Frequency-magnitude Distribution Of Earthquakes Research Articles

Spatio-temporal analysis of b-value prior to 28 April 2021 Assam Earthquake and implications thereof

Here, we report a Spatio-temporal analysis of the frequency magnitude distribution of earthquakes (b-value) before the 28th April 2021 (Mw 6.4) earthquake event observed in northeast India. To esti- mate the average b-value for the study region, a data set of 750 earthquake events with magnitude Mw ≥ 3.9 is extracted from the homogenous part of the earthquake catalog (1950-2021) documented by the United States Geological Survey (USGS) and International seismological center (ISC) in the region. For spatial analysis of the disparities in b-value, the whole study region is subdivided into 16 square grids of dimension 1o×1o and the b-value is calculated for each subsection. In congruence with other studies, this work yields b-values ranging from 0.66 to 1.25. After the calculation of the b-value for each grid, it is observed that the grid with the epicentral location of the 28th April 2021 (6.4) earthquake has a low b-value. Accordingly, the spatial correlation and aberrant pattern between b-value and focal depth have been comprehensively explored. It is observed that the b-value sig- nificantly dips within a depth range of ~15-35 km which implicates high-stress accumulation and crustal homogeneity. The depth-wise variation in b-value infers the antithetical relationship between b-value and crustal stress. Mostly interplate earthquakes are observed in the study region; thereby hinting at intense seismicity at the upper crust.

Revisiting b-value for extended Kopili region of Northeast India and probabilistic estimation of seismic hazard attributes thereof

Kopili fault and its' surroundings of northeast India have remained witness to some devastating earthquakes. The complex tectonic feature of this region incurs a high seismicity rate relative to other regions. Very recently, an earthquake of magnitude 6.4 (MW) has jolted this region. Considering the persistently rising seismicity pattern of this region, the present study intends to integrate the frequency magnitude distribution of earthquakes with other pertinent seismic hazard parameters. This work presents the estimation of the return period and probability of non-exceedance of medium to large earthquakes for the Kopili fault and its' surroundings via the extreme value method. Two statistical methods, namely the Gutenberg-Richter(G-R) power law and Gumbel extreme value method (GEVM), are applied for the correct estimation of the b-value for the Kopili fault and its' surrounding region. The uniform and homogenous magnitude scale dataset consisting of 550 events observed within a limit of 350 km from the Kopili fault (1964 to 2022) are used in the study. The b-value computed using GR relation was found to be unsatisfactory (0.73) due to an inadequate dataset—thus Gumbel's extreme value method is used for the estimation of the b-value (0.97). Gumbel's extreme value method is used for the calculation of the probability of occurrence of earthquakes, return period, and the most probable earthquake magnitude for the study region. The most probable annual earthquake is found to be ∼ MW 4.99. The return period and probability of occurrence of design-based earthquakes are also calculated. The correlation between b-value, seismic moment, and fractal dimension for seismic hazard analysis is also investigated. The antithetical relationship between seismic moment and b-value is observed for the region while the deep-root character of the Kopili fault and the strong, unfractured character of the Shillong plateau is highlighted with the spatial distribution of fractal dimension in the study region.

Frequency-magnitude distribution of earthquakes at Etna volcano unravels critical stress changes along magma pathways

The high seismic productivity of volcanic areas provides the chance to investigate the local stress conditions with great resolution, by analysing the slope of the frequency-magnitude distribution of earthquakes, namely the b-value. Here we investigated the seismicity of Mt. Etna between 2005 and 2019, focusing on one of the largest known episodes of unrest in December 2018, when most of the intruding magma aborted, rather oddly, its ascent inside the volcano. We found a possible stress concentration zone along magma pathways, which may have inhibited the occurrence of a larger eruption. If the origin of such hypothetical loaded region is related to tectonic forces, one must consider the possibility that geodynamic processes can locally result in such rapid crustal strain as to perturb the release of magma. Strong b-value time-variations occurred a few days before the unrest event, suggesting new possibilities for investigating the volcano state and impending eruptions.

