Introduction

The seismicity parameter b-value is an important parameter in the empirical magnitude-frequency formula, \(LgN = a - bM\), which was derived from the study of global seismicity patterns (Gutenberg and Richter 1944). In the formula, N denotes the frequency of earthquakes above certain magnitude M, a denotes the level of seismicity in the region, and b denotes the proportional relationship between earthquakes of different magnitudes in the region.

Mogi (1962), based on the results of rock simulation tests, suggested that the inhomogeneity of the medium is the main factor leading to the change of b-value, and the larger the b-value is, the more significant the inhomogeneity of the medium is. Scholz (1968), based on the results of the uniaxial and triaxial rock pressure experiments, suggested that the stress state of the rock determines the change of the b-value, i.e., the lower the b-value is, the higher the stress. Wyss (1973) proposed that the increase in effective stress decreases the b-value. Li (1987) verified the conclusions of Mogi and Scholz and concluded that stress and medium inhomogeneity are important influences on the b-value. Wiemer et al. (1998) proposed that the b-value is affected by stress and medium inhomogeneity. EI-Isa and Eaton (2014) concluded that the b-value has a strong relationship with tectonic stress. Based on the relationship between b-value and stress state mentioned above, it is obtained that b-value is inversely proportional to the magnitude of stress, and that low b-value regions have high stress accumulations, In particular, there is a clear trend of decreasing b-values before a fault rupture or fault slip or an earthquake occurrence, but not all of them have decreasing b-value before an earthquake. For example, in the years before a major earthquake, the b-value increases, and after the main shock, the b-value decreases, which is supported by the studies of Fiedler (1974) and Smith (1981) in Venezuela, and Smith (1981, 1986) in New Zealand and California. However, Gulia and Wiemer (2019) showed that after a mainshock, the b-value increased, by 20%.

Many researchers have studied the physical significance of the b-value in depth, and many case studies have corroborated the phenomenon of lower b-values often occurring in the source and neighboring areas before earthquakes. For example, Nanjo and Yoshida (2021) studied the variation in the b-value in and around the source area of M = 6.9 and M = 6.8 earthquakes off the coast of Miyagi Prefecture, Japan. The authors found that the earthquakes occurred near a region with very small b-values, its value is approximately 0.4, even after the earthquake. Zeng et al. (2020) studied the changes in the b-values before Ms = 6.0 earthquake in Changning, China, and found that low b-value anomalies (≤ 0.85) were present in the epicenter area and adjacent areas before the Changning earthquake, with a decrease in the b-values near the epicenter five months before the earthquake. Jiang and Feng (2021) studied the characteristics of the pre-earthquake b-value anomaly in Jiuzhaigou, China, for a M = 7.0 earthquake and found that this region had significantly low b-value anomalies (0.82>b>0.75) before the earthquake.

Furthermore, Xie et al. (2022) studied the pre-seismic b-value variation of a M = 6.0 earthquake in Luxian, China, and found that anomalous low b-value (0.917>b>0.493) features occurred in and around the source area. In summary, a long-term b-value decrease in a certain area reflects the high internal stress of Earth’s crust. The possibility of a significant rupture and a strong earthquake increases in these cases, so special attention should be paid to these areas. Spatiotemporal b-value scan results can provide information about the locations of future strong earthquakes and can be used to analyze the likelihood of strong earthquakes. They can reveal and infer the relative levels of stress accumulation in active ruptures at different stages, outline possible asperity or closed fracture segments, and further determine the substantial earthquake hazard posed by active ruptures. The b-value has gradually become an effective tool for assessing the level of regional stress, inferring the foci of future strong earthquakes, and determining potential earthquake source areas.

The study of b-values has been controversial, among other things, whether b-values really vary in time and space, and if there is a variation, whether the interpretation of this variation is of practical significance, i.e., whether there is really a geologic activity that corresponds to the temporal and spatial variations in b-values, and whether b-values can be used to predict earthquakes.

The opposing view is that the b-value is almost the same in different tectonic regions, and even if there are some temporal and spatial variations, they are partly originate from anthropogenic factors (Frohich and Davis 1993; Jackson and Kagan 1999; Amorese et al. 2010). e.g., the choice of threshold magnitude, the binning and the measurement errors of magnitudes, the sample size, the width of the range covered by magnitude data, the mixing of magnitude types, the lack of statistical rigor in the analysis. Indeed, there are many elements that affect the results of b-value calculation, but we can constantly optimize the above methods to make the b-value closer to the real value.

The positive view is that the b-value not only varies in different regions and at different times, but also reflects a certain extent the seismicity of the region during the time period, which is one of the commonly used parameters for the delineation of potential seismicity zones and the estimation of seismicity trends (Wiemer and Benoit 1996; Mori and Abercrombie 1997; Power et al. 1998; Rydelek et al. 2002; Bridges and Gao 2006; Murru et al. 2007a, b). Wiemer and Wyss (2002) affirmed the practical significance of the temporal and spatial variations of the b-value, and after a careful study in this paper, the practical significance of the temporal and spatial variations of the b-value is likewise affirmed, and the b-value can predict earthquakes to some extent.

