Epidemic-type Aftershock Sequence Model Research Articles

Statistics between mainshocks and foreshocks in Italy and Southern California

The most used and accepted models for daily forecasts are based on short‐term space and time earthquake clustering for occurrence rates and on the Gutenberg‐Richter law for the frequency‐magnitude. These models have been demonstrated to produce reliable prospective space‐time‐magnitude forecasts during an aftershock sequence, but their skill in forecasting mainshocks is still under discussion. This paper studies the foreshock statistics of the Italian and Californian seismicity in two ways: i) we compare the foreshock activity observed in real seismic catalogs and in synthetic catalogs derived from a pure Epidemic‐Type Aftershock Sequence (ETAS) model; ii) we analyze the triggering capability of earthquakes using different ETAS parameterizations, in order to check whether large events are triggered in the same way as regular earthquakes. The results indicate that the foreshock activity observed in the real catalogs is compatible with what is expected by the ETAS model. Moreover, we find that the empirical foreshock rates have an intrinsic variability due to limited sampling that may explain most of the differences found so far in different seismic catalogs.

Statistic analysis of swarm activities around the Boso Peninsula, Japan: Slow slip events beneath Tokyo Bay?

Seismicity reflects underground stress states, satisfying scaling laws such as Gutenberg-Richter law and Omori-Utsu law. Standard seismicity models based on these scaling laws, such as the Epidemic Type Aftershock Sequence (ETAS) model, are useful to identify swarm anomalies in seismicity catalogs. Llenos et al. (2009) applied the ETAS model to swarms triggered by slow slip events (SSEs) and found that stressing rate controls the background seismicity μ suggesting that swarms can be utilized to monitor stress change due to various aseismic processes. Following their work, we analyze the 2002 and 2007 Boso swarms triggered by the Boso SSEs (Ozawa et al., 2007) and a swarm beneath Tokyo Bay, in June 2005. A single ETAS model cannot explain the high seismicity during a swarm. Although a combination of three ETAS models for pre-swarm, swarm, and post-swarm periods better explains the data, a simpler model with an ETAS model and a boxcar function is even better. Similarity of the seismicity model, together with the locations and focal mechanisms, suggests that three swarms share a common source of stress, and the possibility of undetected SSE beneath Tokyo Bay.

Masking of earthquake triggering behavior by a high background rate and implications for epidemic-type aftershock sequence inversions

[1] We examine the effects of the spontaneous background event rate and aftershock triggering characteristics on the temporal statistics of seismicity in the epidemic-type aftershock sequence model. Recent work has shown that the earthquake interevent time distribution is generally bimodal: a superposition of a gamma component from triggered aftershocks at short time intervals and an exponential component at longer intervals from spontaneous events and the overlapping of independent aftershock sequences. The relative size of these two components varies between catalogs, so there is no simple, universal scaling; at the extreme of high spontaneous rate, e.g., in large regions, the high probability of temporally overlapping aftershock sequences causes the exponential component to dominate. Here we further explore the effects of both the spontaneous rate and the aftershock triggering parameters. We show that the analytical theory of Saichev and Sornette (2007), although valid under their assumptions, gives the impression of a more “universal” behavior if used outside its stated range of applicability. We also show that within the high-overlap (high-spontaneous rate) regime, a maximum likelihood inversion of the model's temporal parameters is both less accurate and biased; specifically, the background rate is systematically overestimated. This has implications on the suitable range of region sizes for which parameter inversion may be reliable and must therefore be taken into account in any inversion for temporal variations in background rate in time-dependent hazard calculation.

PI forecast with or without de-clustering: an experiment for the Sichuan-Yunnan region

Abstract. Pattern Informatics (PI) algorithm uses earthquake catalogues for estimating the increase of the probability of strong earthquakes. The main measure in the algorithm is the number of earthquakes above a threshold magnitude. Since aftershocks occupy a significant proportion of the total number of earthquakes, whether de-clustering affects the performance of the forecast is one of the concerns in the application of this algorithm. This problem is of special interest after a great earthquake, when aftershocks become predominant in regional seismic activity. To investigate this problem, the PI forecasts are systematically analyzed for the Sichuan-Yunnan region of southwest China. In this region there have occurred some earthquakes larger than MS 7.0, including the 2008 Wenchuan earthquake. In the analysis, the epidemic-type aftershock sequences (ETAS) model was used for de-clustering. The PI algorithm was revised to consider de-clustering, by replacing the number of earthquakes by the sum of the ETAS-assessed probability for an event to be a "background event" or a "clustering event". Case studies indicate that when an intense aftershock sequence is included in the "sliding time window", the hotspot picture may vary, and the variation lasts for about one year. PI forecasts seem to be affected by the aftershock sequence included in the "anomaly identifying window", and the PI forecast using "background events" seems to have a better performance.

