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We investigate earthquake clustering, a prominent feature of seismic catalogs, in terms of distribution of the number of triggered events as described by a branching process (Kagan and Knopoff, Phys. Earth Planet. Inter., 1976; Saichev et al., Pure Appl. Geophys., 2005, and references therein). According to recent literature (e.g. Shebalin et al., Geophys. J. Int., 2020, and references therein), the productivity of a magnitude m event is defined as the number of triggered events of magnitude above m-Δ, where Δ is a positive default value. For a magnitude m event, we distinguish between the number of its direct descendants and the total number of its descendants, denoted respectively by the random variables v and V, both depending on Δ. Empirical analysis often testifies in favor of the identity of the type of distribution of both quantities (v and V) associated with the main event, and hypothetically is exponential. The testing or substantiation of this hypothesis is important for modeling seismicity and presents a serious challenge for seismic statistics. In the standard Epidemic Type Aftershock Sequence – ETAS – model (Ogata, Ann. Inst. Stat. Math., 1998), the distribution of v is Poissonian. Therefore we consider the general ETAS model adapted to any distribution of v and prove that the branching structure of the model excludes the possibility of having a common distribution type (for example, Poisson or exponential) for both v and V at once (Molchan et al., Geophys. J. Int., 2022). The second theoretical result relates to the behaviour of the tails of the productivity distribution. We show that there is a fundamental difference in tail behavior of the V-distributions for general-type clusters and for clusters with a dominant initial magnitude: the tail is heavy in the former case and light in the latter. The real data display similar behavior. Theoretical conclusions are also illustrated through the analysis of a synthetic earthquake catalog.

This study aims at developing new macroseismic intensity attenuation models valid for Italy by exploiting the most updated macroseismic dataset and earthquakes catalogue, as well as the information obtained from a critical analysis of the most recent models in the literature. Several different attenuation models have been calibrated as a function of the moment magnitude (Mw) and epicentral distance from 16.260 intensity data points, that are related to 119 earthquakes occurred after 1900. According to trends and residuals analysis, the preferred calibrated intensity (I) attenuation function is a Log-Linear model for epicentral distance (r in km) and a linear model for Mw as: I = 1.81 - 2.61 LogR - 0.0039R + 1.42Mw with R=(r2+(9.87)2)0.5; the estimated standard deviation is σ=0.75. Also noteworthy is an unusual multiplicative model for macroseismic intensity attenuation which performs as well as the best model and shows higher sensitivity to physical parameters, such as focal depth and magnitude, especially in the epicentral area. Performance of all calibrated models was also checked on an independent set of 15 post-1900 Italian earthquakes.

An adequate characterization of the temporal features of background seismicity, namely after removal of temporally and spatially clustered events (e.g. aftershocks), is a key element in several studies aimed at earthquake forecasting and seismic hazard assessment. In order to investigate the features of background seismicity component, we analyse the rate of background events, that is the rate of main/independent earthquakes as identified by Nearest Neighbour (NN) and Stochastic Declustering (SD) methods. The use of two different declustering methods, which are based on diverse statistical and physical assumptions, allows us to assess whether the identified features depend on the specific definition of background events. In this study, we carry out an in depth analysis of the time changes of background seismicity rate in Northeastern Italy, by means of continuous-time Hidden Markov Models, a stochastic tool that can be used to assess heterogeneity in the temporal pattern of seismicity rates. Specifically, we aim at understanding if the analysed time series can be better described by a homogeneous Poisson model, with unique constant rate, or by a switched Poisson model (i.e. linked to some systematic changes in earthquakes occurrence rates) or whether a basically different model is required. The analysis performed based on Markov modulated Poisson process, and according to Bayesian Information Criterion, shows that a switched Poisson process with three states is the best model describing the background rate identified by SD and NN approaches. The capability of adopted methodology in identifying seismicity rate changes, as well as the sensitivity of the method against the minimum magnitude threshold of analysis, have been verified by applying the method to synthetic catalogues with known properties, namely Poissonian time series with different rates prescribed in specific time intervals. The obtained results suggest that a Poisson model with multiple rates can be used to properly describe background seismicity in Northeastern Italy.

