Wavelet-based Multifractal Analysis of Real and Simulated Time Series of Earthquakes

Synopsis This study introduces a new approach (based on the Continuous Wavelet Transform Modulus Maxima method) to describe qualitatively and quantitatively the complex temporal patterns of seismicity, their multifractal and clustering properties in particular. Firstly, we analyse the temporal characteristics of intermediate depth seismic activity in the Vrancea region, Romania. The second case studied is the shallow, crustal seismicity, which occurred in a relatively large region surrounding the epicentre of the 1995 Kobe earthquake. In both cases we have declustered the earthquake catalogue before analysis. The results obtained in the case of the Vrancea region show that for a relatively large range of scales, the process is nearly monofractal and random (does not display correlations). For the second case, two scaling regions can be readily noticed. At small scales the series display multifractal behaviour, while at larger scales we observe monofractal scaling. The Hölder exponent for the monofractal region is around 0.8, which would indicate the presence of long-range dependence (LRD). This result might be the consequence of the complex oscillatory or power law trends of the analysed time series. In order to clarify the interpretation of the above results, we consider two “artificial” earthquake sequences. Firstly, we generate a “low productivity” earthquake catalogue, by using the ETAS model. The results, as expected, show no significant LRD for this simulated process. We also generate an event sequence by considering a cellular fault embedded in a 3-D elastic halfspace. The series display clear quasi-periodic behaviour, as revealed by simple statistical tests. The result of the wavelet-based multifractal analysis shows several distinct scaling domains. We speculate that each scaling range corresponds to a different periodic trend of the time series.