Frequency-magnitude Distribution Research Articles

The estimation of b-value of the frequency–magnitude distribution and of its 1σ intervals from binned magnitude data

SUMMARY The estimation of the slope (b-value) of the frequency–magnitude distribution of earthquakes is based on a formula derived by Aki decades ago, assuming a continuous exponential distribution. However, as the magnitude is usually provided with a limited resolution, its distribution is not continuous but discrete. In the literature, this problem was initially solved by an empirical correction (due to Utsu) to the minimum magnitude, and later by providing an exact formula such as that by Tinti and Mulargia, based on the geometric distribution theory. A recent paper by van der Elst showed that the b-value can be estimated also by considering the magnitude differences (which are proven to follow an exponential discrete Laplace distribution) and that in this case the estimator is more resilient to the incompleteness of the magnitude data set. In this work, we provide the complete theoretical formulation including (i) the derivation of the means and standard deviations of the discrete exponential and Laplace distributions; (ii) the estimators of the decay parameter of the discrete exponential and trimmed Laplace distributions and (iii) the corresponding formulas for the parameter b. We deduce (iv) the standard 1σ intervals for the estimated b. Moreover, we are able (v) to quantify the error associated with the Utsu minimum-magnitude correction. Furthermore, we have discussed the formulas to produce statistically independent magnitude differences. We tested extensively the b-value estimators on simulated synthetic data sets including complete catalogues as well as catalogues affected by a strong incompleteness degree such as aftershock sequences where the incompleteness is made to vary from one event to the next. We have also analysed the real aftershock sequence of the 30/10/2016 Norcia (central Italy) to integrate the finding of the simulations. To judge the performance of the various estimators we have introduced an index p that can be seen as a non-parametric extension of the Student's t index. The main outcomes of this paper are that (1) the b-value estimators devised for continuous magnitude data are not adequate for binned magnitudes, (2) for complete data sets, estimators based on magnitudes and on magnitude differences provide substantially equivalent results, (3) for incomplete magnitude data sets, estimators based on magnitude differences provide better results and (4) for incomplete aftershock sequences there is no evidence that methods based on positive magnitude differences are superior than other methods using differences. This conclusion is further confirmed by our analysis of the above-mentioned Norcia seismic sequence. This last finding contrasts with the van der Elst's claim that the so called ${{b}_ + }$ method is the most adequate to treat real aftershock sequences.

Probabilistic estimation of the source component of seismic hazard in North-Eastern Brazil

Stable continental regions pose unique challenges for conducting Probabilistic Seismic Hazard Analysis because the earthquake activity driving mechanisms are poorly understood. For instance, the lower seismicity (hence the paucity of data) and the absence of well-defined active fault systems complicate accurately determining seismic source parameters. Northeastern Brazil is a stable continental region exhibiting moderate-size events recorded with significant seismic intensities and provoking the collapse of poorly constructed buildings in the last century. Thus, assessing the seismic hazard is critical for seismic risk mitigation. The seismic hazard depends on three components: source, path, and site, and here, we present the probabilistic seismic hazard analysis of the source component for NE Brazil. Spatial aggregation of earthquake sources outlined four areal seismic zones. A goodness-of-fit test rejected the Gutenberg-Richter model of magnitude frequency distribution in one of the studied seismic zones. For this reason, we estimated the magnitude probability distribution function in that zone using a nonparametric adaptive kernel estimator. In other zones the Gutenberg-Richter magnitude frequency model was applied. In either way of the magnitude probability distribution modelling we considered the upper bound for magnitude equal to 6.6 mR, based on the upper bound of a 95 % confidence interval for the standard normal distribution of palaeoearthquake sizes. Our findings suggests that potentially damaging events are likely to occur, and we cannot neglect chances for the occurrence of earthquakes exceeding 5.2 mR. The calculated mean return periods indicate significantly shorter intervals between consecutive large events than palaeoseismic records.

