Combining a Locally Restricted Anisotropic Spatial Kernel with an ETAS Incomplete Model for Better Forecasts of the 2019 Ridgecrest Sequence

Strong earthquakes cause aftershock sequences that are clustered in time according to a power decay law, and in space along their extended rupture, shaping a typically elongate pattern of aftershock locations. A widely used approach to model seismic clustering is the Epidemic Type Aftershock Sequence (ETAS) model, that shows three major biases: First, the conventional ETAS approach assumes isotropic spatial triggering, which stands in conflict with observations and geophysical arguments for strong earthquakes. Second, the spatial kernel has unlimited extent, allowing smaller events to exert disproportionate trigger potential over an unrealistically large area. Third, the ETAS model assumes complete event records and neglects inevitable short-term aftershock incompleteness as a consequence of overlapping coda waves. These three effects can substantially bias the parameter estimation and particularly lead to underestimated cluster sizes. In this article, we combine the approach of Grimm (2021), which introduced a generalized anisotropic and locally restricted spatial kernel, with the ETAS-Incomplete (ETASI) time model of Hainzl (2021), to define an ETASI space-time model with flexible spatial kernel that solves the abovementioned shortcomings. We apply different model versions to a triad of forecasting experiments of the 2019 Ridgecrest sequence, and evaluate the prediction quality with respect to cluster size, largest aftershock magnitude and spatial distribution. The new model provides the potential of more realistic simulations of on-going aftershock activity, e.g.~allowing better predictions of the probability and location of a strong, damaging aftershock, which might be beneficial for short term risk assessment and desaster response.

ETAS Space–Time Modeling of Chile Triggered Seismicity Using Covariates: Some Preliminary Results

Chilean seismic activity is one of the strongest in the world. As already shown in previous papers, seismic activity can be usefully described by a space–time branching process, such as the ETAS (Epidemic Type Aftershock Sequences) model, 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 use of covariates can improve the description of triggered seismicity in the ETAS model, so in this paper, we study the Chilean seismicity separately for the North and South area, using some GPS-related data observed together with ordinary catalog data. Our results show evidence that the use of some covariates can improve the fitting of the ETAS model.

Improving Earthquake Doublet Frequency Predictions by Modified Spatial Trigger Kernels in the Epidemic-Type Aftershock Sequence (ETAS) Model

ABSTRACT Earthquake sequences add a substantial hazard beyond the solely declustered perspective of common probabilistic seismic hazard analysis. A particularly strong driver for both social and economic losses are so-called earthquake doublets (more generally multiplets), that is, sequences of two (or more) comparatively large events in spatial and temporal proximity. Without differentiating between foreshocks and aftershocks, we hypothesize three main influencing factors of doublet occurrence: (1) the number of direct and secondary aftershocks triggered by an earthquake; (2) the occurrence of independent clusters and seismic background events in the same time–space window; and (3) the magnitude size distribution of triggered events (in contrast to independent events). We tested synthetic catalogs simulated by a standard epidemic-type aftershock sequence (ETAS) model for both Japan and southern California. Our findings show that the common ETAS approach significantly underestimates doublet frequencies compared with observations in historical catalogs. In combination with that the simulated catalogs show a smoother spatiotemporal clustering compared with the observed counterparts. Focusing on the impact on direct aftershock productivity and total cluster sizes, we propose two modifications of the ETAS spatial kernel to improve doublet rate predictions: (a) a restriction of the spatial function to a maximum distance of 2.5 estimated rupture lengths and (b) an anisotropic function with contour lines constructed by a box with two semicircular ends around the estimated rupture segment. These modifications shift the triggering potential from weaker to stronger events and consequently improve doublet rate predictions for larger events, despite still underestimating historic doublet occurrence rates. Besides, the results for the restricted spatial functions fulfill better the empirical Båth’s law for the maximum aftershock magnitude. The tested clustering properties of strong events are not sufficiently incorporated in typically used global catalog scale measures, such as log-likelihood values, which would favor the conventional, unrestricted models.

Probabilistic Forecasting of Hydraulic Fracturing-Induced Seismicity Using an Injection-Rate Driven ETAS Model

The development of robust forecasts of human-induced seismicity is highly desirable to mitigate the effects of disturbing or damaging earthquakes. We assess the performance of a well-established statistical model, the epidemic-type aftershock sequence (ETAS) model, with a catalog of ∼93,000 microearthquakes observed at the Preston New Road (PNR, United Kingdom) unconventional shale gas site during, and after hydraulic fracturing of the PNR-1z and PNR-2 wells. Because ETAS was developed for slower loading rate tectonic seismicity, to account for seismicity caused by pressurized fluid, we also generate three modified ETAS with background rates proportional to injection rates. We find that (1) the standard ETAS captures low seismicity between and after injections but is outperformed by the modified model during high-seismicity periods, and (2) the injection-rate driven ETAS substantially improves when the forecast is calibrated on sleeve-specific pumping data. We finally forecast out-of-sample the PNR-2 seismicity using the average response to injection observed at PNR-1z, achieving better predictive skills than the in-sample standard ETAS. The insights from this study contribute toward producing informative seismicity forecasts for real-time decision making and risk mitigation techniques during unconventional shale gas development.

