Simulated envelopes of non-isotropically scattered body waves as compared to observed ones: Another manifestation of fractal heterogeneity

Envelopes of scalar waves are simulated at various distances from an instant point source embedded in a random uniformly scattering medium by means of direct Monte-Carlo modelling of wave-energy transport. Three types of scattering radiation pattern ('indicatrix') are studied, for media specified by (1) a Gaussian autocorrelation function of inhomogeneities, (2) a power-law ('fractal', k(-alpha)) inhomogeneity spectrum and (3) the mix of case (1) and the isotropic indicatrix (very small + large inhomogeneities). We look for a model that can qualitatively reproduce the two most characteristic features of real S-wave envelopes of near earthquakes, namely (1) the broadening of the 'direct' wave group with distance and (2) the monotonously decaying shape of the coda envelope that does not deviate strongly from that expected in the isotropic scattering case. Both properties are observed for any band over a wide frequency range (1-40 Hz). The well-studied isotropic scattering model realistically predicts the appearance of codas but fails to predict pulse broadening. The model of large-scale inhomogeneity realistically predicts the mode of pulse broadening but fails to predict codas. We have found that, for a particular frequency band, within each class of inhomogeneity studied, both requirements can be qualitatively satisfied by a certain choice of parameters. In the Gaussian-ACF case, however, this match can be obtained only for a narrow frequency range. In contrast, the fractal case (with a value of exponent alpha of about 3.5-4) reproduces qualitatively the observed wide-band behaviour, and we consider it a reasonable representation of the gross properties of the earth medium.