Comparison of branching models for seismicity and likelihood maximization through simulated annealing

Stochastic branching models provide a good description of some aspects of the temporal organization in seismicity. Generally, they assume that magnitudes are independent of history, as in the widely used Epidemic Type Aftershock Sequence (ETAS) model. Here, we consider a recent epidemic-like model where time-magnitude and magnitude-magnitude correlations are introduced via a dynamical scaling (DS) hypothesis, namely, the magnitude difference between earthquakes fixes the time scale for correlations. We also consider a variation of the ETAS model where the c parameter of the Omori law is not fixed but depends on the parent magnitude. We develop a novel procedure to maximize the log likelihood of the different models. This method is based on a Monte Carlo sampling in the parameter space with a variable step size during the evolution to converge to the given accuracy. The log likelihood indicates that the DS model provides the best fit for the major California sequences and the whole catalog, setting as lower magnitude thresholds M(c) >= 3.5. For M(c) = 3, the best fit is obtained by a model with c depending on both the parent magnitude and goes into M(c) as in the generalized Omori law. The better performance of the DS model with respect to the ETAS model is attributed to correlations in magnitudes.