Carbon Loss from Earthquake Induced Landslides in Fiordland

Landslides triggered by two earthquakes in Fiordland, New Zealand, were mapped using Google Earth Pro retro satellite imagery (eye altitude of 1.5-2.5 km, enabling ± 50 m precision), then catalogued and classified in a GIS inventory. Ground acceleration during the 2003 Mw 7.2 Secretary Island Earthquake resulted in at least 1852 landslides, but the larger Mw 7.8 2009 Dusky Sound Earthquake had lower accelerations and produced only 313. Both events dislodged large swaths of native forest, grassland vegetation and soil from Fiordland’s steep slopes in similar proportions, some landing in rivers and fiords. The landslide maps and magnitude-frequency distributions show close similarity to a published Global Forest Loss dataset derived from 2001-2022 satellite imagery. Assuming forest loss in this wilderness is predominantly landslide-related, it enables the more-precise earthquake-induced landslide mapping to be placed in a longer-term context of other vegetation loss, mostly rainfall-induced landslides, during the past two decades. The total area of forest loss during the 2003 earthquake was anomalous, whereas during 2009 the earthquake losses were similar to areas of forest loss assumed to be rainfall-induced landslides throughout Fiordland each year. Carbon concentrations of landslide vegetation and soil were calculated by defining vegetation types from a published Land Cover Database, and soil organic carbon concentration from a 1 km resolution raster dataset. Total carbon loss from earthquake-induced landslides amounts to 2.05E+06 tonnes for the 2003 Mw7.2 and 2.17E+05 tonnes for the 2009 Mw 7.8. By way of comparison, New Zealand’s total annual carbon sequestration was 6.3E+06 tonnes in 2020. Earthquake-induced and rainfall-induced landslides are shown to account for significant changes in stored carbon and available carbon sequestration, especially in areas of densely vegetated land, such as Fiordland. Processes of landscape disturbance should be included in carbon accounting and estimates of national greenhouse gas emissions.
 
 Supervised by: Caroline Orchiston
 Scholarship Project Funded by: Centre for Sustainability

Relation between earthquake swarm activity and tides in the Noto region, Japan

The Noto region in Japan has been experiencing earthquake swarm activity since mid-2018, with repeated rises and falls in activity. Crustal deformation (expansion and uplift) observed there since the end of 2020 has been connected to crustal fluids. Studies in other regions have suggested that tides are related to earthquake swarm activities associated with crustal fluids. Therefore, we investigated whether tides are also involved in the Noto earthquake swarm activity. Our results suggest a tidal correlation only at greater depths in the southern part of the analyzed area (‘region Sd’). There, we inferred that an increase in pore fluid pressure caused by the inflow of deep fluids may have led to a decrease in fault fracture strength, making the local seismicity relatively susceptible to the effects of tidal forces. The significantly high value in region Sd of the scaling parameter b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b$$\\end{document} of the Gutenberg–Richter law (describing the earthquake magnitude–frequency distribution) and observed crustal deformation are consistent with this interpretation.Graphical

18-years of high-Alpine rock wall monitoring using terrestrial laser scanning at the Tour Ronde east face, Mont-Blanc massif

Since the end of the 20th century, each decade has been warmer than the previous one in the European Alps. As a consequence, Alpine rock walls are generally facing high rockfall activity, likely due to permafrost degradation. We use a unique terrestrial laser scanning derived rockfall catalog over 18 years (2005–2022) compared with photographs (1859–2022) to quantify the evolution of the east face of Tour Ronde (3440–3792 m a.s.l.) in the Mont-Blanc massif (western European Alps) that is permafrost-affected. Overall, 210 rockfalls were identified, from 1 to 15 500 m3. Forty-five events were >100 m3 while cumulated volume of events <10 m3 represents <1% of the fallen rocks. The rockfall magnitude-frequency distribution of the overall inventory follows a power law, with a mean exponent b of 0.44 ± 0.03, characterizing a high contribution of large rockfalls. The depth of failure ranges from a few centimeters to more than 20 m while 95% of the rockfalls depth is <5 m, highlighting the role of the active layer. The mean rock wall erosion rate is 18.3 ± 0.2 mm yr−1 for the 2005–2022 period and ranks in the top range of reported values in the Alps. It has greatly increased between the periods 2006–2014 and 2016–2022, probably in relation to a series of summer heat waves. The exceptional erosion rate of 2015 is driven by one large rockfall in August. Since 2006, an ice apron that covered 16 100 m2 has now almost vanished, and the surface of the glacier du Géant at the rock wall foot has lowered by several tens of meters. The retreat of these two ice masses contributed to the rock wall instability as more than 35% of the rockfall volume detached from the deglaciated surfaces.