Statistics of the earthquakes in the central Himalaya and its vicinity in last 56 years, with an emphasis in the 25 April 2015 Gorkha, Nepal earthquake

To understand the variation of stress levels in the region 80°E – 89°E and 26°N – 31°N, the statistical analysis of earthquake frequency-magnitude distribution and spatio-temporal variation of fractal correlation dimension of earthquake epicenter distribution are estimated. The analysis is carried out on declusterised catalogue containing 1185 events of 56 years from February 1964 to November 2020. The study area is divided into three regions the western Nepal and vicinity (Region A), central Nepal and vicinity (Region B) and eastern Nepal and vicinity (Region C), respectively. The magnitude of completeness (Mc) varies from 3.6 to 4.0 for the study period. The spatial fractal dimension (Dc) and b-value are calculated as 1.89 ± 0.02 and 0.68 ± 0.03 for the western Nepal, 1.76 ± 0.01 and 0.60 ± 0.05 for the central Nepal, whereas they are estimated as 1.85 ± 0.02 and 0.63 ± 0.03 for the eastern part of the Nepal. The b-values obtained for all three regions are very low comparing to global average value of 1. The time clustering of the events in the respective regions are 0.26 ± 0.003, 0.31 ± 0.004 and 0.26 ± 0.02 as indicated by temporal fractal dimension (Dt). The higher Dc, lower b and Dt values associated with the regions indicate high stress concentration and stronger epicenter clustering in these regions. The strongly increasing trend of fractal dimension and strongly decreasing trend of b-value show the high probabilities of occurring the large earthquake in both central Nepal (82.5°E – 85.5°E and 27.5°N – 30°N) and eastern Nepal (85.5°E – 88.2°E and 26.45°N – 28.6°N) as compared to western Nepal (80°E – 82.5°E and 28°N – 30.5°N). This statistical analysis of spatial and temporal characteristics of the earthquake activity may give significant signs of the future seismic hazard along central Himalaya region.

Spatiotemporal Evolution of Ground-Motion Intensity at the Irpinia Near-Fault Observatory, Southern Italy

ABSTRACT Fault zones are major sources of hazard for many populated regions around the world. Earthquakes still occur unanticipated, and research has started to observe fault properties with increasing spatial and temporal resolution, having the goal of detecting signs of stress accumulation and strength weakening that may anticipate the rupture. The common practice is monitoring source parameters retrieved from measurements; however, model dependence and strong uncertainty propagation hamper their usage for small and microearthquakes. Here, we decipher the ground motion (i.e., ground shaking) variability associated with microseismicity detected by dense seismic networks at a near-fault observatory in Irpinia, Southern Italy, and obtain an unprecedentedly sharp picture of the fault properties evolution both in time and space. We discuss the link between the ground-motion intensity and the source parameters of the considered microseismicity, showing a coherent spatial distribution of the ground-motion intensity with that of corner frequency, stress drop, and radiation efficiency. Our analysis reveals that the ground-motion intensity presents an annual cycle in agreement with independent geodetic displacement observations from two Global Navigation Satellite System stations in the area. The temporal and spatial analyses also reveal a heterogeneous behavior of adjacent fault segments in a high seismic risk Italian area. Concerning the temporal evolution of fault properties, we highlight that the fault segment where the 1980 Ms 6.9 Irpinia earthquake nucleated shows changes in the event-specific signature of ground-motion signals since 2013, suggesting changes in their frictional properties. This evidence, combined with complementary information on the earthquake frequency–magnitude distribution, reveals differences in fault segment response to tectonic loading, suggesting rupture scenarios of future moderate and large earthquakes for seismic hazard assessment.

Seismotectonics and Spatio-temporal Variations in Seismicity Rates along Andaman Trench

The earthquake Frequency Magnitude Distribution (FMD) or b values have been meticulously used in the last two decades as a deterministic earthquake precursor or indicator, in earthquake forecasting or probabilistic seismic hazard assessment studies, in a variety of tectonic and stress regimes. In the present study, temporal fluctuations of b values were analysed by means of 185 well located (Mw ≥ 5) earthquakes, in the region of Andaman and Nicobar Islands (N10° to 14°) for a time period of ∼100 years (1918 to 2018). Variation in b value rates before and after the 2004 great Sumatra-Andaman earthquake (Mw 9.3) was analysed, preceded by a comprehensive geodynamic and seismotectonic- crustal deformation overview along the Andaman trench section. The analysis suggest a precursory drop in b values from 1.05 ± 0.05 (1918 to 2004) to 0.90 ± 0.05 post 2004 earthquake (2004 to 2018) reflecting a greater slip deficit and strain localization, that exceeds considerably the pattern of energy release in this region. Further, the region of Andaman and Nicobar Islands have witnessed a poor slip distribution (on-fault reactivation) from the great earthquakes that nucleated near the Sumatra segment, and have ruptured few large earthquakes, with no accounts of great earthquake (>Mw 8) during the last century. This emphasizes that, strain accumulation has been concentrated over a long period of time, and can contribute a destructive deformation event in near future.

