Scale‐dependent dynamic wave propagation in heterogeneous media: II. theory

Abstract

The measurable travel times of seismic events propagating in heterogeneous media depend on the geologic scale, the seismic wavelength, and the propagation distance. In general, the velocity inferred from arrival times is slower when the wavelength is longer than the scale of heterogeneity and faster when the wavelength is shorter. For normal incidence propagation in stratified media, this is the difference between averaging seismic slownesses, in the short wavelength limit, and averaging elastic compliances, in the long wavelength limit. In two and three dimensions there is also the path effect. Shorter wavelengths tend to find faster paths, thus biasing the travel times to lower values. In the short wavelength limit, the slowness inferred from the average travel time is smaller than the mean slowness of the medium. When the propagation distance is much larger than the scale of the heterogeneity the path effect causes the velocity increase from long to short wavelength to be much larger in two dimensions than in one dimension, and even larger in three dimensions. The scale-dependent travel time has several implications for seismic measurement and interpretation: The measured velocity depends not just on the rock properties, but on the scale of the measurement relative to the scale of the geology. The amount of scale-dependent dispersion depends on the spatial resolution of the receiver array. When comparing measurements made at different scales, for example logs and surface seismic, it is not always correct simply to apply the Backus average; the correct procedure will vary from case to case with the scale of the geology. Scale effects must be included with other viscoelastic mechanisms of dispersion when comparing measurements made at different frequencies.