Gutenberg-Richter Law Research Articles

Seeing through noise in power laws.

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto's law of wealth and the Gutenberg-Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov-Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

Fracture Electromagnetic Radiation Induced by a Seismic Active Zone (in the Vicinity of Eilat City, Southern Israel)

This paper deals with the quantitative analysis of measured fracture-induced electromagnetic radiation (FEMR) near the Dead Sea Transform using the Angel-M1 instrument, which enables the recording of FEMR signals in a 3D manner. The results showed both the possibility of estimating the sizes of micro-fractures that are the sources of radiation and assessing the direction of the fractures’ locations to the measuring device, as well as the range of magnitude (Mw) of the impending “events” (EQs) associated with the FEMR measurements. Moreover, the relation between the measured FEMR activity (the number of FEMR hits per unit of time) and the FEMR event magnitudes showed consistency with the Gutenberg–Richter relationship for the region. Such measurements could therefore constitute a preliminary ‘field reinforcement’ towards a valid EMR method for a real earthquake forecast, which would provide much earlier warnings than seismic methods. The observed FEMR measurements could only be used to assess the stress concentrations and micro-fracturing in the region since they related to the very initial nucleation phase of a “virtual” earthquake. Nonetheless, they provide the necessary feasibility test for a forecasting method since all of the lab-measured FEMR features were confirmed in the field.

Testing Hypotheses for Geological Controls on Hydraulic-Fracturing-Induced Seismicity in the Montney Formation, Canada

Hydraulic fracturing (HF) can trigger induced seismicity, but documented occurrences tend to be localized compared with the regional extent of industry operations. Factors that determine intrinsic geological susceptibility of a given region to induced seismicity remain incompletely understood. To address this uncertainty, we have developed a stochastic modeling approach to enable statistical testing of hypotheses regarding the distribution of induced seismicity. For reference, we adopted a null hypothesis that HF-induced seismic events are randomly associated with HF wells. Realizations of synthetic induced-seismicity catalogs are generated based on the Gutenberg–Richter relationship for magnitudes and explicit assumed spatial relationship(s) between HF wells and other known features, such as mapped structural corridors. Uncertainties in observed event locations and magnitudes are also considered. Based on 1000 independent realizations for each test scenario, normalized correlation coefficients, Bayesian information criteria and other statistical measures are used to quantify the similarity of synthetic catalogs to the observed seismicity distribution. We applied this approach to induced seismicity associated with HF operations within the Montney Formation, in western Canada. Three hypotheses were tested, each showing a statistically significant improvement over the null hypothesis. A previous machine-learning-based model for Seismogenic Activation Potential (SAP) showed the highest correlation between observed and synthetically generated seismicity catalogs. Our method has been developed using cloud-based computing and is easily adapted to other regions and data types.

Estimating the Magnitude of Completeness of Earthquake Catalogs Using a Simple Random Variable Transformation

Abstract The estimation of the magnitude of completeness for a seismic catalog is the starting point for most of the seismological statistical analyses. It is important, in particular, for the estimation of the Gutenberg–Richter (GR) law parameters. This law corresponds to an exponential distribution of the magnitudes. The recent studies have shown that the proper method to estimate the completeness is to check the exponentiality of the data. Here, we try to improve this method using a simple random variable transformation, which makes the method more robust in the case of catalogs with magnitudes coming from exponential distributions with different parameters, that is, catalogs with different b-values of the GR law.

Statistical analogies between earthquakes, micro-quakes in metals and avalanches in the 1D Burridge-Knopoff model

Universalities and intriguing analogies in the statistics of avalanches are revealed for three physical systems defined on largely different length and energy scales. Earthquakes induced by tectonic scale dynamics, micro-scale level quakes observed from slipping crystallographic planes in metals and a one-dimensional, room-scale spring-block type Burridge-Knopoff model is studied from similar statistical viewpoints. The validity of the Gutenberg-Richter law for the probability density of the energies dissipated in the avalanches is proven for all three systems. By analysing data for three different seismic zones and performing acoustic detection for different Zn samples under deformation, universality for the involved scaling exponent is revealed. With proper parameter choices the 1D Burridge-Knopoff model is able to reproduce the same scaling law. The recurrence times of earthquakes and micro-quakes with magnitudes above a given threshold present again similar distributions and striking quantitative similarities. However, the 1D Burridge-Knopoff model cannot account for the correlations observed in such statistics.

