We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in (Saichev and Sornette, 2007), which describes the distribution of inter-event times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of inter-event times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of inter-event times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistics for the specific purpose of identifying the clustering parameter.
Deep Dive into Nonlinear theory and tests of earthquake recurrence times.
We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in (Saichev and Sornette, 2007), which describes the distribution of inter-event times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of inter-event times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of inter-event times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistics f
Most complex systems of interest in the natural and social sciences exhibit intermittent bursts of activity interspersed within long times of reduced activity. A simple metric to characterize this property consists in the distribution of recurrence (also called "waiting" or "inter-event") times between (suitably defined) events. Recently, the literature has undergone itself a burst of publication activity on this topic, motivated by the idea that distributions of recurrence times may be one of the most important complexity measures for both random fields and nonlinear dynamical systems [13]. The applications include recurrence time and anomalous transport [39], waiting times between earthquakes [3,4,28,33,34] and rock fractures [12], time intervals between consecutive e-mails [2] and between web browsing, library visits and stock trading [38].
Much of the recent interest of the statistical physics community focused on applying scaling techniques, which are common tools in the study of critical phenomena, to the statistics of inter-earthquake recurrence times or waiting times [1,3,4,5,6,7,8,9,11,27]. Many of the claims made in these recent articles on recurrence statistics have either been challenged, refuted or explained by previously known facts about earthquake statistics [14,25,26,29,33,34]. In particular, two of us [33,34] have developed a general theory of the statistics of inter-event times in the framework of the general class of self-excited Hawkes conditional Poisson processes [16,17,18] adapted to modeling seismicity. The corresponding model is known as the epidemic-type aftershock sequence (ETAS) model, in which any earthquake may trigger other earthquakes, which in turn may trigger more, and so on. Introduced in slightly different forms by Kagan and Knopoff [24] and Ogata [30], the model describes statistically the spatio-temporal clustering of seismicity. Using three well-known statistical laws of statistical seismicity (the Gutenberg-Richter, the Omori law and the productivity law), the empirical observations on the distribution of earthquake recurrence times can be explained within this model without invoking additional mechanisms other than the well-known fact that earthquakes can trigger other earthquakes [33,34].
A recent development is the proposition that inter-event time distributions may provide a new and more reliable way to measure of the so-called background earthquake activity [14,15]. This question arises as follows: if earthquakes trigger other earthquakes, how much of the observed seismicity is due to past seismicity (endogenous origin) and how much is resulting from an “external” driving source (exogenous origin) often referred to as “background” seismicity thought to reflect the driving tectonic forces at large scales. This question obviously generalizes to any system in which future events may be in part triggered by past events, such as in commercial sales [35] and web browsing activity [10,38].
Within the ETAS framework, the fraction of events in a given catalog which have been triggered by previous events can be shown [21] to be nothing but the so-called branching ratio n, defined mathematically as the average number of first-generation events triggered by a given preceding event [20]. Reciprocally, the fraction of background events is equal to 1-n (note that these models assume that the triggering branching-like processes are sub-critical: n < 1). The degree to which the parameter n can be retrieved from the distribution of inter-event times relies on departure from universality pointed out by Hainzl et al. [14] and two of us [33,34]. In this respect, the ETAS model provides an excellent training ground.
Using synthetic catalogs generated with the ETAS model, Hainzl et al. [14] found that the estimation of n using the distribution of inter-event times is better than from the application of a standard declustering procedure [32].
More progress can be achieved by a better understanding of the sensitivity of the distribution of inter-event times to the branching ratio n. In principle, the theoretical framework based on the technique of probability generating functions developed in Ref. [33,34] provides an ideal approach to this problem. However, this previous effort was limited on two accounts. First, while Saichev and Sornette derived the full exact nonlinear integral equations of the problem, they ended solving their linearized versions in order to derive the distribution of inter-event times. The present paper keeps the full nonlinear integral equations and shows that the linear simplification leads to systematic biases in the estimations of the key parameters of the ETAS model, and in particular of the branching ratio n which has been the focus of recent interest in the seismological community [14,15]. Secondly, only preliminary sensitivity analysis was performed with respect to n. The present paper presents a detailed treatment of the full exact nonlinear equations
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
