
Please use this identifier to cite or link to this item:
http://hdl.handle.net/2122/4885

Authors:  Lolli, B.^{*} Boschi, E.^{*} Gasperini, P.^{*} 
Title:  A comparative analysis of different models of aftershock rate decay by maximum likelihood estimation of simulated sequences 
Title of journal:  Journal of Geophysical Research 
Series/Report no.:  B01/114 (2009) 
Publisher:  AGU 
Issue Date:  Jan2009 
DOI:  10.1029/2008JB005614 
Keywords:  aftershock 
Abstract:  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 bandlimited 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 (1 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. 
Appears in Collections:  04.06.09. Waves and wave analysis 04.06.03. Earthquake source and dynamics Papers Published / Papers in press

Files in This Item:
File 
Size  Format  Visibility 
Article.pdf  1.4 MB  Adobe PDF  View/Open


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.