Bayesian Estimation of the ETAS Model for Earthquake Occurrences

ABSTRACT The epidemic-type aftershock sequence (ETAS) model is widely used in seismic forecasting. However, most studies of ETAS use point estimates for the model parameters, which ignores the inherent uncertainty that arises from estimating these from historical earthquake catalogs, resulting in misleadingly optimistic forecasts. In contrast, Bayesian statistics allows parameter uncertainty to be explicitly represented and fed into the forecast distribution. Despite its growing popularity in seismology, the application of Bayesian statistics to the ETAS model has been limited by the complex nature of the resulting posterior distribution, which makes it infeasible to apply to catalogs containing more than a few hundred earthquakes. To combat this, we develop a new framework for estimating the ETAS model in a fully Bayesian manner, which can be efficiently scaled up to large catalogs containing thousands of earthquakes. We also provide easy-to-use software that implements our method.

Earthquake clusters expected from bare statistics: How bursts and swarms emerge from exogenous and epidemic aftershock processes.

<p>Earthquake catalogs exhibit strong spatio-temporal correlations. As such, earthquakes are often classified into clusters of correlated activity. Clusters themselves are traditionally classified in two different kinds: (i) bursts, with a clear hierarchical structure between a single strong mainshock, preceded by a few foreshocks and followed by a power-law decaying aftershock sequence, and (ii) swarms, exhibiting a non-trivial activity rate that cannot be reduced to such a simple hierarchy between events. </p><p>The Epidemic Aftershock Sequence (ETAS) model is a linear Hawkes point process able to reproduce earthquake clusters from empirical statistical laws [Ogata, 1998]. Although not always explicit, the ETAS model is often interpreted as the outcome of a background activity driven by external forces and a Galton-Watson branching process with one-to-one causal links between events [Saichev et al., 2005]. Declustering techniques based on field observations [Baiesi & Paczuski, 2004] can be used to infer the most likely causal links between events in a cluster. Following this method, Zaliapin and Ben‐Zion (2013) determined the statistical properties of earthquake clusters characterizing bursts and swarms, finding a relationship between the predominant cluster-class and the heat flow in seismic regions.</p><p>Here, I show how the statistical properties of clusters are related to the fundamental statistics of the underlying seismogenic process, modeled in two point-process paradigms [Baró, 2020].</p><p>The classification of clusters into bursts and swarms appears naturally in the standard ETAS model with homogeneous rates and are determined by the average branching ratio (nb) and the ratio between exponents α and b characterizing the production of aftershocks and the distribution of magnitudes, respectively. The scale-free ETAS model, equivalent to the BASS model [Turcotte, et al., 2007], and usual in cold active tectonic regions, is imposed by α=b and reproduces bursts. In contrast, by imposing α<0.5b, we recover the properties of swarms, characteristic of regions with high heat flow. </p><p>Alternatively, the same declustering methodology applied to a non-homogeneous Poisson process with a non-factorizable intensity, i.e. in absence of causal links, recovers swarms with α=0, i.e. a Poisson Galton-Watson process, with similar statistical properties to the ETAS model in the regime α<0.5b.</p><p>Therefore, while bursts are likely to represent actual causal links between events, swarms can either denote causal links with low α/b ratio or variations of the background rate caused by exogenous processes introducing local and transient stress changes. Furthermore, the redundancy in the statistical laws can be used to test the hypotheses posed by the ETAS model as a memory‐less branching process. </p><p>References:</p><ul><li> <p>Baiesi, M., & Paczuski, M. (2004). <em>Physical Review E</em>, 69, 66,106. doi:10.1103/PhysRevE.69.066106.</p> </li> <li> <p>Baró, J. (2020).  <em>Journal of Geophysical Research: Solid Earth,</em> 125, e2019JB018530. doi:10.1029/2019JB018530.</p> </li> <li> <p>Ogata, Y. (1998) <em>Annals of the Institute of Statistical Mathematics,</em> 50(2), 379–402. doi:10.1023/A:1003403601725.</p> </li> <li> <p>Saichev, A., Helmstetter, A. & Sornette, D. (2005) <em>Pure appl. geophys.</em> 162, 1113–1134. doi:10.1007/s00024-004-2663-6.</p> </li> <li> <p>Turcotte, D. L., Holliday, J. R., and Rundle, J. B. (2007), <em>Geophys. Res. Lett.</em>, 34, L12303, doi:10.1029/2007GL029696.</p> </li> <li> <p>Zaliapin, I., and Ben‐Zion, Y. (2013), <em>J. Geophys. Res. Solid Earth</em>, 118, 2865– 2877, doi:10.1002/jgrb.50178.</p> </li> </ul>

