About Earthquake Forecasting by Markov Renewal Processes

Abstract

We propose and validate a new method for the evaluation of seismic hazard. In particular, our aim is to model large earthquakes consistently with the underlying geophysics. Therefore we propose a non-Poisson model, which takes into account occurrence history, improved with some physical constraints. Among the prevalent non-Poisson models, we chose the Markov renewal process, which is expected to be sufficient to capture the main characteristics, maintaining simplicity in analysis. However, due to the introduction of some physical constraint, our process differs significantly from others already presented in literature. A mixture of exponential + Weibull distributions is proposed for the waiting times and their parameters are estimated following the likelihood method. We validated our model, using data of earthquakes of high severity occurred in Turkey during the 20 th century. Our results exhibit a good accordance with the real events.

Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models

“…More recent studies include the application of Weibull semi-Markov process through a new parametric estimation method (Alvarez, 2005). A similar application based on a mixture of exponential and Weibull distribution for sojourn times was later presented by Garavaglia and Pavani (2009). Parametric methods provide estimators with several attractive asymptotic properties; however, these estimators present inconvenience when the sample size is small.…”

A parametric Markov renewal model for predicting tropical cyclones in Bangladesh

“…The Poisson model or some of its variants are considered in several studies for describing cyclone occurrences, for instance, Lu and Garrido (2005) applied a doubly periodic non-homogeneous Poisson model, and Jagger et al (2002) proposed a space-time model for hurricane data. Among other modelling approaches that are available in the literature to model natural hazards, Markov renewal process (MRP) and semi-Markov process (SMP) have received considerable focus (e.g., Gregory et al, 1993;Lardet and Obled, 1994;Alvarez, 2005;Garavaglia and Pavani, 2011;Masala, 2012a). …”

“…In seismic hazard analysis Alvarez (2005) argued that Poisson model may not be appropriate for large earthquakes as the process is inherently memoryless, and proposed a Markov renewal model for studying earthquake occurrences in North Anatolian Fault Zone in Turkey. Using the same data Garavaglia and Pavani (2011) proposed an MRP where a mixture of exponential and Weibull distributions were used for the waiting time distribution of the large earthquakes. Then the so-called 'crossing state probabilities' are estimated.…”

“…where the model was validated using a hurricane database, and cross state predictions were performed. However, the parameters of transition probability matrices were first estimated assuming a sample from a multinomial distribution, and then using these estimates, sojourn time parameter estimation was performed in Garavaglia andPavani (2011), Masala (2012a), Masala (2012b). Recently, a non-parametric approach of a semi-Markov model was adopted by Votsi et al (2012) for the estimation of earthquake occurrences in the Northern Aegean Sea in Greece.…”

“…It provides a detail of estimation of parameters for an MRP, and while estimating parameters, likelihood function is maximized both for parameters of the transition probability matrix and sojourn time distributions simultaneously unlike Garavaglia andPavani (2011), Masala (2012a) and Masala (2012b). Model building strategies were performed by testing a series of nested hypotheses whether we can further reduce the sojourn time parameters to find an optimal model.…”