On September 5, 2022, at 12:52, a M = 6.8 earthquake occurred in Luding County, Ganzi Prefecture, Sichuan Province, China, at a depth of 16 km. The epicenter was located near the Moxi Fault in the southeastern section of the Xianshui River Fault Zone (29.59°N, 102.08°E), and the earthquake was a mainshock–aftershock type earthquake (A large activity occurs followed by a series of smaller earthquakes called aftershocks) with a maximum intensity of IX. This earthquake caused 97 deaths, and 20 people were missing. The direct economic losses amounted to 154.80 billion yuan. Located on the eastern edge of the Qinghai-Tibet Plateau, the Xianshui River Fault Zone is a northwest-trending arc-shaped left-lateral strike-slip Fault Zone, see Fig. 1. This zone is also the active boundary between the Bayankara Block and the Sichuan–Yunnan Block, starting in the north near the east valley of Ganzi and extending in the northwest-southeast direction, with a dip angle of roughly 55–80°. The zone passes through Luhuo, Daofu, Qianning, and Kangding and intersects with the Longmenshan Fault Zone and the Anning River Fault near asbestos, having a total length of approximately 400 km.

Fig. 1
figure 1

Tectonic background and active faults in the source area and adjacent areas

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The tectonic background and historical earthquake overview of this earthquake

The Xianshui River Fault Zone is divided into two major segments, northwest and southeast, with the Huiyuan Temple La Division Basin as the boundary. The northwest segment has a slip rate of less than 8.4 mm/yr; while, the southeast segment has a slip rate of 4.0–5.2 mm/yr (Liang 2019). The northwest segment can be subdivided into the Luhuo segment (slip rate of 9.13 mm/yr), the Daofu segment (slip rate of 8.57 mm/yr), and the Qianning segment (slip rate of 7.67 mm/yr), with a relatively homogeneous geometry. The southeast segment can be subdivided into the Kangding segment (slip rate of 6.14 mm/yr) and the Moxi segment (slip rate of 4.41 mm/a) (Li et al. 2019). The structure of the Kangding segment is relatively complex, and it is mainly composed of three faults, including the Seraha faults (slip rate of 1.2 mm/yr), Foldotang faults (slip rate of 1.3–3.4 mm/yr), and Yala River faults (slip rate of 0.7–1.0 mm/yr), which are nearly parallel to each other.

The seismic activity within the Xianshui River Rift Zone is intense, with more than 50 earthquakes with M >  = 5.0 occurring since 1700, including eight earthquakes with magnitudes M >  = 7.0 (Fig. 1). Entire sections of the faults are prone to earthquakes and cause the ground to rupture. The most recent strong earthquake with magnitudes of 5.8 or 6.3 occurred on November 22, 2014, more than a year before the strong earthquake in Luding. The Sichuan–Yunnan rhombic and Bayan-Ka-La massifs exhibit frequent occurrence of strong earthquakes. There have been a number of earthquakes in the Sichuan–Yunnan rhombic massif and in the boundary or interior of the Bayankara massif: On May 21, 2021, there occurred M = 5.8, M = 6.4 earthquakes in Yangbi County, Dali, Yunnan. On May 22, 2021, there occurred a M = 7.4 earthquake in Maduo County, Qinghai. On January 2, 2022, there occurred a M = 5.5 earthquake in Ninglang County, Lijiang, Yunnan. On June 1, 2022, there occurred a M = 6.1 earthquake in Lushan County, Yaan, Sichuan and on June 10, 2022, there occurred a M = 6 earthquake in Markang, Sichuan.

In this respect, we calculated the spatial and temporal b-value variations of the earthquake catalog before the Luding 6.8-magnitude earthquake in a large-scale region (100–103.1°E, 29–32°N) and a small-scale region (29–30.5°N, 101.5–103°E). Based on error analysis, we also analyzed the spatial and temporal variation characteristics of the pre-earthquake b-values and the influence of different scales on the spatial and temporal variation characteristics of the pre-earthquake b-value. In addition, the spatiotemporal picture characteristics of the b-value were interpreted in relation to the distribution of moderate to strong earthquakes and active faults. The low b-value zones were further delineated to determine the potential seismic hazard zones, Let everyone know that these potential areas are more likely to experience the next major earthquake, attracting people's attention.

As mentioned above, Nanjo and Yoshida (2021) discovered that the earthquake occurred in a low b-value area (approximately 0.4); Zeng et al. (2020) found that the b-value before the earthquake was at a low value (≤ 0.85); Jiang and Feng (2021) found that there was a significant low b-value anomaly before the earthquake (0.82 > b > 0.75); Xie et al. (2022) found that low b-value anomalies (0.917 > b > 0.493) are characteristic of the seismic source area and its neighboring areas. In this paper, we obtained b-values between 0.68 and 1.2 with an average value of 0.93, and in the spatial distribution of b-values, the b-value before the earthquake decreases to less than 0.9, and the b-value further decreases with the change of time. In summary, this paper defines low b-value anomalies, which can be considered as low b-value anomalies when the b-value is less than 0.9.

Data and methods

Data sources and preprocessing

The study area is the source area of the M = 6.8 Luding earthquake and the neighboring areas (100–103.1°E, 29–32°N), and the seismic catalogs used in this study were all obtained from the Sichuan Earthquake Network Center between January 1, 2020, and September 17, 2010. A total number of 47,719 earthquakes with an M ≥ 0.1 were collected in the catalog, including four earthquakes with magnitudes of 6.0 or higher; 10 earthquakes with magnitudes of 5.0–5.9; 71 earthquakes with magnitudes of 4.0–4.9; 554 earthquakes with magnitudes of 3.0–3.9; 3,915 earthquakes with magnitudes of 2.0–2.9; 22,261 earthquakes with magnitudes of 1.0–1.9; and 20,904 earthquakes with magnitudes of 1.0 or less. The empirical relationship between MS and ML in Chinese mainland was obtained from the statistical regression of Wang et al. (2010) as ML = MS. Therefore, in this paper, MS and ML are not converted into each other, and the earthquake magnitude is uniform in terms of its scale (Fig. 2).

Fig. 2
figure 2

Historical seismicity and active rupture distribution in the neighboring area of the M = 6.8 Luding earthquake

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The "background earthquakes", which represent the "normal" seismic activity, are the basic material for the study of seismic activity anomalies and the determination of seismic hazards. Usually, the earthquake catalogs are mixed with aftershock sequences, foreshock sequences, earthquake swarms, and other seismic clusters, and such mixing will bring greater interference to the scientific research with the background earthquakes as the main object of the study, so the removal of the clusters not only conforms to the needs of the laws of seismic physics itself, but also is the need for the technical treatment of specific seismic activity research. In short, it is important to decluster and to analyze the minimum magnitude of completeness (Mc) of the seismic catalog to obtain a reasonable and reliable seismic catalog for further study without the influence of data selection. In this study, we first selected the empirical relationship based on the time and distance windows of the seismic events proposed by Uhrhammer (1986) to decluster and obtain the independent mainshock catalog, can be used to calculate Mc for the study area. After declustering, a total of 20,749 earthquakes with magnitudes of M ≥ 0.1 were reported.

Currently, the commonly used methods for calculating regional Mc are: (i) the Entire Magnitude Range method (EMR) (Ogata and Katsura 1993), (ii) the Maximum Curvature method (MAXC) (Wiemer and Wyss 2000), (iii) the Goodness-of-Fit Test (GFT) (Wiemer and Wyss 2000) and (iv) Mc by b-value stability (MBS) (Cao and Gao 2002). Among the four methods mentioned above, the MAXC method and the GFT method have simple applicability, but the resulting Mc is relatively low and often underestimated, in this paper, the Mc obtained using the MAXC method is 1.2 and the Mc obtained using the GFC method is 1.2. Furthermore, the Mc obtained by the MBS method is relatively high, and the MBS method is more influenced by the sample size and has a higher uncertainty, in this paper, the Mc obtained using the MBS method is 1.6. In contrast, the EMR method produces a stable and reliable Mc that is closer to the true value (Huang et al. 2016; Woessner and Wiemer 2005). In addition, Woessner and Wiemer (2005) improved the traditional EMR method by introducing a new method for determining the magnitude of completeness Mc and its uncertainty. This method models the entire magnitude range (EMR method), thus providing a comprehensive model of seismic activity. Finally, the Entire Magnitude Range method (EMR) (Woessner and Wiemer 2005) is used in this paper to calculate the completeness magnitude Mc in the study area. The total number of earthquakes required for the calculation is 22,974. The result obtained is closer to the real value, which is obtained from Fig. 3 the Mc is 1.4.

Fig. 3
figure 3

Minimum integrity magnitude analysis of the earthquake catalog in the study area

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Method of calculating b-value

The maximum likelihood method (MLE) (Aki 1965; Utsu 1965; Shi and Bolt 1982) and the least squares method (LSM) (Okal and Kirby 1995; Main 2000; Zöller et al. 2002) are the two commonly used methods for calculating b-values. The MLE method compared to the LSM method, MLE method places more emphasis on the weight of small and moderate earthquakes and is rarely affected by the sample size, So, in this paper, this method is used to calculate the b-value before the main shock. With the continuous deepening of research on b value, the calculation method of b value is constantly being optimized. Recently, a new method called b-positive estimator (Van der Elst 2021; Lippiello and Petrillo 2023) has been proposed. This method is not affected by short-term aftershock incompleteness when calculating b-values, and can obtain stable and accurate b-values after the main shock. Therefore, in this paper, b-positive estimator method is used to calculate the b-value after the main shock.