Complex network theory has been recently applied to get new insights and a different perspective in the study of earthquake patterns. Several studies (see for instance Daskalaki et al., J Seismol, 2016; Telesca, Phys. Chem. Earth, 2015; Varini et al., J Geophys Res, 2020; Ebrahimi et al., Chaos Solitons Fractals, 2021, and references therein) were based on the preliminary mapping of the time series of earthquakes into networks, by applying visibility graph method or other clustering algorithms. In a second step, the topological properties of the obtained networks were analyze by exploiting tools of complex network theory with the aim of discovering possible precursory signatures of strong earthquakes or other features relevant to hazard assessment.In this study we investigated the earthquake clusters extracted by two data-driven declustering algorithms: the nearest-neighbor, which classifies the earthquakes on the basis of a nearest-neighbor distance between events in the space-time-energy domain (Zaliapin and Ben-Zion, J Geophys Res, 2013), and the stochastic declustering, which is based on the space-time ETAS point process model (Zhuang et al., J Geophys Res, 2004). Case studies from selected sequences, occurred in Central Italy from 1985 to 2021, are examined in some detail.The earthquake clusters extracted by the two declustering algorithms are compared by different tools, so as to assess the similarities and differences in their classification and characterization (Varini et al., J Geophys Res, 2020). The connections between events forming a cluster, as defined by the considered declustering method, allow us representing its hierarchical structure by means of a tree graph. The topological structure of the clusters is then investigated by means of centrality measures in the frame of Network analysis.

In recent years, new approaches for developing earthquake rupture forecasts (ERFs) have been proposed to be used as an input for probabilistic seismic hazard assessment (PSHA). Zone- based approaches with seismicity rates derived from earthquake catalogs are commonly used in many countries as the standard for national seismic hazard models. In Italy, a single zone- based ERF is currently the basis for the official seismic hazard model. In this contribution, we present eleven new ERFs, including five zone-based, two smoothed seismicity-based, two fault- based, and two geodetic-based, used for a new PSH model in Italy. The ERFs were tested against observed seismicity and were subject to an elicitation procedure by a panel of PSHA experts to verify the scientific robustness and consistency of the forecasts with respect to the observations. Tests and elicitation were finalized to weight the ERFs. The results show a good response to the new inputs to observed seismicity in the last few centuries. The entire approach was a first attempt to build a community-based set of ERFs for an Italian PSHA model. The project involved a large number of seismic hazard practitioners, with their knowledge and experience, and the development of different models to capture and explore a large range of epistemic uncertainties in building ERFs, and represents an important step forward for the new national seismic hazard model.

We investigate the statistical properties of declustered catalogs as obtained from the application of two different data-driven declustering algorithms, namely the nearest-neighbor method and the stochastic declustering method (Benali et al., Stoch. Environ Res Risk Assess, 2020). The nearest-neighbor method partitions earthquakes into background and clustered components, based on nearest-neighbor distances between earthquakes in the space-time-magnitude domain (Zaliapin and Ben-Zion, J Geophys Res, 2013); the stochastic declustering method classifies earthquakes into background and clustered components through a probabilistic procedure based on the estimation of the space-time ETAS model (Zhuang et al., J Geophys Res, 2004).Two Italian case studies are considered: North-Eastern Italy (data from OGS Bulletins) and Central Italy (data from ISIDe catalog). For both case studies, the time series of background seismicity are obtained from the two declustering methods. Then we investigate the general assumption according to which the temporal sequence of background seismicity is suitably modelled by the stationary Poisson model. For this purpose, several features and statistical tests are considered to verify the main properties that characterize Poisson processes (e.g. events are independent, exponential inter-arrival times, etc.). Whenever the Poissonian hypothesis is rejected, we get evidence of certain heterogeneity in the background sequence, which leads us to rule out the simple Poisson process for background seismicity modeling. As a simple and more suitable alternative, we consider here the Markov Modulated Poisson Process (MMPP model), which allows the Poisson seismicity rate to change over time according to a finite (unknown) number of states of the system. The MMPP model turns out suitable for identifying and quantifying heterogeneities in background seismicity, as well as for comparing them against the two considered declustering algorithms.

Earthquake clustering is a significant feature of seismic catalogs, both in time and space. Several methodologies for earthquake cluster identification have been proposed in the literature in order to characterize clustering properties and to analyze background seismicity. We consider two recent data-driven declustering techniques, one based on nearest-neighbor distance and the other on a stochastic point process. These two methods use different underlying assumptions and lead to different classifications of earthquakes into background events and clustered events. We investigated the classification similarities by exploiting graph representations of earthquake clusters and tools from network analysis. We found that the two declustering algorithms produce similar partitions of the earthquake catalog into background events and earthquake clusters, but they may differ in the identified topological structure of the clusters. Especially the clusters obtained from the stochastic method have a deeper complexity than the clusters from the nearest-neighbor method. All of these similarities and differences can be robustly recognized and quantified by the outdegree centrality and closeness centrality measures from network analysis.

