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From a statistical point of view, a series of seismic events is seen as the n-element sample from a stochastic process. If this process is stationary and memoryless then the complete information on the parameters of seismic events is preserved in the joint probability distribution functions of the parameters. In common practice the joint probability description is replaced with marginal distributions of a particular parameter. Such an approach through the marginal distributions splits the seismic process into a number of respective processes of the parameters of seismic events. The parameters can either be measurable, for example, the occurrence time, epicenter coordinates, focal depth, magnitude etc., or derived, for example, inter-event time, inter-event distance, maximum magnitude etc. Each of these parameter processes can be studied separately. This paper reviews recent investigations into the statistical properties of the parameter series of seismic events from mines. Because the seismic process in mines is controlled by complex and changeable anthropogenic factors it is not surprising that the differences in statistical properties between mining-induced seismicity (MIS) and natural seismicity are substantial. Unlike tectonic seismicity, which is permanent in both space and time, MIS is transient and forms time-space zones or clusters that correlate in space and time with mining works. The event occurrence process in mines is not Poissonian. It is time-dependent and, at best, can be regarded as quasi-stationary. Information on non-stationarity is stored mainly in smaller and more frequent events. The event magnitude processes in MIS is also time-dependent. Consequently, the Gutenberg b-value, the return period and other hazard parameters vary in time. Furthermore, for the majority of mining event parameterisations, their stochastic features cannot be ignored. The interval estimation of autocorrelation function used to study short-term interrelations, and the Hurst rescaled-range analysis applied to investigate long-term clustering, show that the occurrence process, the magnitude process and likely the event location process, have memory of both the long- and the short-type. Inter-event times, inter-event distances and magnitudes are internally interrelated. One of the possible ways for these interactions among mining-induced seismic events is the static Coulomb stress transfer. The subsequent events tend to locate within the areas of increased Coulomb stress due to previous events. Finally, the magnitude distribution of mining seismic events is complex and often multi-modal. Populations of magnitudes consist of at least two components. The magnitude distribution cannot be accurately approximated with the Gutenberg–Richter model. The unique features of the MIS process presented: time-dependence, memory, inter-relations, multi-modality etc., show that the process is complex. This complexity complicates the practice of the statistical analysis of mining seismic data. On the other hand, however, the non-stationarity and inter-relations mean that the mining seismic event generation process is intrinsically predictable.

Some Unique Statistical Properties of the Seismic Process in Mines