A Bayesian Framework for Aftershock Forecasting and Testing

<p>Earthquakes trigger subsequent earthquakes. They form clusters and swarms in space and in time. This is a direct manifestation of the non-Poisson behavior in the occurrence of earthquakes, where earthquake magnitudes and time intervals between successive events are not independent and are influenced by past seismicity. As a result, the distribution of the number of earthquakes is no longer strictly Poisson and the statistics of the largest events deviate from the GEV distribution. In statistical seismology, the occurrence of earthquakes is typically approximated by a stochastic marked point process. Among different models, the ETAS model is the most successful in reproducing several key aspects of seismicity. Recent analysis suggests that the ETAS model generates sequences of events which are not Poisson. This becomes important when the ETAS based models are used for earthquake forecasting (Shcherbakov et al., Nature Comms., 2019). In this work, I consider the Bayesian framework combined with the ETAS model to constrain the magnitudes of the largest expected aftershocks during a future forecasting time interval. This includes the MCMC sampling of the posterior distribution of the ETAS parameters and computation of the Bayesian predictive distribution for the magnitudes of the largest expected events. To validate the forecasts, the statistical tests developed by the CSEP are reformulated for the Bayesian framework. In addition, I define and compute the Bayesian p-value to evaluate the consistency of the forecasted extreme earthquakes during each forecasting time interval. The Bayesian p-value gives the probability that the largest forecasted earthquake can be more extreme than the observed one. The suggested approach is applied to the recent 2019 Ridgecrest earthquake sequence to forecast retrospectively the occurrence of the largest aftershocks (Shcherbakov, JGR, 2021). The results indicate that the Bayesian approach combined with the ETAS model outperformed the approach based on the Poisson assumption, which uses the extreme value distribution and the Omori law.</p>

Testing physics and statics based hybrid ETAS models

<p>Currently, the Epidemic Type Aftershock Sequence (ETAS) model is state-of-the-art for forecasting aftershocks. However, the under-performance of ETAS in forecasting the spatial distribution of aftershocks following a large earthquake make us adopt alternative approaches for the modelling of the spatial ETAS-kernel. Here we develop a hybrid physics and statics based forecasting model. The model uses stress changes, calculated from inverted slip models of large earthquakes, as the basis of the spatial kernel in the ETAS model in order to get more reliable estimates of spatiotemporal distribution of aftershocks. We evaluate six alternative approaches of stress-based ETAS-kernels and rank their performance against the base ETAS model. In all cases, an expectation maximization (EM) algorithm is used to estimate the ETAS parameters. The model approach has been tested on synthetic data to check if the known parameters can be inverted successfully. We apply the proposed method to forecast aftershocks of mainshocks available in SRCMOD database, which includes 192 mainshocks with magnitudes in the range between 4.1 and 9.2 occurred from 1906 to 2020. The probabilistic earthquake forecasts generated by the hybrid model have been tested using established CSEP test metrics and procedures. We show that the additional stress information, provided to estimate the spatial probability distribution, leads to more reliable spatiotemporal ETAS-forecasts of aftershocks as compared to the base ETAS model.</p>

ETAS Space time modelling of Chile induced seismicity using covariates.

<p>Chilean seismic activity is among the strongest ones in the world. As already shown in previous papers, seismic activity can be usefully described by a space-time branching process, like the ETAS (Epidemic Type Aftershock Sequences) model, which is a semiparametric model with a large time scale component for the background seismicity and a small time scale component for the induced 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); the induced seismicity is modelled with a parametric space-time function. In classical ETAS models the expected number of induced events depends only on the magnitude of the main event. From a statistical point of view, forecast of induced 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, in space and time domain, of a seismic sequence. Together with magnitude, to explain the expected number of induced events we also used other covariates. According to this formulation, the expected number of events induced by event E<sub>i </sub>is a function of a linear predictor η<sub>i</sub>=<strong>x<sub>i</sub></strong><strong>β, </strong>where <strong>x<sub>i </sub></strong>is the vector of covariates observed for the i-th event (the first is usually the magnitude m<sub>i</sub>), and <strong>β</strong> is a vector of parameters to be estimated together with the other parametric and nonparametric components of the ETAS model. We obtained some interesting result using some covariates related to the depth of events and to some GPS measurement, corresponding to earth movement observed before main events. We find that some of these models can improve the description and the forecasting of the induced seismicity in Chile, after a subdivision of the country in different spatial regions. We used open-source software (R package etasFLP) to perform the semiparametric estimation of the ETAS model with covariates.</p>