Nowaday, macroseismic data are still essential for the seismic hazard assessment in several regions because they provide important knowledge on preinstrumental earthquakes, nedeed to compile historical earthquake catalogs. This is especially true for Italy, which boasts a large and accurate macroseismic database, DBMI15, composed by 122701 macroseismic records related to 3212 earthquakes occurred from 1000 up to 2014. It should be noted that some records are incomplete or the available information is insufficient for the assignment of the intensity at a given site (e.g. intensity IX-X denotes that the level of damage at that site is uncertain and evaluated IX or X with a probability of 50% each). In order to respect both the ordinal nature of macroseismic intensity and its tendency to decrease with distance from the epicentre, we consider the beta-binomial model by Rotondi and Zonno (Ann. Geophys., 2004; Rotondi et al., Bull. Earthq. Eng., 2016) which describes the probability distribution of the intensity at a site, conditioned on the epicentral intensity and on the epicentre-to-site distance. The application of the beta-binomial model typically requires rounding-up or -down the observed intensities to the nearest integer values. We propose an extension of the beta-binomial model in order to include in the stochastic modelling the uncertainty in the assignment of the intensities. Then we exploit the advantages of the Bayesian approach for uncertainty quantification both in the estimation procedure and in the forecast of damage scenarios.

Physics-based models focus on the generation process of individual earthquakes but the strong space-time interaction existing among the events of seismic sequences requires that the seismic activity be studied as a whole through statistics-based models in order to forecast its future trend. Properties such as long-range interactions, power law distributions, fractal geometries are common to all complex systems and are also shared by earthquakes and fault systems. Over recent years, on the one hand many studies have shown the inability of classical statistical mechanics to treat complex systems in exhaustive manner, and, on the other hand, the application of the Tsallis entropy Sq - a generalization of the classical Boltzmann-Gibbs entropy in non-extensive sense – has led to long-tailed power-law distributions typical of complex systems (Vallianatos et al. 2018); it seems hence that the non-extensive statistical physics can offer an appropriate framework of investigation for complex phenomena. In this work we follow this approach giving a detailed statistical treatment of its application to Italian earthquake sequences covering a period of some years; each data set was partitioned in moving time windows.Given a continuous variable X with probability distribution f(X), by maximizing the Tsallis entropy under appropriate constraints, such as the generalized expectation value and the normalization constant, it turns out that f(X) is a q-exponential distribution. The q entropic index can assume positive values less or larger than 1: in the former case the system is in a super-additive state and f(X) is defined on a finite domain depending on model parameters, in the latter case the system is in a sub-additive state and f(X) is defined on R+. Through a variable transformation required by the fragment-asperity model for earthquake generation, one derives the probability distribution of the magnitude from the two versions of f(X). Following the Bayesian approach we have estimated the parameters by generating random samples from the posterior distributions through the Metropolis-Hasting algorithm; moreover, in each time window, we have evaluated the Tsallis entropy and compared the performance of the two versions of the magnitude distribution in terms of marginal posterior likelihood. The temporal variations of the q-index and of the entropy Sqcan be helpful in identifying in which dynamics regime the system is, and therefore in improving our ability to forecast seismicity evolution. Some of the results achieved partially disagree with those present in the literature (Vallianatos et al. 2018); what seems reasonable is to consider the change of one of these variables, rather than a specific trend, as index of a phase change of the physical system.References Vallianatos F., Michas G. and G. Papadakis (2018) Nonextensive statistical seismology: An overview, from: Complexity and Seismic Time Series. Measurement and Application, eds. Chelidze T., Vallianatos F., Telesca L., Elsevier, 25-59

The aim of this paper is to identify the ground motion models (GMMs), applicable in active shallow crustal regions (ASCRs) and subduction zones (SZs), to be used for the new release of the seismic hazard model of Italy (MPS19) for peak ground acceleration and 12 ordinates of the acceleration response spectra (5% damping) in the period range 0.05–4 s. The steps to achieve such goal are: (1) a pre-selection of the GMMs that takes into account the suitability for the application to the Italian territory and the fulfillment of the new hazard model requirements; (2) the assessment of a proper scoring of the pre-selected GMMs using strong-motion data recorded in Italy; (3) the selection of the GMMs to be used in the hazard calculation. The final set of GMMs describes satisfactorily the epistemic uncertainty of the ground shaking process, privileging the simplicity and flexibility of the functional form. Finally, the weights of the selected GMMs are assigned combining the results of the scoring and the weights obtained through an experts’ elicitation.