A statistical framework for detection of b-value anomalies in Italy

SUMMARY This study presents a new robust statistical framework, in which to measure relative differences, or deviations from a hypothetical reference value, of Gutenberg–Richter b-value. Moreover, it applies this method to recent seismicity in Italy, to find possible changes of earthquake magnitude distribution in time and space. The method uses bootstrap techniques, which have no prior assumptions about the distribution of data, keeping their basic features. Excluding Central Italy, no significative b-value variation is found, revealing that the frequency–magnitude distribution exponent is substantially stable or that data are not able to reveal hidden variations. Considering the small size of examined magnitude samples, we cannot definitively decide if the higher b-values in Central Italy, consistently founded by all applied tests, have a physical origin or result from a statistical bias. In any case, they indicate short-lived excursions which have a temporary nature and, therefore, cannot be associated solely to spatial variations in tectonic framework. Both the methodological issues and the results of the application to seismicity in Italy show that a correct assessing of b-value changes requests appropriate statistics, that accurately quantify the low accuracy and precision of b-value estimation for small magnitude samples.

On the b-value dependency of injection-induced seismicity on geomechanical parameters

Variability in the b-value, which describes the frequency distribution of earthquake magnitudes, is usually attributed to variations in differential stress in the setting of natural and laboratory earthquakes. However, differential stress is unlikely to explain b-value variations on the reservoir scale of injection-induced seismicity cases where significant differential stress variability can hardly be expected. We investigate the responsible geomechanical parameters for the b-value reduction observed in injection-induced seismicity at the Basel EGS field in Switzerland, for which a measured in-situ stress model and fault orientations of numerous microseismic events are available. We estimate the shear and normal stresses along faults, differential stress, pore pressure increase at failure, and the Coulomb failure stress, for each event. Event magnitude and shear stress display the most systematic and clear correlation between each other, while other parameters do not show a clear correlation with event magnitude. We further examine the relationship between the b-value and these geomechanical parameters. We discover that the b-value systematically decreases with increasing shear stress. Again, other geomechanical parameters do not show a clear correlation with the b-value. We conclude that b-value variability is explained by variations in shear stress in the injection-induced seismicity setting, where near constant differential stress conditions are expected. Furthermore, we observe that b-value reduction with time also correlates with an increasing number of events along faults having high shear stress, which strongly supports our conclusions. Thus, we discovered a profound physical mechanism behind b-value variation in injection-induced seismicity beyond general understanding of b-value variation.

Statistical Analysis of Mt. Vesuvius Earthquakes Highlights Pitfalls in Magnitude Estimation

Here, we characterize the statistical behaviour of the Mt. Vesuvius seismicity using distinct available catalogues. Our analysis confirms that for this area, the GR distribution exhibited two scaling regimes of the b-value, not commonly observed for the standard frequency-magnitude distribution of earthquakes. By assuming a physical cause, we tested four different hypotheses for the source of the break in the scaling: finite size effect, depth variations in the b-value, radial dependence in the b-value, and different b-values for swarm and non-swarm events. None of the above reasons are able to explain the observation. Thus, we investigated the possibility of some pitfalls in magnitude estimation. Based on our analysis, we suggest there is a bias in the duration magnitude the catalogues are based on. This is due to the arbitrary extrapolation to smaller magnitudes of a linear regression derived for earthquakes with m≥3.0. When a suitable correction is applied to the estimated magnitude, the GR distribution assumes the usual shape, with a b-value closer to that usually observed in volcanic areas. Finally, the analysis of the time variation of some statistical parameters reveals that the state of the volcano appears to be stationary over the entire analysed period, possibly with only a slight decrease in the b-value, indicating a small reduction in differential stress.