Method of calculating b-value

Common methods used to calculate the b-value include the maximum likelihood method (MLE) and the least squares method (LSM). When the least squares method is used to calculate the b-value, the cumulative frequency is generally used to reduce the effect of the earthquake binning, emphasizing the role of earthquakes containing a larger amount of rich information and providing a small weighting for earthquakes with lower magnitudes. The maximum likelihood method for calculating b-values was proposed by Aki (1965). The Aki method for the estimation of b-values is not introduced for gridded data, but is a general one, and it is capable to be applied on an entire regional earthquake catalog as the least square method. This method for calculating its b-value is expressed as follows:

$$b = \frac{\lg e}{{\overline{M} - \left( {Mc - \frac{\Delta M}{2}} \right)}}.$$
(1)

The standard deviation of the b-values is calculated as follows (Shi and Bolt 1982):

$$\delta b = 2.30b^{2} \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {M_{i} - \overline{M}} \right)^{2} }}{{n\left( {n - 1} \right)}}},$$
(2)

where Mc is the completeness magnitude and ΔM is the magnitude binning.

In this paper, the magnitude interval is taken as 0.1, \(\overline{M }=\frac{1}{N}\sum_{i=1}^{N}{M}_{i}\) is the average magnitude. e is the natural constant, and n is the sample size used to calculate the b-value. Using Eqs. (1) and (2), the maximum likelihood method averages the magnitudes of all earthquakes with the same weight, equivalent to weighting the information of a larger number of small earthquakes. Since this paper primarily analyzes the regional seismic activity using the small-earthquake catalog, the maximum likelihood method was chosen to calculate the seismic activity parameters, i.e., the a-value and b-value.

Furthermore, estimating b-values with finite seismic samples (n) introduces a systematic bias, but this bias can be corrected, and by using the Central Limit Theorem, a corrected expression for the b-value caused by the finite samples and the deviation from the GR law is obtained (Godano et al. 2024):

$$b_{N} = b\left( {1 + \frac{1}{n}} \right),$$
(3)

where n is the number of seismic samples, b is the true value, and \({b}_{N}\) is the actual value. The above equation shows that when a sample of earthquakes is extracted from a simulated catalog, the evaluated b value is overestimated by a term inversely proportional to the number of events n used for its evaluation.

Then the true value of b is:

$$b = \frac{{b_{N} }}{{1 + \frac{1}{n}}}.$$
(4)

The “b-positive” approach

In recent years, it has been discovered that the main cause of instrumental catalog incompleteness is the overlap of aftershock tails (Arcangelis et al. 2018), which affects the calculation of Mc and b-values, and for this reason, Van der Elst (2021) proposes an improved method of measuring b-values that is not subject to transient variations of the catalog completeness, is not subject to detection problems, and does not require a data window, the new method, called "b-positive estimator". The authors define the Laplace estimator as:

$$\beta ^{ + } = \left( {m' - M_{c} '} \right)^{{ - 1}} m' \ge M_{c} ',$$
(5)
$$\beta ^{ - } = \left( { - m' - M_{c} '} \right)^{{ - 1}} m' \le - M_{c} ',$$
(6)

\(m'\) is the sample mean of the magnitude differences, \(M_{c} '\) is a minimum magnitude difference. Lippiello and Petrillo optimized this method in 2023. The authors point out that this robustness can be further improved by considering positive differences in magnitude, not only between consecutive earthquakes, but also between different earthquake pairs. This approach is called the "b-more positive estimator".

B-positive estimator significant advantage lies in the behavior of the magnitude difference distribution, that if evaluated for earthquakes pairs that are sufficiently close in both space and time, is independent of the value of Mc, and even Mc = -Infinity can be considered. The b-positive estimator stands out as the most accurate approach for estimating the b-value in incomplete catalogs.

Due to the overlap of aftershock tails, the magnitude completeness can change over time, and the b-value can change after large earthquakes, which can lead to incomplete earthquake catalogs and affect the results of b-value calculations. Therefore, in this paper, according to the new method of calculating the b-value mentioned above, get a \({\beta }^{+}\) Estimator value of 0.95 after the main shock, Compared with the b-value before the main shock, the difference is not significant and within a reasonable margin of error.

Calculation of b-values in a regional scale

Characteristics of the temporal variations of the b-values

The calculated b-values for the study area from 2010 to 2022 range from 0.689 to 1.169, with a mean value of 0.928. The solid blue line in Fig. 4a shows the location of the mean b-value. The analysis of the earthquake catalog in the study area during the period when the b-value was greater than the average and the graph of the magnitude variation over time in the study area (Fig. 4b) show that the earthquakes in the study area were mainly of small magnitude, with only a few moderate to strong earthquakes. Most of the moderate to strong earthquakes occurred during the period when the b-value was less than the average. Most of the small earthquakes occurred during the period when the b-value was above the average.