The long-term recurrence of strong earthquakes is often modeled according to stationary Poisson processes for the sake of simplicity. However, renewal and self-correcting point processes (with nondecreasing hazard functions) are more appropriate. Short-term models mainly fit earthquake clusters due to the tendency of an earthquake to trigger other earthquakes. In this case, self-exciting point processes with nonincreasing hazard are especially suitable. To provide a unified framework for analysis of earthquake catalogs, Schoenberg and Bolt proposed the short-term exciting long-term correcting model in 2000, and in 2005, Varini used a state-space model to estimate the different phases of a seismic cycle. Both of these analyses are combinations of long-term and short-term models, and the results are not completely satisfactory, due to the different scales at which these models appear to operate. In this study, we propose alternative modeling. First, we split a seismic sequence into two groups: the leader events, non-secondary events the magnitudes of which exceed a fixed threshold; and the remaining events, which are considered as subordinate. The leader events are assumed to follow the well-known self-correcting point process known as the stress-release model. In the interval between two subsequent leader events, subordinate events are expected to cluster at the beginning (aftershocks) and at the end (foreshocks) of that interval; hence, they are modeled by a failure process that allows bathtub-shaped hazard functions. In particular, we examined generalized Weibull distributions, as a large family that contains distributions with different bathtub-shaped hazards, as well as the standard Weibull distribution. The model is fit to a dataset of Italian historical earthquakes, and the results of Bayesian inference based on the Metropolis–Hastings algorithm are shown.

According to the idea now widespread that macroseismic intensity should be expressed in probabilistic terms, a beta-binomial model has been proposed in the literature to estimate the probability of the intensity at site in the Bayesian framework and a clustering procedure has been adopted to define learning sets of macroseismic fields required to assign prior distributions of the model parameters. This article presents the results concerning the learning sets obtained by exploiting the large Italian macroseismic database DBM1I11 (Locati et al. in DBMI11, the 2011 version of the Italian Macroseismic Database, 2011. http://emidius.mi.ingv.it/DBMI11/ ) and discusses the problems related to their use in probabilistic modelling of the attenuation in seismic regions of the European countries partners of the UPStrat-MAFA project (2012), namely South Iceland, Portugal, SE Spain and Mt Etna volcano area (Italy). Anisotropy and the presence of offshore earthquakes are some of the problems faced. All the work has been carried out in the framework of the Task B of the project.

The macroseismic intensity is an ordinal variable that describes the seismic damage effects of an earthquake. The collection of intensity values recorded at sites of an area hit by an earthquake is called macroseismic field; it constitutes the only information on historical earthquakes for which no instrumental recordings are available. Around the area of the epicenter, lines bounding points of equal seismic intensity (isoseismal lines) are used to represent the spatial distribution of the macroseismic intensities, and their shapes can suggest the characteristics of the earthquake source. Our aim is to identify clusters of macroseismic fields according to the size and shape of their isoseismal lines, or in seismological terms, according to the trend of macroseismic attenuation. First, the isoseismal lines of some fields are approximated by convex hulls. Then, fixed an intensity value, the set of the corresponding convex hulls are analyzed on the basis of statistical data depth functions. This nonparametric method ranks the convex hulls according to their statistical depth values, which in the present study are defined by the modified local half-region depth function. A similarity measure based on the same depth function allows us to compare all pairs of hulls and build a dissimilarity matrix to which a clustering procedure is applied in order to detect clusters of fields homogenoeus from the attenuation viewpoint. This method is illustrated on both simulated and real macroseismic data.

A procedure for seismic risk assessment is applied to the Mt. Etna area (eastern Sicily, Italy) through assessment of urban system dysfunction following the occurrence of an earthquake. The tool used is based on the Disruption Index as a concept implemented in Simulator QuakeIST, which defines urban disruption following a natural disaster. The first element of the procedure is the definition of the seismic input, which is based on information about historical seismicity and seismogenic faults. The second element is computation of seismic impact on the building stock and infrastructure in the area considered. Information on urban-scale vulnerability was collected and a geographic information system was used to organise the data relating to buildings and network systems (e.g., building stock, schools, strategic structures, lifelines). The central idea underlying the definition of the Disruption Index is identification and evaluation of the impact on a target community through the physical elements that most contribute to severe disruption. The procedure applied in this study (i.e., software and data) constitutes a very useful operational tool to drive the development of strategies to minimise risks from earthquakes.