Next-day earthquake forecasts for the Japan region generated by the ETAS model

This paper gives the technical solutions of implementing the space-time epidemic-type aftershock sequence (ETAS) model for short-term (1-day) earthquake forecasts for the all-Japan region in the Collaboratory for the Study of Earthquake Predictability (CSEP) project in Japan. For illustration, a retrospective forecasting experiment is carried out to forecast the seismicity in the Japan region before and after the Tokachi-Oki earthquake (M 8.0) at 19:50:07 (UTC) on 25 September 2003, in the format of contour images. The optimal model parameters used for the forecasts are estimated by fitting the model to the observation records up to the starting time of the forecasting period, and the probabilities of earthquake occurrences are obtained through simulations. To tackle the difficulty of heavy computations in fitting a complicated point-process to a huge dataset, an “off-line optimization” and “online forecasting” scheme is proposed to keep both the estimates of model parameters and forecasts updated according to the most recent observations. The results show that the forecasts have captured the spatial distribution and temporal evolution of the features of future seismicity. These forecasts are tested against the reference Poisson model that is stationary in time but spatially inhomogeneous.

Significant improvements of the space-time ETAS model for forecasting of accurate baseline seismicity

The space-time version of the epidemic type aftershock sequence (ETAS) model is based on the empirical laws for aftershocks, and constructed with a certain space-time function for earthquake clustering. For more accurate seismic prediction, we modify it to deal with not only anisotropic clustering but also regionally distinct characteristics of seismicity. The former needs a quasi-real-time cluster analysis that identifies the aftershock centroids and correlation coefficient of a cluster distribution. The latter needs the space-time ETAS model with location dependent parameters. Together with the Gutenberg-Richter’s magnitude-frequency law with location-dependent b-values, the elaborated model is applied for short-term, intermediate-term and long-term forecasting of baseline seismic activity.

Comparison of branching models for seismicity and likelihood maximization through simulated annealing

[1] Stochastic branching models provide a good description of some aspects of the temporal organization in seismicity. Generally, they assume that magnitudes are independent of history, as in the widely used Epidemic Type Aftershock Sequence (ETAS) model. Here, we consider a recent epidemic‐like model where time‐magnitude and magnitude‐magnitude correlations are introduced via a dynamical scaling (DS) hypothesis, namely, the magnitude difference between earthquakes fixes the time scale for correlations. We also consider a variation of the ETAS model where the c parameter of the Omori law is not fixed but depends on the parent magnitude. We develop a novel procedure to maximize the log likelihood of the different models. This method is based on a Monte Carlo sampling in the parameter space with a variable step size during the evolution to converge to the given accuracy. The log likelihood indicates that the DS model provides the best fit for the major California sequences and the whole catalog, setting as lower magnitude thresholds Mc ≥ 3.5. For Mc = 3, the best fit is obtained by a model with c depending on both the parent magnitude and goes into Mc as in the generalized Omori law. The better performance of the DS model with respect to the ETAS model is attributed to correlations in magnitudes.

A random effects epidemic-type aftershock sequence model

A random effects epidemic-type aftershock sequence model

The ETAS model for daily forecasting of Italian seismicity in the CSEP experiment

This study investigates the basic properties of the recent shallow seismicity in Italy, through stochastic modeling and statistical methods. Assuming that earthquakes are the realization of a stochastic point process, we have modeled the occurrence rate density in space, time and magnitude using an epidemic-type aftershock sequence model. By applying the maximum likelihood procedure, we estimated the parameters of the model that best fit the Italian instrumental catalog, as recorded by the Istituto Nazionale di Geofisica e Vulcanologia (INGV) from April 16, 2005, to June 1, 2009. Then we applied the estimated model to a second independent dataset (June 1, 2009, to September 1, 2009). We show that the model performed well on this second database, through the relevant statistical tests. The model proposed in the present study is suitable for computing earthquake occurrence probability in real time, and to take part in international initiatives such as the Collaboratory Study for Earthquake Predictability, where we have submitted this model for daily forecasting of Italian seismicity for ML>4.0.

