Extrapolation of Time-Dependent Waveforms along their Path of Propagation

Abstract

Summary Time-dependent waveforms are commonly extrapolated in space by means of rays and occasionally by means of diffraction integrals. It is possible to extrapolate time-dependent waves in space with a partial differential equation derived from the wave equation. There are stable numerical approximations. An example illustrates a mechanism for ' signal-generated noise ' which is consistent with observations. When a wave propagates in an inhomogeneous medium the waveform changes. Given that the wave has been observed at a suitable number of points in space we may attempt to solve two types of problems. First we may attempt to ascertain the nature of material inhomogeneity along the wave paths, and second, we may attempt to extrapolate the disturbance back to the source in an attempt to discover the nature of the source. With few exceptions, the methods used during the past decade for doing this kind of geophysical work may be summarized as follows: when wave equations are to be used, separability is achieved by considering cases in which the material inhomogeneity is a function of only one spatial co-ordinate. When higherdimensional inhomogeneity is so severe that it cannot be ignored, then the wave equations are almost always specialized to ray theory. Ray theory is especially useful when only the travel time is required. Although the amplitude may also be obtained by ray theory it is often of marginal utility because amplitude measurement is made ambiguous by changing waveforms. What we develop in this paper is a finite difference approach to the wave equation which tracks the time dependent waveform of a travelling wave in two-dimensionally inhomogeneous material. This is an extension of earlier work done by one of the authors in the frequency domain. Although the Fourier transform relates time-domain solutions to frequency-domain solutions, there are several compelling practical factors which give impetus to this study. When a waveform is small at certain times of interest and large at times which are not of interest, then a satisfactory approximation to the Fourier integral may be difficult to obtain even if values are obtained with good accuracy at many frequencies: an example is the head wave. Another example occurs in reflection seismology where the most interesting part of the waveform is the late-arriving weak echoes. Another example is when the time function is of long duration but only a small portion of it is of interest; this is usually the case with short-period earthquake seismograms where there is never any hope of interpreting more than the wave packets which come from identifiable phases.