This study presents a series of self-correctingmodels that are obtained by integrating information about seismicity and fault sources in Italy. Four versions of the stress release model are analyzed, in which the evolution of the system over time is represented by the level of strain, moment, seismic energy, or energy scaled by the moment. Wecarry out the analysis on a regional basis by subdividing the study area into eight tectonically coherent regions. In each region, we reconstruct the seismic history and statistically evaluate the completeness of the resulting seismic catalog. Following the Bayesian paradigm, we applyMarkov chainMonte Carlomethods to obtain parameter estimates and a measure of their uncertainty expressed by the simulated posterior distribution. The comparison of the four models through the Bayes factor and an information criterion provides evidence (to different degrees depending on the region) in favor of the stress release model based on the energy and the scaled energy. Therefore, among the quantities considered, this turns out to be the measure of the size of an earthquake to use in stress release models. At any instant, the time to the next event turns out to followaGompertz distribution, with a shape parameter that depends on time through the value of the conditional intensity at that instant. In light of this result, the issue of forecasting is tackled through both retrospective and prospective approaches. Retrospectively, the forecasting procedure is carried out on the occurrence times of the events recorded in each region, to determine whether the stress release model reproduces the observations used in the estimation procedure. Prospectively, the estimates of the time to the next event are compared with the dates of the earthquakes that occurred after the end of the learning catalog, in the 2003-2012 decade.

The author provides an approximated solution for the filtering of a state-space model, where the hidden state process is a continuous-time pure jump Markov process and the observations come from marked point processes. Each state k corresponds to a different marked point process, defined by its conditional intensity function λk(t). When a state is visited by the hidden process, the corresponding marked point process is observed. The filtering equations are obtained by applying the innovation method and the integral representation theorem of a point process martingale. Since the filtering equations belong to the family of Kushner–Stratonovich equations, an iterative solution is calculated. The theoretical solution is approximated and a Monte Carlo integration technique employed to implement it. The sequential method has been tested on a simulated data set based on marked point processes widely used in the statistical analysis of seismic sequences: the Poisson model, the stress release model and the Etas model.

SUMMARY In this paper, we evaluate the seismic hazard of a region in southern Italy by analysing stress release models from the Bayesian viewpoint; the data are drawn from the most recent version of the parametric catalogue of Italian earthquakes. For estimation we just use the events up to 1992, then we forecast the date of the next event through a stochastic simulation method and we compare the result with the really occurred shocks in the span 1993–2002. The original version of the stress release model, proposed by Vere-Jones in 1978, transposes Reid's elastic rebound theory in the framework of stochastic point processes. Since the nineties enriched versions of this model have appeared in the literature, applied to historical catalogues from China, Iran, Japan; they envisage the identification of independent or interacting tectonic subunits constituting the region under exam. It follows that the stress release models, designed for regional analyses, are evolving towards studies on fault segments, realizing some degree of convergence to those models that start from an individual fault and, considering the interactions with nearby segments, are driven to studies on regional scale. The optimal performance of the models we consider depends on a set of choices among which: the seismogenic region and possible subzones, the threshold magnitude, the length of the time period. In this paper, we focus our attention on the influence of the subdivision of the region under exam into tectonic units; in the light of the recent studies on the fault segmentation model of Italy we propose a partition of Sannio-Matese-Ofanto-Irpinia, one of the most seismically active region in southern Italy. The results show that the performance of the stress release models improves in terms of both fitting and forecasting when the region is split up into parts including new information about potential seismogenic sources.

Point processes are the stochastic models most suitable for describing physical phenomena that appear at irregularly spaced times, such as the earthquakes. These processes are uniquely characterized by their conditional intensity, that is, by the probability that an event will occur in the infinitesimal interval (t, t+Δt), given the history of the process up tot. The seismic phenomenon displays different behaviours on different time and size scales; in particular, the occurrence of destructive shocks over some centuries in a seismogenic region may be explained by the elastic rebound theory. This theory has inspired the so-called stress release models: their conditional intensity translates the idea that an earthquake produces a sudden decrease in the amount of strain accumulated gradually over time along a fault, and the subsequent event occurs when the stress exceeds the strength of the medium. This study has a double objective: the formulation of these models in the Bayesian framework, and the assignment to each event of a mark, that is its magnitude, modelled through a distribution that depends at timet on the stress level accumulated up to that instant. The resulting parameter space is constrained and dependent on the data, complicating Bayesian computation and analysis. We have resorted to Monte Carlo methods to solve these problems.

