STATISTICS OF TRAVELTIMES AND AMPLITUDES IN RANDOM MEDIA

Abstract

In this study, we build on the results of the study of 3-D wave propagation in weakly heterogeneous media conducted by Baig et al. We measure traveltimes and amplitudes from 'ground-truth' seismograms, computed using a numerical wave propagation code, and compare the measurements with approximate finite-frequency and ray-theoretical values for these quantities. Ray-theoretical traveltimes become invalid whenever the scale length of the 3-D heterogeneity is smaller than half the maximum width of the Fresnel zone; in contrast, ray-theoretical amplitudes have a much more restricted range of validity: the scale length should not be less than one Fresnel-zone maximum width. Finite-frequency theory gives better results for amplitudes, suffering no observable degradation for small-scale media for the weakest heterogeneity considered, but suffering appreciable misfit in more strongly heterogeneous media. Using these finite-frequency expressions for traveltime, we derive expressions for the expected variances of traveltimes and amplitudes that act, in most cases, as extensions to the ray-theoretical expressions. Finally, we propose using the amplitude variance as a criterion for delineating the validity of these linear approximations. For traveltimes, provided that one rejects waveforms that do not yield a good cross-correlation traveltime, the remaining data are linearly related to the model over the values of theoretical amplitude variance that we probe in this experiment. Amplitudes do not behave as well: when the theoretical amplitude variance rises above 0.1, significant non-linearities start to invalidate our linear approximation.