RESUMEN
Earthquakes follow an exponential distribution referred to as the Gutenberg-Richter law, which is characterized by the b value that represents a ratio of the number of large earthquakes to that of small earthquakes. Spatial and temporal variation in the b value is important for assessing the probability of a larger earthquake. Conventionally, the b value is obtained by a maximum-likelihood estimation based on past earthquakes with a certain sample size. To properly assess the occurrence of earthquakes and understand their dynamics, determining this parameter with a statistically optimal method is important. Here, we discuss a method that uses a state space model and a particle filter, as a framework for time-series data, to estimate temporal variation in the b value. We then compared our output with that of a conventional method using data of earthquakes that occurred in Tohoku and Kumamoto regions in Japan. Our results indicate that the proposed method has the advantage of estimating temporal variation of the b value and forecasting magnitude. Moreover, our research suggests no heightened probability of a large earthquake in the Tohoku region, in contrast to previous studies. Simultaneously, there is the potential of a large earthquake in the Kumamoto region, emphasizing the need for enhanced monitoring.
RESUMEN
The relation between the size of an earthquake mainshock preparation zone and the magnitude of the forthcoming mainshock is different between nucleation and domino-like cascade models. The former model indicates that magnitude is predictable before an earthquake's mainshock because the preparation zone is related to the rupture area. In contrast, the latter indicates that magnitude is substantially unpredictable because it is practically impossible to predict the size of final rupture, which likely consists of a sequence of smaller earthquakes. As this proposal is still controversial, we discuss both models statistically, comparing their spatial occurrence rates between foreshocks and aftershocks. Using earthquake catalogs from three regions, California, Japan, and Taiwan, we showed that the spatial occurrence rates of foreshocks and aftershocks displayed a similar behavior, although this feature did not vary between these regions. An interpretation of this result, which was based on statistical analyses, indicates that the nucleation model is dominant.
RESUMEN
To quantify symmetry and entropy inherent in the discrete patterns such as spatial self-organization in cell sorting and mussel bed ecosystems, we introduce the discrete Walsh analysis. This analysis enables us to estimate the degree of the complicated symmetry, and to extract the symmetry from the pattern that seems to be asymmetric. The results obtained in this paper are summarized as follows. (I) The geometrical patterns of the cell sorting become systematic with the predominance of the particular symmetry. This implies that not only the entropy but also the particular symmetry can decrease in the biological process. (II) The magnitude of the symmetry is related to the absolute value of the adhesion, and the type of the symmetry is related to the sign of the adhesion. That is, centro-symmetry dominates in the cell sorting pattern caused by large negative adhesion, and double symmetry dominates in the pattern caused by large positive adhesion. (III) Spatial self-organization in mussel bed is accompanied by the decreasing of the centro-symmetry. This implies that the positive "adhesion" between mussel individuals increases with time. (IV) In the biological process, the Curie symmetry breaking occurs at intervals.