Stable estimation of the Gutenberg–Richter b-values by the b-positive method: a case study of aftershock zones for magnitude-7 class earthquakes

Abstract

The traditional approach for estimating the b-value of the Gutenberg–Richter law, which is posited to inversely correlate with differential stress, has historically relied on the maximum likelihood technique, utilizing data from earthquakes exceeding a magnitude cutoff, Mc. This traditional approach is significantly influenced by the value of Mc, leading to extensive research focused on methods for determining Mc with greater accuracy. However, a recent study introduced a novel method based on the frequency distribution of magnitude difference, termed the b-positive method. This innovative method could enable more robust b-value estimations, even in scenarios where Mc may vary spatially and temporally. Our study concentrated on analyzing aftershocks, related to 25 magnitude-7 class earthquakes surrounding the Japanese archipelago. We estimated the b-values using both the goodness-of-fit test for Mc, a traditional approach, and the b-positive method. The aftershock data were examined over two distinct time frames: the initial 10 days following each mainshock and an extended period of 1000 days. Our findings indicated that the estimates produced by the b-positive method showed negligible variation between the 10-day and 1000-day aftershock periods (correlation coefficient of 0.95), whereas the traditional approach tended to yield lower b-values for the 10-day aftershocks compared to those from the 1000-day period. Variations in b-values, when analyzed using the traditional approach, could be inaccurately ascribed to temporal fluctuations in differential stress that may not actually be present. The b-positive method offers a vital solution to prevent these erroneous interpretations, serving as an essential alternative.

Graphical abstract

1 Introduction

The Gutenberg–Richter b-value (Gutenberg and Richter 1944) serves as a critical measure for understanding local stress environments through seismicity analysis. This value, derived from the relationship ${\text{log}}_{10}N\propto -bM$, where $N$ is the number of earthquakes exceeding magnitude $M$, inversely correlates with differential stress, suggesting its utility in assessing fault stress conditions as supported by numerous studies (Scholz 1968; Wyss 1973; Schorlemmer et al. 2005; Nanjo et al. 2012; Tormann et al. 2015; Chiba 2020).

When comparing with actual seismicity data, the Gutenberg–Richter law can be formulated as a probability density function that characterizes the likelihood of earthquakes surpassing a cut-off magnitude ${M}_{c}$:

$$f\left(x\right)=\beta \text{exp}(-\beta x)\, \quad \text{for } x\ge 0,$$

(1)

where $x=M-{M}_{c}$ and $\beta =b\text{ln}\left(10\right).$ ${M}_{c}$ varies spatiotemporally in accordance with the conditions of observation (e.g., Ogata and Katsura 1993; Nanjo et al. 2010; Mignan 2012; Nakamura 2022).

The traditional approach to estimating the b-value in the Gutenberg–Richter law for earthquake magnitude–frequency distribution has long relied on the maximum likelihood estimation technique using earthquakes greater than the cut-off magnitude ${M}_{c}$ (Aki 1965). The estimated b-values based on the traditional approach is significantly influenced by the value of ${M}_{c}$ (e.g., Nakamura 2022), leading to extensive research on methods to determine ${M}_{c}$ more accurately (e.g., Wiemer and Wyss 2000; Mignan and Woessner 2012). The estimation of ${M}_{c}$ is conducted using methods such as the goodness-of-fit to magnitude histograms. However, there are cases where the histogram shape is not simple, rendering it not always possible to achieve a robust estimation.

A recent study (van der Elst 2021) proposed a new method based on the frequency distribution of magnitude differences, named the b-positive method. This new approach does not require the estimation of ${M}_{c}$, potentially allowing for more robust b-value estimation.

In this study, we apply both the goodness-of-fit test for ${M}_{c}$, a traditional approach, and the b-positive method to seismicity data that undergoes spatiotemporal variations of ${M}_{c}$, specifically focusing on aftershock activity following major earthquakes. We then conduct a comparative analysis of the estimated b-values derived from each method. The dataset used is similar to that in our previous study on the expansion of aftershock zone (Mitsui et al., 2024). Our findings reveal that the b-positive method consistently produces stable b-values across different aftershock time windows, whereas the traditional approach tends to yield more variable b-values.

