Y. Y. Kagan INVERSE GAUSSIAN DISTRIBUTION AND ITS APPLICATION TO EARTHQUAKE OCCURRENCE 18-JUN-2007 As Matthews et al. (2002) indicate, the Inverse Gaussian Distribution (IGD) has been known since 1915. However, the distribution only acquired its name in 1945. This is the name found in recent statistical encyclopedias and books (for example, Kotz et al., 2006; Seshadri, 1999), on Wolfram website, and Wikipedia, while the "Brownian passage-time" (BPT) is unknown. Kagan and Knopoff (1987a) proposed to use this distribution (without calling it IGD) to describe the inter-earthquake time distribution. The major impetus was to explain aftershock statistics (Omori's law) by stress dynamics. The IGD is controlled by two parameters and has two periods -- a scale-invariant, power-law part for small time (with the pdf exponent 1.5) and a quasi-exponential tail for longer time intervals (i.e., the Poisson distribution). There are several ways to define the parameters, and this contributes to the difficulties of understanding the distribution and its analysis. By selecting certain parameter values one can convert the IGD into a more-or-less power-law (Levy stable) or into a quasi-periodic Gaussian distribution without the power-law component (Matthews et al., 2002, are effectively doing the latter). Kagan and Knopoff (1981) showed that the power-law with the exponent value of 1.5 reproduces major statistical properties of the shallow earthquake clustering. Thus, Kagan and Knopoff (1981; 1987a) used the IGD to find a clue to the real phenomenon -- the aftershock occurrence which follows Omori's law, whereas Matthews et al. (2002) applied this distribution to explain earthquake quasi-periodicity, the feature that probably does not exist. All examples of earthquake periodic behavior are the result of retrospective selection from available data. As Jackson (2007) shows all the tests of the validity of quasi-periodic earthquakes, using new data, ended in failure. A more consistent analysis should use the general IGD form in order to explain earthquake short-term clustering and obtain large earthquake recurrence statistics without infinities (Kagan and Knopoff, 1987a, p. 725, line 13). An analogy with the Gutenberg-Richter relation (G-R) can be used here. The pure G-R for magnitudes is equivalent to the Pareto (power-law) distribution for the seismic moment with the pdf exponent of about 1.6 (Kagan, 1991). Such distribution has an infinite moment rate, thus, as Wyss (1973) suggested, it needs to be capped at the high moment end. The introduction of the upper moment cutoff or a taper allows to quantitatively compare plate-tectonic, geodetic, and seismological data (Molnar, 1979; Anderson, 1979; Kagan, 1991; Bird and Kagan, 2004). Presently, it is possible to evaluate the temporal probability of earthquake occurrence using both seismic data and tectonic/geodetic deformation results. The former approach allows to forecast the earthquake rate for short-time intervals, where Omori's law controls earthquake occurrence, and for intermediate time scales, comparable to a catalog length (Kagan and Jackson, 2000). The deformation data allow to extend the time interval to much longer periods (Bird and Kagan, 2004). For longer time periods, we do not have a model to go beyond the time-independent Poisson process. However, similarly to the G-R results, we can hope that introducing an upper taper on the power-law inter-earthquake time distribution would allow us to significantly expand the time analysis possibilities. However, here we encounter a problem. The IGD may explain the inter-event time statistics only for the first generation (parent-child) of clustered earthquakes. Most of the observed aftershocks or clustered events are generally separated from their `parents' by multiple generations. For example, the present aftershocks of the 1811-12 New Madrid earthquakes or the 1952 Kern County, California, earthquake are mostly distant relatives of their progenitors (mainshocks). For the G-R this fact is mostly irrelevant, since the size distribution of clustered events is largely independent of their history. The time distribution has a different behavior: this can be seen from the exponent values of Omori's law (about 1.0) and the IGD (1.5). In the Omori law we count all aftershocks, disregarding their parentage, whereas the IGD describes the time distribution for the NEXT event, i.e., when the stress level reaches a critical value to trigger a new rupture. When we count all events, we decrease the exponent from 1.5 to 1.0 (Kagan and Knopoff, 1987a). By using the stochastic reconstruction (based on earthquake branching models -- Kagan and Knopoff, 1981; 1987b; Ogata, 1988), we can attempt to establish the interdependence structure of earthquake clusters, but such an analysis cannot yet be carried out to its full extent. In this respect, the paleo-seismic or historic earthquake data may present a unique opportunity to study inter-event statistics and compare the data to the IGD. Since the data are limited to large earthquakes, the clustering or interdependence is likely to be restricted to the first generation. For instance, our analysis of the CMT catalog with the magnitude threshold M5.8 (Kagan and Jackson, 2000) suggests that only about 20% of events are dependent (clustered). However, the paleo-seismic data are likely to be severely biased for small time interval statistics. The statistical analysis of earthquake catalogs (Kagan and Jackson, 1999) suggests that even strong earthquakes (M>7.5) are clustered in time (http://moho.ess.ucla.edu/~kagan/paleo.txt). We may be able to combine these earthquake catalog results with the paleo-seismic data to infer the properties of the recurrence time distribution for large earthquakes. The short-time earthquake statistics can be viewed as a mixture of two distributions: the renewal-type IGD for dependent earthquakes plus the Poisson sequence of independent events. The paleo-seismic data refer to the rupture recurrence at one site. However, for paleo-seismic data the hypocenter or the centroid can be far away or even on another fault (Kagan, 2005; Field, 2007). All these effects need to be quantitatively studied in order for the paleo-seismic results to be comparable to the catalog data. Moreover, to be effective, the paleo-seismic data need to be corrected for small time bias, and for other systematic and random effects (see more in http://moho.ess.ucla.edu/~kagan/paleo.txt). In addition, the distribution of earthquake size vs its depth and a probability of surface rupture need to be studied in order to take into account the influence of the surface on earthquake statistics, see http://moho.ess.ucla.edu/~kagan/surface_slip.txt To understand the long-term temporal behavior of earthquakes, we could also examine the time statistics of regions with different tectonic deformation rates. If the deformation rate is slow, aftershock sequences have a long time history (the 1891 Nobi earthquake, the 1811-12 New Madrid earthquakes are good illustrations, see also Ebel et al., 2000). In subduction zones, aftershock sequences look much shorter. A statistical study of the dependence of aftershock sequence duration vs the tectonic rate may be quite useful. Dieterich (1994 and thereafter) investigated the properties of aftershock sequences. Such studies need to be extended to multiple regions and they need to be statistically rigorous. In conclusion, two classical statistical earthquake distributions to a large degree governed our approach to the seismic hazard analysis: Omori's law and the Gutenberg-Richter relation. As explained above, the recent developments in earthquake size statistics considerably improved our understanding of earthquake occurrence process and would likely lead to significantly better estimates of seismic hazard. Apparently, similar progress in understanding the earthquake time statistics is much more difficult to achieve: we cannot ignore spatial variables, and the available data are not as extensive, so the task is more complex and challenging. Only by applying rigorous methods, careful analysis of systematic and random effects, and critical testing of models and hypotheses would we be able to solve this problem. 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