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The Gutenberg-Richter relation for earthquake magnitudes is the most famous empirical law in seismology. It states that the frequency of earthquake magnitudes follows an exponential distribution; this has been found to be a robust feature of seismicity above the completeness magnitude, and it is independent of whether global, regional, or local seismicity is analyzed. However, the exponent b of the distribution varies significantly in space and time, which is important for process understanding and seismic hazard assessment; this is particularly true because of the fact that the Gutenberg-Richter b-value acts as a proxy for the stress state and quantifies the ratio of large-to-small earthquakes. In our work, we focus on the automatic detection of statistically significant temporal changes of the b-value in seismicity data. In our approach, we use Bayes factors for model selection and estimate multiple change-points of the frequency-magnitude distribution in time. The method is first applied to synthetic data, showing its capability to detect change-points as function of the size of the sample and the b-value contrast. Finally, we apply this approach to examples of observational data sets for which b-value changes have previously been stated. Our analysis of foreshock and after-shock sequences related to mainshocks, as well as earthquake swarms, shows that only a portion of the b-value changes is statistically significant.
One of the crucial components in seismic hazard analysis is the estimation of the maximum earthquake magnitude and associated uncertainty. In the present study, the uncertainty related to the maximum expected magnitude mu is determined in terms of confidence intervals for an imposed level of confidence. Previous work by Salamat et al. (Pure Appl Geophys 174:763-777, 2017) shows the divergence of the confidence interval of the maximum possible magnitude m(max) for high levels of confidence in six seismotectonic zones of Iran. In this work, the maximum expected earthquake magnitude mu is calculated in a predefined finite time interval and imposed level of confidence. For this, we use a conceptual model based on a doubly truncated Gutenberg-Richter law for magnitudes with constant b-value and calculate the posterior distribution of mu for the time interval T-f in future. We assume a stationary Poisson process in time and a Gutenberg-Richter relation for magnitudes. The upper bound of the magnitude confidence interval is calculated for different time intervals of 30, 50, and 100 years and imposed levels of confidence alpha = 0.5, 0.1, 0.05, and 0.01. The posterior distribution of waiting times T-f to the next earthquake with a given magnitude equal to 6.5, 7.0, and7.5 are calculated in each zone. In order to find the influence of declustering, we use the original and declustered version of the catalog. The earthquake catalog of the territory of Iran and surroundings are subdivided into six seismotectonic zones Alborz, Azerbaijan, Central Iran, Zagros, Kopet Dagh, and Makran. We assume the maximum possible magnitude m(max) = 8.5 and calculate the upper bound of the confidence interval of mu in each zone. The results indicate that for short time intervals equal to 30 and 50 years and imposed levels of confidence 1 - alpha = 0.95 and 0.90, the probability distribution of mu is around mu = 7.16-8.23 in all seismic zones.
Kijko et al. (2016) present various methods to estimate parameters that are relevant for probabilistic seismic-hazard assessment. One of these parameters, although not the most influential, is the maximum possible earthquake magnitude m(max). I show that the proposed estimation of m(max) is based on an erroneous equation related to a misuse of the estimator in Cooke (1979) and leads to unstable results. So far, reported finite estimations of m(max) arise from data selection, because the estimator in Kijko et al. (2016) diverges with finite probability. This finding is independent of the assumed distribution of earthquake magnitudes. For the specific choice of the doubly truncated Gutenberg-Richter distribution, I illustrate the problems by deriving explicit equations. Finally, I conclude that point estimators are generally not a suitable approach to constrain m(max).
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 seismicity generates complex clustering of earthquakes in space and time. Therefore, the problem 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.