A random effects epidemic-type aftershock sequence model

We consider an extension of the temporal epidemic-type aftershock sequence (ETAS) model with random effects as a special case of a well-known doubly stochastic self-exciting point process. The new model arises from a deterministic function that is randomly scaled by a nonnegative random variable, which is unobservable but assumed to follow either positive stable or one-parameter gamma distribution with unit mean. Both random effects models are of interest although the one-parameter gamma random effects model is more popular when modeling associated survival times. Our estimation is based on the maximum likelihood approach with marginalized intensity. The methods are shown to perform well in simulation experiments. When applied to an earthquake sequence on the east coast of Taiwan, the extended model with positive stable random effects provides a better model fit, compared to the original ETAS model and the extended model with one-parameter gamma random effects.

Precursory seismic anomalies and transient crustal deformation prior to the 2008 Mw = 6.9 Iwate‐Miyagi Nairiku, Japan, earthquake

The epidemic‐type aftershock sequence (ETAS) model has been widely used for detecting seismicity anomalies, such as quiescence and activation during aftershock sequences of large earthquakes and background levels of seismicity. The causes of such anomalies are thought to be associated with spatiotemporal changes in stress, even tiny perturbation of local stress. Here we analyze the seismic activity during the decade prior to the 2008 Iwate‐Miyagi inland earthquake of M7.2 (Mw6.9), to fit the ETAS model to the seismicity from various regions around the source over northern Honshu, Japan. From the viewpoint of the analyses with the ETAS model, we find northern Honshu is divided into three areas of distinctive behaviors, increased seismicity, decreased seismicity, and normal seismicity relative to the ETAS prediction. As in other previously published papers, here we hypothesize that Coulomb stress changes caused by a few years of precursory slip of the Iwate‐Miyagi earthquake resulted in the seismicity changes in and around the 2008 source region. The confirmed significant seismic anomalies in respective regions are consistent with the increments of the Coulomb failure stress of the corresponding regions that are calculated by the assumed slow slip on the southern part of the faults of the main shock. The local crustal deformations observed from a dense Global Positioning System network, including a station right above the focal fault, supports that slow slip on the fault had been taking place for about 5 years prior to the occurrence of the focal earthquake and suggests that the slip terminated or migrated to down‐dip extension of the fault around 1 year before the rupture.

Spatiotemporal correlations of aftershock sequences

Aftershock sequences are of particular interest in seismic research because they may condition seismic activity in a given region over long time spans. Although they are typically identified with periods of enhanced seismic activity after a large earthquake as characterized by the Omori law, our knowledge of the spatiotemporal correlations between events in an aftershock sequence is limited. Here, we study the spatiotemporal correlations of two aftershock sequences form California (Parkfield and Hector Mine) using the recently introduced concept of “recurrent” events. We find that both sequences have very similar properties and that most of them are captured by the space‐time epidemic‐type aftershock sequence (ETAS) model if one takes into account catalog incompleteness. However, the stochastic ETAS model does not capture those spatiotemporal correlations that give rise to the observed distribution of recurrent events on small spatial scales. We also find that there is no clear evidence for stress shadows that have been observed for smaller earthquakes.

The Assumption of Poisson Seismic-Rate Variability in CSEP/RELM Experiments

Abstract Evaluating the performances of earthquake forecasting/prediction models is the main rationale behind some recent international efforts such as the Regional Earthquake Likelihood Model (RELM) and the Collaboratory for the Study of Earthquake Predictability (CSEP). Basically, the evaluation process consists of two steps: (1) to run simultaneously all codes to forecast future seismicity in well-defined testing regions; (2) to compare the forecasts through a suite of statistical tests. The tests are based on the likelihood score, and they check both the time and space performances. All these tests rely on some basic assumptions that have never been deeply discussed and analyzed. In particular, models are required to specify a rate in space-time-magnitude bins; it is assumed that these rates are independent and characterized by Poisson uncertainty. In this work we have explored in detail these assumptions and their impact on CSEP testing procedures when applied to a widely used class of models, that is, the Epidemic-Type Aftershock Sequence (ETAS) models. Our results show that, if an ETAS model is an accurate representation of seismicity, the same right model is rejected by the current CSEP testing procedures a number of times significantly higher than expected. We show that this deficiency is due to the fact that the ETAS models produce forecasts with a variability significantly higher than that of a Poisson process, invalidating one of the main assumptions that stands behind the CSEP/RELM evaluation process. Certainly, this shortcoming does not negate the paramount importance of the CSEP experiments as a whole, but it does call for a specific revision of the testing procedures to allow a better understanding of the results of such experiments.

