etas_solve: A robust program to estimate the ETAS parameters

Abstract:

The epidemic-type aftershock sequence (ETAS) model introduced by Ogata (1988) has been widely used to quantitatively describe seismicity (e.g. Ogata, 1992; Llenos et al., 2009). However, only a few programs for estimation of the ETAS parameters are publicly available, and it is difficult to automatically apply some of them to observed data due to initial value dependence (e.g. Ogata, 2006). A robust ETAS estimation program is required to meet the recent enhancement of earthquake catalogs. In this study, we developed a new program, etas_solve, that is based on Newton’s method and calculates exact gradient and Hessian by using the automatic differentiation technique (Griewank, 1989). The program also supports auxiliary window in time and magnitude (Wang et al., 2010).
To demonstrate robustness of the developed program, we tested the dependence of estimated parameters on the choice of initial value by running the program from 1,024 randomly chosen initial values, and then compared the results with that of SAPP (Ogata 2006). We used aftershock data of 26th July 2003 earthquake of M6.2 at the northern Miyagi japan, which is shipped with SAPP, as a testing data. We found that estimation values with etas_solve were independent of the initial value for the testing data, while that with SAPP were varied with the initial value. Although there was initial value dependence in the SAPP's results, the estimated values by SAPP with small (≤10^-5) gradient coincided with the solution by etas_solve. etas_solve took longer computation time per iteration than SAPP due to the exact Hessian calculation, but total execution time was comparable to that of SAPP since less number of iterations for convergence was required. In addition, etas_solve was faster than SAPP on multicore machines (around 8-fold speed up with a 16 core machine) since etas_solve is parallelized by OpenMP.
etas_solve is written in Fortran and distributed under GNU General Public License at https://github.com/kshramt/fortran_lib and https://bitbucket.org/kshramt/fortran_lib.