Natural Time Analysis of Seismicity

Abstract

Assuming that a mainshock may be considered as a new phase, the natural time analysis of the seismicity reveals that the normalized power spectrum Π(φ) for small φ (φ → 0) or the quantity K1(= _χ2_–_χ_2) may be considered as an order parameter for seismicity. The probability distribution P(K1) of this order parameter is obtained from the calculation of the variance K1 when a time window of length l (= number of consecutive events) is sliding through an earthquake catalog. The K1 value at which this probability distribution P(K1) maximizes is designated by K1,p. By using P(K1), we find: first, studying the order parameter fluctuations relative to the standard deviation of its distribution, we observe that (a) the scaled distributions of different seismic areas (as well as that of the worldwide seismicity) fall on a universal curve and (b) this curve exhibits an “exponential tail” similar to that observed in certain non-equilibrium systems (e.g. 3D turbulent flow) as well as in several equilibrium critical phenomena, e.g., 2D Ising, 3D Ising, 2D XY. Second, the constant b in the Gutenberg–Richter (G-R) law for EQs, N(=M) = 10a-bM, is determined from the Maximum Entropy Principle which leads to b ≈ 1 in accordance with the b value obtained from real seismic data. Third, by analyzing either the original earthquake catalog or a shuffled one the following results are obtained for the Southern California Earthquake Catalog (SCEC) as well as for the Japanese Meteorological Agency Earthquake Catalog (Japan). Concerning the K1,p values, we find K1,p = 0.066 for the original data, while K1,p = 0.064 for the randomly shuffled data (with possible uncertainty of ±0.001). Both these K1,p values, the difference of which is shown to be associated with temporal correlations between the EQ magnitudes M, differ markedly from the value ? u = 1/12(≈ 0.083) of the “uniform” distribution, which is interpreted as reflecting that the process’s increments’ infinite variance contributes significantly to self-similarity. Fourth, upon employing multifractal cascades (generalized Cantor sets) in natural time an interconnection between K1,p and the parameter b of the G-R law is obtained which for b ≈ 1 leads to K1,p = 0.064 that coincides with the K1,p value obtained from the (randomly) shuffled earthquake data of Japan and SCEC. Fifth, by applying DFA to the earthquake magnitude time series of the SCEC and Japan data, we confirm that temporal correlations exist between EQ magnitudes. Sixth, focusing on the order parameter fluctuations of seismicity before and after mainshocks, we find the following. The P(K1) versus K1 plot before mainshocks exhibits a significant bimodal feature which is reminiscent of the bimodal feature observed in the pdf of the order parameter when approaching (from below) Tc in equilibrium critical phenomena. Finally, the G-R law or its generalization in the frame of the nonextensive statistical mechanics, if combined with natural time, which captures the temporal correlations between EQ magnitudes, can reproduce the features of real seismic data.