Fig. 4
figure 4

Variations in a b-value with time and b magnitude with time in large-scale region

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Figure 4a shows that the b-values in the study area fluctuated widely, and the b-values suddenly decreased and increased several times. The periods with significant b-value changes in the region were selected and analyzed in conjunction with the moderate to strong earthquakes. Several moderate to strong earthquakes (indicated by the solid red line in Fig. 4a were preceded by an abrupt decrease in the b-values, which dropped to a minimum and then gradually increased to a maximum peak. The earthquakes occurred during the period from the minimum to the maximum peak in the b-value, including the earthquake investigated in this study. However, the magnitude of the drop in the b-values before each of the moderate to strong earthquakes was different. For the earthquakes with serial numbers 1, 3, 4, and 7, small decreases in the b-values occurred three months before these earthquakes. For those with serial numbers 2, 5, 6, and 8, large decreases in the b-values occurred three months before the earthquakes. The relationship between the decrease in the b-values and the magnitude of the moderate to strong earthquakes is still unclear and requires further study. At this stage, we can only note whether the pre-earthquake b-value decreases during a specific period to determine whether an earthquake has occurred. The decrease in the b-values was not evident one to two months before the earthquake with serial number 4, but it was apparent starting one year before the earthquake. Therefore, when analyzing the b-value variations before an earthquake, choosing the right time to study is very important. In addition, for the earthquake with serial number 7, the b-value decreased insignificantly more than one month before the earthquake, but six months before the earthquake, the b-value decreased significantly.

Spatial variations characteristics of the b-value

To analyze the spatial variations in the b-values for different periods and regions in the source and neighboring areas of the 6.8-magnitude Luding earthquake, we define six periods for b-value spatial scanning: I (January 1, 2010–December 31, 2012), II (January 1, 2013–December 31, 2015), III (January 1, 2016–December 31, 2019), IV (January 1, 2020–September 5, 2022), V (January 1, 2010–September 5, 2022), and VI (September 5, 2022–September 17, 2022, after the Luding earthquake).

In this study, a spatial interval of 0.05° × 0. 05° is used for gridding. Taking each grid cell’s node as the center of a circle with a radius of 50 km, earthquakes with M ≥ Mc were selected. The b-value spatial scan was performed using the MLE method. Grid cells with fewer than 50 earthquakes were not included in the calculation to ensure sufficient samples and are shown as blank areas in Fig. 5. In Fig. 5, the different colors represent different b-values, and the spatial scan results show that the b-values of this region are between 0.6 and 1.5. Considering the systematic bias brought by the estimation of b-value with limited seismic samples (n), n is chosen as 50 in this paper, then according to Eq. (3) \(b=\frac{{b}_{N}}{1.02}\), then the real b-value in this region is 0.588–1.471, which is not much different from the actual b-value, and does not affect the results of the subsequent study. In this section, the spatial and temporal variations in the b-values in the source and neighboring areas and the Xianshui River Rift Zone before this earthquake are analyzed in terms of both temporal and spatial evolution. The Kangding earthquake on November 22, 2014, which occurred in the Xianshui River Fault Zone, as well as the Lushan earthquake on April 20, 2013, and the Lushan earthquake in the Longmenshan Fault Zone on June 1, 2022, were also selected for analysis of the spatial and temporal variations in the b-values in the source and neighboring areas before the earthquakes.

Fig. 5
figure 5

Spatial distribution of b-values for different time periods in the large-scale region: a 2010/1/1–2012/12/31, b 2013/1/1–2015/12/31, c 2016/1/1–2019/12/31, d 2020/1/1–2022/9/5, e 2010/1/1–2022/9/5, f 2022/9/5–2022/9/17

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From the spatial variation of Fig. 5a–f, it can be seen that different ruptures correspond to different b-values, which are inversely proportional to the stress, reflecting the difference in the level of stress accumulation of different ruptures, and that earthquakes will occur when the stress accumulates to a certain level, thus reflecting the spatial variability of seismic hazard. The results show slightly higher b-values in the southeast section of the Xianshui River Fault Zone than in the northwest segment. As can be seen from the temporal evolution shown in Fig. 5a–f, the b-values differ in the different periods. Before the Luding earthquake, the b-values of the source area and adjacent areas gradually decreased from 1.15 in period I (Fig. 5a) to 0.9 in period IV (Fig. 5d). The b-values in the source and adjacent areas further decreased shortly after the earthquake, and their values dropped to 0.8 (Fig. 5f).

Similarly, the b-value of the Xianshuihe Fault Zone gradually decreased and is now at a lower b-value, indicating that this fault zone is prone to accumulating stress and experiencing moderate to strong earthquakes. In this Study, we conclude that the slip rate of the northwest section of the Xianshui River Fault Zone is higher than that of the southeast section, such that the northwest section moves toward the southeast section at a higher rate. This compressional force caused by the northwest section accumulates at a decreasing rate interval, resulting in a higher stress accumulating in the southeast section. In addition, the slip rate of the Kangding section in the southeast section is 6.14 mm/yr, and the slip rate of the Moxi section is 4.41 mm/yr. The Moxi section accumulates a higher stress and has a higher possibility of rupture, which makes it more prone to earthquakes. Hence, the M = 6.8 Luding earthquake occurred in the Moxi section of the southeast section of the Xianshui River Fault Zone.

The spatial distribution of the b-values in Fig. 5 is block-like or band-like, which correlates well with the distribution of the regional faults, especially the Xianshui River Fault Zone and Longmenshan Fault Zone. Studies have shown that asperity or closed fracture segments with high stress accumulation in active Fault Zone are prone to strong earthquakes (Aki 1984; Wiemer and Wyss 1997; Wyss et al. 2000). A asperity can be interpreted as the area of stress concentration and strength on the fault surface before the earthquake. This area is the starting point of a rupture, or else the most severe damage occurs at this location during an earthquake (Xu et al. 2022). The closed fracture section is where strain energy or stress concentration is easily accumulated. Therefore, according to the spatial variations in the b-value with a block-like or band-like distribution, the region can be delineated as asperity or as a closed fracture section. A block-like or band-like is a shape in a graph with a distinct area and boundary that can be easily identified as a graphical feature. In the b-value spatial distribution graph, the b-value spatial differences can be more intuitively seen, especially near the fault zone this difference is more obvious, and the low b-value anomaly has been defined above. To find block-like or band-like in the graph, the following principles are followed: (i) looking near the fault zone, easy to find; (ii) in the low b-value area; (iii) with the passage of time, the low b-value area changes significantly. Based on the above three principles, the asperity, or bumps, with serial numbers 1, 2, and 3 can be clearly outlined from Fig. 5a–f. The asperity’s b-value size and spatial location change with time.

The No. 1 bump is located in the Longmenshan Fault Zone, where the M6.1 earthquake on June 1, 2022, and the M7.1 earthquake on April 20, 2013, occurred. The pre-earthquake b-values dropped to less than 0.7 for the M6.1 earthquake and less than 0.9 for the M7.1 earthquake. This is also consistent with the above conclusion that it is unclear how the magnitude of the b-value drop relates to the magnitude of moderately strong earthquakes, but the pre-earthquake b-values were both in the low b-value region.

The No. 2 bump is located in the source area and neighboring areas of the Luding earthquake. In addition, the southeast section of the Xianshui River Fault Zone is prone to accumulate stress and can be classified as a closed fracture section. The No. 3 bump is located to the left of the M6.3 Kangding earthquake that occurred on November 22, 2014, and a small spatial change in the bump’s location has occurred. As can be seen from Fig. 5a–d, the b-value of the bump changed abruptly, reflecting the abrupt change in stress, indicating that the concave-bump body activity was intense and should be taken seriously. The b-value decreased before the M6.3 earthquake, dropping to less than 0.7.

In summary, the pre-earthquake b-values decreased to less than 1 for the above three strong earthquakes. The above three asperities suggest the locations of future strong earthquakes and require continuous attention. However, from the overall perspective (Fig. 5e), the b-values of the southeast section of the Xianshui River Fault Zone, the source area, and the neighboring areas are around 1, and their anomalous features are not very obvious. Therefore, selecting data for the appropriate year for the b-value spatial scan is important. Usually, we divide the data into periods based on the monitoring capability of the seismic network, the occurrence of large earthquakes in that period, and whether the spatial and temporal variations in the b-value are apparent.

Spatial scan error estimation of b-values

Factors such as the starting magnitude of the sample, sampling interval, sample size, completeness magnitude, magnitude binning, and the spatial and temporal extents of the sampling all directly affect the maximum likelihood error of the b-value. In order to check whether the b-value calculation results are reliable, Eq. (2) is used to calculate the b-value fitting error in this paper. Generally, the more sample sizes involved in the calculation within each grid area, the smaller the b-value error is. Therefore, the sample size used to fit the b-value of each grid point requires no less than 50 samples (Fig. 6). The results show that most earthquakes have b-value errors in the range of 0.05–0.1, and only individual grid points have larger b-value errors (> 0.2), suggesting high confidence in the information.

Fig. 6
figure 6

Error distribution of b-values for different time periods in the large-scale region. a 2010/1/1–2012/12/31, b 2013/1/1–2015/12/31, c 2016/1/1–2019/12/31, d 2020/1/1–2022/9/5, e 2010/1/1–2022/9/5, f 2022/9/5–2022/9/17

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Calculation of b-values in a local scale

Characteristics of the temporal variations of the b-values

In this paper, we further analyze the b-values in a small-scale area near the source (29–30.5°N, 101.5–103°E) to obtain a more obvious relationship between the b-value changes and this earthquake and to verify the conclusions drawn from the large-scale area. Figure 7a shows that from 2010 to 2022, the b-values in the study area ranged from 0.694 to 1.223, with a mean value of 0.925. The solid blue line in Fig. 7a shows the location of the mean b-value, which is not significantly different from the large-scale regional mean b-value of 0.928 (Fig. 4a). In conjunction with Fig. 7b, it can be seen that all of the earthquakes with M ≥ 4.0 in this region occurred at low b-values (less than the average b-value). The b-value also dropped back to a lower value before the Luding earthquake; before the moderately strong earthquakes, the b-values tended to be below the average value. This is consistent with the conclusion obtained for the large-scale region, but the phenomenon observed in the small-scale region is more pronounced. In addition, the b-values in the study area fluctuate widely, and the b-values have suddenly decreased and increased many times. Before the occurrence of several moderate to strong earthquakes, the b-values suddenly decreased; after the earthquakes, the b-values slowly rebounded. Combined with the temporal distribution characteristics of the large-scale b-values, it can be determined that the b-value changes can reflect the changes in the stress state. The abnormal characteristics can be used as a sign of the occurrence of moderate to strong earthquakes in the region.

