Self-exciting point process in modeling earthquake occurrences

In this paper, we present a procedure for modeling earthquake based on spatial-temporal point process. The magnitude distribution is expressed as truncated exponential and the event frequency is modeled with a spatial-temporal point process that is characterized uniquely by its associated conditional intensity process. The earthquakes can be regarded as point patterns that have a temporal clustering feature so we use self-exciting point process for modeling the conditional intensity function. The choice of main shocks is conducted via window algorithm by Gardner and Knopoff and the model can be fitted by maximum likelihood method for three random variables.


Introduction
A spatio-temporal point process is a random collection of points, where each point represents the time and location of an event. Examples of events include incidence of disease, sightings or births of a species, or the occurrences of fires, earthquakes, lightning strikes, tsunamis, or volcanic eruptions. Typically the spatial locations are recorded in three spatial coordinates, e.g. longitude, latitude, and height or depth, though sometimes only one or two spatial coordinates are available or of interest. One particularly useful approach to modeling spatio-temporal point processes is through the conditional intensity function which depend on the location and times of past events occurring in the time interval (0, t) [9]. In this paper we analyze earthquake model in Java island using a spatial-temporal point process approach. We present self-exciting point process where the earthquake magnitude as the mark. Point process and its notation are defined in second section. Third section defines the truncated exponential distribution and fourth section defines the spatio-temporal point precess and the self-exciting point process. We consider the distributions of earthquake magnitudes, mainshocks, and aftershocks in estimating the intensity function. Finally we apply the model to the earthquake data in Java island in the last section.
This means that any compact set K must not contain infinitely many points x i . Let In other words, a point process N is random element or a random function which assumes point measures as values: for every ω ∈ Ω the value m(·) = N (·, ω) is a point measure. In particular, N (K) < ∞ for compact sets K ⊂ E. The next result gives a justification of the fact that we may interpret a point process N as a collection (N (A)) A∈E of the random variables

Truncated Exponential Distribution
The truncated exponential model is based on Gutenberg-Richter magnitude recurrence relation. The Gutenberg-Richter recurrence relation is expressed as: where N(M) is the mean number of earthquakes per unit volume and per unit time having magnitude greater than M [4]. The a-value represents the number of events per unit time with magnitudes greater than zero, which describes the seismic activity rate in the area of interest. The b-value defines the ratio of small and large magnitude events and typically takes a value between 0.8-1.0 [3], [5]. Equation (1)  magnitude. Therefore, exponential probability distribution would describe the probability of occurrence rate of earthquake.
Although the standard Gutenberg-Richter recurrence relation can be applied to an infinite range of magnitudes, it is common to apply bounds at minimum and maximum magnitude values. This is because seismic sources are usually associated with a capacity to produce some maximum magnitude and for engineering purposes earthquakes of very small magnitudes that do not cause damage to structures are not of interest. If parameter of magnitude distribution is β then the corresponding probability density function using these minimum and maximum values is expressed below in its bounded form [3]: M min is the smallest magnitude considered in the model (here M min = 5) and M max is the largest magnitude event which can be expected in the region. Different M max values may be assigned in different regions.

Self Exciting Point Process
A spatio-temporal point process N is mathematically defined as a random measure on a region S ⊆ R x R 3 of space and time, taking values in the non-negative integers Z + . In this framework the measure N (A) represents the number of points falling in the subset A of S. Any analytical spatiotemporal point process is characterized uniquely by its associated conditional intensity process λ [6]. λ(t, x, y|H t ) may be thought of as the frequency with which events are expected to occur around a particular location (t, x, y) in spacetime, conditional on the prior history, H t , of the point process up to time t [11]. A self exciting point processes can be used for modeling events that are clustered together in time and space. Consider events is time of i -th earthquake, m i > m 0 is magnitude of i -th earthquake, m 0 is cut off magnitude. The self-exciting point process captures the tendency for earthquakes to spawn aftershocks, clustered about their parents [10]. The conditional intensity function is where (i) the function µ(x, y) represents the deterministic background rate and is assumed to take the semi-parametric form where µ is an unknown constant and u(x, y) is an unknown function (ii) mainshocks of magnitude m produce a number of aftershock with distribution k(m) (iii) aftershocks are spatially distributed about their parents according to the bivariate density function v(·) (iv) aftershocks are temporally distributed about their parents according to a nonnegative function g(·).
In application we make use of the specific function for number of aftershocks. The temporal aftershocks follow Omori's Law: the frequency of aftershocks at time t after the main shock decays hyperbolically [12]: and the spatial distribution of aftershock is determined by the Gaussian density function [7]: v(x, y|m) = 1 2πde α(m−m 0 ) exp{− 1 2 For determining the intensity function (3) the declustering algorithm is applied. We focus on the window algorithms, where magnitude-dependent windows in space and time are used to identify earthquakes of the same cluster. Here we use the window method by Gardner and Knopoff to identify the mainshocks (Table 1) [1], [2]. Furthermore we should find the background intensity u(x, y) first to estimate the conditional intensity function. An estimator of the background intensity can be estimated using thinning procedure and the model parameters θ = (µ, A, c, α, p, d, q, γ) are estimated simultaneously using an iterative approach [13], [14], [15]. The parameter vector θ is estimated by maximizing the log-likelihood function

Earthquake modeling
In this research we use the earthquake data in Java Island obtained from U.S. Geological Survey. The data was restricted to events with time period January 1973 -December 2010 from the rectangular area 5.  Figure 1 describes the frequency of earthquakes as a power of magnitude. Therefore, exponential probability distribution would describe the probability of occurrence rate of earthquakes. We take M min = 5 and M max = 7.9 and estimate β in equation (2)  Calculation is done for events ij having magnitude ≤ m i and satisfy window parameters (Table 1), i.e. △r ij ≤ △R and △t ij ≤ △T . (iii) Stop step (ii) if Gardner-Knopoff criteria is not satisfied, go to step (i).   Figure 2 while Table 2 shows the maximum likelihood estimates for parameters of the self-exciting point process obtained from Algorithm 2. This estimates have log likelihood = -1589.4533 and AIC = 3194.9. From the probability ρ j we can see that most of the events are either independent events or aftershocks. About 36.46% of the events have probability of being an aftershock. Figure 3 shows Although we obtain maximum likelihood estimates for the parameters of the model, there remains some room for further improvement. In this model, information on shocks less than 5 in magnitude was not available. Modifications of (4) - (6) can be made to model variability in mean aftershock number and the spatio-temporal clustering of aftershocks. However we should be care to avoid over parametrization of model.

Conclusion
A probabilistic approach for earthquake data in this research is self-exciting point process. This model consist of three variables and eight parameters to be estimated simultaneously.

Acknowledgments
We are grateful to Universitas Sebelas Maret for financial support through Grant of Maintenance Research Group 2015. We also thank referees for their critical and helpful comments.