Self-organized criticality and earthquake predictability: A long-standing question in the light of natural time analysis

After the Bak-Tang-Wisenfeld seminal work on self-organized criticality (SOC), the following claim appeared by other workers in the 1990s: Earthquakes (EQs) cannot be predicted, since the Earth is in a state of SOC and hence any small earthquake has some probability of cascading into a large event. Here, we discuss that such claims do not stand in the light of natural time analysis, which was shown at the beginning of the 2000s to extract the maximum information possible from complex systems time series. A useful quantity to identify the approach of a dynamical system to criticality is the variance $\kappa_1(\equiv \langle \chi^2 \rangle - \langle \chi \rangle^2)$ of natural time χ, which becomes equal to 0.070 at the critical state for a variety of dynamical systems. This also holds for experimental results of critical phenomena such as growth of ricepiles, seismic electric signals activities, and the subsequent seismicity before the associated main shock. Another useful quantity is the change $\Delta S$ of the dynamic entropy $S (=\langle \chi \ln \chi \rangle - \langle \chi \rangle \ln \langle \chi \rangle)$ under time reversal, which is minimized before a large avalanche upon analyzing the Olami-Feder-Christensen model for EQs in natural time. Such a minimum actually occurred on 22 December 2010, well before the M9 Tohoku EQ in Japan on 11 March 2011, being accompanied by increases of both the complexity measure of the $\Delta S$ fluctuations and the variability of the order parameter of seismicity (which was minimized two weeks later). These increases conform to the seminal work on phase transitions by Lifshitz and Slyozov and independently by Wagner as well as to more recent work by Penrose et al. In addition, the evolution of the complexity measure of the $\Delta S$ fluctuations reveals a reliable estimation of the occurrence time of this M9 EQ.