Standard Errors of Parameter Estimates in the ETAS Model

Abstract Point process models such as the epidemic-type aftershock sequence (ETAS) model have been widely used to analyze and describe seismic catalogs and to perform short-term forecasting. The standard errors of parameter estimates in the ETAS model are significant and cannot be ignored. This paper uses simulations to explore the accuracy of conventional standard error estimates based on the Hessian matrix of the log-likelihood function of the ETAS model. We show that such error estimates are not accurate when the observed space-time window is small. One must only cautiously trust the Hessian-based standard error estimates for this model when using local datasets with time windows of several years in length. The standard errors for all parameter estimates introduced by magnitude errors in typical earthquake catalogs are found to be smaller than those introduced by choosing a finite time window. However, neither effect is insignificant.

On the Increase of Background Seismicity Rate during the 1997-1998 Umbria-Marche, Central Italy, Sequence: Apparent Variation or Fluid-Driven Triggering?

Abstract We investigate the temporal evolution of background seismicity rate in the Umbria-Marche sector of the northern Apennines that was struck by the 1997–1998 Colfiorito seismic sequence. Specifically, we apply the Epidemic-Type Aftershock Sequences (ETAS) model to separate the background seismicity rate from the coseismic triggered rate of earthquake production. Analyzed data are extracted from the Catalogo della Sismicita Italiana (CSI1.1, catalog of Italian seismicity) 1981–2002, which contains for the study area 12.163 events with M L >1.5. The capability of the ETAS model to match the observed seismicity rate is tested by analyzing the model residuals and by applying two nonparametric statistical tests (the runs and the Kolmogorov-Smirnov tests) to verify the fit of residuals to Poisson hypothesis. We first apply the ETAS model to the seismicity that occurred in the study area during the whole period covered by the CSI1.1 catalog. Our results show that the ETAS model does not explain the temporal evolution of seismicity in a time interval defined by change points identified from time-evolution of residuals and encompassing the Colfiorito seismic sequence. We therefore restrict our analysis to this period and analyze only those events belonging to the 1997–1998 seismic sequence. We again obtain the inadequacy of a stationary ETAS model with constant background rate to reproduce the temporal pattern of observed seismicity. We verify that the failure of the ETAS model to fit the observed data is caused by the increase of the background seismicity rate associated with the repeated Colfiorito mainshocks. We interpret the inferred increase of background rate as a consequence of the perturbation to the coseismic stress field caused by fluid flow and/or pore-pressure relaxation. In particular, we show that the transient perturbation caused by poroelastic relaxation can explain the temporal increase of background rate that therefore represents a fluid signal in the seismicity pattern.

On the relationship between lower magnitude thresholds and bias in epidemic‐type aftershock sequence parameter estimates

Modern earthquake catalogs are often described using spatial‐temporal point process models such as the epidemic‐type aftershock sequence (ETAS) models of Ogata (1998). Earthquake catalogs often have issues of incompleteness and other inaccuracies for earthquakes of magnitude below a certain threshold, and such earthquakes are typically removed prior to fitting a point process model. This paper investigates the bias in the parameters in ETAS models introduced by the removal of the smallest events. It is shown that in the case of most of the ETAS parameters, the bias increases approximately exponentially as a function of the lower magnitude cutoff.

Hybrid kernel estimates of space–time earthquake occurrence rates using the epidemic-type aftershock sequence model

The following steps are suggested for smoothing the occurrence patterns in a clustered space–time process, in particular the data from an earthquake catalogue. First, the original data is fitted by a temporal version of the ETAS model, and the occurrence times are transformed by using the cumulative form of the fitted ETAS model. Then the transformed data (transformed times and original locations) is smoothed by a space–time kernel with bandwidth obtained by optimizing a naive likelihood cross-validation. Finally, the estimated intensity for the original data is obtained by back-transforming the estimated intensity for the transformed data. This technique is used to estimate the intensity for earthquake occurrence data for associated with complex sequences of events off the East Coast of Tohoku district, northern Japan. The intensity so obtained is compared to the conditional intensity estimated from a full space–time ETAS model for the same data.

