Scale‐dependent seismic velocity in heterogeneous media

The measurable traveltimes 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 traveltimes to lower values. In the short wavelength limit, the slowness inferred from the average traveltime 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 wavelengths to be much larger in two dimensions than in one dimension, and even larger in three dimensions. The amount of velocity dispersion can be understood theoretically, but there is some discrepancy between theory and experiment as to what ratio of wavelength to heterogeneity scale separates the long and short wavelength limits. The scale‐dependent traveltime implies that a measured velocity depends not just on rock properties, but also on the scale of the measurement relative to the scale of the geology. When comparing measurements made at different scales, for example logs and surface seismic, it is not always correct to simply 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. The amount of observed scale‐dependent dispersion also depends on the spatial resolution of the receiver array. For example, the first‐break time of the average trace from a stack, a large group array, or a large laboratory transducer may be earlier than the average of first‐break times measured with individual small‐scale receivers.