ARTICLE https://doi.org/10.1038/s41467-019-11958-4 OPEN Forecasting the magnitude of the largest expected earthquake Robert Shcherbakov 1,2, Jiancang Zhuang 3, Gert Zöller 4 & Yosihiko Ogata3 The majority of earthquakes occur unexpectedly and can trigger subsequent sequences of events that can culminate in more powerful earthquakes. This self-exciting nature of seis- micity generates complex clustering of earthquakes in space and time. Therefore, the pro- 1234567890():,; blem of constraining the magnitude of the largest expected earthquake during a future time interval is of critical importance in mitigating earthquake hazard. We address this problem by developing a methodology to compute the probabilities for such extreme earthquakes to be above certain magnitudes. We combine the Bayesian methods with the extreme value theory and assume that the occurrence of earthquakes can be described by the Epidemic Type Aftershock Sequence process. We analyze in detail the application of this methodology to the 2016 Kumamoto, Japan, earthquake sequence. We are able to estimate retrospectively the probabilities of having large subsequent earthquakes during several stages of the evolution of this sequence. 1 Department of Earth Sciences, University of Western Ontario, London, ON N6A 5B7, Canada. 2 Department of Physics and Astronomy, University of Western Ontario, London, ON N6A 3K7, Canada. 3 Institute of Statistical Mathematics, 10-3 Midori-Cho, Tachikawa-Shi, Tokyo 190-8562, Japan. 4 Institute of Mathematics, University of Potsdam, 14476 Potsdam-Golm, Germany. Correspondence and requests for materials should be addressed to R.S. (email: [email protected]) NATURE COMMUNICATIONS | (2019) 10:4051 | https://doi.org/10.1038/s41467-019-11958-4 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11958-4 ayesian methods offer indispensable tools in studies of aftershocks9,32–34. As a result, the retrospective analysis is typi- natural hazards, particularly, earthquakes1–3. They can cally performed by stacking many foreshock sequences associated B 33,35–40 provide a suite of approaches to analyze statistical aspects with particular regions . Therefore, the incorporation of of natural seismicity, where the occurrence of earthquakes is a the seismicity prior to the occurrence of large mainshocks is single realization of an underlying stochastic process. They also important for short-term earthquake forecasting approaches. can be used to provide inferences about future evolution of In this work, we developed a framework to constrain the seismicity. Specifically, one can compute predictive distributions magnitudes of the largest expected earthquakes to occur during a by studying past seismicity to provide probabilistic constraints, future time interval by analyzing past seismicity. To accomplish for example, on the occurrence of subsequent large this, we combined the Bayesian analysis with extreme value sta- earthquakes2,4. The occurrence of large magnitude earthquakes is tistics to compute the Bayesian predictive distribution for the a direct manifestation of multi-scale physical processes operating magnitude of the largest expected event to exceed a certain value in the crust and upper mantle5–8. However, the comprehensive in the near future. In the analysis, we assume that the earthquake understanding of the mechanisms and underlying stochastic occurrence rate can be modeled by the ETAS process, where each dynamics that control the occurrence of earthquakes remains one earthquake is capable of triggering subsequent events10,11.We of the challenging tasks in earthquake seismology. Therefore, the also assume that the distribution of earthquake magnitudes can problem of formulating a reliable and operational scheme for the be approximated by the left-truncated exponential distribution41. estimation of the magnitudes of the extreme earthquakes to occur To model the uncertainties of the model parameters, we is of critical importance. employed the Markov Chain Monte Carlo (MCMC) method to Earthquakes form clusters in space and in time. This clustering is sample the posterior distribution of the model parameters and to a result of several physical mechanisms operating in the seismo- use the generated chain of the parameters to simulate forward in genic crust. One of them is the triggering by preceding earthquakes time an ensemble of the ETAS processes. In addition, we showed that can lead to a cascade of events with a complicated branching that the extreme value distribution for the magnitudes of the structure9. To account for such a clustering, the Epidemic Type largest events in the ETAS model deviates from the Gumbel Aftershock Sequence (ETAS) model offers a realistic and quantifi- distribution. To illustrate our