Correlations of Seismicity Patterns in Southern California with Surface Heat Flow Data

We investigate the relations between properties of seismicity patterns in Southern California and the surface heat flow using a relocated earthquake catalog. We first search for earthquake sequences that are well separated in time and space from other seismicity and then determine the epidemic type aftershock sequence (ETAS) model parameters for the sequences with a sufficient number of events. We focus on the productivity parameter α of the ETAS model that quantifies the relative efficiency of an earthquake with magnitude M to produce aftershocks. By stacking sequences with relatively small and relatively large α values separately, we observed clear differences between the two groups. Sequences with a smaller α have a relatively large number of foreshocks and relatively small number of after- shocks. In contrast, more typical sequences with larger α have relatively few fore- shocks and larger number of aftershocks. The stacked premainshock activity for the more typical latter sequences has a clear increase in the day before the occurrence of the main event. The spatial distribution of the α values correlates well with the surface heat flow: areas of high heat flow are characterized by relatively small α, indicating that in such regions the swarm-type earthquake activity is more common. Our results are compatible with a damage rheology model that predicts swarm-type seismic activity in areas with relatively high heat flow and more typical foreshock- mainshock-aftershock sequences in regions with normal or low surface heat flow. The high variability of α in regions with either high or low heat flow values indicates that at local scales additional factors (e.g., fluid content and rock type) may influence the seismicity generation process.

Probability gains of an epidemic-type aftershock sequence model in retrospective forecasting of M ≥ 5 earthquakes in Italy

A stochastic triggering (epidemic) model incorporating short-term clustering was fitted to the instrumental earthquake catalog of Italy for event with local magnitudes 2.6 and greater to optimize its ability to retrospectively forecast 33 target events of magnitude 5.0 and greater that occurred in the period 1990–2006. To obtain an unbiased evaluation of the information value of the model, forecasts of each event use parameter values obtained from data up to the end of the year preceding the target event. The results of the test are given in terms of the probability gain of the epidemic-type aftershock sequence (ETAS) model relative to a time-invariant Poisson model for each of the 33 target events. These probability gains range from 0.93 to 32000, with ten of the target events yielding a probability gain of at least 10. As the forecasting capability of the ETAS model is based on seismic activity recorded prior to the target earthquakes, the highest probability gains are associated with the occurrence of secondary mainshocks during seismic sequences. However, in nine of these cases, the largest mainshock of the sequence was marked by a probability gain larger than 50, having been preceded by previous smaller magnitude earthquakes. The overall evaluation of the performance of the epidemic model has been carried out by means of four popular statistical criteria: the relative operating characteristic diagram, the R score, the probability gain, and the log-likelihood ratio. These tests confirm the superior performance of the method with respect to a spatially varying, time-invariant Poisson model. Nevertheless, this method is characterized by a high false alarm rate, which would make its application in real circumstances problematic.

Modeling seismic swarms triggered by aseismic transients

The rate of earthquake occurrence varies by many orders of magnitude in a given region due to variations in the stress state of the crust. Our focus here is on variations in seismicity rate triggered by transient aseismic processes such as fluid flow, fault creep or magma intrusion. While these processes have been shown to trigger earthquakes, converting observed seismicity variations into estimates of stress rate variations has been challenging. Essentially aftershock sequences often obscure changes in the background seismicity rate resulting from aseismic processes. Two common approaches for estimating the time dependence of the underlying driving mechanisms are the stochastic Epidemic Type Aftershock Sequence model (ETAS) [Ogata, Y., (1988), Statistical models for earthquake occurrences and residual analysis for point processes, J. Am. Stat. Assoc., 83, 9–27.] and a physical approach based on the rate- and state-model of fault friction [Dieterich, J., (1994), A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res., 99, 2601–2618.]. The models have different strengths that could be combined to allow more quantitative studies of earthquake triggering. To accomplish this, we identify the parameters that relate to one another in the two models and examine their dependence on stressing rate. A particular conflict arises because the rate–state model predicts that aftershock productivity scales with stressing rate while the ETAS model assumes that it is time independent. To resolve this issue, we estimate triggering parameters for 4 earthquake swarms contemporaneous with geodetically observed deformation transients in various tectonic environments. We find that stressing rate transients increase the background seismicity rate without affecting aftershock productivity. We then specify a combined model for seismicity rate variations that will allow future studies to invert seismicity catalogs for variations in aseismic stressing rates.