Modeling and forecasting earthquakes is challenging due to the complex interplay and clustering of main-shocks and aftershocks. The epidemic-type aftershock sequence (ETAS) model represents the conditional intensity of earthquakes as the superposition of a background and aftershock rate which allows for the declustering of the earthquakes. Its success has led to the development of numerous versions of the ETAS model. Among these extensions is the renewal ETAS (RETAS) model, which has shown promising potential. The RETAS model endows the main-shock arrival process with a renewal process, which serves as an alternative to the homogeneous Poisson process. Model fitting is performed using likelihood-based estimation by directly optimizing the exact likelihood. However, inferring the branching structure from the fitted RETAS model remains a challenging task since the declustering algorithm that is currently available for the ETAS model is not directly applicable. Therefore, this article develops an iterative algorithm to calculate the smoothed main- and aftershock probabilities, conditional on all available information contained in the catalog. Consequently, an estimate of the background spatial intensity function and model parameters can be obtained using an iterative semiparametric procedure with the smoothing parameters selected using information criteria. The methods proposed herein are illustrated on simulated data and a New Zealand earthquake catalog. 

Stochastic declustering of earthquakes with the spatiotemporal renewal ETAS model

EM algorithm for the estimation of the RETAS model*

Abstract The paper focuses on the evaluation of hailstorms’ and thunderstorms winds’ events in the United States of America, in the period from 1996 to 2022, under the marked spatio-temporal self-exciting point processes point of view. The aim of the present article is the assessment and description of the spatio-temporal spontaneous and reproducing activity of severe hailstorms’ and thunderstorms winds’ processes. The present application shows how the spatio-temporal pattern is well-fitted and clearly explainable, according to the flexible semi-parametric ETAS model fitting. 

Severe convective storms’ reproduction: empirical analysis from the marked self-exciting point processes point of view

Pertemuan lempeng Eurasia, lempeng Indo-Australia, dan lempeng Pasifik terjadi di sekitar Pulau Sulawesi dan Maluku. Pertemuan ketiga lempeng ini menyebabkan sering terjadi gempa bumi. Faktor-faktor yang mempengaruhi gempa bumi secara geologis adalah jarak terhadap zona subduksi, gunung berapi, dan sesar aktif. Untuk keperluan mitigasi risiko bencana gempa bumi, perlu dilakukan prediksi risiko terjadinya gempa bumi. Pada penelitian ini menggunakan model inhomogeneous thomas cluster process yang digunakan untuk memprediksi risiko kejadian gempa bumi. Variabel yang digunakan dalam penelitian ini adalah koordinat zona subduksi, sesar aktif, dan gunung api di pulau Sulawesi dan Maluku. Hasil yang diperoleh berdasarkan fungsi K gempa bumi di pulau Sulawesi dan Maluku membentuk cluster. Uji chi-square menyimpulkan kejadian gempa bumi di Pulau Sulawesi dan Maluku bersifat inhomogen (tidak homogen). Berdasarkan model estimasi inhomogeneous thomas cluster process, variabel zona subduksi, sesar aktif, dan gunung berapi mempunyai pengaruh yang signifikan terhadap terjadinya gempa bumi. Dari ketiga variabel yang mempengaruhi terjadinya gempa terbesar (1,8 kali) adalah variabel letak gunung api. Berdasarkan hasil prediksi, Provinsi Gorontalo, Sulawesi Utara, Sulawesi Tengah, Maluku, dan Maluku Utara memiliki risiko gempa bumi yang tinggi.

Prediksi resiko gempa menggunakan model spatial point process

Abstract We propose a spatiotemporal point process model that enhances the classical Epidemic-Type Aftershock Sequence (ETAS) model. This is achieved with the introduction of a renewal main-shock arrival process and we call this extension the renewal ETAS (RETAS) model. This modification is similar in spirit to the renewal Hawkes (RHawkes) process but the conditional intensity process supports a spatial component. It empowers the main-shock intensity to reset upon the arrival of main-shocks. This allows for heavier clustering of main-shocks than the classical spatiotemporal ETAS model. We introduce a likelihood evaluation algorithm for parameter estimation and provide a novel procedure to evaluate the fitted model's goodness-of-fit (GOF) based on a sequential application of the Rosenblatt transformation. A simulation algorithm for the RETAS model is outlined and used to validate the numerical performance of the likelihood evaluation algorithm and GOF test procedure. We illustrate the proposed model and methods on various earthquake catalogues around the world each with distinctly different seismic activity. These catalogues demonstrate the RETAS model's additional flexibility in comparison to the classical spatiotemporal ETAS model and emphasizes the potential for superior modelling and forecasting of seismicity. 

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Spatiotemporal ETAS Model with a Renewal Main-Shock Arrival Process

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Spatiotemporal ETAS Model with a Renewal Main-Shock Arrival Process