New Estimates of Magnitude‐Frequency Distribution and b‐Value Using Relative Magnitudes for the 2011 Prague, Oklahoma Earthquake Sequence

AbstractThe magnitude‐frequency distribution (MFD) describes the relative proportion of earthquake magnitudes and provides vital information for seismic hazard assessment. The b‐value, derived from the MFD, is commonly used to estimate the probability that a future earthquake will exceed a specified magnitude threshold. Improved MFD and b‐value estimates are of great importance in the central and eastern United States where high volumes of fluid injection have contributed to a significant rise in seismicity over the last decade. In this study, we recalculate the magnitudes of 8,775 events for the 2011 Prague, Oklahoma sequence using a relative magnitude approach that depends only on waveform data to calculate magnitudes. We also compare the distribution of successive magnitude differences to the MFD and show that a combination of the magnitude difference distribution (MDFD) and relative magnitudes yields a reliable estimate of b‐value. Using the MDFD and relative magnitudes, we examine the temporal and spatial variations in the b‐value and show that b‐value ranges between ∼0.6 and 0.85 during the aftershock sequence for at least 5 months after the M 5.7 mainshock, though areas surrounding the northeast part of the sequence experience higher b‐values (0.7–0.85) than the southwestern part of the Meeker‐Prague fault where b‐value is the lowest (0.6–0.7). We also identify a cluster of off‐fault events with the highest b‐values in the catalog (0.85). These new estimates of MFD and b‐value will contribute to understanding of the relations between induced and tectonic earthquake sequences and promote discussion regarding the use of b‐value in induced seismic hazard estimation.

How Injection History Can Affect Hydraulic Fracturing–Induced Seismicity: Insights from Downhole Monitoring at Preston New Road, United Kingdom

ABSTRACT In August 2019, a multistage hydraulic fracturing (HF) operation was carried out at Preston New Road, United Kingdom. HF caused abundant seismic activity that culminated with an ML 2.9 event. The seismic activity was recorded by a downhole array of 12 sensors located in a nearby monitoring well. About 55,556 events were detected and located in real time during the operation by a service company. In this study, we first improve the number of detections by applying template matching and later calculate the moment magnitude of the associated earthquakes. Then we show that by separately analyzing the periods during and immediately after injection, distinct patterns can be identified. We observe an increase in the delay and decrease in amplitude of peak seismicity during subsequent phases of injection. After injection, the seismicity decay can be described by the Omori–Utsu law. The decay rate tends to slow with each successive injection, in particular during the later injection stages. In addition, the frequency–magnitude distribution evolves from a tapered distribution (lack of large events) to a bilinear distribution (excess of large events). This evolution is gradual, with the corner magnitude increasing with each injection. We interpret these patterns as the result of the combined effect of two factors: (1) the stimulated volume becoming increasingly aseismic and (2) the gradual increase in its size, which increases the probability of triggered events on preexisting faults. More generally, these patterns suggest that seismic activity during injection is strongly influenced by the injection history and is modulated by local conditions such as stress state, fault structure, and permeability.