Quantifying the probability and uncertainty of multiple-structure rupture for Taiwan

This study identifies structure pairs with the potential for simultaneous rupture in a coseismic period and quantifies their rupture recurrence intervals. To assess the potential for a multiple-structure rupture, we calculated the probability of Coulomb stress triggering between seismogenic structures in Taiwan. We assumed that a multiple-structure rupture would occur if two structures could trigger each other by enhancing the plane with thresholds of a Coulomb stress increase and the distance between the structures and identified various sets of seismogenic structure pairs accordingly. We discussed the uncertainty of multiple-structure pair identification from various thresholds of stress change and structure distances, effective friction coefficient, and rotation of rake angles. To estimate the recurrence intervals for multiple-structure ruptures, we implemented a scaling law and the Gutenberg-Richter law in which the slip rate could be partitioned based on the magnitudes of the individual structure and multiple-structure ruptures. Considering that one structure may be involved in multiple cases of multiple-structure ruptures, we developed new formulas for slip partitioning in a complex fault system. By implementing the range of rupture area and slip rate of each structure, the magnitudes and recurrence intervals of multiple-structure ruptures could be estimated. We discussed the epistemic uncertainties of recurrence interval from deviations of slip rate and rupture area, various empirical formula of rupture parameters. The multiple-structure rupture with a larger characteristic magnitude would be crucial for the safety evaluation of infrastructures.

Clustering Analysis of Seismicity in the Anatolian Region with Implications for Seismic Hazard.

The Anatolian region is one of the most seismically active tectonic settings in the world. Here, we perform a clustering analysis of Turkish seismicity using an updated version of the Turkish Homogenized Earthquake Catalogue (TURHEC), which contains the recent developments of the still ongoing Kahramanmaraş seismic sequence. We show that some statistical properties of seismic activity are related to the regional seismogenic potential. Mapping the local and global coefficients of variation of inter-event times of crustal seismicity which occurred during the last three decades, we find that territories prone to major seismic events during the last century usually host globally clustered and locally Poissonian seismic activity. We suggest that regions with seismicity associated with higher values of the global coefficient of variation of inter-event times, CV, are likely to be more prone to hosting large earthquakes in the near future than other regions characterized by lower values, if their largest seismic events have the same magnitude. If our hypothesis is confirmed, clustering properties should be considered as a possible additional information source for the assessment of seismic hazard. We also find positive correlations between global clustering properties, the maximum magnitude and the seismic rate, while the b-value of the Gutenberg-Richter law is weakly correlated with them. Finally, we identify possible changes in such parameters before and during the 2023 Kahramanmaraş seismic sequence.

Modeling Exact Frequency-Energy Distribution for Quakes by a Probabilistic Cellular Automaton.

We develop the notion of Random Domino Automaton, a simple probabilistic cellular automaton model for earthquake statistics, in order to provide a mechanistic basis for the interrelation of Gutenberg-Richter law and Omori law with the waiting time distribution for earthquakes. In this work, we provide a general algebraic solution to the inverse problem for the model and apply the proposed procedure to seismic data recorded in the Legnica-Głogów Copper District in Poland, which demonstrate the adequacy of the method. The solution of the inverse problem enables adjustment of the model to localization-dependent seismic properties manifested by deviations from Gutenberg-Richter law.

Visibility Graph Analysis of the Seismic Activity of Three Areas of the Cocos Plate Mexican Subduction Where the Last Three Large Earthquakes (M > 7) Occurred in 2017 and 2022.

The understanding of the dynamical behavior of seismic phenomena is currently an open problem, mainly because seismic series can be considered to be produced by phenomena exhibiting dynamic phase transitions; that is, with some complexity. For this purpose, the Middle America Trench in central Mexico is considered a natural laboratory for examining subduction because of its heterogenous natural structure. In this study, the Visibility Graph method was applied to study the seismic activity of three regions within the Cocos plate: the Tehuantepec Isthmus, the Flat slab and Michoacan, each one with a different level of seismicity. The method maps time series into graphs, and it is possible to connect the topological properties of the graph with the dynamical features underlying the time series. The seismicity analyzed was monitored in the three areas studied between 2010 and 2022. At the Flat Slab and Tehuantepec Isthmus, two intense earthquakes occurred on 7 and 19 September 2017, respectively, and, on 19 September 2022, another one occurred at Michoacan. The aim of this study was to determine the dynamical features and the possible differences between the three areas by applying the following method. First, the time evolution of the a- and b-values in the Gutenberg-Richter law was analyzed, followed by the relationship between the seismic properties and topological features using the VG method, the k-M slope and the characterization of the temporal correlations from the γ-exponent of the power law distribution, P(k) ∼ k-γ, and its relationship with the Hurst parameter, which allowed us to identify the correlation and persistence of each zone.