Study of Temporal Analysis onEarthquake Data and Statistical Seismology of Nepal

An earthquake is the greatest terrifying natural disaster that can do significant damages. A 7.8 magnitude earthquake struck Barpak, which is located about 81 kilometers northwest of Kathmandu killing thousands and affecting millions of citizens on 25th April 2015. This is the most powerful disaster to strike Nepal since an earthquake in 1934. The aim of this paper is to study temporal analysis on earthquake data of Nepal and to provide some new analytical insights on the importance of statistical seismology. The data of aftershock occurrence followed by the most catastrophic natural disaster were studied using earthquake catalogue produced by the National seismological center Nepal and the United States geological survey. The catalog covers the time span from 1994 to 2015 and includes about 821 earthquakes having magnitude (ML) four or greater. Descriptive statistics are used to describe the basic features of the earthquake data for 22 years using Excel. The frequency-magnitude distribution of earthquakes has been analyzed in view of its application in seismic pattern study. The relationship between a number of events, time, magnitude, depth, latitude and longitude were studied and interpreted. People want to know the prediction of an earthquake but neither physical nor statistical models are sufficient for predictions and other analyses of a seismic pattern. The development of effective procedures and models for this issue may well require a fusion of statistical and physical ideas in ways, which differ radically from standard practice in either field.

Seismic response to evolving injection at the Rotokawa geothermal field, New Zealand

Catalogs of microseismicity are routinely compiled at geothermal reservoirs and provide valuable insights into reservoir structure and fluid movement. Hypocentral locations are typically used to infer the orientations of structures and constrain the extent of the permeable reservoir. However, frequency-magnitude distributions may contain additional, and underused, information about the distribution of pressure. Here, we present a four-year catalog of seismicity for the Rotokawa geothermal field in the central Taupō Volcanic Zone, New Zealand starting two years after the commissioning of the 140 MWe Nga Awa Purua power station. Using waveform-correlation-based signal detection we double the size of the previous earthquake catalog, refine the location and orientation of two reservoir faults and identify a new structure. We find the rate of seismicity to be insensitive to major changes in injection strategy during the study period, including the injectivity decline and shift of injection away from the dominant injector, RK24. We also map the spatial distribution of the earthquake frequency-magnitude distribution, or b-value, and show that it increases from ∼1.0 to ∼1.5 with increasing depth below the reservoir. As has been proposed at other reservoirs, we infer that these spatial variations reflect the distribution of pressure in the reservoir, where areas of high b-value correspond to areas of high pore-fluid pressure and a broad distribution of activated fractures. This analysis is not routinely conducted by geothermal operators but shows promise for using earthquake b-value as an additional tool for reservoir monitoring and management.

Generalized earthquake frequency–magnitude distribution described by asymmetric Laplace mixture modelling

SUMMARYThe complete part of the earthquake frequency–magnitude distribution, above the completeness magnitude mc, is well described by the Gutenberg–Richter law. On the other hand, incomplete data does not follow any specific law, since the shape of the frequency–magnitude distribution below max(mc) is function of mc heterogeneities that depend on the seismic network spatiotemporal configuration. This paper attempts to solve this problem by presenting an asymmetric Laplace mixture model, defined as the weighted sum of Laplace (or double exponential) distribution components of constant mc, where the inverse scale parameter of the exponential function is the detection parameter κ below mc, and the Gutenberg–Richter β-value above mc. Using a variant of the Expectation-Maximization algorithm, the mixture model confirms the ontology proposed by Mignan [2012, https://doi.org/10.1029/2012JB009347], which states that the shape of the earthquake frequency–magnitude distribution shifts from angular (in log-linear space) in a homogeneous space–time volume of constant mc to rounded in a heterogeneous volume corresponding to the union of smaller homogeneous volumes. The performance of the proposed mixture model is analysed, with encouraging results obtained in simulations and in eight real earthquake catalogues that represent different seismic network spatial configurations. We find that k = κ/ln(10) ≈ 3 in most earthquake catalogues (compared to b = β/ln(10) ≈ 1), suggesting a common detection capability of different seismic networks. Although simpler algorithms may be preferred on pragmatic grounds to estimate mc and the b-value, other methods so far fail to model the angular distributions observed in homogeneous space-time volumes. Mixture modelling is a promising strategy to model the full earthquake magnitude range, hence potentially increasing seismicity data availability tenfold, since ca. 90 per cent of earthquake catalogue events are below max(mc).