approach, we analyzed one recent able approximation10–12. Specifically, it can model the earthquake prominent sequence, the 2016 Kumamoto, Japan, earthquake rate before and after strong earthquakes. This, in turn, allows to sequence, where we were able to compute the probabilities of quantify the increased earthquake hazard after a mainshock by having the largest expected events above certain magnitudes to incorporating the triggering ability of foreshocks, a mainshock, and occur during several stages of the sequence. We also applied the subsequent aftershocks. It also can be used for short-term fore- analysis to several past major aftershock sequences and compared casting of large earthquakes by studying past seismicity12–16. the obtained probabilities with the ones computed using several The problem of estimating the largest expected magnitude of methods based on the Omori–Utsu approximation of the after- an earthquake during a future time interval has been considered shock rate. As a main result of this work, we developed an in the past17,18. In addition, it was also addressed in case of inference procedure to estimate the probabilities of having largest aftershocks19,20. A closely related problem is the estimation of the expected events during an earthquake sequence described by the absolute maximum magnitude of an earthquake for a given ETAS process. The suggested approach can be implemented in seismogenic zone1,21–23. In this regard, the Bayesian approach current or future operational earthquake forecasting schemes, combined with extreme value statistics was used to analyze this where the constraints on the magnitudes of future large earth- problem by assuming that the occurrence of earthquakes can be quakes are taken into account. approximated by a homogeneous Poisson process2,24. The Bayesian methods were also used to constrain the magnitudes of the largest expected aftershocks, where the earthquake rate was Results modeled by the Omori–Utsu law and the events were assumed to Bayesian predictive distribution. The occurrence of earthquakes follow a non-homogeneous Poisson process4,25. However, those can be modeled as a stochastic marked point process in time42.In approaches do not fully incorporate the complicated triggering this description, the earthquakes are characterized by their structure of earthquake sequences, which are important for short- occurrence times ti and the earthquake magnitudes mi represent term earthquake forecasting. marks. The seismicity during a specified time interval can be = = … A notable exception is the application of a spatio-temporal described by an ordered set S {(ti, mi)}:i 1, . A reasonable version of the ETAS model to the retrospective forecasting of approximation is to assume that earthquake magnitudes can be early seismicity associated with the 2016 sequence of several large described by a parametric distribution function Fθ(m), where θ is earthquakes in Amatrice, Italy14. In that work, the MCMC a set of parameters. In addition, the occurrence of earthquakes is λ approach was used to sample the posterior distribution of the controlled by a conditional time-dependent rate, ωðtjHtÞ, where ω ETAS parameters, in order to estimate the rates of seismicity is a set of model parameters and Ht represents the history of during short forecasting time intervals. However, the parameters past seismicity up to time t. of the ETAS model were subjected to certain constraints in order In the analysis that follows, we consider past earthquakes to facilitate the sampling. during a training time interval [T0, Te]. In order to take into The expected large aftershocks can significantly increase the account the effect of early events, we use the earthquakes in the 19,26 fi earthquake hazard after the occurrence of a mainshock . rst part of this interval [T0, Ts] to estimate properly the 16,27,28 Several approaches were proposed to address this problem . conditional earthquake rate during the target time interval [Ts, θ ω In this context, the issue of the catalog early incompleteness was Te], for which the model parameters and can be inferred. We 20,29–31 + Δ also incorporated in a number of studies . In the majority also consider a forecasting time interval [Te, Te T] during of these studies, the point estimators for the model parameters which we compute the probabilities of having extreme events were used to compute the associated rates or probabilities. As a above a certain magnitude (Fig. 1). result, the incorporation of the uncertainties of the model para- The power of the Bayesian approach is in its ability to provide meters was not fully addressed. predictions for an unobserved quantity of interest3,43. It goes In earthquake clustering, foreshocks, which precede