This paper concerns the problem of predicting the maximum expected earthquake magnitude μ in a future time interval Tf given a catalog covering a time period T in the past. Different studies show the divergence of the confidence interval of the maximum possible earthquake magnitude m_{ max } for high levels of confidence (Salamat et al. 2017). Therefore, m_{ max } should be better replaced by μ (Holschneider et al. 2011). In a previous study (Salamat et al. 2018), μ is estimated for an instrumental earthquake catalog of Iran from 1900 onwards with a constant level of completeness ( {m0 = 5.5} ). In the current study, the Bayesian methodology developed by Zöller et al. (2014, 2015) is applied for the purpose of predicting μ based on the catalog consisting of both historical and instrumental parts. The catalog is first subdivided into six subcatalogs corresponding to six seismotectonic zones, and each of those zone catalogs is subsequently subdivided according to changes in completeness level and magnitude uncertainty. For this, broad and small error distributions are considered for historical and instrumental earthquakes, respectively. We assume that earthquakes follow a Poisson process in time and Gutenberg-Richter law in the magnitude domain with a priori unknown a and b values which are first estimated by Bayes' theorem and subsequently used to estimate μ. Imposing different values of m_{ max } for different seismotectonic zones namely Alborz, Azerbaijan, Central Iran, Zagros, Kopet Dagh and Makran, the results show considerable probabilities for the occurrence of earthquakes with Mw ≥ 7.5 in short Tf , whereas for long Tf, μ is almost equal to m_{ max }
Introduction to special issue: Dynamics of seismicity patterns and earthquake triggering - Preface
(2006)
Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake.
The Coulomb failure stress (CFS) criterion is the most commonly used method for predicting spatial distributions of aftershocks following large earthquakes. However, large uncertainties are always associated with the calculation of Coulomb stress change. The uncertainties mainly arise due to nonunique slip inversions and unknown receiver faults; especially for the latter, results are highly dependent on the choice of the assumed receiver mechanism. Based on binary tests (aftershocks yes/no), recent studies suggest that alternative stress quantities, a distance-slip probabilistic model as well as deep neural network (DNN) approaches, all are superior to CFS with predefined receiver mechanism. To challenge this conclusion, which might have large implications, we use 289 slip inversions from SRCMOD database to calculate more realistic CFS values for a layered half-space and variable receiver mechanisms. We also analyze the effect of the magnitude cutoff, grid size variation, and aftershock duration to verify the use of receiver operating characteristic (ROC) analysis for the ranking of stress metrics. The observations suggest that introducing a layered half-space does not improve the stress maps and ROC curves. However, results significantly improve for larger aftershocks and shorter time periods but without changing the ranking. We also go beyond binary testing and apply alternative statistics to test the ability to estimate aftershock numbers, which confirm that simple stress metrics perform better than the classic Coulomb failure stress calculations and are also better than the distance-slip probabilistic model.
The knowledge of the largest expected earthquake magnitude in a region is one of the key issues in probabilistic seismic hazard calculations and the estimation of worst-case scenarios. Earthquake catalogues are the most informative source of information for the inference of earthquake magnitudes. We analysed the earthquake catalogue for Central Asia with respect to the largest expected magnitudes m(T) in a pre-defined time horizon T-f using a recently developed statistical methodology, extended by the explicit probabilistic consideration of magnitude errors. For this aim, we assumed broad error distributions for historical events, whereas the magnitudes of recently recorded instrumental earthquakes had smaller errors. The results indicate high probabilities for the occurrence of large events (M >= 8), even in short time intervals of a few decades. The expected magnitudes relative to the assumed maximum possible magnitude are generally higher for intermediate-depth earthquakes (51-300 km) than for shallow events (0-50 km). For long future time horizons, for example, a few hundred years, earthquakes with M >= 8.5 have to be taken into account, although, apart from the 1889 Chilik earthquake, it is probable that no such event occurred during the observation period of the catalogue.