The correlation of b-value in the earthquake frequency-magnitude distribution, heat flow and gravity data in the Sivas Basin, central eastern Turkey

In this study, the seismicity in the Sivas Basin, central eastern Turkey from 1903 to 2018 is investigated by the Gutenberg and Richter relationship (1944) and the maximum likelihood method (Aki, 1965) to explore the b-value. The b-value is regarded as one of the important parameters representing the nature of the occurrence of earthquakes. Particularly, the b-value characterizes the state of stress in the crust. The Sivas Basin exhibits the low b-value (0.52 and 0.9), moderate/high heat flow values (70-80 mWm-2), large negative anomalies owing to sedimentary basin and low seismicity and the epicenter distributions of earthquakes are located at the upper crust and along the Deliler-Tecer and Yukari Kizilirmak Fault Zones in the study region.

Regional variations and earthquake frequency–magnitude distribution and fractal dimension in the North of Central-East Iran Blocks (NCEIB)

The Gutenberg–Richter parameters (a and b), fractal dimension (D C ), and relationships between these parameters are calculated for different regions of the North of Central-East Iran Blocks (NCEIB). The whole examined area (between 34°–36° N and 55°–61° E) is divided into 55 equal square grids. Both the a and b values for the frequency–magnitude distribution (FMD) and D C are investigated simultaneously from 55 equal square grids. By using the completeness earthquake dataset for earthquakes of the instrumental period from 1976 to 2015, it is concluded that calculated values of a, b, and D C imply variations of seismotectonic stress. The most vulnerable regions for occurrence of the large earthquakes in the NCEIB are considering the computed lowest b values and the highest D C values. The relationships among D C –b and D C –(a/b) are used to classify the level of earthquake hazards for individual seismic source zones, in which the calibration curves illustrate a positive correlation among the D C and b values (Dc = 0.609 b − 0.008) and also a positive correlation among the D C and a/b ratios (Dc = 0.069 a/b + 0.215) having similar regression coefficients (R2 = 0.80 to 0.87) for both regressions. It is observed that the relationship among a/b and D C may be used for evaluation of seismicity and earthquake hazard assessment because of the high value for correlation coefficients and limited scattering of the calculated parameters.

Spatial variation in the frequency-magnitude distribution of earthquakes under the tectonic framework in the Middle East

Spatial variations of seismic energy released and b-value over the Middle East region are investigated based on a seismicity catalog from 1995 to 2007. The goal is to use these seismic parameters and based on other geodetic and geophysical observations, such as GPS measurements, strain rate model, fault distribution, focal mechanism, crustal model, Q model, and gravity measurements, etc., to uncover spatial patterns that seem anomalous. Areas of high energy released (cumulative) seem to correspond to the areas of relatively high b-values. Areas of high energy released and high b-values seem to correspond very well with the location of continental collision where earthquake activities are high. The divergent boundary between Arabia and African plates and subduction zone at Makran seem to correspond to low to moderate energy release.Northern Pamir, Azerbaijan-Caucasus, the lower part of Zagros Mountains, eastern Turkey, Owen Fracture Zone, Strait of Bob-el-Mandeb, and south of the Sulaiman Shear Zone seem to correspond to high cumulative energy-released, high strain rate, high b-values, and high fault density. While, the central and eastern Iran, southern Zagros, northern Pakistan, Gulf of Aden, Alborz, southwest of the Caspian Sea, western Caucasus and Kopeh-Dagh seem to correspond with lower b-values.The cross-section map for Hindu-Kush shows general decreasing of the b-values with depth, however, a region of high b-value is observed underneath Pamir at depths from 170 to 230km. This anomaly region can be due to dehydration of Pamir crustal slab at depth.

