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Spatial–temporal variations of bvalues prior to mediumtolarge earthquakes in Taiwan and the feasibility of realtime precursor monitoring
Earth, Planets and Space volume 76, Article number: 119 (2024)
Abstract
This study explores the bvalue variations prior to M ≥ 6.0 earthquakes in Taiwan, examining their potential as earthquake precursors. Focusing on the 2018 Hualien earthquake and others between 1999 and 2021, we found that many large earthquakes occurred in areas with low bvalues a year prior, although there were no significant temporal changes near the epicenters. However, for more accurate earthquake precursors, incorporating additional factors is recommended to minimize uncertainty.
Graphical Abstract
1 Introduction
The Gutenberg–Richter law (also known as ‘G–R law’, Gutenberg and Richter 1944) describes the relationship between earthquake magnitude and the number of earthquakes greater than or equal to that magnitude. It can be expressed as:
where M is the earthquake magnitude, N is the number of earthquakes with a magnitude greater than M, and a and b are constants. This relationship has been widely applied in seismological research, e.g., in assessing the frequency of earthquakes of different magnitudes (Wang et al. 2016), spatiotemporal variations of bvalues before some M ≥ 6.0 earthquakes (Chan et al. 2012), the relationship between bvalues and stress states (Wu et al. 2018), and using bvalues as indicators for earthquake precursors (e.g., Gulia and Wiemer 2019; Wang 2021; Wang et al 2015; 2016).
Previous studies (Lay and Wallace 1995) indicated that the global average bvalue is approximately 1.0. However, the bvalue is spatial heterogeneous due to differences in regional geological structures and the earthquake data used. For example, Smith (1981) observed that regions near the epicenter of M ≥ 6.0 earthquakes initially showed an increase in the bvalue, which then returned to normal. Conversely, some studies have shown that bvalues decrease before M ≥ 6.0 earthquakes (e.g., Chan et al. 2012). As a result, the mechanism behind the variations of bvalues in areas adjacent to impending earthquakes remains controversial.
Recent advances in integrated seismic networks have led to an improvement in the quality of earthquake detection by Taiwan’s regional observation network, making the earthquake catalog more comprehensive (Chang et al. 2012; Lai et al. 2016). Furthermore, due to the high seismic activity in Taiwan, there is a wealth of earthquake data available, providing a more detailed basis for precursor research. Previous studies (e.g., Chan et al. 2012; Chen et al. 1990; Lin 2010; Wu and Chiao 2006) investigated the variations in bvalues before large earthquakes up to 2009. However, that research did not include post2010 earthquakes. Moreover, the Central Weather Administration Seismic Network was upgraded in 2012, capable of recording more small earthquakes and providing a more precise earthquake catalog (Chang et al. 2012; Lai et al. 2016), potentially enhancing the quality of precursor research.
Given this context, this study aims to explore the spatiotemporal variations of bvalues before moderate to large earthquakes with hypocentral depth less than 40 km in Taiwan using a more comprehensive and complete dataset. Initially, the 2018 M_{w} 6.3 Hualien earthquake sequence was investigated to understand the spatiotemporal variations of its preceding bvalues and to clarify the feasibility of using bvalues as a precursor index. Based on the experience from the Hualien case, this research further discusses the spatiotemporal variations of bvalues before earthquakes larger than magnitude 6.0 in Taiwan from 1999 to 2021. Lastly, the study examined the March 2022 Yuli earthquake and the September 2022 Chihshang earthquake sequence as case studies to assess the feasibility of realtime monitoring of bvalue earthquake precursors.
2 Bvalue estimation and earthquake catalog
The G–R law illustrates seismic activity in a region. In this study, we obtained bvalues by using the maximum likelihood method (MLE) to regress an earthquake catalog (Aki 1965; Kijko 1988; Utsu 1965, 1999). MLE is a fundamental statistical method used to estimate the parameters of a statistical model given observed data. The MLE method involves maximizing the likelihood function, which represents the conditional probability of observing the given data under specific model parameters, denoted as \(\theta\). Under the G–R law, the formula calculated using the likelihood function is expressed as:
where \(f\left(y\theta \right)\) is the probability density function of the residuals \(y\), defined as the difference between the observed and modeled values (\(y={y}_{i}\widehat{{y}_{i}}\)), given the parameter \(\theta\). Assuming that the residuals are normally distributed, the parameters \(\theta\) are typically the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the distribution.
The goal of MLE is to find the parameter values \(\theta\) that maximize the likelihood function:
For normally distributed residuals, the probability density function is given by:
Taking the natural logarithm of this function simplifies the computation and leads to the loglikelihood function, which is commonly used in MLE due to its computational advantages:
In practical applications, maximizing the loglikelihood function is equivalent to minimizing the negative loglikelihood function. This can be efficiently performed using Python’s ‘scipy.optimize.minimize’ function, which minimizes a scalar function of one or more variables. The loglikelihood function can be minimized as follows:
This minimization is using the limitedmemory BFGS (LBFGSB) algorithm, implemented in the ‘scipy.optimize.minimize’ function. Upon successful minimization, the results yield the most likely a and b value given the residuals (\(y\)).
To validate regression quality, we reported corresponding standard deviation of the regression (Aki 1965) as:
where \(n\) is number of events, \({y}_{i}\) and \({\widehat{y}}_{i}\) are the observed and modeled values for magnitude \({m}_{i}\), \(\overline{m}\) is the averaged magnitude for the catalog.
Investigating the bvalue evolution relies on highquality earthquake records. In Taiwan, the broadcast of earthquake information is officially overseen by the Central Weather Administration (CWA), formerly known as the '’Central Weather Bureau’ before September, 2023. The Central Weather Administration Seismographic Network (CWASN), established in 1991 (Shin 1993), became fully operational by 1993. Before 2012, the CWASN primarily used shortperiod sensors and some strongmotion sensors with 12bit resolution (represented as triangles in Fig. 1c). Since 2012, these sensors have been upgraded to 24bit resolution. To enhance the detection of both strong and subtle seismic activity, a variety of sensors have been deployed, including shortperiod, broadband, and strongmotion types. These sensors were strategically placed at different locations: on the surface, in boreholes, and even on the ocean floor (Hsiao et al. 2014). The network's coverage and density were further enhanced by integrating the ‘Broadband Array in Taiwan for Seismology’ (https://bats.earth.sinica.edu.tw) network, managed by Academia Sinica, and a Japanese station named YOJ (represented as triangles in Fig. 1d). In the CWASN catalog, the magnitude scale of each earthquake is in local magnitude (Shin 1993). We are aware that the local magnitude scale might be saturated for large events. However, most of the events used in our bvalue calculations are smaller than magnitude 6.0 (events with M ≥ 6.0 were considered as target events for the precursory index).
Considering the complexity of Taiwan’s tectonic setting, which includes two subduction systems, we focused on shallow seismic activity in the crust. We selected earthquakes with depths shallower than 40 km, corresponding to the average depth of the isovelocity line of the Pwave velocity of 7.5 km/s from tomography (Huang et al. 2014).
Earthquakes with smaller magnitude occur more frequently (Gutenberg and Richter 1944). However, due to observational quality constraints, small magnitude earthquakes are difficult to be recorded comprehensively. The smallest earthquake magnitude that a network can observe completely is referred to as the minimum complete earthquake magnitude. To ensure the quality of the earthquake catalog, we followed the procedure of the maximum curvature method proposed by Wiemer and Wyss (2000) that assessed the minimum magnitude at which all events are recorded (known as the magnitude of completeness or “M_{c}”) across various observation periods. To determine M_{c}, we counted the number of events in each magnitude bin and identified the magnitude with the highest event count as M_{c}.
To understand the spatial distribution of M_{c}, we divided the study region into grid points with latitude and longitude intervals of 0.1°. We then searched for earthquake data within 0.15° of the center of each grid point, presenting the spatial distribution of M_{c} for the periods 1994–2011 and 2012–2021 (presented in Fig. 1a and b, respectively). During 1994–2011, the M_{c} for inland Taiwan was mostly around 2.0, while it approached 3.0 further away from the island (Fig. 1a). From 2012 to 2021, due to the improvement in seismograph quality and station coverage (Fig. 1d), the M_{c} for inland Taiwan was mostly below 1.5, and around 2.0 off the coast, with only a few areas nearing 3.0 (Fig. 1b). Overall, the M_{c} on inland Taiwan is lower than that off the island, and there was a noticeable decrease in M_{c} after 2012. This indicates that the quality of observations improved after the integration of the seismic network. Based on the above analysis, this study will use the earthquake catalog, considering the completeness over time, space, and magnitude, to calculate the bvalue.
3 Examining the 2018 Hualien earthquake sequence
To understand the spatiotemporal evolution of the bvalue prior to M ≥ 6.0 earthquakes, we first examined the 2018 M_{w} 6.3 Hualien earthquake sequence. The mainshock of this sequence occurred on February 6, 2018, at 15:50 (UTC). Prior to this earthquake, there were seven foreshocks with M_{L} greater than 5.0. The earliest of these foreshocks took place on February 4, 2018, at 13:12 (UTC) with M_{L} 5.3, and the largest foreshock (M_{L} 5.9) occurred on February 4, 2018, at 13:56 (UTC). Considering that this earthquake sequence encompasses foreshocks, the mainshock, and aftershocks, it presents an ideal case for studying the behavior preceding a large earthquake.
