A comparative analysis of different models of aftershock rate decay by maximum likelihood estimation of simulated sequences

Most empirical studies on the decay of the rate of aftershock with time after a main shock assume the simple power law described by the modified Omori model (MOM). A couple of alternative models, also including an exponential decay at long times, have been proposed in the last decades: the modified stretched exponential (MSE) model and the band-limited power law (LPL). The first aim of this work is to discuss the functional properties of such models and the relations existing on their parameters. In particular, we derive, in term of common transcendental functions, the analytical integrals of the LPL and MSE rate functions, which are useful to simplify and speed up computations. We also define, as a function of the parameters of the LPL, two characteristic times tb and ta, which correspond approximately to the delay time c of the MOM and the exponential decay relaxation time t0 of the MSE, respectively. Hence, the MOM, the MSE, and the LPL models can be compared among each other in terms of the same set of four general parameters: (1) the power law exponent, (2) the initial delay time, (3) the exponential relaxation time (∞ for the MOM), and (4) a normalization parameter, which can be related in some cases to the total number of potential aftershocks. A second aim of this paper is to test the ability of maximum likelihood methods to detecting the most appropriate decay model among alternatives. By the analysis of sequences simulated according to a MSE or a LPL we show that if the assumed exponential decay relaxation time is shorter than the time window over which the sequence is observed, the Akaike and Bayesian information criteria select correctly the true model (MSE or LPL). Conversely, when the relaxation time is definitely longer than the observing window, the information criteria usually prefer the MOM, but the maximum likelihood of the true model is higher, and if the data set of shocks is sufficiently large, the order of magnitude of the simulated characteristic time of the exponential decay can be estimated quite consistently. Hence, when analyzing real sequences, the possible emergence of the exponential decay might be hidden by the short duration of the time window if the standard information criteria are considered. Moreover, when the relaxation time is short, the estimated power law exponent p for the MOM results definitely higher than that assumed in the simulation. This indicates that high values of p (>1.5–2.0), sometimes observed in real sequences, might be due to the (not modeled) early startup of the negative exponential decay. Our analysis cannot decide which model is the most appropriate in describing the temporal behavior of aftershock rate after a main shock but suggests that the assumption of a model also including the exponential decay is generally preferable as it allows capture of all of the features of sequence decay.