the Poisson time-independent model (dlogL). For the likelihood estimation, we have adopted the values obtained for Tr and Cv reported in Table 2.4 for each fault segment.
Table 2.3. Features of the 2,000 years synthetic catalog
Figure 2.15. Frequency magnitude distribution of the 2,000 years synthetic catalog
Both Figure 2.16 and Table 2.4 show, as expected, that the most active segments are those characterized by higher slip rate (such as A, B and C). The simulation also shows that for the less active segments (E), interevent times longer than 400 years are possible. The coefficient of variation Cv is typically close to 0.3, which can be associated with remarkably time-predictable behavior of the seismicity. The log-likelihood difference denotes a better performance of the renewal model against the time-independent hypothesis. As the simulator algorithm allows the computing of the stress on all the cells constituting the seismic structure adopted in the model, we may build up the stress history on each cell and display it in a sort of animation (Figure 2.17).
Figure 2.16. Interevent time distribution from a simulation of 2,000 years of seismic activity across the Nankai mega-thrust fault system. For a color version of this figure, see www.iste.co.uk/limnios/statistical.zip
Table 2.4. Statistical parameters of the 2,000 years synthetic catalog of the Nankai mega-thrust fault system
Figure 2.18(a) shows (in a normalized scale) the time variation of the average stress computed on all the cells of each segment from A to E in a time span of 2,000 years. Vertical bars indicate the occurrence time of earthquakes whose ruptures significantly modified the stress of more than one segment. The increase of stress is mostly due to tectonic loading, but is also due to stress transfer from the cells of a segment to the others. The average stress drops at the time of each earthquake according to the size of the rupture on the specific segment participating in the earthquake.
Figure 2.18(b) shows the standard deviation (again in a normalized scale) of stress on the cells of each segment. Figure 2.18(c) shows the ratio between the values of the average stress and its standard deviation. This parameter increases always and only before a strong earthquake. Assuming that the trend shown in Figure 2.17 before large earthquakes (i.e. stress becomes higher and more uniform on a relevant patch of the fault system) is a phenomenon that happens in reality, it could be regarded as a promising tool for forecasting strong events. Unfortunately, it is not feasible to map the stress on the fault surface in the real environment with the necessary resolution. However, some seismicity patterns observable by means of modern high-quality seismic networks, such as spatio-temporal changes of b-value, could be associated with stress variation in the crust and applied in an operational forecasting system (Montuori et al. 2016; Gulia et al. 2016; Gulia and Wiemer 2019).
Figure 2.17. Stress time history on the Nankai seismogenic structure for the first 76 years. For a color version of this figure, see www.iste.co.uk/limnios/statistical.zip
2.5. Appendix 1: Relations among source parameters adopted in the simulation model
Figure 2.18. (a) The time history (in a normalized scale) of the average stress computed on all the cells of each segment from A to E in a time span of 2,000 years. Vertical bars indicate the occurrence time of earthquakes whose ruptures significantly modified the stress of more than one segment. The increase in stress is mostly due to tectonic loading, but is also due to stress transfer from the cells of a segment to the others. The average stress drops at the time of each earthquake according to the size of the rupture on the specific segment participating in the earthquake. (b) The same as in the top panel for the standard deviation of stress on the cells of each segment. The standard deviation decreases (i.e. the stress becomes more uniform) when the occurrence time of earthquakes approaches. (c) Ratio between the values of the average stress and its standard deviation. This parameter increases always and only before a strong earthquake, as a possible precursor of large size events. For a color version of this figure, see www.iste.co.uk/limnios/statistical.zip
This appendix provides the theoretical framework of the model adopted in the simulation algorithms described in section 2.3. Because the magnitudes of the earthquakes of the synthetic catalog generated by the algorithm are obtained from their seismic moments, and the number of these earthquakes per unit time must fit the constraint of the slip rate assigned to any fault segment of the model, it is necessary to adopt relationships between the magnitude and both the rupture area and the average slip of each event.
The scalar seismic moment of an earthquake is defined as:
where μ is the shear modulus of the elastic medium,
where Δσ is the static stress drop of the earthquake. From [2.5] and [2.6], we obtain:
