Gert Zöller

Observation of growing correlation length as an indicator for critical point behavior prior to large earthquakes

We test the critical point concept for earthquakes in terms of the spatial correlation length. A system near a critical point is associated with a diverging correlation length following a power law time-to-failure relation. We estimate the correlation length directly from an earthquake catalog using single-link cluster analysis. Therefore we assume that the distribution of moderate earthquakes reflects the state of the regional stress field. The parameters of the analysis are determined by an optimization procedure, and the results are tested against a Poisson process with realistic distributions of epicenters, magnitudes, and aftershocks. A systematic analysis of all earthquakes with M≥6.5 in California since 1952 is conducted. In fact, we observe growing correlation lengths in most cases. The null hypothesis that this behavior can be found in random data is rejected with a confidence level of more than 99%. Furthermore, we find a scaling relation log R∼0.7M (log〈ξmax〉 ∼ 0.5M), between the mainshock magnitude M and the critical region R (the correlation length 〈ξmax〉 before the mainshock), which is in good agreement with theoretical values.

Similar power laws for foreshock and aftershock sequences in a spring‐block model for earthquakes

We introduce a crust relaxation process in a continuous cellular automaton version of the Burridge and Knopoff [1967] model. The most important model parameters are the level of conservation and the ratio of the crust relaxation time to the tectonic reloading time. In correspondence with the original spring-block model, the modified model displays a robust power law distribution of event sizes. The principal new result obtained with our model is the spatiotemporal clustering of events exhibiting several characteristics of earthquakes in nature. Large events are followed by aftershock sequences obeying the Omori [1894] law and preceded by localized foreshocks, which are initiated after a time period of seismic quiescence. While we observe a considerable variability of precursory seismicity, we find that the rate of foreshocks increases on average, according to a power law with an exponent q, which is in good agreement with the exponent p of the Omori law. In contrast to other events, the distribution of foreshock sizes is characterized by a significantly smaller Richter B value. Our model reproduces simultaneously the empirically observed values of the power law exponents, the Richter B, p and q, and their variability.

Seismic quiescence as an indicator for large earthquakes in a system of self‐organized criticality

Seismically active fault systems may be in a state of self-organized criticality (SOC). Investigations of simple SOC models have suggested that earthquakes might be inherently unpredictable. In this paper, we analyze the question of predictability in a more complex and realistic SOC model, which consists of a spring-block system with transient creep characteristics. Additionally to the power law distribution of earthquake sizes, this model reproduces also foreshock and aftershock sequences. Aside from a short-term increase of seismicity immediately prior to large model earthquakes, these events are preceded on average by an intermediate-term period of reduced seismicity. The stronger and the longer the duration of this period, the larger on average is the subsequent mainshock. We find that the detection of seismic quiescence can improve the time—independent hazard assessment. The improvement is most significant for the largest target events.

Estimation of the Maximum Possible Magnitude in the Framework of a Doubly Truncated Gutenberg–Richter Model

We discuss to what extent a given earthquake catalog and the assumption of a doubly truncated Gutenberg–Richter distribution for the earthquake magnitudes allow for the calculation of confidence intervals for the maximum possible magnitude M . We show that, without further assumptions such as the existence of an upper bound of M , only very limited information may be obtained. In a frequentist formulation, for each confidence level α the confidence interval diverges with finite probability. In a Bayesian formulation, the posterior distribution of the upper magnitude is not normalizable. We conclude that the common approach to derive confidence intervals from the variance of a point estimator fails. Technically, this problem can be overcome by introducing an upper bound ![Graphic][1] for the maximum magnitude. Then the Bayesian posterior distribution can be normalized, and its variance decreases with the number of observed events. However, because the posterior depends significantly on the choice of the unknown value of ![Graphic][2] , the resulting confidence intervals are essentially meaningless. The use of an informative prior distribution accounting for preknowledge of M is also of little use, because the prior is only modified in the case of the occurrence of an extreme event. Our results suggest that the maximum possible magnitude M should be better replaced by M T , the maximum expected magnitude in a given time interval T , for which the calculation of exact confidence intervals becomes straightforward. From a physical point of view, numerical models of the earthquake process adjusted to specific fault regions may be a powerful alternative to overcome the shortcomings of purely statistical inference. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif

Recurrence Time Distributions of Large Earthquakes in a Stochastic Model for Coupled Fault Systems: The Role of Fault Interaction

We study the effect of fault interaction on the recurrence time distribution of large earthquakes on the same fault. A single isolated fault is modeled by a Brownian relaxation oscillator leading to a Brownian passage-time distribution for the recurrence intervals. Interaction between different faults is imposed in terms of stress increase and decrease resulting in three possible ways: the occurrence of an earthquake is advanced or delayed, or the earthquake is triggered instantaneously. The results indicate the existence of two regimes: for weakly coupled faults, the recurrence time distribution of earthquakes on one fault follows mostly the Brownian passage-time distribution. For a strongly coupled system, the faults are synchronized and the effect of instantaneous triggering becomes dominant: the recurrence time distribution follows a Gamma or a Weibull distribution. The transition from weak to strong coupling is abrupt and behaves like a phase transition. It occurs when the stress transfer equals the average stress deficit. The results are interpreted in terms of a phase diagram. This diagram includes a regime, where the distribution of recurrence times is similar to a numerical model for California. We claim that the emergence of the Gamma and the Weibull distribution can be considered as an effect of fault interaction.

