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 nonparametric
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.