Integrated microseismic and geomechanical analysis of hydraulic fracturing induced fault reactivation: a case study in Sichuan Basin, Southwest China

Because of the special geographical location adjacent to the Tibetan Plateau, the geological setting of the Sichuan Basin is very complex, which brings a great challenge to shale gas development. In this study, we conducted an integrated microseismic and geomechanical analysis on a refracturing well in the Sichuan Basin. We observed that casing deformation occurred in the area of fault reactivation. However, the reactivation of pre-existing faults does not always cause casing deformation. In addition, a cluster of high-magnitude microseismic events will be triggered during the fault reactivation, which is also helpful to recognize the reactivated faults. Furthermore, according to the definition of b-value in the Gutenberg–Richter law, a group of high-magnitude microseismic events usually shows a low b-value and vice versa. Based on the analysis of treatment data and microseismicity, we found that there is a positive relationship between b-value and instantaneous shut-in pressure (ISIP): lower b-value represents lower ISIP. The investigation of the stress shadow effect based on the theoretical solution of a plane strain fracture is also convincing to prove the relationship between b-value and ISIP. Therefore, a significant relationship between hydraulic fracturing, microseismicity, and fault reactivation is established. Importantly, some useful implications of this relationship could be utilized for early warning of fault reactivation (probably casing deformation) and optimization of refracturing design for unconventional shale plays.

Estimating Earthquake-Induced Tsunami Height Probabilities without Sampling

Given a distribution of earthquake-induced seafloor elevations, we present a method to compute the probability of the resulting tsunamis reaching a certain size on shore. Instead of sampling, the proposed method relies on optimization to compute the most likely fault slips that result in a seafloor deformation inducing a large tsunami wave. We model tsunamis induced by bathymetry change using the shallow water equations on an idealized slice through the sea. The earthquake slip model is based on a sum of multivariate log-normal distributions, and follows the Gutenberg-Richter law for seismic moment magnitudes ranging from 7 to 9. For a model problem inspired by the Tohoku-Oki 2011 earthquake and tsunami, we quantify annual probabilities of differently sized tsunami waves. Our method also identifies the most effective tsunami mechanisms. These mechanisms have smoothly varying fault slip patches that lead to an expansive but moderately large bathymetry change. The resulting tsunami waves are compressed as they approach shore and reach close-to-vertical leading wave edge close to shore.

Cluster-based foreshock discrimination model with flexible time horizon and mainshock magnitudes

Foreshock detection before mainshock occurrence is an important challenge limiting the short-term forecasts of large earthquakes. Various models for predicting mainshocks based on discrimination of foreshocks activity have been proposed, but many of them work in restricted scenarios and neglect foreshocks and mainshocks out of their scope. In disaster prevention, it is often necessary to change the forecast period and the magnitude of target mainshocks. This paper presents a cluster-based statistical discrimination of foreshocks which is applicable all over Japan and adjustable with respect to forecasting time span and mainshock magnitudes. Using the single-link clustering method, the model updates the expanding seismic clusters and determines in real time the probabilities that larger subsequent events will occur. The foreshock clusters and the others show different trends of certain feature statistics with respect to their magnitudes and spatiotemporal distances. Based on those features and the epicentral location, a nonlinear logistic regression model is used to evaluate the probabilities that growing seismic clusters are foreshocks that will trigger mainshocks within 30 days. The log of odds is estimated between the foreshock clusters and other clusters for respective feature values as nonlinear spline functions from a Japanese hypocenter catalog for the period 1926–1999. Based on the estimated odds functions, foreshock clusters tend to have smaller differences in their two largest magnitudes, shorter time durations, and slightly longer epicentral distances within the individual clusters. Given a potential foreshock cluster, its mainshock magnitude can be predicted by the Gutenberg–Richter law over the largest foreshock magnitude. The timing of mainshock occurrences from foreshocks is also predicted by multiplying the portion of mainshocks within a shorter span from those within 30 days by the evaluated foreshock probabilities. The predictive performance of our model is validated by the holdout method using a Japanese hypocenter catalog before and after 2000. The evaluated foreshock probabilities are roughly consistent with the actual portion of foreshocks in the validation catalog and could serve as an alert for large mainshocks.

Global versus local clustering of seismicity: Implications with earthquake prediction

The estimation of the maximum expected magnitude is crucial for seismic hazard assessment. It is usually inferred via Bayesian analysis; alternatively, the size of the largest possible event can be roughly obtained from the extent of the seismogenic source and the depth of the brittle–ductile transition. However, the effectiveness of the first approach is strongly limited by catalog completeness and the intensity of recorded seismicity, so that it can be of practical use only for aftershocks, while the second is affected by extremely large uncertainties. In this article, we investigate whether it may be possible to assess the magnitude of the largest event using some statistical properties of seismic activity. Our analysis shows that, while local features are not appropriate for modeling the emergence of peaks of seismicity, some global properties (e.g., the global coefficient of variation of interevent times and the fractal dimension of epicenters) seem correlated with the largest magnitude. Unlike several scientific articles suggest, the b-value of the Gutenberg–Richter law is not observed to have a predictive power in this case, which can be explained in the light of heterogeneous tectonic settings hosting fault systems with different extension.

TheK−MSlope: A Potential Supplement forb-Value

AbstractThe b-value of the Gutenberg–Richter law describes the relationship of size and frequency distribution of earthquakes. Its variations may be related to stress state, hence has been used for short-term earthquake forecasting. However, estimation of b-value faces many uncertainties, making it difficult to interpret b-value changes as tectonic signals or statistical artifacts. Some recent studies have suggested that the b-values of some seismic catalogs are proportional to the K−M slope (KMS) obtained from the visibility graph analysis. If confirmed, the KMS may provide additional constraints to the b-value and its variations. In this study, we used synthetic seismic catalogs that obey various probability distributions to demonstrate that the proportional relationship between KMS and b-value is universal and stable, and that the KMS/b ratio is dependent on the catalog size. We found that the KMS estimation can perform better than the commonly used b-value estimation methods, especially when the catalogs are incomplete or have variations in the magnitude–frequency relations. We proposed an improved KMS method to ensure that the KMS value provides pure magnitude information and is not affected by the occurrence time or orders of the events. We used the KMS to estimate the spatiotemporal variations of b-value in the northeastern Tibetan plateau; the results are consistent with those derived from traditional b-value estimation methods. We suggest that the KMS can be used as a verification of or a supplement to the b-value.

Real time Gutenberg–Richterb-value estimation for an ongoing seismic sequence: an application to the 2022 marche offshore earthquake sequence (ML 5.7 central Italy)

SUMMARYWe estimate the b-value parameter of the Gutenberg–Richter law for earthquake magnitudes in the early stage of the Costa Marchigiana (Italy) seismic sequence, starting on 2022 November 9, with an ML 5.7 event in the Adriatic sea. In particular, we estimate both the completeness magnitude Mc and the b-value within the first 4 and 7 d after the initial strong event in the sequence. Our work represents a practical example of b-value estimation in ‘true’ real time, that is, during the seismic sequence, and its possible interpretation in terms of short-term forecasting. We highlight some critical issues to consider both in estimating/intepreting the b-value, and in evaluating the real time estimation of Mc. These issues are mainly due to the fact that preliminary catalogues available in real time are quite different from the revised ones, which are usually delivered after a few months. The criticalities are linked to the raw data recorded at an early-stage, an unreliable evaluation of the Mc with statistical approaches, the Short Term Aftershock Incompleteness entailed after the initial strong event, and the magnitude binning. Our results show that real time estimation of the b-value can give insights into the evolution of an ongoing seismic sequence, when attention is paid to data quality and quantity.

Earthquake magnitude distribution and aftershocks: A statistical geometry explanation.

The emergence of a power-law distribution for the energy released during an earthquake is investigated in several models. Generic features are identified which are based on the self-affine behavior of the stress field prior to an event. This field behaves at large scale as a random trajectory in one dimension of space and a random surface in two dimensions. Using concepts of statistical mechanics and results on the properties of these random objects, several predictions are obtained and verified, in particular the value of the power-law exponent of the earthquake energy distribution (the Gutenberg-Richter law) as well as a mechanism for the existence of aftershocks after a large earthquake (the Omori law).

