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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.

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.

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.

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.

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.