Long memory in earthquake time series: The case study of the geysers geothermal field
The present study aims at proving the existence of long memory (or long-range dependence) in the earthquake process through the analysis of time series of induced seismicity. Specifically, we apply alternative statistical techniques borrowed from econometrics to the seismic catalog of The Geysers geothermal field (California), the world's largest geothermal field. The choice of the study area is essentially guided by the completeness of the seismic catalog at smaller magnitudes (a drawback of conventional catalogs of natural seismicity). Contrary to previous studies, where the long-memory property was examined by using non-parametric approaches (e.g., rescaled range analysis), we assume a fractional integration model for which the degree of memory is defined by a real parameter d, which is related to the best known Hurst exponent. In particular, long-memory behavior is observed for d > 0. We estimate and test the value of d (i.e., the hypothesis of long memory) by applying parametric, semi-parametric, and non-parametric approaches to time series describing the daily number of earthquakes and the logarithm of the (total) seismic moment released per day. Attention is also paid to examining the sensitivity of the results to the uncertainty in the completeness magnitude of the catalog, and to investigating to what extent temporal fluctuations in seismic activity induced by injection operations affect the value of d. Temporal variations in the values of d are analyzed together with those of the b-value of the Gutenberg and Richter law. Our results indicate strong evidence of long memory, with d mostly constrained between 0 and 0.5. We observe that the value of d tends to decrease with increasing the magnitude completeness threshold, and therefore appears to be influenced by the number of information in the chain of intervening related events. Moreover, we find a moderate but significant negative correlation between d and the b-value. A negative, albeit weaker correlation is found between d and the fluid injection, as well as between d and the annual number of earthquakes.