Non-extensive statistical physics applied to heat flow and the earthquake frequency–magnitude distribution in Greece

This study investigates seismicity in Greece and its relation to heat flow, based on the science of complex systems. Greece is characterised by a complex tectonic setting, which is represented mainly by active subduction, lithospheric extension and volcanism. The non-extensive statistical physics formalism is a generalisation of Boltzmann–Gibbs statistical physics and has been successfully used for the analysis of a variety of complex systems, where fractality and long-range interactions are important. Consequently, in this study, the frequency–magnitude distribution analysis was performed in a non-extensive statistical physics context, and the non-extensive parameter, qM, which is related to the frequency–magnitude distribution, was used as an index of the physical state of the studied area. Examination of the spatial distribution of qM revealed its relation to the spatial distribution of seismicity during the period 1976–2009. For focal depths ≤40 km, we observe that strong earthquakes coincide with high qM values. In addition, heat flow anomalies in Greece are known to be strongly related to crustal thickness; a thin crust and significant heat flow anomalies characterise the central Aegean region. Moreover, the data studied indicate that high heat flow is consistent with the absence of strong events and consequently with low qM values (high b-values) in the central Aegean region and around the volcanic arc. However, the eastern part of the volcanic arc exhibits strong earthquakes and high qM values whereas low qM values are found along the North Aegean Trough and southwest of Crete, despite the fact that strong events are present during the period 1976–2009 in both areas.

The Spatial Scale of Detected Seismicity

An experimental method for the spatial resolution analysis of the earthquake frequency-magnitude distribution is introduced in order to identify the intrinsic spatial scale of the detected seismicity phenomenon. We consider the unbounded magnitude range m ∈ (−∞, +∞), which includes incomplete data below the completeness magnitude m c. By analyzing a relocated earthquake catalog of Taiwan, we find that the detected seismicity phenomenon is scale-variant for m ∈ (−∞, +∞) with its spatial grain a function of the configuration of the seismic network, while seismicity is known to be scale invariant for m ∈ [m c, +∞). Correction for data incompleteness for m < m c based on the knowledge of the spatial scale of the process allows extending the analysis of the Gutenberg–Richter law and of the fractal dimension to lower magnitudes. This shall allow verifying the continuity of universality of these parameters over a wider magnitude range. Our results also suggest that the commonly accepted Gaussian model of earthquake detection might be an artifact of observation.

CU-PSHA: A MATLAB Software for Probabilistic Seismic Hazard Analysis

In this study, an open source MATLAB software, called CU-PSHA, is developed in order to analyze probabilistic earthquake hazards. This software aims to provide a user friendly and flexible tool for evaluating reliable earthquake hazard estimates. With the CU-PSHA, the probability of distances between the earthquake sources and the study site can be estimated. Two choices for the estimation of earthquake frequency–magnitude distribution, the exponential magnitude distribution and the characteristic earthquake models, are provided. Some strong ground–motion attenuation models are available for both shallow crustal and subduction zone earthquakes. The probability of exceedance of any individual given ground shaking value can be obtained, allowing the display of a seismic hazard curve. In addition with the supplementary MATLAB scripts, this CU-PSHA software can be employed in general seismic hazard mapping, for both ground shaking level and probability of occurrence, in any specific given time span.

Earthquake frequency-magnitude distribution and fractal dimension in mainland Southeast Asia

The 2004 Sumatra and 2011 Tohoku earthquakes highlighted the need for a more accurate understanding of earthquake characteristics in both regions. In this study, both the a and b values of the frequency-magnitude distribution (FMD) and the fractal dimension (D C ) were investigated simultaneously from 13 seismic source zones recognized in mainland Southeast Asia (MLSEA). By using the completeness earthquake dataset, the calculated values of b and D C were found to imply variations in seismotectonic stress. The relationships of D C -b and D C -(a/b) were investigated to categorize the level of earthquake hazards of individual seismic source zones, where the calibration curves illustrate a negative correlation between the D C and b values (D c = 2.80 - 1.22b) and a positive correlation between the D C and a/b ratios (D c = 0.27(a/b) - 0.01) with similar regression coefficients (R2 = 0.65 to 0.68) for both regressions. According to the obtained relationships, the Hsenwi-Nanting and Red River fault zones revealed low-stress accumulations. Conversely, the Sumatra-Andaman interplate and intraslab, the Andaman Basin, and the Sumatra fault zone were defined as high-tectonic stress regions that may pose risks of generating large earthquakes in the future.

