[1] Assuming that earthquakes are the realization of a stochastic point process and that the magnitude distribution of all earthquakes is described by the Gutenberg-Richter law with a constant b value, we model the occurrence rate density of earthquakes in space and time by means of an epidemic model. The occurrence rate density is computed by the sum of two terms, one representing the independent, or spontaneous activity, and the other representing the activity induced by previous earthquakes. While the first term depends only on space, the second one is factored into three terms that include the magnitude, time, and location, respectively, of the past earthquakes. In this paper we use the modified Omori law for the time term, focusing our investigation on the magnitude and space terms. We formulate two different hypotheses for each of them, and we find the respective maximum likelihood parameters on the basis of the catalog of instrumental seismicity recorded in Italy from 1987 to 2000. The comparison of the respective likelihood computed for the seismicity recorded in 2001 provides a way for choosing the best model. The confidence level of our choice is then assessed by means of a Monte Carlo simulation on the varioushypotheses. Our study shows that an inverse power density function is more reliable than a normal density function for the space distribution and that the hypothesis of scale invariance of aftershock productivity with respect to magnitude can be rejected with high confidence level. The final model is suitable for computing earthquake occurrence probability in real circumstances.INDEX TERMS: 7223 Seismology: Seismic hazard assessment and prediction; 7230 Seismology: Seismicity and seismotectonics; 7260 Seismology: Theory and modeling; KEYWORDS: earthquake clustering, aftershocks, stochastic processes, hypothesis test, Italian seismicity Citation: Console, R., M. Murru, and A. M. Lombardi, Refining earthquake clustering models,

Abstract. Earthquakes are regarded as the realization of a point process modeled by a generalized Poisson distribution. We assume that the Gutenberg-Richter law describes the magnitude distribution of all the earthquakes in a sample, with a constant b value. We model the occurrence rate density of earthquakes in space and time as the sum of two terms, one representing the independent, or spontaneous, activity and the other representing the activity induced by previous earthquakes. The first term depends only on space and is modeled by a continuous function of the geometrical coordinates, obtained by smoothing the discrete distribution of the past instrumental seismicity. The second term also depends on time, and it is factorized in two terms that depend on the space distance (according to an isotropic normal distribution) and on the time difference (according to the generalized Omori law), respectively, from the past earthquakes. Knowing the expected rate density, the likelihood of any realization of the process (actually represented by an earthquake catalog) can be computed straightforwardly. This algorithm was used in two ways: (1) during the learning phase, for the maximum likelihood estimate of the few free parameters of the model, and (2)

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[1] We revisit the issue of the so-called Båth's law concerning the difference D 1 between the magnitude of the main shock and the second largest shock in the same sequence. A mathematical formulation of the problem is developed with the only assumption being that all the events belong to the same self-similar set of earthquakes following the GutenbergRichter magnitude distribution. This model shows a substantial dependence of D 1 on the magnitude thresholds chosen for the main shocks and the aftershocks and in this way partly explains the large D 1 values reported in the past. Analysis of the New Zealand and Preliminary Determination of Epicenters (PDE) catalogs of shallow earthquakes demonstrates a rough agreement between the average D 1 values predicted by the theoretical model and those observed. Limiting our attention to the average D 1 values, Båth's law does not seem to strongly contradict the Gutenberg-Richter law. Nevertheless, a detailed analysis of the D 1 distribution shows that the Gutenberg-Richter hypothesis with a constant b-value does not fully explain the experimental observations. The theoretical distribution has a larger proportion of low D 1 values and a smaller proportion of high D 1 values than the experimental observations. Thus, Båth's law and the Gutenberg-Richter law cannot be completely reconciled, although based on this analysis the mismatch is not as great as has sometimes been supposed.

We forecast time‐independent and time‐dependent earthquake ruptures in the Marmara region of Turkey for the next 30 years using a new fault segmentation model. We also augment time‐dependent Brownian passage time (BPT) probability with static Coulomb stress changes (ΔCFF) from interacting faults. We calculate Mw > 6.5 probability from 26 individual fault sources in the Marmara region. We also consider a multisegment rupture model that allows higher‐magnitude ruptures over some segments of the northern branch of the North Anatolian Fault Zone beneath the Marmara Sea. A total of 10 different Mw = 7.0 to Mw = 8.0 multisegment ruptures are combined with the other regional faults at rates that balance the overall moment accumulation. We use Gaussian random distributions to treat parameter uncertainties (e.g., aperiodicity, maximum expected magnitude, slip rate, and consequently mean recurrence time) of the statistical distributions associated with each fault source. We then estimate uncertainties of the 30 year probability values for the next characteristic event obtained from three different models (Poisson, BPT, and BPT + ΔCFF) using a Monte Carlo procedure. The Gerede fault segment located at the eastern end of the Marmara region shows the highest 30 year probability, with a Poisson value of 29% and a time‐dependent interaction probability of 48%. We find an aggregated 30 year Poisson probability of M > 7.3 earthquakes at Istanbul of 35%, which increases to 47% if time dependence and stress transfer are considered. We calculate a twofold probability gain (ratio time dependent to time independent) on the southern strands of the North Anatolian Fault Zone.

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[1] In this study, we compute the effect of stress change due to previous historical earthquakes on the probability of occurrence of future earthquakes on neighboring faults. Following a methodology developed in the last decade, we start from the estimate of the probability of occurrence in the next 50 years for a characteristic earthquake on known seismogenic structures, based on a time-dependent renewal model. Then a physical model for the Coulomb stress change caused by previous earthquakes on these structures is applied. The influence of this stress change on the occurrence rate of characteristic earthquakes is computed, taking into account both permanent (clock advance) and temporary (rate-and-state) perturbations. We apply this method to the computation of earthquake hazard of the main seismogenic structures recognized in the Central and Southern Apennines region, for which both historical and paleoseismological data are available. This study provides the opportunity of reviewing the problems connected with the estimate of the parameters of a renewal model in case of characteristic earthquakes characterized by return times longer than the time spanned by the available catalogues and the applicability of the concept of characteristic earthquake itself. The results show that the estimated effect of earthquake interaction in this region is small compared with the uncertainties affecting the statistical model used for the basic time-dependent hazard assessment.

Here we give a brief outline of epidemic models incorporating the short-term clustering properties of earthquakes.Assuming the validity of the Gutenberg-Richter law, the occurrence-rate density of seismic events triggered by previous earthquakes can be described as: