The effect of random velocity functions on the travel times and amplitudes of seismic waves

Abstract

Small-scale variations in composition and anisotropy in rocks and minerals will cause corresponding small-scale variations in the seismic velocity. Because of these variations the velocity can be described as a fractal or, in other words, a function with random variations and no derivative. Seismic waves on travelling through the Earth act as a low pass filter and smooth out the variations. Because of these effects the observed instantaneous seismic velocities will have a small indeterministic component on its deterministic component which, in this case, will be the average velocity. The magnitude and scale of the indeterministic component will depend on the size characteristics of the smoothing filter. In the real Earth this is a three-dimensional spatial filter. A seismic ray on passing through a homogeneous earth will follow a smooth minimum-time path, while a ray on passing through the Earth with a small indeterministic velocity component will encounter velocity perturbations along its ray path which cause the direction to fluctuate slightly. This has the effect of lengthening the path such that the travel times are biased to a larger value than the corresponding times for the homogeneous earth. It will also have the effect of focusing the rays to cause local large amplitudes and defocusing the rays to cause regions of low amplitude signals. A set of numerical experiments was performed to study the travel-time fluctuations. Rays were traced through earth models consisting of slightly random smoothed velocity perturbations superimposed on ones composed of simple vertical gradients. An interesting result of these experiments was the fact that the time delays occurred in such a way as to break up a smooth travel-time line into one with short line segments giving one the impression that the Earth is layered. The depths to the apparent discontinuities were directly related to the correlation distance of a smoothing filter used in the experiments. The experiments also confirmed the expected erratic amplitude behaviour of the main signals and showed that the position of the PmP cusp is very sensitive to the nature of the lateral and vertical heterogeneities. Comparisons of the results of the numerical experiments were made with a record section from a long-range refraction profile recorded in Saskatchewan in 1979 by the Canadian COCRUST group. A non-layered interpretation of the observations indicated that an rms velocity deviation of only 1 per cent at any depth and a correlation distance of 5 km were sufficient to explain the irregularities in the observations.