Synthetic seismograms produced by the finite difference method are used to study the scattering of elastic and acoustic waves in two-dimensional media with random spatial variations in seismic velocity. The results of this study provide important insights about the propagation of short-period (5), the self-similar medium is characterized by a scattering Q that is constant with frequency, whereas theory predicts that the apparent Q in an exponential medium is proportional to frequency. These alternative models of crustal heterogeneity can thus be tested by improved measurements of the frequency dependence of crustal Q at frequencies greater than about 1 Hz, assuming that scattering is responsible for most of the attenuation at these frequencies. Measurements of the time decay of the synthetic coda waves clearly show that the single scattering model of coda decay is not appropriate in the presence of moderate amounts of scattering attenuation (scattering Q ≥ 200).

In these cases, Q values derived from the coda decay rate using the single scattering theory do not correspond to the transmission Q of the medium. The cross correlation of synthetic waveforms observed for an array of receivers along the free surface is observed to be dependent on the correlation of synthetic distance of the medium. The self-similar random medium proposed here for the crust produces waveform variations at high frequencies (15-30 Hz) similar to those reported for actual small-scale seismic arrays with apertures of hundreds of meters.