mentioned media.

Body Wave Amplitude Calculations--II

A modification of asymptotic ray theory that replaces the ‘energy propagation in tubes’ determination of amplitude by a ‘disk superposition model of the wavefront’ leads to a simple construction technique for synthetic seismograms. The disk-ray theory (DRT) method generates arrivals based solely on the distance function Δ(p). The DRT computed arrivals compare very well with wave theory computations for caustics, reflections and refractions. The approximation of head-waves is adequate only at high frequencies. No approximation at all is made for the effects of waveguides, tunnelling or diffraction at spherical interfaces.

Modeling of atmospheric nuclear explosions over a mountainous region by vertical and horizontal single forces

Observed P and S waves from the respective Soviet and Chinese atmospheric explosions of October 30, and 1961 and October 14, 1970 are found to differ both in their S/P amplitude ratios and the form of the individual wavelets. This difference is attributed to the effect of the local topography on the ensuing ground motion at the source. Asymptotic ray theory indicates that an additional horizontal force component was acting at the source of the Chinese event while the Soviet event could be modeled by a vertical force only. Generalized ray theory is then applied to calculate the body wave forms in each case. The agreement between the theoretical results and the observations is excellent.

Teleseismic Response of a Multi-layered Dipping Media using Asymptotic Ray Theory

A plane wave solution in a multi-layered dipping medium with arbitrary dips is not available in practical form, but we show that algorithms based on a partial ray expansion using asymptotic ray theory presents no difficulties. The selection of multiple and converted phases is done automatically by the computer to minimize the error in the partial ray expansion below any desired level. For modest dips, the peak positions in the ratio of the vertical to the horizontal crustal transfer functions do not shift significantly. However, the relative peak amplitudes are modified considerably. Therefore, it seems feasible to use the ratio of transfer functions to obtain a suitable layered model, and dip determinations may be carried out as a secondary stage of iteration to improve the agreement between field experiment and model theoretical computations. As an example we obtain a crustal model, 35 km thick, near the Edmonton observatory with a steeply dipping sill, 2 km thick, at a depth of about 19 km and most likely made up of high-velocity material.

Body Wave Amplitude Calculations

The exact wave-theory response for a radially heterogeneous sphere has been used to find approximate formulae for treating seismic body wave amplitudes. A simple approximate formula has been found which preserves the simplicity of ray theory computations and the accuracy of exact wave solutions for band-limited signals. The new formula reduces to asymptotic ray theory predictions for simple velocity structures, but describes the effects of caustics far more accurately. In addition, the predicted amplitudes are not sensitive to small discontinuities in the first or second derivative of the velocity. The implementation of this formula requires the construction of synthetic seismograms. Thus the interference properties of various arrivals can be studied directly.

Partial Ray Expansion Required to Suitably Approximate the Exact Wave Solution

Summary Whenever synthetic seismograms are computed for layered media with the aid of ray theories only a partial ray expansion is possible. We present criteria that may be incorporated in a computer algorithm to obtain an automatic selection of rays which will minimize the error in the partial ray expansion below a preselected level. The program, results of which are presented in this paper, can also be used for the study of multiple reflected and converted phases or for media with dipping interfaces. 1. Introduction Various ray theories which decompose the total wave field in a layered medium into contributions attributed to individual rays have been used to advantage in the generation and analysis of synthetic seismograms. Among these are generalized ray theory (Spencer 1960) in which the displacement field is decomposed into contributions from an infinite set of rays, each ray being evaluated by a numerical solution of the impulse response (Bortfield 1967; Muller 1968, 1970) or by the Cagniard-de Hoop method (Helmberger 1968; Gilbert & Helmberger 1972). The solution for each ray is exact but the computations are very complex so that only a limited number of rays are used in the solution. An alternative method is asymptotic ray theory (Hron & Kanasewich 1971; Hron 1972) in which the field quantities are expressed as an infinite power series of reciprocal frequency combined with a spacedependent vector which is independent of frequency. The displacement field also becomes decomposed into an infinite set of rays but a greater number may be taken into the approximation by limiting the power series of each to the first non-zero term in the expansion. Although different techniques are used to evaluate the ray amplitudes, the final result is always a synthetic seismogram in which the kinematic and dynamic characteristics of particular rays may be studied. Ray theories are limited invariably by computational difficulties associated with the selection of the phases in the partial ray expansion. Very often the selection is based on the personal intuition or experience of the seismologist. I€ the selection is poor then the seismograms are misleading for rays with significant amplitudes will be omitted. An initial attempt at producing a computer algorithm that would generate systematically a set of rays, determine their kinematic and dynamic analogues and test for significant amplitude levels was reported by Hron & Kanasewich (1971). The algorithm for ray generation proved to be very efficient so that seismograms based on the evaluation of more than 150 000 individual rays were not exceptional (Hron 1972). However, no estimate of the accuracy of this partial ray expansion was possible because the convergence of the complete expansion had not been established. This

The importance of S-wave precursors in shear-wave studies

Precursors to the S or shear wave can cause serious error in interpretation of seismograms, particularly if only the vertical component is available. A series of synthetic seismograms are constructed using either asymptotic ray theory or the Fourier transform of a crustal transfer function obtained from a Haskell-Thomson matrix formulation to show the contribution of S - to P -wave conversions as precursors to the shear wave. Errors of 1 to 6 sec may occur if only the vertical component of seismic displacement is available. Errors may occasionally occur if horizontal seismograms are used, particularly when seismic stations are situated on thick sedimentary basins. In this case, particle-motion studies and arrays of recording instruments are required to avoid errors which may be as large as 2 sec. The problem arises at all distances and is of particular importance when ts − tp times are used to determine epicentral distances, first-motion studies, and in the study of premonitory phenomena using the ratio ts / tp for earthquake prediction.

Synthetic seismograms for deep seismic sounding studies using asymptotic ray theory

In asymptotic ray theory, the solution for particle motion is assumed to be an infinite power series of inverse frequency and a vector amplitude, Ān ( x, y, z ), independent of frequency. A point source with any desired impulse response and radiation pattern is easily incorporated. A synthetic seismogram computer program has been written for a plane-layered homogeneous elastic media using the first or second terms of the expansion where necessary. Multiply converted refracted and reflected phases and also head waves at distances away from the critical angle are included. In addition, the phases are all identified and their amplitude-distance function plotted if desired. The synthetic seismograms are calculated for a model in southern Alberta and another in northwestern Ontario as obtained by deep seismic sounding programs. It is found that reflected phases dominate the seismograms and they are at least as important as head waves in the interpretation of experimental results.