– it is increased at every time step (e.g. one day) through equation [2.18], according to the long-term slip rate (tectonic loading) mainly constrained by geodetic measurements;
– it is decreased at the occurrence time of every rupture, by a given amount (e.g. 3.3 MPa); the same cell can rupture more than once in the same earthquake;
– it is increased by a Coulomb stress change associated with a point source at the center of any other cell that ruptures during an earthquake; as all cells are assumed to rupture approximately with the same mechanism and the same fault plane, this stress change is always positive (see the appendix in section 2.6 for details).
Figure 2.3. Flow chart of the computer code for earthquake simulation adopted in this study (based on Console et al. 2015). For a color version of this figure, see www.iste.co.uk/limnios/statistical.zip
The events are initiated one by one on the cell with the largest stress budget, but only if it exceeds a given stress threshold. This is assumed to be spatially constant through the entire source area for the sake of simplicity. The second ruptured cell of the specific event that is chosen is that of the largest stress budget among the eight cells surrounding the nucleation cell, and so on for the next ruptured cells, until the stopping condition is met, when none of the cells, including and surrounding the cells previously ruptured in the same event, has a stress budget exceeding the threshold (Figure 2.4).
Particular attention has been given to the part of the simulator code that tunes the conditions of stopping an already initiated rupture. We obtained reasonable results by introducing a pair of “heuristic” rules to modulate the stress threshold to be exceeded for expanding an ongoing event into new cells or repeating the slip on an already ruptured cell. These rules, which have a relevant impact on the magnitude distribution of the synthetic catalogs, are:
1) The stress threshold adopted for the nucleation of an event is decreased, after the initial rupture of the nucleation cell, by a quantity which is proportional to the square root of the number of the already ruptured cells, multiplied by a free parameter called the “strength reduction coefficient” (S-R). This feature mimics the sharp decrease of strength at the edges of an expanding rupture, through a sort of weakening mechanism. Increasing this parameter encourages the growth of ruptures, thus decreasing the b-value in the frequency-magnitude distribution. This parameter has a similar role to the η free parameter in the Virtual Quake simulator developed for California (Schultz et al. 2017).
2) The square root of the number of already ruptured cells used in the previous rule is limited to a number equal to the width of the fault system, divided by the size of a cell, and multiplied by a free parameter called “fault aspect ratio” (A-R).
Although the first of these two empirical rules enhances the capability of an already nucleated event to expand into a larger rupture, the second one limits this enhancement to a size that does not exceed by many times the width of the fault system. As has been proved by numerous tests, the strength reduction coefficient (of the order of a few percent) influences the proportion of the seismic moment released by small and by large earthquakes: the smaller this parameter, the larger the number of small events. On the contrary, for the fault aspect ratio, it has no influence on the magnitude distribution of the background activity, but affects the shape of the magnitude distribution in the large magnitude range. This simple algorithm ensures a stable process, during which the stress budget is maintained below the nucleation threshold and never vanishes, if a suitable initial value is chosen. The earthquake rate is modulated by the slip rate assigned to each fault segment.
The smallest magnitude produced by the simulator is that corresponding to the rupture of a single cell; however, the computer code allows the user to arbitrarily choose the smallest magnitude reported in the synthetic catalog (which cannot be smaller than the magnitude associated with an event rupturing an area equal to only one cell). Moreover, the computer code includes an option for running in warm-up mode for the desired number of years in order to reach a stable situation before the real start of the synthetic catalog. It was proved by many tests that the values arbitrarily assigned to the initial stress budget of any cell do not affect the statistical properties of the synthetic catalogs if the selected warm-up time is long enough (e.g. 1,000 years).
Figure 2.4. Scheme of rupture nucleation and propagation adopted in this earthquake simulator (based on Console et al. 2017). For a color version of this figure, see www.iste.co.uk/limnios/statistical.zip
The simulation algorithm, in its simplicity, provides preference for new ruptures to nucleate at the points of the fault where the stress budget is higher, i.e. where the time elapsed since the latest event is longer. Once it is nucleated, the rupture expands in the directions where the stress budget is still higher, thus simulating a preference for filling pre-existing gaps and epicenter migration. Moreover, because of the stress transfer included in the model, earthquakes are more likely to occur close to the borders of the rupture of a preceding large earthquake, simulating a feature similar to aftershock production.
2.3.2. Frequency-magnitude distribution of the simulated catalog (2015)
In the application of the simulator, the rectangular source area representing the CGFS was discretized in 6,288 cells of 0.5 × 0.5 km. We chose for the synthetic catalogs a minimum magnitude of 4.0, which is produced approximately by the rupture of six cells. According to the “Ellsworth B” relation between magnitude and fault area (WGCEP 2003), the magnitude of earthquakes rupturing the entire CGFS (the area of which is 1,572 km2) would be 7.4. The duration of all the synthetic catalogs was 100,000 years. In order to explore the effect of the two free parameters, S-R and A-R separately, we carried out a series of eight simulations, each time using different combinations of these parameters. This effect is illustrated in Figure 2.5 by the magnitude-frequency density distribution of the earthquakes contained in the respective synthetic catalogs. Figures 2.5(a) and 2.5(b) show that the first free parameter, S-R, does have influence on the total number of earthquakes contained in the catalogs. The difference clearly comes from the substantial contribution given to the moment released by the high magnitude range, maintaining the same constraint for the moment rate released by the entire fault system.