The Disruption Index is used here for the assessment of urban disruption in the Mt. Etna area after a natural disaster. The first element of the procedure is the definition of the seismic input, which is based on information about the historical seismicity and seismogenic faults. The second element is the computation of the seismic impact on the building stock and infrastructure in the region considered. Information on urban-scale vulnerability was collected and a geographic information system was used to organize the data relating to buildings and network systems (e.g., typologies, schools, strategic structures, lifelines). The central idea underlying the definition of the Disruption Index is the identification and evaluation of the impacts on a target community, considering the physical elements that contribute most to the severe disruption. The results of this study are therefore very useful for earthquake preparedness planning and for the development of strategies to minimize the risks from earthquakes. This study is a product of the European “Urban Disaster Prevention Strategies using Macroseismic Fields and Fault Sources” project (UPStrat-MAFA European project 2013).

This communication aims to show some results on the distribution of the waiting time of a self-correcting point process that is known in seismology as the stress release model, and in actuarial sciences as the classical risk model. By assuming this model and conditional on the history of the process up to time $$t$$ , the waiting time of the next event from $$t$$ follows a Gompertz distribution, in which the shape parameter is history dependent. This reveals some new features of the model: (1) Just after a major event, a high probability can be attributed to very short waiting times, and not exclusively to long ones; (2) By updating the history while waiting for the next event, the expected waiting time is postponed in time and its variance decreases. We illustrate the application of a stress release model to a sequence of Italian earthquakes. A Bayesian approach is proposed to consider the uncertainty of the model parameters.

It is a widely shared opinion that not only secondary earthquakes (aftershocks) but also main earthquakes tend to occur in time-space clusters. The importance of this assumption requires the application of statistical tools to objectively evaluate its coherence with the reality at different scales of size-space-time. Global tests allow us to select the data sets with significant space-time clustering in order to perform more in-depth analyses to detect cluster locations. According to different fixed magnitude thresholds, we perform two global statistical tests, the Knox test and the Jacquez test, based on the space-time distance between pairs of earthquakes under the null hypothesis of uniform distribution in time and space, and evaluate the significance of the possible clusters. We analyze subsets of historical Italian earthquakes drawn from the Parametric Catalog of Italian Earthquakes (CPTI04) with magnitude thresholds 4.5, 5.3 and 6.0, associated with the composite seismogenic sources of the Database of Individual Seismogenic Sources. Each subset is related to one of the eight tectonically homogeneous macroregions in which the Italian territory has been divided. Significant space-time clustering is found for all sets with a magnitude threshold of 4.5. This tendency decreases drastically or disappears when the cut off rises to 5.3, with the exception of two macroregions located in the Eastern Alps and the Calabrian Arc, respectively, where evidence of space-time interaction may refer to stress transfer among consecutive or adjacent faults. The link between clustering effect and tectonic behavior could guide the choice of different stochastic point processes to model the seismic activity.

We consider continuous-time models where the observed process depends on an unob- served jump Markov Process. We develop a sequential Monte Carlo algorithm which makes it possible to lter and smooth this latent process, and compute the likelihood pointwise. We develop a Rao- Blackwellisation technique which allows to signican tly reduce the Monte Carlo noise of this algorithm. Possible extensions of our algorithm and further directions of research are discussed.

We consider point processes defined on the space–time domain which model physical processes characterized qualitatively by the gradual increase over time in some energy until a threshold is reached, after which, an event causing the loss of energy occurs. The risk function will, therefore, increase piecewise with sudden drops in correspondence to each event. This kind of behaviour is described by Reid's theory of elastic rebound in the earthquake generating process where the quantity that is accumulated is the strain energy or stress due to the relative movement of tectonic plates. The complexity and the intrinsic randomness of the phenomenon call for probabilistic models; in particular the stochastic translation of Reid's theory is given by stress release models. In this article we use such models to assess the time-dependent seismic hazard of the seismogenic zone of the Corinthos Gulf. For each event we consider the occurrence time and the magnitude, which is modelled by a probability distribution depending on the stress level present in the region at any instant. Hence we are dealing here with a marked point process. We perform the Bayesian analysis of this model by applying the stochastic simulation methods based on the generation of Markov chains, the so called Markov chain Monte Carlo (MCMC) methods, which allow one to reconcile the model's complexity with the computational burden of the inferential procedure. Stress release and Poisson models are compared on the basis of the Bayes factor.