This study utilizes the complete part of earthquake catalog for precise bvalue assessment in each analysis. We determined the M_{c} for events in the grid cell, including those with M ≥ M_{c} in the calculation (Fig. 2). The bvalue was reported for cells with over 100 events. From the bvalue map of the year before the Hualien mainshock (Fig. 2a), the bvalue at the epicenter cell of the Hualien earthquake was found to be 0.73, ranking in the lowest 15th percentile in Taiwan. This illustrates a relatively low spatial value prior to the significant seismic event, below the global average of 1.0 as noted by Lay and Wallace (1995).
To investigate the temporal distribution of the bvalue before the Hualien earthquake, we initially calculated the time variation of the bvalue over the 5 years prior (Fig. 3), aiming to understand if the bvalue at the epicenter changed systematically over time. The results revealed a significant drop in the bvalue over the 5 years leading up to the earthquake (from 0.99 to 0.86). The value then stabilized (within the standard deviation range) until 1 year before the earthquake, when the bvalue sharply decreased again (from 0.87 to 0.73). To further validate significance of the temporal variations, we followed Utsu (1999) and implemented ∆AIC, represented as:
where N_{1} and N_{2} are numbers of events in two groups, b_{1} and b_{2} are bvalues in two groups. Typically, a ∆AIC exceeding approximately 2 is viewed as indicating significant variation, and a ∆AIC of 5 or more is considered highly significant, indicating that the two groups have different bvalues. According to the temporal distribution of bvalue, standard deviation, and ∆AIC (Fig. 3), there was a decrease in the bvalue during the 5 years preceding the earthquake, particularly noticeable in the fourth and final years, which exhibited significantly high ∆AICs values of 4.06 and 55.98, respectively. This analysis considered yearly earthquake catalogs, which lacked finer temporal resolution. Consequently, the activity characteristics of the foreshock sequence were not discerned.
To gain insight into the finer temporal variations of the bvalue and its feasibility as an earthquake precursor indicator, we computed the bvalue for every 100 (green lines in Fig. 4) and 200 (black lines in Fig. 4) earthquakes within 30 km of the Hualien earthquake epicenter in the year leading up to the mainshock, shifting by one earthquake each time, and analyzed the continuous time variation of the bvalue (Fig. 4). Although larger bvalue variations were obtained for the calculation using smaller number of events, their temporal trends are similar with those using larger number of events, that earthquakes greater than magnitudes 4.0 (blue stars in Fig. 4) or 5.0 (orange stars in Fig. 4) could lead to rapid drops in the bvalue, even though most of these earthquakes were not followed by larger ones. Approximately 3 days before the mainshock (on February 3rd), a series of intense foreshocks initiated (Fig. 4b). During this foreshock period, the lowest bvalue appeared near the M_{w}5.9 foreshock. For most of the time leading up to the main shock, the bvalue was significantly below 1.0.
In the case of the Hualien earthquake, the bvalue continuously declined over the 5 years preceding the earthquake (Fig. 3), with a further decrease during the foreshock sequence (Fig. 4b). Spatially, the epicenter appeared to consistently exhibit a low bvalue before the earthquake (Fig. 2a). This case suggests that earthquake sequences with foreshocks, or those showing temporal and spatial characteristics of low bvalues, may act as earthquake precursors. To determine whether this hypothesis applies to most large earthquakes, we applied this procedure to other earthquakes with magnitudes greater than M_{w} 6.0, as detailed below.
4 Spatiotemporal variation of bvalues prior to M ≥ 6.0 earthquakes in Taiwan from 1999 to 2021
To summarize the spatiotemporal characteristics of the bvalues in preseismic periods of large earthquakes, this study considers earthquakes in Taiwan with magnitudes greater than 6.0 that occurred between 1999 and 2021. In this period, Taiwan recorded 62 earthquakes exceeding magnitude 6.0. During this timeframe, if an earthquake of magnitude greater than 6.0 happened within 1 month and 30 km of a prior one, it was classified as an aftershock of the earlier quake and omitted from the analysis. Consequently, after removing these aftershocks, 46 earthquakes were subject to analysis (Table 1 and Fig. 5a). To enhance the reliability of bvalue calculations, we have chosen to present only 31 events (highlighted in italic in Table 1) that are located in cells with an event count exceeding 100.