The September 26, 2019 Silivri Earthquake (Mw=5.6), NW Türkiye

The September 26, 2019 Silivri earthquake (MW=5.6-5.8) occurred along the North Anatolian Fault Zone segments extending beneath the Marmara Sea. In the present study the teleseismic P waveforms and 20-year long background seismicity of the earthquake (MW=5.6-5.8) have been analyzed. Point-source inversion of the teleseismic P waveforms revealed that the earthquake was due to oblique faulting and released a seismic moment of 3.2 x 1017 Nm (MW=5.6). The frequency-magnitude distributions (FMDs) for the background seismicity have been calculated for 5-year long time windows after the 1999 İzmit earthquake. The considerable decrease of b-value of the FMDs just before the 2019 Silivri earthquake has been interpreted as stress increase along the fault segments which provides a reasonable clue for the occurrence of the earthquake. The FMD distribution for the 5 year-long time windows before the 2019 Silivri earthquake suggests a recurrence time interval of 168 years and occurrence probability of %16 within the next 30 years for a Mw=7.5 earthquake.

A note on two alternative probabilistic methods to address parametric uncertainty in magnitude frequency distribution (MFD) logic trees (BEEE-D-22-00704)

A note on two alternative probabilistic methods to address parametric uncertainty in magnitude frequency distribution (MFD) logic trees (BEEE-D-22-00704)

Spatiotemporal Variations of the Frequency-Magnitude Distribution in the 2019 Mw 7.1 Ridgecrest, California, Earthquake Sequence.

Significant seismic activity has been witnessed in the area of Ridgecrest (Southern California) over the past 40 years, with the largest being the Mw 5.8 event on 20 September 1995. In July 2019, a strong earthquake of Mw 7.1, preceded by a Mw 6.4 foreshock, impacted Ridgecrest. The mainshock triggered thousands of aftershocks that were thoroughly documented along the activated faults. In this study, we analyzed the spatiotemporal variations of the frequency-magnitude distribution in the area of Ridgecrest using the fragment-asperity model derived within the framework of non-extensive statistical physics (NESP), which is well-suited for investigating complex dynamic systems with scale-invariant properties, multi-fractality, and long-range interactions. Analysis was performed for the entire duration, as well as within various time windows during 1981-2022, in order to estimate the qM parameter and to investigate how these variations are related to the dynamic evolution of seismic activity. In addition, we analyzed the spatiotemporal qM value distributions along the activated fault zone during 1981-2019 and during each month after the occurrence of the Mw 7.1 Ridgecrest earthquake. The results indicate a significant increase in the qM parameter when large-magnitude earthquakes occur, suggesting the system's transition in an out-of-equilibrium phase and its preparation for seismic energy release.

Characteristic versus Gutenberg–Richter Nucleation-Based Magnitude–Frequency Distributions in the New Zealand National Seismic Hazard Model 2022

Abstract Probabilistic seismic hazard analysis requires a seismicity rate model, or in other words, a forecast of earthquake rates. In the New Zealand National Seismic Hazard Model 2022, the seismicity rate model is constructed through independent forecasts of earthquakes on mapped faults and earthquakes distributed over cells in a spatial grid. Here, we explore the seismicity rate model for upper plate (hypocenter ≥ 40 km) events, to investigate the shape of magnitude–frequency distributions (MFDs) considering events nucleating (or for which the hypocenters are located) within individual fault zone. We find that more than 80% of the fault zones have MFDs that are better described by a Gutenberg–Richter (GR) distribution, instead of a characteristic distribution (i.e., rates of larger magnitudes much higher than the GR trend). Furthermore, the MFD classifications are neither influenced by time-dependent (and time-independent) considerations nor directly affected by the size (or area) of the fault zones. Fault zones with faster slip rates (&amp;gt;20 mm/yr) exhibit characteristic MFDs, whereas those with slower slip rates may or may not. Although multifault ruptures are prevalent in the characteristic distributions, large maximum magnitude (Mw &amp;gt;8.0) plays a pivotal role producing a characteristic MFD. On the other hand, physically unconnected multifault ruptures (i.e., involving rupture jumps ≥ 10 km) are mostly observed with GR distributions.

