ANALYSIS OF SEISMIC WAVE DYNAMICS BY MEANS OF INTEGRAL REPRESENTATIONS AND THE METHOD OF DISCONTINUITIES

Abstract

We analyze the dynamics (amplitudes and phase distortions) of seismic waves as they propagate along the ray. Our analysis is performed via a ray series approximation in the time domain. That is, we concentrate on characterizing the sharp changes (discontinuities) of the signal that are localized near the wavefront. After convolution of the terms of such a series with a proper temporally short (high-frequency) wavelet, one obtains a synthetic seismic signal at a given point of interest. We present an outline of the proposed technique that yields integrals describing the wavefield. These integrals are similar to oscillatory integrals in the frequency domain. This description is uniformly valid near caustics, allowing the calculation of higher order terms of the ray series approximation. Practical use of the technique is illustrated by several examples which show two possible uses of the technique: general understanding of what is happening during wave propagation and practical calculations. First, we show how the structure of the ray decomposition changes near the simple caustic, and then we calculate a synthetic signal near the cusp caustic. The advantage of the technique is that the problem of seismic wave calculation is technically reduced to a problem of double integration of a Dirac g-function; thus, it is computationally effective.