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

Abstract

[1] 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.Citation: Bottiglieri, M., E. Lippiello, C. Godano, and L. de Arcangelis (2011), Comparison of branching models for seismicity and likelihood maximization through simulated annealing,

The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study

“…In our study we take explicitly into account the aftershock incompleteness adopting the same procedure developed in [ 40 ] to reproduce both the non-trivial dependence of the c -value in the OU law on the mainshock magnitude [ 8 , 41 , 42 , 43 , 44 , 45 , 46 ], as well as the non trivial magnitude correlations between subsequent earthquakes [ 11 , 47 , 48 , 49 , 50 , 51 , 52 , 53 ]. The model, defined as ETASI2 model, implements aftershock incompleteness by multiplying the occurrence rate in Equation ( 1 ) by a detection rate function of the magnitude represented by the cumulative distribution function of the normal distribution.…”

The Overlap of Aftershock Coda Waves and Short‐Term Postseismic Forecasting

“…We attribute these deviations to the cascading process that can produce a more gradual decrease of the aftershock number from the initial plateau compared to the situation when higher‐order generation aftershocks are not taken into account (Hainzl, ). A better fit for F ( x ) in numerical and instrumental catalogs is provided by obtained by Lippiello, Bottiglieri, et al () under a dynamical scaling assumption (Bottiglieri et al, , , ; de Arcangelis et al, ; Lippiello, Godano, & de Arcangelis, ; Lippiello et al, , ). This choice of F ( x ) reduces the value of the χ ‐squared by a factor 5 with respect to the OU law, even if the agreement with experimental data is still very poor according to a Pearson's χ ‐squared test.…”

“…obtained by Lippiello, Bottiglieri, et al (2007) under a dynamical scaling assumption (Bottiglieri et al, 2009(Bottiglieri et al, , 2010(Bottiglieri et al, , 2011de Arcangelis et al, 2016;Lippiello et al, 2009Lippiello et al, , 2013. This choice of F(x) reduces the value of the -squared by a factor 5 with respect to the OU law, even if the agreement with experimental data is still very poor according to a Pearson's -squared test.…”

Post Seismic Catalog Incompleteness and Aftershock Forecasting

“…We attribute these deviations to the cascading process which can produce a more gradual decrease of the aftershock number from the initial plateau compared to the situation when higher order generation aftershocks are not taken into account [12]. A better fit for F(x) in numerical and instrumental catalogs is provided by F(x) = A log (1 + Bx −p ) obtained by Lippiello et al [46] under a dynamical scaling assumption [38,45,[47][48][49][50][51].…”

“…Deviations from the GR law are a natural consequence of this assumption with a completeness magnitude depending on time in agreement with what is observed in experimental data (Equation (2)). The study of the maximum likelihood [51] has shown that this method provides a more accurate description of the aftershock rate decay than the ETAS model.…”