Fig. 7
figure 7

Variations in a b-value with time and b magnitude with time in small-scale region

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Spatial distribution characteristics of b-values

The results of the b-value time scan for the large- and small-scale regions show a correspondence between the increase or decrease in b-values and the occurrence of moderate to strong earthquakes, which can be used as a reference for earthquake prediction in the future. However, the structure of the study area is complex, so the time scan features only reflect the possibility of earthquake occurrence on the time scale of the study area, and the location of earthquake occurrence cannot be discerned. Therefore, a spatial scan of the b-values should be performed for the study area, and an explanation should be given for the areas in the study area where the b-values are in an abnormal state. In this paper, the study area is divided into six time periods when conducting the b-value spatial scan, as in the large-scale region. Since the region’s area is small, the scan radius is reduced to 40 km when the gridding is conducted, and the other parameters are the same as for the large-scale region.

The calculation results are shown in Fig. 8. The small-scale region can better reflect the spatial distribution of the b-values and further verify the conclusions drawn from the results for the large-scale region. In this region, the b-values ranged from 0.7 to 1.6, similarly, the true b-value of the region is 0.686–1.569, which is not far from the actual b-value and does not affect the results of subsequent studies. as seen from the temporal evolution shown in Fig. 8a–f. The b-values in the source area and neighboring regions gradually decreased before this earthquake, from 1.2 in period I (Fig. 8a) to 0.85 in period IV (Fig. 8d). The b-values in the source area and neighboring regions further decreased to 0.75 within a short time after the earthquake (Fig. 8f). These results are not much different from the change in the b-values in the large-scale region, i.e., a zone of significantly low b-values.

Fig. 8
figure 8

Spatial distribution of b-values for different time periods in the small-scale region: a 2010/1/1–2012/12/31, b 2013/1/1–2015/12/31, c 2016/1/1–2019/12/31, d 2020/1/1–2022/9/5, e 2010/1/1–2022/9/5, f 2022/9/5–2022/9/17

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Similarly, the b-value of the Xianshui River Fault Zone has gradually decreased, and it has been at a lower b-value, which needs our attention. In addition, the abrupt change in the b-values in this region is evident from Fig. 8a–b, indicating that the region is highly active. Spatially, the b-value of the southeast section of the Xianshui River Fault Zone is higher than that of the northwest section, and the asperity with serial numbers 1, 2, and 3 can be clearly outlined. Compared with the asperity outlined in the large-scale area, the anomalous characteristics of the asperity in the small-scale area are more pronounced, and it can be seen that the low b-value area migrates farther with time. The low b-value area is usually accompanied by the occurrence of moderate to strong earthquakes; while, the high b-value areas are scattered throughout the study area. Figure 8d, f shows the results of the spatial scan of the b-values two years before and after this earthquake. Before this earthquake, the source area and its nearby areas had low b-values, and the b-value magnitude decreased further in a short time after the earthquake, making this a significant low-anomaly area.

Spatial scan error estimation of b-values

This study adopted Eq. (2) to calculate the b-value fitting error for the small-scale region. Since the area of the region is small, the sample size used in the fitting of the b-value of each grid point is required to be at least 40 (Fig. 9). The results show that the error of the b-values for most of the earthquakes is between 0.05 and 0.15. The error of the b-values calculated for individual grid points is significant (> 0.2), this corresponds to the large-scale regional findings with high confidence in the information.

Fig. 9
figure 9

Error distribution of b-values for different time periods in small-scale region: a 2010/1/1–2012/12/31, b 2013/1/1–2015/12/31, c 2016/1/1–2019/12/31, d 2020/1/1–2022/9/5, e 2010/1/1–2022/9/5, f 2022/9/5–2022/9/17

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Relationship between the b-value and moderate earthquakes and ruptures

The determination of the low b-value zone and the moderate-strong earthquake risk zone depends to a large extent on the knowledge of the active rupture in the region. Therefore, a comprehensive understanding of the development degree of the active ruptures in the region and their activity and tectonic system is a necessary prerequisite for studying the b-value hazard zones and analyzing moderate to strong earthquake hazards. The study region is located at the Tibetan Plateau’s eastern edge, a strong earthquake activity area. Fault tectonics are developed in the region. The Holocene active faults mainly include the Xianshui River Fault Zone, Longmen Mountain Fault Zone, Anning River Fault, Daliang Mountain Fault, Yulongxi Fault, Ganzi–Litang Fault, and Anning River Fault, which were intensely active during the Quaternary, and especially since the Late Quaternary. These faults are seismic faults for strong and large earthquakes. Table 1 presents a list of the features of the main active faults in the region. Figure 10 shows the spatial distribution of the major ruptures, earthquakes with magnitudes of 3 or higher, and b-values within the region. The data further illustrate the relationship between the b-value and moderate to strong earthquakes and ruptures.

