two methods rapidly deteriorates. This effect, known as “grid dispersion,” must be taken into account in order to avoid erroneous interpretation of seismograms obtained by finite-difference techniques. Both seconti-order accuracy and fourth-order accuracy finite-difference algorithms are considered. For the second-order scheme, a good rule of thumb is that the ratio of the upper half-power wavelength of the source to the grid interval should be of the order of ten or more. For the fourth-order scheme, it is found that the grid can be twice as coarse (five or more grid points per upper half-power wavelength) and good results are still obtained. Analytical predictions of the effect of grid dispersion are presented; these seem to be in agreement with the experimental results. ence methods to that obtained by classical analytical methods for the simple case of an infinite twodimensional 90.degree wedge (quarter space) in an otherwise infinite homogeneous medium. The source field was that due to a line source distribution located parallel to the corner of the wedge (see Figure I). For analytical simplicity, the acoustic velocity of the wedge medium is taken to be zero. This is equivalent to having a perfectlj “soft” wedge medium such as a vacuum.

/pdf/accuracy-of-finite-difference-modeling-of-the-acoustic-wave-1t2mtwotr6.pdf

Accuracy of finite-difference modeling of the acoustic wave equation

Summary A finite difference scheme is developed for seismological problems, containing a new treatment of surface and interface boundary conditions. The new representations of boundary conditions are associated with truncation errors of the second order, while previous representations were of the first order only. We show by comparison with analytical solutions that increasing the accuracy of the representation of boundary conditions increases the order of accuracy of the whole solution. The present scheme can be programmed easily and applied to seismological problems. We illustrate this by solving the problem of pulse propagation in two models of rift valley, and in the vicinity of an elastic wedge embedded in another elastic space.

/pdf/an-improved-representation-of-boundary-conditions-in-finite-27zynzijrj.pdf

An Improved Representation of Boundary Conditions in Finite Difference Schemes for Seismological Problems

This paper reviews applications of the finite-difference and finite-element methods to the study of seismic wave scattering in both simple and complex velocity models. These numerical simulations have improved our understanding of seismic scattering in portions of the earth where there is significant lateral heterogeneity, such as the crust. The methods propagate complete seismic wavefields through highly complex media and include multiply scattered waves and converted phases (e.g.,P toSV, SV toP, body wave to surface wave). The numerical methods have been especially useful in cases of moderate and strong scattering in complex media where multiple scattering becomes important. Progress has been made with numerical methods in understanding how near-surface, low-velocity basin structures scatter surface waves and vertically-incident body waves. The numerical methods have proven useful in evaluating scattering of surface waves and body waves from topography of both the free surface and interfaces buried at depth. Numerical studies have demonstrated the importance of conversions from body waves to surface waves (andvice versa) when lateral heterogeneities and topographic relief are present in the uppermost crust. Recently, several investigations have applied numerical methods to study seismic wave propagation in velocity models which vary randomly in space. This stochastic approach seeks to understand the effects of small-scale complexity in the earth which cannot be resolved deterministically. These experiments have quantified the relationships between the statistical properties of the random heterogeneity and the measurable properties of high-frequency (≥1 Hz) seismograms. These simulations have been applied to the study of many features observed in actual high-frequency seismic waves, including: the amplitude and time decay of seismic coda, the apparent attenuation from scattering, the dispersion of waveforms, and the travel time and waveform variations across arrays of receivers.

A review of numerical experiments on seismic wave scattering

On the interaction of elastic waves with buried land mines :an investigation using the finite-difference time-domain method

In this paper, the stability behavior of the first-order finite-difference time-domain algorithm for elastodynamics at the interface between two different materials is investigated. A necessary condition for stability is established, which, dependent on the material properties of the two media, might be more restrictive than the well-known Courant condition. It is shown that this more restrictive stability condition can be avoided if the material properties are averaged on the boundary.

On the stability of the FDTD algorithm for elastic media at a material interface

This paper deals with elastic-wave propagation in two evenly-welded quarter-spaces. A compressional line source can be located at any point within either medium. The numerical solutions to this problem have been obtained by using the finite difference method. A computer program has been written to obtain synthetic seismograms of the horizontal and vertical displacements at all nodes of the superimposed grid, for the following cases: (a) elastic-wave propagation in a quarter-space, and (b) elastic-wave propagation in two quarter-spaces. Reflected, converted, transmitted, and diffracted phases are identified and interpreted. Surface and interface waves, originated at the corner by diffraction of the source pulse, are investigated as a function of the rigidity contrast and the velocity contrast between the two media and of the position of the source. Two-dimensional seismic modeling techniques have been used to provide a qualitative experimental verification of the numerical results.

Elastic-wave propagation in two evenly-welded quarter-spaces

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Elastic-wave propagation in two evenly-welded quarter-spaces