Finite difference simulations of seismic scattering: Implications for the propagation of short‐period seismic waves in the crust and models of crustal heterogeneity

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 (< 1 s) seismic waves in the earth's crust and place significant constraints on the fluctuation spectrum of crustal heterogeneity on length scales from tens of kilometers to tens of meters. The synthetic seismograms are analyzed to determine the variation in travel times and waveforms across arrays of receivers. The apparent attenuation caused by scattering and the time decay and amplitude of the seismic coda are also quantified with the numerical simulations. Random media with Gaussian and exponential correlation functions are considered, as well as a self‐similar medium with equal variations in seismic velocity over a broad range of length scales. These media differ in the spectral falloff of their velocity fluctuations at wavelengths smaller than 2π times the correlation distance a . The synthetic seismograms demonstrate that a random medium with self‐similar velocity fluctuations at length scales less than about 50 km ( a ≥ 10 km) can explain both travel time anomalies reported for teleseismic arrivals across large‐scale seismic arrays (e.g., LASA and NORSAR) and the presence of seismic coda at frequencies of 30 Hz and greater commonly observed in microearthquake waveforms. Media with Gaussian and exponential correlation functions in velocity do not account for both sets of observations for reasonable standard deviations in velocity (≤10%). The scattering attenuation ( Q −1 ) observed in the simulations for Gaussian media is peaked at ka between 1 and 2, where k is the seismic wave number. The observed attenuation in exponential media increases with frequency for ka < 1 and remains about constant for 1 ≤ ka ≤ 5.6. At high frequencies ( ka > 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 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.