Continuous-cyclic variations in the b-value of the earthquake frequency-magnitude distribution

Seismicity of the Earth (M ≥ 4.5) was compiled from NEIC, IRIS and ISC catalogues and used to compute b-value based on various time windows. It is found that continuous cyclic b-variations occur on both long and short time scales, the latter being of much higher value and sometimes in excess of 0.7 of the absolute b-value. These variations occur not only yearly or monthly, but also daily. Before the occurrence of large earthquakes, b-values start increasing with variable gradients that are affected by foreshocks. In some cases, the gradient is reduced to zero or to a negative value a few days before the earthquake occurrence. In general, calculated b-values attain maxima 1 day before large earthquakes and minima soon after their occurrence. Both linear regression and maximum likelihood methods give correlatable, but variable results. It is found that an expanding time window technique from a fixed starting point is more effective in the study of b-variations. The calculated b-variations for the whole Earth, its hemispheres, quadrants and the epicentral regions of some large earthquakes are of both local and regional character, which may indicate that in such cases, the geodynamic processes acting within a certain region have a much regional effect within the Earth. The b-variations have long been known to vary with a number of local and regional factors including tectonic stresses. The results reported here indicate that geotectonic stress remains the most significant factor that controls b-variations. It is found that for earthquakes with Mw ≥ 7, an increase of about 0.20 in the b-value implies a stress increase that will result in an earthquake with a magnitude one unit higher.

Multiscale Mapping of Completeness Magnitude of Earthquake Catalogs

We propose a multiscale method to map the spatial variations of the com- pleteness magnitude Mc of earthquake catalogs. The Gutenberg-Richter law describ- ing the earthquake frequency-magnitude distribution (FMD) might not hold over the entire magnitude range, and small areas may exhibit a specific type of seismicity, especially in volcanotectonic contexts. For these reasons, any scaling relation should be obtained by adapting the dimension of the studied zone to the range of the event magnitude. Here, we associate ranges of larger magnitudes with increasing areas for data selection based on empirical relations in seismotectonics. Then, for each point in space, we document the earthquake FMD at all length scales within the corresponding earthquake magnitude ranges. High resolution of the Mc-value is achieved through the determination of the smallest space-magnitude scale in which the Gutenberg-Richter law is verified. The multiscale procedure isolates the magnitude range that meets the best local seismicity and local record capacity. Using artificial catalogs and earthquake catalogs of the Lesser Antilles arc, this Mc-mapping method is shown to be efficient in regions with mixed types of seismicity, a variable density of epicenters, and various levels of registration.

Characteristics of seismic activity in the Western, Central and Eastern parts of the North Anatolian Fault Zone, Turkey: Temporal and spatial analysis

Characteristics of seismic activity along the North Anatolian Fault Zone are analyzed between 1970 and 2010. Magnitude completeness changes between 2.7 and 2.9 in the North Anatolian Fault Zone. The frequency-magnitude distribution of earthquakes is well represented with a b-value typically close to 1. A clear decrease in temporal distribution of b-value is observed before the strong main shocks. Correlation dimension values are relatively large and the seismic activity is more clustered at larger scales in the North Anatolian Fault Zone.

Mesh dependence and slip complexity in earthquake fault models

AbstractWe study the origins of mesh dependence in earthquake dynamics and show that mesh sensitivity in numerical models of earthquake rupture can unearth new, exciting physical phenomena, and provide hidden clues to discovering the physics underlying earthquake complexity. We show (in agreement with previous numerical studies of dynamic rupture at faults) that coarse meshes or discrete models produce more interesting physics because these numerical simulations produce richer spatio‐temporal complexity at faults over multiple earthquake cycles. These discrete models are desirable as they reproduce the observed Gutenberg–Richter power‐law frequency magnitude distribution of earthquakes more accurately. However, this complexity is lost as the mesh size is refined, which is undersirable from a physical point of view. We investigate this mesh sensitivity by analysing the higher‐order perturbative terms introduced into numerical models with coarse meshes. The introduction of these higher‐order, nonlinear terms into standard continuum models generally used to describe earthquake dynamics may provide a key to reproducing earthquake complexity while partially removing the associated mesh dependence without the need to modify the underlying mesh. This is a first step towards deriving a physical law capable of reproducing the scale‐invariant behaviour in earthquake sizes and reconciling continuum and discrete models of earthquake rupture. Copyright © 2009 John Wiley &amp; Sons, Ltd.