We investigate the influence of spatial heterogeneities on various aspects of brittle failure and seismicity in a model of a large strike-slip fault. The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of regions around the fault, static/kinetic friction laws, creep with depth-dependent coefficients, and 3-D elastic stress transfer. The dynamic rupture is approximated on a continuous time scale using a finite stress propagation velocity ("quasidynamic model''). The model produces a "brittle- ductile'' transition at a depth of about 12.5 km, realistic hypocenter distributions, and other features of seismicity compatible with observations. Previous work suggested that the range of size scales in the distribution of strength-stress heterogeneities acts as a tuning parameter of the dynamics. Here we test this hypothesis by performing a systematic parameter-space study with different forms of heterogeneities. In particular, we analyze spatial heterogeneities that can be tuned by a single parameter in two distributions: ( 1) high stress drop barriers in near- vertical directions and ( 2) spatial heterogeneities with fractal properties and variable fractal dimension. The results indicate that the first form of heterogeneities provides an effective means of tuning the behavior while the second does not. In relatively homogeneous cases, the fault self-organizes to large-scale patches and big events are associated with inward failure of individual patches and sequential failures of different patches. The frequency-size event statistics in such cases are compatible with the characteristic earthquake distribution and large events are quasi-periodic in time. In strongly heterogeneous or near-critical cases, the rupture histories are highly discontinuous and consist of complex migration patterns of slip on the fault. In such cases, the frequency-size and temporal statistics follow approximately power-law relations
We show that realistic aftershock sequences with space-time characteristics compatible with observations are generated by a model consisting of brittle fault segments separated by creeping zones. The dynamics of the brittle regions is governed by static/kinetic friction, 3D elastic stress transfer and small creep deformation. The creeping parts are characterized by high ongoing creep velocities. These regions store stress during earthquake failures and then release it in the interseismic periods. The resulting postseismic deformation leads to aftershock sequences following the modified Omori law. The ratio of creep coefficients in the brittle and creeping sections determines the duration of the postseismic transients and the exponent p of the modified Omori law
The problem of estimating the maximum possible earthquake magnitude m(max) has attracted growing attention in recent years. Due to sparse data, the role of uncertainties becomes crucial. In this work, we determine the uncertainties related to the maximum magnitude in terms of confidence intervals. Using an earthquake catalog of Iran, m(max) is estimated for different predefined levels of confidence in six seismotectonic zones. Assuming the doubly truncated Gutenberg-Richter distribution as a statistical model for earthquake magnitudes, confidence intervals for the maximum possible magnitude of earthquakes are calculated in each zone. While the lower limit of the confidence interval is the magnitude of the maximum observed event, the upper limit is calculated from the catalog and the statistical model. For this aim, we use the original catalog which no declustering methods applied on as well as a declustered version of the catalog. Based on the study by Holschneider et al. (Bull Seismol Soc Am 101(4): 1649-1659, 2011), the confidence interval for m(max) is frequently unbounded, especially if high levels of confidence are required. In this case, no information is gained from the data. Therefore, we elaborate for which settings finite confidence levels are obtained. In this work, Iran is divided into six seismotectonic zones, namely Alborz, Azerbaijan, Zagros, Makran, Kopet Dagh, Central Iran. Although calculations of the confidence interval in Central Iran and Zagros seismotectonic zones are relatively acceptable for meaningful levels of confidence, results in Kopet Dagh, Alborz, Azerbaijan and Makran are not that much promising. The results indicate that estimating mmax from an earthquake catalog for reasonable levels of confidence alone is almost impossible.
Paleoearthquakes and historic earthquakes are the most important source of information for the estimation of long-term earthquake recurrence intervals in fault zones, because corresponding sequences cover more than one seismic cycle. However, these events are often rare, dating uncertainties are enormous, and missing or misinterpreted events lead to additional problems. In the present study, I assume that the time to the next major earthquake depends on the rate of small and intermediate events between the large ones in terms of a clock change model. Mathematically, this leads to a Brownian passage time distribution for recurrence intervals. I take advantage of an earlier finding that under certain assumptions the aperiodicity of this distribution can be related to the Gutenberg-Richter b value, which can be estimated easily from instrumental seismicity in the region under consideration. In this way, both parameters of the Brownian passage time distribution can be attributed with accessible seismological quantities. This allows to reduce the uncertainties in the estimation of the mean recurrence interval, especially for short paleoearthquake sequences and high dating errors. Using a Bayesian framework for parameter estimation results in a statistical model for earthquake recurrence intervals that assimilates in a simple way paleoearthquake sequences and instrumental data. I present illustrative case studies from Southern California and compare the method with the commonly used approach of exponentially distributed recurrence times based on a stationary Poisson process.
