Introduction

"Act with awareness"

- Pittacus of Mytilene

I.1. Motivation and objectives

Earthquakes constitute one of the most lethal natural hazards resulting in more than 200.000 casualties worldwide each decade. They can become vastly devastating and life-threatening, as in the cases of the recent 2011 M8.9 Japan, the 2008 M8.0 Sichuan China and the 2004 M9.3 Sumatra earthquakes. Earthquake forecasting is therefore a social demand, and scientific efforts have to be intensified for this scope. The quest for earthquake prediction dates back to times when superstition prevailed, and prediction was the domain of occultism and this search is still ever-present. Despite more than a century of research, research on earthquake prediction has undergone broad criticism and skepticism, is continuously debatable and continuously remains as an insolvable but highly attractive scientific problem.

At the outset, clarification is needed regarding the usage of the term "prediction" in seismology. Reliable earthquake predictions are considered the ones that provide a space-time-magnitude range, including the magnitude scale (i.e. local magnitude, moment magnitude, etc.) and the number of earthquakes expected in this range (i.e. zero, one, at least one, etc.). The forecast or prediction of an earthquake is a statement about time, hypocenter location, magnitude and the probability of occurrence of an individual future event within reasonable error ranges [ZÖL 09].

The prediction was continuously expressed as the occurrence probability of an earthquake, in a given time, space and magnitude range. The definition of this range constitutes a scientific target by itself. The techniques developed for this scope were diverse, and thus, earthquake prediction was discriminated in a short term, when the referred time interval concerned a day to a few hundred days before a strong earthquake, an intermediate term covering the interval from about one year to one decade and a long term for intervals longer than a decade [KNO 96].

I.2. Seismic hazard assessment

For the evaluation of seismic hazard, a set of parameters are used that express the intensity of ground motion. Thus, the probability of exceedance of predefined parameter values in a specified exposure time needs to be calculated. For the seismic hazard assessment at a specific site, either the deterministic approach or the stochastic approach is followed. In the deterministic approach, the ground shaking at the site is estimated from one or more earthquakes of a specified location and magnitude. Deterministic earthquake scenarios may be based on the actual occurrences of past events, or they may be postulated scenarios backed by analysis of seismological and geological data. The other approach is the probabilistic method, in which the contributions from all possible earthquakes around the site are integrated to find the shaking to not overpass a certain probability estimate at that place in some time period.

Both approaches exhibit strong and weak points. "The deterministic approach provides a clear and trackable method of computing seismic hazard, whose assumptions are easily discerned. It provides understandable scenarios that can be related to the problem at hand. However, it has no way for accounting for uncertainty. Conclusions based on deterministic analysis can be easily upset by the occurrence of new earthquakes". The probabilistic approach to seismic hazard calculations originally proposed by Cornell [COR 68] uses an integration of the anticipated ground motion produced by all earthquake sources and magnitudes comprised in a specific area around the site of interest, for calculating the probabilities of certain levels of the ground motion there. In this way, the probabilistic method provides the potential of the specific ground motion exceedance during some time period. "The probabilistic approach is capable of integrating a wide range of information and uncertainties into a flexible framework. Unfortunately, its highly integrated framework can obscure those elements that drive the results and its highly quantitative nature can lead to false impressions of accuracy".

I.3. Earthquake occurrence models

Comparing the deterministic approach for seismic hazard assessment with the probabilistic one, we should note that the latter is the most favorable today, relying on stochastic models for estimating the probabilities of generation of strong earthquakes. Deterministic earthquake prediction is still far from becoming feasible for practical applications, whereas the probabilistic one is realistic.

Besides data analysis, the modeling of the earthquake process is essential for a deep understanding and potential forecasts of the earthquake process. Progress in earthquake modeling can be assessed by examining different model classes. The two main classes are stochastic models and physics-based models [HAI 09]. Here, we focus on stochastic models serving as a tool for probabilistic seismic hazard assessment. Let us first provide the fundamental difference between a stochastic model and a physical model. The main difference between a stochastic model and a physical model is that the former, in contrast to the latter, considers that the physical process depends on some random aspects and therefore could not be fully understood. These random aspects are taken into account in the stochastic modeling and are expressed by means of parameters or associated stochastic processes. The stochastic models could enable us to quantify the parts of the physical process that are accessible to direct measurement, the parts that are due to its randomness and the associated uncertainties. On the other hand, the physical models aim to achieve full understanding and prediction of the physical process. The strict discrimination between the stochastic and physical models, however, cannot be unambiguously performed, since a large percentage of the models comprise physical, stochastic and empirical components.

Stochastic models play two main roles in their diverse fields of application [VER 10b, VER 10a]. First, in statistical mechanics, stochastic models aim to understand the associated physical process itself. Second, they aim to achieve planification, decision-making and/or prediction. Earthquake occurrence models are further divided into time-dependent and time-independent ones. The main assumption of the time-independent earthquake occurrence models is that the number of earthquake occurrences follows the Poisson distribution. In this case, the only information that is needed in order to calculate the associated probabilities is the mean recurrence times. The most common time-independent stochastic model of earthquake occurrences is the Poisson model, which assumes that earthquake occurrence does not depend on time. This model considers that the epicenters and times of earthquakes that exceed a certain threshold magnitude correspond to the realization of a temporally homogeneous Poisson process and serve as a test bed for comparisons with more complicated models.

In contrast, according to the time-dependent earthquake occurrence models, the probability of an earthquake occurrence is not independent of the other event times. These models require not only the mean recurrence times of earthquakes but also the variance of the frequency of earthquakes and the time since the last event. Given that the memoryless property that characterizes the Poisson process is quite restrictive, other stochastic models have been considered. The time-dependent models are particularly appealing since they provide results that are consistent with the elastic rebound theory of earthquakes. Renewal, Markov and semi-Markov models belong to this category.

Earthquake occurrence models, either time-independent or time-dependent, are based on assumptions regarding the magnitude-frequency distribution. The simplest among them is the "characteristic earthquake model", in which all strong earthquakes associated with a certain fault segment are assigned similar magnitudes, average displacements and rupture lengths, while the Gutenberg-Richter magnitude-frequency distributions and multi-segment ruptures involve more complexities. The time-dependent models are more complicated with more input parameters and assumptions. The stochastic models constitute an extended version of physical models since they aim to explain the variability of the observations and the hidden features underlying the physical process. Let us now describe some of the most important stochastic models that have been used to model earthquake occurrences. Each model provides a different type of insight into the physical process and its randomness. The merit of models lies in the degree to which they can explain a composite phenomenon.

I.3.1. Stress release models

Point process modeling for earthquake data was introduced by Vere-Jones [VER 66, VER 70, VER 78]. In this type of modeling, a seismic sequence is interpreted as a realization of a point process. The statistical inference of point processes is based on the conditional intensity function, which characterizes entirely a point process model. The corresponding models, known as stress release models, constitute the probabilistic translation of Reid's theory of the elastic rebound [REI 10], widely used in the analysis of the historical catalogues for China, Japan and Iran. These models are based on the Cramér-Lundberg model,...