Assessment of the seismic hazard in North China by combining micro-seismicity records and geodetic observations

North China has suffered numerous devastating historical and recent earthquakes, causing huge loss of life and property. Therefore, it is vital to identify the regions with the greatest potential to generate large earthquakes in the future. Previous studies mainly evaluated the seismic hazard in North China using earthquake records or geodetic methods separately. Here, we evaluated the seismic records and geodetic observations in combination to comprehensively assess the earthquake potential. We first evaluated the earthquake catalog and determined that the completeness range of magnitude is 1.9 to 2.8 based on the b-values estimated by magnitude-frequency distribution using catalogs with different minimum-magnitude cutoffs. We then obtained the spatial distributions of three strain rate parameters, including the maximum shear, the principal, and second invariant of the strain rates, using interseismic Global Navigation Satellite System (GNSS) observations. The robustness of the strain rate model was validated by comparing previous studies. We then analyzed the seismic hazard in North China based on the strain rates and b-values and determined the regions with relatively high seismic hazard potentials. The results suggested that large strain rates are mainly concentrated on the western border of the Ordos Block and the Tangshan fault zone. Different strain rate parameters yielded similar spatial distributions of recurrence intervals with different magnitudes. We further revealed that the western border of the Ordos Block and Tangshan fault regions exhibit relatively short (∼7000–15,000 year) ML 7.5 earthquake recurrence times, compared with those of other regions in North China. Although the results obtained in this study were limited by the completeness of earthquake records, the spatial distribution of the GNSS observation, the assumption of elastic strain accumulation, and the complexity of the mechanisms of intraplate earthquakes, our findings will contribute to the assessment of seismic hazards in North China.

A Seismogenic Slab Source Model for Aotearoa New Zealand

Abstract Intraslab seismicity within the Hikurangi and Puysegur subduction zones constitutes &amp;gt;50% of recorded (Mw≥4.0 events) earthquakes in Aotearoa New Zealand. Here, we develop a source model for intraslab seismicity using recently augmented datasets including models of subduction interface geometries, an earthquake catalog, and a regional moment tensor catalog. For the areal zones of uniform seismicity, we consider the whole of each slab, as well as demarcations between shallower (depth ≤40 km) and deeper regions. Thereafter, we evaluate the magnitude–frequency distributions in each zone. To compute smoothed seismicity distributions, we apply a novel quasi-3D approach that involve: (1) delineation of midslab surfaces (defined by regions of maximum earthquake density), (2) orthogonal projections of hypocenters onto the midslab profiles, (3) uniform gridding of 0.1° down-dip on the midslab, and (4) application of smoothing kernel on the projected hypocenters. We also develop a model to characterize the focal mechanisms of the intraslab earthquakes using the regional moment tensor catalog. This model has median strike angles subparallel to subduction trenches and median dip angles ≥60° in both the subduction zones. The distribution of rake angles suggests that the Hikurangi slab has an extensional regime in the shallower parts but a compressional regime in the deeper parts, indicative of slab flexure. In contrast, the Puysegur slab predominantly exhibits a compressional regime.

Fractal and Spectral Analysis of Seismicity in the Lai Chau Area (Vietnam)

The time dynamics of the instrumental seismicity recorded in the area of the Lai Chau reservoir (Vietnam) between 2015 and 2021 were analyzed in this study. The Gutenberg–Richter analysis of the frequency–magnitude distribution has revealed that the seismic catalog is complete for events with magnitudes larger or equal to 0.6. The fractal method of the Allan Factor applied to the series of the occurrence times suggests that the seismic series is characterized by time-clustering behavior with rather large degrees of clustering, as indicated by the value of the fractal exponent α≈0.55. The time-clustering of the time distribution of the earthquakes is also confirmed by a global coefficient of variation value of 1.9 for the interevent times. The application of the correlogram-based periodogram, which is a robust method used to estimate the power spectrum of short series, has revealed three main cycles with a significance level of p&lt;0.01 (of 10 months, 1 year, and 2 years) in the monthly variation of the mean water level of the reservoir, and two main periodicities with a significance level of p&lt;0.01 (at 6 months and 2 years) in the monthly number of earthquakes. By decomposing the monthly earthquake counts into intrinsic mode functions (IMFs) using the empirical decomposition method (EMD), we identified two IMFs characterized by cycles of 10 months and 2 years, significant at the 1% level, and one cycle of 1 year, significant at the 5% level. The cycles identified in these two IMFs are consistent with those detected in the water level, showing that, in a rigorously statistical manner, the seismic process occurring in the Lai Chau area might be triggered by the loading–unloading operational cycles of the reservoir.