2 Data

We utilize hypocenter catalog data from the Japan Meteorological Agency (JMA), selectively incorporating only those entries flagged with "high precision" in hypocenter determination. Our dataset encompasses earthquakes with JMA magnitudes of 2.0 or above, recorded from 2000 to 2021. Following our previous study (Mitsui et al., 2024), 25 earthquakes with magnitudes ranging from 6.9 to 7.4 near the Japanese archipelago are considered “mainshocks” (Table 1). We do not include earthquakes occurring within one day of magnitude 8 (the 2003 Tokachi-oki earthquake on Sep. 26, 2003) or 9 (the 2011 Tohoku earthquake on Mar. 11, 2011) events and outer-rise earthquakes. Figure 1 illustrates the spatial distribution of 25 mainshocks. In Table 1, there are mainshocks that occurred in close spatial and temporal proximity, resulting in overlapping aftershock zones (e.g., Mainshocks 3 and 4). Moreover, for Mainshock 14 (a possible foreshock of the magnitude 9 Tohoku earthquake), only aftershocks up to approximately two days before the Tohoku earthquake are considered. For Mainshocks 24 and 25, due to the limitations of the catalog, only aftershocks up to the end of 2021 are considered. While all mainshocks are included in our analysis, readers conducting their own analyses might consider excluding one of them to avoid potential statistical biases.

Table 1 Magnitude-7 class “mainshocks” around the Japanese archipelago

Fig. 1

Magnitude-7 class “mainshocks” in this study. The beachballs depict the double-couple component of the centroid moment tensors provided by JMA. The light orange lines delineate the trenches. In the magnified section of the map, aftershocks occurring within a 1000-day period following Mainshock 9 are presented as an example

Defining aftershocks as earthquakes occurring near a mainshock, we utilize the three-dimensional distance from the mainshock hypocenter as an isotropic spatial coordinate for analysis (Mitsui et al., 2024). The spatial definition of the aftershock zone is based on the well-established methodology (Gardner and Knopoff 1974; van Stiphout et al. 2012). It is specifically delineated by a spatial window, defined as the distance $d$ [km] from the hypocenter of the mainshock:

$$d={10}^{0.1238M+0.983},$$

(2)

where $M$ is the magnitude of the mainshock. The spatial parameter $d$ ranges from 68.7 km to 79.3 km in this study.

The temporal window for the analysis is defined by the elapsed time $t$ [days] since the occurrence of the mainshock. To facilitate comparison, we employ both a longer period of 1000 days and a shorter period of 10 days. This temporal window also closely follows the precedent set by our previous study on the expansion of aftershock zone (Mitsui et al. 2024). Figure 1 shows an example of the aftershocks for the mainshock.

3 Methods

To ascertain the cut-off magnitude ${M}_{c}$ preceding the calculation of the b-value in the traditional approach, we compute the $R$ parameter for the goodness-of-fit (Wiemer and Wyss 2000) in the cumulative count of earthquakes across each magnitude bin:

$$R=1-\left(\frac{\sum_{{M}_{i}}^{{M}_{\text{max}}}\left|{C}_{i}-{S}_{i}\right|}{\sum_{i}{C}_{i}}\right),$$

(3)

where ${C}_{i}$ represents the observed cumulative number, and ${S}_{i}$ signifies the synthetic cumulative number derived from the Gutenberg–Richter law across each magnitude bin (with a bin width of 0.1). The catalog’s minimum magnitude of completeness, denoted as ${M}_{c}$, is identified as the lowest magnitude at which $R$ achieves a predetermined threshold. Following a previous study (Mignan and Woessner 2012), this threshold is initially established at 0.95. Should this threshold not be attained, it is subsequently adjusted to 0.90. In the event that the threshold remains unmet, ${M}_{c}$ is then determined based on the mode of the frequency–magnitude distribution.