Earthquake rates are driven by tectonic stress buildup, earthquake-induced stress changes, and transient aseismic processes. Although the origin of the first two sources is known, transient aseismic processes are more difficult to detect. However, the knowledge of the associated changes of the earthquake activity is of great interest, because it might help identify natural aseismic deformation patterns such as slow-slip events, as well as the occurrence of induced seismicity related to human activities. For this goal, we develop a Bayesian approach to identify change-points in seismicity data automatically. Using the Bayes factor, we select a suitable model, estimate possible change-points, and we additionally use a likelihood ratio test to calculate the significance of the change of the intensity. The approach is extended to spatiotemporal data to detect the area in which the changes occur. The method is first applied to synthetic data showing its capability to detect real change-points. Finally, we apply this approach to observational data from Oklahoma and observe statistical significant changes of seismicity in space and time.
We show how the maximum magnitude within a predefined future time horizon may be estimated from an earthquake catalog within the context of Gutenberg-Richter statistics. The aim is to carry out a rigorous uncertainty assessment, and calculate precise confidence intervals based on an imposed level of confidence a. In detail, we present a model for the estimation of the maximum magnitude to occur in a time interval T-f in the future, given a complete earthquake catalog for a time period T in the past and, if available, paleoseismic events. For this goal, we solely assume that earthquakes follow a stationary Poisson process in time with unknown productivity Lambda and obey the Gutenberg-Richter law in magnitude domain with unknown b-value. The random variables. and b are estimated by means of Bayes theorem with noninformative prior distributions. Results based on synthetic catalogs and on retrospective calculations of historic catalogs from the highly active area of Japan and the low-seismicity, but high-risk region lower Rhine embayment (LRE) in Germany indicate that the estimated magnitudes are close to the true values. Finally, we discuss whether the techniques can be extended to meet the safety requirements for critical facilities such as nuclear power plants. For this aim, the maximum magnitude for all times has to be considered. In agreement with earlier work, we find that this parameter is not a useful quantity from the viewpoint of statistical inference.
Time-dependent probabilistic seismic hazard assessment requires a stochastic description of earthquake occurrences. While short-term seismicity models are well-constrained by observations, the recurrences of characteristic on-fault earthquakes are only derived from theoretical considerations, uncertain palaeo-events or proxy data. Despite the involved uncertainties and complexity, simple statistical models for a quasi-period recurrence of on-fault events are implemented in seismic hazard assessments. To test the applicability of statistical models, such as the Brownian relaxation oscillator or the stress release model, we perform a systematic comparison with deterministic simulations based on rate- and state-dependent friction, high-resolution representations of fault systems and quasi-dynamic rupture propagation. For the specific fault network of the Lower Rhine Embayment, Germany, we run both stochastic and deterministic model simulations based on the same fault geometries and stress interactions. Our results indicate that the stochastic simulators are able to reproduce the first-order characteristics of the major earthquakes on isolated faults as well as for coupled faults with moderate stress interactions. However, we find that all tested statistical models fail to reproduce the characteristics of strongly coupled faults, because multisegment rupturing resulting from a spatiotemporally correlated stress field is underestimated in the stochastic simulators. Our results suggest that stochastic models have to be extended by multirupture probability distributions to provide more reliable results.
Convergence of the frequency-magnitude distribution of global earthquakes - maybe in 200 years
(2013)
I study the ability to estimate the tail of the frequency-magnitude distribution of global earthquakes. While power-law scaling for small earthquakes is accepted by support of data, the tail remains speculative. In a recent study, Bell et al. (2013) claim that the frequency-magnitude distribution of global earthquakes converges to a tapered Pareto distribution. I show that this finding results from data fitting errors, namely from the biased maximum likelihood estimation of the corner magnitude theta in strongly undersampled models. In particular, the estimation of theta depends solely on the few largest events in the catalog. Taking this into account, I compare various state-of-the-art models for the global frequency-magnitude distribution. After discarding undersampled models, the remaining ones, including the unbounded Gutenberg-Richter distribution, perform all equally well and are, therefore, indistinguishable. Convergence to a specific distribution, if it ever takes place, requires about 200 years homogeneous recording of global seismicity, at least.