On the relation between rain, clouds, and cosmic rays

ABSTRACT We investigate the correlation between precipitation, clouds and cosmic rays based on field observations to develop a new short-term forecasting tool for their extremely high values. The results show that the cosmic ray intensity appears to be highly correlated with the one-year time-lagged rainfall amount with the cosmic rays leading. A plausible explanation is given for this, based on the modulation of cosmic rays by the solar cycle and an earlier finding of a strong correlation between the stratospheric - lower mesospheric temperature fluctuations and the sunspot number cycle with a time lag of one year (sun leading). This suggests that cosmic rays are not the trigger, but rather signals sunspot activity that with a delay of one year may affect cloud formation and therefore rainfall. In addition, high rates of rainfall exhibit a power-law behavior, such as the Gutenberg-Richter law characterizing the large area and perimeter size of rainfall and clouds, which have been proposed recently. Finally, the new tool for the nowcasting the extremely high cosmic ray intensity values can be used for nowcasting the extremely high rainfall values. Their combination can be used to study the modulation of cosmic ray properties by climate change parameters.

How to Choose a Magnitude Interval to Evaluate the Slope of the Magnitude-frequency Graph

In modern seismological practice, to describe the distribution of magnitudes, the Gutenberg-Richter law is widely used, one of the parameters of which is the b-value (the slope of the magnitude-frequency graph on a log scale). Authors propose new approaches to the problem of adequate and efficient statistical estimation of this parameter. The problem of the correct choice of the magnitude interval is discussed, on which the straightness of the Gutenberg-Richter law is observed with an acceptable degree of accuracy and which should be used to estimate the b-value. An efficient method of accounting for discreteness and aggregation of magnitudes in earthquake catalogs (the maximum likelihood method for discrete distributions) is proposed. The problem of changes in time of the lower limit of representative earthquakes registration is considered and a statistical approach is proposed for their description.

Temporal Variations of Seismicity Rates and Gutenberg–Richterb-Values for a Stochastic Declustered Catalog: An Example in Central Italy

AbstractOne important aspect of the seismicity is the spatiotemporal clustering; hence, the distinction between independent and triggered events is a critical part of the analysis of seismic catalogs. Stochastic declustering of seismicity allows a probabilistic distinction between these two kinds of events. Such an approach, usually performed with the epidemic-type aftershock sequence (ETAS) model, avoids the bias in the estimation of the frequency–magnitude distribution parameters if we consider a subset of the catalog, that is, only the independent or the triggered events. In this article, we present a framework to properly include the probabilities of any event to be independent (or triggered) both in the temporal variation of the seismic rates and in the estimation of the b-value of the Gutenberg–Richter law. This framework is then applied to a high-definition seismic catalog in the central part of Italy covering the period from April 2010 to December 2015. The results of our analysis show that the seismic activity from the beginning of the catalog to March 2013 is characterized by a low degree of clustering and a relatively high b-value, whereas the following period exhibits a higher degree of clustering and a smaller b-value.

Seismic hazard analysis of Silchar city located in North East India

North East India is a highly seismically active zone of India due to the collision tectonics between the Indian and the Eurasian plate in the north and subduction tectonics along the Indo-Myanmar range (IMR) in the east. It is utmost important to investigate the seismicity of this region along with uniform hazard spectra (UHS). In this paper, seismic hazard study has been performed at the Silchar district headquarters of Assam state, located in North East India. The earthquake data have been collected from various agencies from 1762 to 2020 and the fault data has been collected from seismotectonic atlas (SEISAT). Earthquake and fault data are superimposed on the Northeast India map using MapInfo software. The fault zone is defined based upon the earthquake’s density near the fault. Seismicity parameters and maximum magnitude have been estimated using Gutenberg-Richter (G-R) relationship. The best-fitted ground motion equation is considered. The outcome is represented in terms of controlling source, seismic hazard curve and UHS for Silchar city with varying ground motions. Finally, the results are presented as PGA based upon DSHA and PSHA methods. The effect of Importance Factor (I) and Response reduction factor (R) have been investigated by comparing UHS with IS: 1893-2016 code response spectra.