# ETAS Space-Time Modeling of Chile Triggered Seismicity Using Covariates: Some Preliminary Results
## 1 Introduction  
Chile is considered one of the most active seismic countries in the world due to its particular location. In particular, Chile is located on three great tectonic plates: Nazca, South American, and Antarctic. The seismic events in Chile result from the convergence of the oceanic lithosphere of the Nazca plate beneath the South American continental plate at a rate of about 6.5 cm/year [1]. The strong coupling between the two plates produces the most intense seismicity and the largest earthquakes in the country [2; 3; 4]. Seismic activity induces various types of ground movements, as seismic waves (P and S-waves), which can cause significant damages to structures, and consequently losses of lives [5; 6]. The velocity of the waves can be different according to the depth [7; 8].  
Seismic activity can be described by a spatio-temporal branching process, such as the ETAS (Epidemic Type Aftershock Sequences) model [9], which is a semiparametric model with a large time-scale component for the background seismicity and a small time-scale component for the triggered seismicity. The large-scale component intensity function is usually estimated by nonparametric techniques, specifically in our paper, we used the Forward Likelihood Predictive approach (FLP) approach [10; 11; 12]; the triggered seismicity is modeled with a parametric space-time function. In classical ETAS models, the expected number of triggered events depends only on the magnitude of the main event. In previous papers [13; 14] we already used the classical ETAS model to describe Chilean seismicity, also using different models for different Chilean areas.  
From a statistical point of view, forecast of triggered seismicity can be performed in the days following a big event; of course, the estimation of this component is very important to forecast the evolution of a seismic sequence in space and time domain.  
In the classical ETAS model [9], the average number of future triggered events just depends on the magnitude of the previous event. Although that represents a very comfortable and useful model, in many situations, it may be too simple to describe the aftershocksdynamics, as often highlighted by diagnostic analysis. Furthermore, the high seismicity of Chile, together with the recent increasing GPS data availability, allowed us to consider a much more complex model for describing and analyzing the Chilean triggered activity.  
In this paper, we provide a description of the triggered seismicity in Chile after a subdivision of the country into two spatial areas. In particular, based on the methodology proposed by [15], we suggest the use of a specific branching-type model for earthquake description (the ETAS model) in a regression-oriented version, accounting also for external covariates, related to the depth of events and to some GPS measurements. For further examples on applications incorporating the external information in self-exciting models, see [16; 17; 18; 19].  
Interesting and encouraging results, though not completely satisfactory, are provided, accounting for covariates related to the depth of events and to some GPS measurements, corresponding to the Earth movement observed until the time of main events. In addition, some of the proposed models can improve the description and the forecasting of the triggered seismicity in Chile, if compared to the classical ETAS model.  
All the reported results are obtained using the open-source software developed in R [20; 21].  
This paper is organized as follows. In Section 2 a description of data is provided jointly to some preliminary analysis. Section 3 reports some details of the theoretical background, recalling some basic theory of space-time point processes and the ETAS model, with an extension of the FLP approach with the inclusion of covariates. The choice of covariates is described in Section 4. A seismic application to the considered data is discussed in Section 5. Section 6 is devoted to discussion of results. Conclusions are reported in Section 7.