Fault Source Models Show Slip Rates Measured across the Width of the Entire Fault Zone Best Represent the Observed Seismicity of the Pallatanga–Puna Fault, Ecuador

AbstractWe explore how variation of slip rates in fault source models affect computed earthquake rates of the Pallatanga–Puna fault system in Ecuador. Determining which slip rates best represent fault-zone seismicity is vital for use in probabilistic seismic hazard assessment (PSHA). However, given the variable spatial and temporal scales slip rates are measured over, significantly different rates can be observed along the same fault. The Pallatanga–Puna fault in southern Ecuador exemplifies a fault where different slip rates have been measured using methods spanning different spatial and temporal scales, and in which historical data and paleoseismic studies provide a record of large earthquakes over a relatively long time span. We use fault source models to calculate earthquake rates using different slip rates and geometries for the Pallatanga–Puna fault, and compare the computed magnitude–frequency distributions (MFDs) to earthquake catalog MFDs from the fault zone. We show that slip rates measured across the entire width of the fault zone, either based on geodesy or long-term geomorphic offsets, produce computed MFDs that compare more favorably with the catalog data. Moreover, we show that the computed MFDs fit the earthquake catalog data best when they follow a hybrid-characteristic MFD shape. These results support hypotheses that slip rates derived from a single fault strand of a fault system do not represent seismicity produced by the entire fault zone.

Enhanced b-value time-series calculation method using data-driven approach

SUMMARY The temporal evolution analysis of b-value of the magnitude–frequency distribution (MFD) is essential for seismic risk analysis. However, ensuring the accuracy and rationality of these analyses depends on various factors, including data quality, data selection and the appropriate computation period partitioning. This study extends the data-driven b-value time-series calculation method, TbDD-BIC, by exploring different model selection techniques. To evaluate the method's effectiveness, we conducted assessments using both synthetic earthquake catalogues and actual seismic data. Our results indicate that selecting a proportion of optimal models (e.g. 5 per cent of the total number of models) using the Akaike information criterion (AIC) and computing the ensemble median yields accurate b0-values of the synthetic earthquake catalogue. This proposed method offers objective calculation rules and precise identification of abrupt b-value changes, enhancing seismicity simulation and seismic hazard analysis.

A New Statistical Perspective on Båth’s Law

Abstract The empirical Båth’s law states that the average magnitude difference (ΔM) between a mainshock and its strongest aftershock is ∼1.2, independent of the size of the mainshock. Although this observation can generally be explained by a scaling of aftershock productivity with mainshock magnitude in combination with a Gutenberg–Richter frequency–magnitude distribution, estimates of ΔM may be preferable because they are directly related to the most interesting information, namely the magnitudes of the main events, without relying on assumptions. However, a major challenge in calculating this value is the bias introduced by missing data points when the strongest aftershock is below the observed cut-off magnitude. Ignoring missing values leads to a systematic error because the data points removed are those with particularly large magnitude differences ΔM. The error can be minimized by restricting the statistics to mainshocks that are at least 2 magnitude units above the cutoff, but then the sample size is strongly reduced. This work provides an innovative approach for modeling ΔM by adapting methods for time-to-event data, which often suffer from incomplete observations (censoring). In doing so, we adequately account for unobserved values and estimate a fully parametric distribution of the magnitude differences ΔM for mainshocks in a global earthquake catalog. Our results suggest that magnitude differences are best modeled by the Gompertz distribution and that larger ΔM are expected at increasing depths and higher heat flows.