The average bvalue at the epicenters of these earthquakes showed a slight decreasing trend over the 5 years leading up to each earthquake (Fig. 5b). The bvalue 1 year before the earthquakes was the lowest at 0.82, a difference of 0.09 from the bvalue 5 years prior. However, when considering the standard deviation (approximately ± 0.2), this decreasing trend in bvalues is not statistically significant.
Although the temporal changes in the bvalues at the epicenters over 5 years were not particularly noticeable, spatial variations were more significant. The average bvalue of 0.82, obtained 1 year prior to the earthquakes, is lower than the global average of 1.0, as documented by Lay and Wallace in 1995. To further illustrate the distribution characteristics of bvalues at the earthquake epicenters and across other regions in Taiwan in the year leading up to each quake, we employed the Molchan diagram (Molchan 1990). This was done to statistically investigate the significance of low bvalues in the epicentral regions. In the Molchan diagram (Fig. 6), the horizontalaxis represents the alert rate (fraction of alarmoccupied space), i.e., the bvalue percentiles of the grid locations of each earthquake. The verticalaxis shows the success rate (fraction of forecasting), which is the ratio of the number of successfully predicted and actually occurring earthquakes to the total number of actual earthquakes. By marking different forecasting results on this diagram, we can compare the efficacy of various forecasting methods. The diagram is ordered according to the percentiles of the epicenter bvalues across Taiwan, from lowest to highest. Ideally, a point in the upper left side of the diagram indicates a high success rate at a low alert rate. Conversely, a random or nonpredictive method would have data points along the diagonal. The Molchan diagram offers a clear and intuitive way to compare the applicability of different forecasting models.
To take the case of the 2018 M_{w} 6.2 Hualien earthquake as an example, the bvalue in the epicentral cell was 0.73, placing it in the lowest 14.8 percentile across Taiwan (Fig. 2a). Therefore, the alarmoccupied space fraction for this location is 14.8%, making it the 9th lowest among 31 events. This equates to a forecasting fraction of 27.4% (shown as red rectangle in Fig. 6). Drawing from the Hualien case experience, we computed the bvalue for the cell associated with each M ≥ 6.0 earthquake. The results of the 31 events (Fig. 6) indicate that most points are located in the upper left side, suggesting that earthquakes tend to occur in areas with low bvalues. About 70% of M ≥ 6.0 earthquake epicenters are situated within the lowest 50th percentile throughout Taiwan, while approximately 50% fall within the lowest 30th percentile. This suggests that most of large earthquakes are in the areas with low bvalues, implying the feasibility of using bvalue spatial distribution as a precursor indicator for earthquakes. By comparing our results with the 99.9% confidence interval curve of the null hypothesis (blue curve in Fig. 6), we found that most events fall outside this range, indicating that the null hypothesis can be rejected with 99.9% confidence.
5 Discussion
Although we found that large earthquakes tend to occur in regions with low bvalues (Fig. 6), the findings of this research indicate that during the 5year period leading up to the earthquake, bvalues near the epicenters of the M ≥ 6.0 earthquakes exhibited no general trend over time, given that this decrease remains within the limits of calculation uncertainty (Fig. 5b). This observation suggests that large earthquakes tend to primarily occur in regions with low bvalues (Fig. 5a), irrespective of their temporal evolution.
To more precisely analyze distribution characteristics, we utilized the Molchan diagram, which shows the percentile of the bvalue from the 1999 to 2021 catalog (red rectangles in Fig. 7). Comparing this with the 99.9% confidence interval curve of the null hypothesis (blue curves in Fig. 7), 22 out of 45 events pass the null hypothesis. When we juxtapose these findings with those derived from the catalog 1 year prior to the earthquake (green rectangles in Fig. 7, identical to Fig. 6), the latter demonstrates slightly improved performance for events with a smaller fraction of alarmoccupied space. This suggests that considering a more recent catalog before a large earthquake could better indicate earthquake precursors, corresponding to the temporally decreasing trend in bvalues (as shown in Fig. 5b).
This study showed that the temporal changes in bvalues are insignificant (Fig. 5b), consistent with findings from previous research (e.g., Schorlemmer et al. 2004; Nanjo et al. 2016). The lack of a significant temporal trend in bvalue (Supplementary Material 1) might be attributed to the unique nucleation process of each large earthquake. For example, as stress accumulates continuously, it may lead to a decrease in bvalue, in line with Suyehiro’s (1966) theory. Additionally, seismic behavior might conform to the Knopoff–Burridge springslider model, where the bvalue initially rises, reaches a peak, and then falls prior to a large earthquake, consistent with the conclusions of Wang et al. (2016).
