Algorithm of micro-seismic source localization based on asymptotic probability distribution of phase difference between two random stationary Gaussian processes

The article is devoted to the problem of estimating the coordinates of micro-seismic sources (localizing the sources) using multichannel data recorded by a surface seismic array. A new statistical algorithm is proposed for the source localization, which is mainly based on the phases of the Discrete Finite Fourier Transforms of the array sensor seismograms. This algorithm was constructed using the Maximum Likelihood concept under the following constraints: (a) noise components of the array seismograms are statistically independent stationary Gaussian processes with different power spectral densities; (b) the signal-to-noise ratios in the array seismograms are small, but the duration of signals generated by a micro-seismic source in the sensors is quite large; (c) the time function of a micro-seismic source can be approximated by a stationary Gaussian random process. The asymptotic probability density function was obtained in the paper for the phase differences of two Gaussian stationary random time series. This function provided a theoretical basis for constructing the new statistical phase algorithm. The algorithm requires evaluation of the coherence functions for all pairs of the sensor seismograms. For this reason, it inquires more calculations for the source localization than the known phase algorithms. But Monte Carlo simulation has shown that the new phase algorithm provides a more accurate estimation of micro-seismic source coordinates compared to the most popular phase algorithm.