Table 1 List of main active faults in the region
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Fig. 10
figure 10

Spatial distribution of moderate and strong earthquakes, ruptures, and b-values

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As can be seen from Fig. 10, there are apparent differences in the spatial distributions of the moderate and strong earthquakes and the b-values along each rupture zone. Most of the moderate and strong earthquakes in the region occurred in the rupture and its adjacent areas. In particular, the most earthquakes occurred in the Xianshui River Rupture Zone, Longmenshan Rupture Zone, and Anning River Rupture, reflecting that these three rupture zones are the most active, strongest, and most dangerous rupture zones in Sichuan Province. The b-value of the Xianshui River Rift Zone is less than 1, and this is an area with many moderate to strong earthquakes. The moderate to strong earthquakes in the Xianshui River Rift Zone are mainly concentrated in the southeast section and the moderate to strong earthquakes in the southeast section are mainly concentrated in the Moxi section. This is also consistent with the conclusion that the sliding rate of the northwest section of the Xianshui River Rift Zone is higher than that of the southeast section and the Moxi section.

The Longmenshan Fault Zone is composed of the Maowen-Wenchuan Fault, the Beichuan-Yingxiu Fault, and the Jiangyou-Guanxian Fault. It is a boundary fault between the Sichuan-Qinghai Block of the Qinghai-Tibet Plateau and the Sichuan Basin in Southern China extending along the Longmen Mountains, with a total length of about 500 km and a width of 40–50 km. The area affected by this fault can be divided into an asperity, and the b-value of the asperity is less than 0.8, making this an area prone to moderate to strong earthquakes. The An Ninghe Fault Zone only spreads to the northern end of its northern section, and its characteristics are not as obvious as those of the Xianshuihe Fault Zone and the Longmenshan Fault Zone, the b-values of which are less than 1. However, it is also an area prone to moderate to strong earthquakes. The three ruptures mentioned above need to be focused on, and they are likely to produce moderate to strong earthquakes in the future and are strong earthquake danger zones. The Daliangshan Fault, Yulongxi Fault, Ganzi-Litang Fault, Anning River Fault, and Daliangshan Fault are prone to small earthquakes, are not prone to moderate to strong earthquakes, and their b-values are approximately 1.

Conclusions

This paper analyzes and discusses the earthquake catalogs before and after the M=6.8 Luding earthquake on two regional scales, large and small, based on the earthquake catalogs provided by the Sichuan Earthquake Network Center for 2010/01/01–2022/09/17. The following conclusions can be drawn from this study.

(1) The characteristics of the temporal variations of the b-values show that the b-values in the regional scale ranged from 0.689 to 1.169, with a mean value of 0.928; whereas, the b-values in the small-scale region ranged from 0.694 to 1.223, with a mean value of 0.925. where 0.928 and 0.925 are the average b-values over a period of time (with different sample sizes in different intervals); while, the fitted b-value of 0.86 is obtained in Fig. 3, and 0.86 is the total value for all earthquakes. The b-values were below the average value before the occurrence of strong earthquakes in the study area, and all of them experienced a sudden decrease, a decrease to a minimum, and then a gradual increase to a maximum value.

(2) The spatial distribution characteristics of the b-values show that the b-values of the southeast section of the Xianshui River Fault Zone are slightly higher than those of the northwest section, and they further outline the asperity or closed fracture sections with serial numbers 1, 2, and 3 in the region according to the spatial variation of the b-values with block-like or band-like distribution characteristic. The No. 1 bump is located in the Longmenshan Fault Zone, the No. 2 bump is located in the source area and neighboring areas of this earthquake, and the No. 3 bump is located to the left of the 2014/11/22 M6.3 Kangding earthquake. The b-value size and spatial location of the bump change over time and may be the location of future strong earthquakes, so it requires continuous attention.

(3) The spatial low b-value anomaly can be used to determine the potential risk area for future moderate to strong earthquakes. The Xianshui River Fault Zone, Longmen Mountain Fault Zone, and Anning River Fault have many small and moderate to strong earthquakes with b-values of less than 1. They are currently in a high stress locking state (with asperity or a locking fracture section), have a high probability of producing moderate to strong earthquakes in the future, and require continuous attention. The Daliangshan Fault, Yulongxi Fault, Ganzi-Litang Fault, Anning River Fault, and Daliangshan Fault are in the high b-value region of the study area, and they currently have relatively low-stress accumulation levels, with minor earthquake activity dominating in the future.

(4) The error of the b-values for most of the earthquakes in the large- and small-scale regions is in the range of 0.05–0.15, and only the errors of the b-values of individual grid points are larger (> 0.2). This information is highly reliable. When the b-values are calculated via spatial and temporal scanning, the analysis of a large-scale area is beneficial to identifying more b-value anomalous areas, but it masks the anomalies in local areas. The analysis of a small-scale area can better represent the b-value variation characteristics of the study area and can more clearly outline the anomalous characteristics of the asperity. Therefore, when performing b-value calculations, an appropriate study area should be selected to avoid missing b-value anomalies.