Quasi-static and Quasi-dynamic Modeling of Earthquake Failure at Intermediate Scales

We present a model for earthquake failure at intermediate scales (space: 100 m-100 km, time: 100 m/vshear 1000’s of years). The model consists of a segmented strike–slip fault embedded in a 3-D elastic solid as in the framework of Ben-Zion and Rice (1993). The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of the regions around the fault, static/ kinetic friction laws with possible gradual healing, and stress transfer based on the solution of Chinnery (1963) for static dislocations in an elastic half-space. As a new ingredient, we approximate the dynamic rupture on a continuous time scale using a finite stress propagation velocity (quasi–dynamic model) instead of instantaneous stress transfer (quasi–static model). We compare the quasi–dynamic model with the quasi–static version and its mean field approximation, and discuss the conditions for the occurrence of frequency-size statistics of the Gutenberg–Richter type, the characteristic earthquake type, and the possibility of a spontaneous mode switching from one distribution to the other. We find that the ability of the system to undergo a spontaneous mode switching depends on the range of stress transfer interaction, the cell size, and the level of strength heterogeneities. We also introduce time-dependent log (t) healing and show that the results can be interpreted in the phase diagram framework. To have a flexible computational environment, we have implemented the model in a modular C + + class library.

The Maximum Earthquake Magnitude in a Time Horizon: Theory and Case Studies

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 α . 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 Λ and obey the Gutenberg–Richter law in magnitude domain with unknown b ‐value. The random variables Λ 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.

Recurrent Large Earthquakes in a Fault Region: What Can Be Inferred from Small and Intermediate Events?

We present a renewal model for the recurrence of large earthquakes in a fault zone consisting of a major fault and surrounding smaller faults with Gutenberg–Richter-type seismicity represented by seismic moment release drawn from a truncated power-law distribution. The recurrence times of characteristic earthquakes for the major fault are explored. It is continuously loaded (plate motion) and undergoes positive and negative fluctuations due to adjacent smaller faults, with a large number N eq of such changes between two major earthquakes. Because the distribution has a finite variance, in the limit N eq→∞ the central limit theorem implies that the recurrence times follow a Brownian passage-time (BPT) distribution. This allows us to calculate individual recurrence-time distributions for specific fault zones without tuning free parameters: the mean recurrence time can be estimated from geological or paleoseismic data, and the standard deviation is determined from the frequency-size distribution, namely, the Richter b -value, of an earthquake catalog. The approach is demonstrated for the Parkfield segment of the San Andreas fault in California as well as for a long simulation of a numerical fault model. Assuming power-law distributed earthquake magnitudes up to the size of the recurrent Parkfield event ( M 6), we find a coefficient of variation that is higher than the value obtained by a direct fit of the BPT distribution to seven large earthquakes. We argue that the BPT distribution is a reasonable choice for seismic hazard assessment because it governs not only Brownian motion with drift but also models with power-law statistics for the recurrence of large earthquakes in an asymptotic limit. Finally, we find that the condition b =1 separates two regimes: in the first ( b 1) it is greater.

Can we test for the maximum possible earthquake magnitude

We explore the concept of maximum possible earthquake magnitude, M, in a region represented by an earthquake catalog from the viewpoint of statistical testing. For this aim, we assume that earthquake magnitudes are independent events that follow a doubly truncated Gutenberg-Richter distribution and focus on the upper truncation M. In earlier work, it has been shown that the value of M cannot be well constrained from earthquake catalogs alone. However, for two hypothesized values M and M′, alternative statistical tests may address the question: Which value is more consistent with the data? In other words, is it possible to reject a magnitude within reasonable errors, i.e., the error of the first and the error of the second kind? The results for realistic settings indicate that either the error of the first kind or the error of the second kind is intolerably large. We conclude that it is essentially impossible to infer M in terms of alternative testing with sufficient confidence from an earthquake catalog alone, even in regions like Japan with excellent data availability. These findings are also valid for frequency-magnitude distributions with different tail behavior, e.g., exponential tapering. Finally, we emphasize that different data may only be useful to provide additional constraints for M, if they do not correlate with the earthquake catalog, i.e., if they have not been recorded in the same observational period. In particular, long-term geological assessments might be suitable to reduce the errors, while GPS measurements provide overall the same information as the catalogs.

The Earthquake History in a Fault Zone Tells Us Almost Nothing about mmax

In the present study, we summarize and evaluate the endeavors from recent years to estimate the maximum possible earthquake magnitude mmax from observed data. In particular, we use basic and physically motivated assumptions to identify best cases and worst cases in terms of lowest and highest degree of uncertainty of mmax. In a general framework, we demonstrate that earthquake data and earthquake proxy data recorded in a fault zone provide almost no information about mmax unless reliable and homogeneous data of a long time interval, including several earthquakes with magnitude close to mmax, are available. Even if detailed earthquake information from some centuries including historic and paleoearthquakes are given, only very few, namely the largest events, will contribute at all to the estimation of mmax, and this results in unacceptably high uncertainties. As a consequence, estimators of mmax in a fault zone, which are based solely on earthquake‐related information from this region, have to be dismissed.