According to previous studies (e.g., Scholz 2015), changes in the bvalue are negatively correlated with stress accumulation. To more accurately predict future earthquake activity characteristics, it is recommended to include other factors, e.g., the Coulomb stress model (King et al. 1994; Toda et al. 2011), for combined assessment. Ruptures on fault planes lead to deformations in the vicinity, which can translate to changes in Coulomb stress. An increase in Coulomb stress may trigger aftershocks or even subsequent larger earthquakes. Incorporating the Coulomb stress model into the analysis might provide a deeper understanding of the likelihood of earthquakes and reduce uncertainties in perception.
Our study utilizes the earthquake catalog from the CWASN, which records a larger number of smaller earthquakes, providing a more comprehensive catalog. To implement realtime bvalue analysis for earthquake precursor evaluation requires a rapid report catalog. It should be noted that using the realtime system, identifying smaller earthquakes becomes challenging, leading to increase of magnitude of completeness M_{c}. Previous studies utilized the rapid report catalog of the CWA (https://www.cwa.gov.tw/V8/C/E/index.html) to produce significant bvalue anomalies using the same calculation process for the Taiwan region. These studies conducted an indepth examination of the 2022 M_{w} 6.7 Yuli earthquake and the 2022 M_{w} 6.8 Chihshang earthquake, as proposed by Chen et al. (2022) and Chen and Chan (2022), respectively. These cases demonstrate that monitoring bvalue evolution can provide insights into the progression of an earthquake sequence and indicate the potential for larger mainshocks during the foreshock period.
6 Conclusions
The estimated bvalue could easily vary depending on the lower boundary of the magnitude M_{c} used in the earthquake catalog. Thus, this study chose to compute M_{c} on a gridbygrid basis (that examples are shown in Figs. 2a, 7a) to minimize biases in the bvalue calculations. After verifying the feasibility of the forecast method based on bvalues with the 2018 Hualien earthquake, this study sequentially computed the bvalue changes before earthquakes with magnitudes greater than 6.0 that occurred in Taiwan from 1999 to 2021. The Molchan diagram (Fig. 6) showed that 70% of earthquake epicenters from the previous year are located within the lowest 50th percentile of all of Taiwan, and 50% are within the lowest 30th percentile. Additionally, most earthquakes with a magnitude of 6.0 or higher occur in areas with lower bvalues than the curve, within a 99.9% confidence interval of the null hypothesis. This suggests that most earthquake epicenters were located in regions with low bvalues.
Availability of data and materials
The earthquake catalog was accessed via: https://gdmsn.cwa.gov.tw. The rapid earthquake report catalog was accessed via: https://www.cwa.gov.tw/V8/C/E/index.html.
Abbreviations
 CWA:

Central Weather Administration
 CWASN:

Central Weather Administration Seismological Network
 G–R law:

Gutenberg–Richter law
 M _{c} :

Magnitude of completeness
 \({M}_{\text{L}}\) :

Local magnitude
 \({M}_{\text{w}}\) :

Moment magnitude
 UTC:

Universal time coordinated
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Acknowledgements
We are grateful for the valuable feedback offered by the editor and two anonymous reviewers. This study was supported by the Ministry of Science and Technology in Taiwan under the grants MOST 1122116M008015MY2 and MOST 1122124M865001. This work is financially supported by the Central Weather Administration and the EarthquakeDisaster & Risk Evaluation and Management Center (EDREaM) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education in Taiwan.
Funding
This study was supported by the Ministry of Science and Technology in Taiwan under the grants MOST 1122116M008015MY2 and MOST 1122124M865001. This work is financially supported by the Central Weather Administration and the EarthquakeDisaster & Risk Evaluation and Management Center (EDREaM) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education in Taiwan.
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Supplementary Information
40623_2024_2065_MOESM1_ESM.xlsx
Supplementary Material 1. The temporal evolutions of the bvalues, ranked in percentiles, and ∆AIC for the 31 events with magnitudes greater than 6.0 that occurred between 1999 and 2021. The events marked in gray are those located in areas with an insufficient number of earthquakes to obtain reliable bvalues and are excluded from further analysis. The remaining events are denoted by a series number that corresponds to the numbering in Table 1.
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Chan, CH., Kao, JC. & Chen, DY. Spatial–temporal variations of bvalues prior to mediumtolarge earthquakes in Taiwan and the feasibility of realtime precursor monitoring. Earth Planets Space 76, 119 (2024). https://doi.org/10.1186/s4062302402065w
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DOI: https://doi.org/10.1186/s4062302402065w