Optimization of rolling window approach to analyze earthquake time series and identify possible precursors

Analysis of seismic precursors is crucial for understanding the spatio-temporal evolution of seismicity and assessing whether a system is approaching an unstable state. Precursors are indicators that are deemed to be related to the processes leading to crustal rupture. Therefore, their real-time monitoring can provide insights into the imminent occurrence of earthquakes. Precursors yield robust results only when analyzed using appropriate techniques. Specifically, the measurement of real-time precursor parameters and the analysis of their temporal trends is highly sensitive to data processing and depends heavily on the characteristics of the seismic data under study. Therefore, careful data management is essential to avoid inappropriate conclusions.

This study examines two seismic precursors: (1) b-value (i.e., slope of the Gutenberg and Richter law), which characterizes the relative likelihood of small versus large earthquakes within a population of events; (2) Hurst exponent, an indicator of the "memory" in time series and, consequently, of the type of stochastic process underlying them (i.e., random, persistent, or anti persistent). While the b-value has paramount importance in earthquake forecasting since its variation (which is deemed to be related to stress conditions of faults) can act as a first-order discriminator between conventional aftershock sequences and sequences including multiplets (i.e., two or more mainshocks that are closely associated in time and space), the Hurst exponent is widely used in econometrics to detect trends and mean reversion in financial data.

The aim of this study is to determine the optimal data-windowing configuration (window size and overlapping percentages), within the framework of a moving window approach, that produces results in agreement with theoretical expectations and effectively captures the characteristics of the seismic data under study. The methodology involves analyzing these precursors (i.e., b-value and Hurst Exponent) along with additional seismic metrics such as: (1) number of earthquakes above a given magnitude threshold, (2) maximum magnitude, and (3) strain energy. Correlation coefficients (e.g., Pearson, Spearman, Kendall) are computed to evaluate the relationships between these parameters under various windowing configurations, including a novel adaptive window approach. The process can be summarized as follows:

• For a given configuration (e.g., window size = 500 years, overlap = 50%), the window slides over time, and parameters are calculated at each step within the time window.
• Correlation coefficients (between pairs of parameters) are computed using various statistical methods (e.g., Pearson, Spearman, Kendall).
• This procedure is repeated across all configurations to identify the setting that maximizes correlations (i.e., higher correlation coefficients and lower p-values).

Analyzing synthetic time series shows that correlations between parameters are sensitive to the adopted configuration, as different data-windowing configurations may capture distinct seismicity patterns. Therefore, the selection of the most effective configuration is strictly study-specific. To enhance the reliability of the results, application of this methodology to other seismic parameters (e.g., Vp/Vs ratio) requires future consideration, especially to validate results against known seismic sequences. The first step towards this direction is the application of the method to time series associated with natural or induced (or triggered) seismicity (e.g., seismic activity in the Geysers geothermal field).