Recursive formula for the peak delay time with travel distance in von Kármán type non-uniform random media on the basis of the Markov approximation

Direct waves of microearthquakes in the high-frequency range (>1 Hz) strongly reflect the random inhomogeneities near their ray paths. This study conducts numerical simulations of envelope broadening of impulsively radiated wavelet assuming spatially non-uniform distribution of random inhomogeneities. We assume plural von Kármán type power spectral density functions (PSDF) for random inhomogeneity to clarify how the non-uniformly distributed random media affect the frequency dependence of envelope broadening. We employ the stochastic ray path method based on the Markov approximation for the mutual coherence function. This method is appropriate to simulate multiple forward scattering during the wave propagation. We mainly examine the travel distance and frequency dependence of the peak delay time in relation to the parameters characterizing the PSDFs. The peak delay time, which is defined as the time lag from the direct-wave onset to the maximum amplitude arrival of its envelope, is the best parameter reflecting the accumulated scattering effect in random media and is quite insensitive to the intrinsic attenuation. According to the numerical simulations in various non-uniform random media, we find some remarkable features in travel distance and frequency dependence, which cannot be found in uniform random media. For example, the frequency dependence in uniform random media is uniquely determined by the spectral gradient of PSDF for arbitrary travel distance; however, that in non-uniform media gradually changes as travel distance increases if the waves have experienced a change of spectral gradient in PSDF. Considering the results of our simulation, we propose a simple recursive formula to calculate the peak delay time in non-uniform random media. This recursive formula can predict the simulation results appropriately and relate the peak delay times to two parameters quantifying the von Kármán type PSDF in short wavelengths. It will become a mathematical base for the inversion of peak delay times of bandpass filtered traces to estimate the spatial distribution of random inhomogeneity spectra.