In the traditional approach, we determine the b-value for each aftershock sequence utilizing the maximum likelihood estimation technique (Aki 1965). The calculated b-value is expressed as follows:

$$b=\frac{{\text{log}}_{10}\left(e\right)}{\overline{M }-({M}_{c}-0.05)},$$

(4)

where $\overline{M }$ represents the average magnitude of all earthquakes exceeding ${M}_{c}$, and 0.05 denotes the catalog's resolution (Guo and Ogata 1997). The 90% confidence interval is derived by examining 500 bootstrap samples. Figure 2a exhibits an example of the histogram of the magnitude with the estimated b-value and its 90% confidence interval. Supplementary Data S1 and S2 show the histograms of the aftershock magnitudes for each mainshock in the 1000-day and 10-day periods, respectively.

Fig. 2

Example of the histogram of the 1000-day aftershocks (Mainshock 18) with estimated b-value and its 90% confidence interval. a For the traditional approach. The dotted vertical line represents ${M}_{c}$. b For the b-positive method

By contrast, the b-positive method (van der Elst 2021) leverages the magnitude difference, symbolized as ${M}{^\prime}$, instead of relying on the magnitude $M$. Specifically, it capitalizes on the magnitude increments (${M}{^\prime}\ge 0.1)$ observed between consecutive earthquakes, where the magnitude of the latter event surpasses that of the former. This approach culminates in the following equation:

$$f\left({M}{^\prime}\right)=\beta \text{exp}\left(-\beta {M}{^\prime}\right).$$

(5)

This formulation is based on the mathematical principle that the difference between parameters following an exponential distribution (here represented by Eq. (1)) adheres to a Laplace distribution (double exponential distribution).

The b-value associated with each aftershock sequence can be determined through the application of the maximum likelihood estimation technique, resulting in the following formula for the estimated b-value:

$$b=\frac{{\text{log}}_{10}\left(e\right)}{\overline{{M }{^\prime}}-(0.1-0.05)},$$

(6)

where $\overline{{M }{^\prime}}$ represents the average magnitude increment and 0.05 denotes the catalog's resolution. The 90% confidence interval is derived by examining 500 bootstrap samples. A cutoff similar to ${M}_{c}$ in the traditional approach is not considered, because given the ${M}{^\prime}$ histogram in this study (Supplementary Data S3 and S4) show no significant need for it. This point will be illustrated later in Supplementary Figure S1. Figure 2(b) exhibits an example of the histogram of the magnitude difference with the estimated b-value and its 90% confidence interval. Supplementary Data S3 and S4 show the histograms of the magnitude differences for each mainshock in the 1000-day and 10-day periods, respectively.

4 Results

Table 2 aggregates the estimated results of the b-values derived from both the traditional approach (with ${M}_{c}$) and the b-positive method across the two timeframes: the longer period of 1000 days and the shorter period of 10 days following each mainshock. We describe the characteristics of these results below.

Table 2 Estimates of the b-values based on the traditional approach with ${M}_{c}$ and the b-positive method

Figure 3 presents a comparison of the b-values between the traditional approach and the b-positive method, for both the long-term (1000-day) and short-term (10-day) aftershock periods. For the long-term analysis, the estimated b-values between both methods do not significantly differ; however, for the short-term analysis, it is evident from the position relative to the blue dashed line (Fig. 3) that the traditional approach systematically yields smaller estimates. When testing the difference in mean b-values between the two methods using the Wilcoxon test, the null hypothesis that both are equal is rejected for the 10-day aftershocks, with a p-value of less than 0.01% (for reference, the p-value for the 1000-day aftershocks is 8%). This observation could support that the goodness-of-fit test has a tendency to underestimate the b-value (Woessner and Wiemer 2005). Additionally, Supplementary Figure S1 includes results from the b-positive method using ${M}{^\prime}\ge 0.2$ and ${M}{^\prime}\ge 0.3$. These variations in conditions did not affect the conclusions.

Fig. 3

Comparison of the b-values between the traditional approach and the b-positive method. The blue dashed line represents $y=x$, a straight line with a slope of 1. a For the 1000-day (longer) aftershock period. b For the 10-day (shorter) aftershock period

To further investigate the above-mentioned aspects, Fig. 4 shows a comparison of the b-values between the 10-day aftershocks and the 1000-day aftershocks, for both the traditional approach and the b-positive method. The analysis reveals that b-values derived from the b-positive method are less affected by the selection of the aftershock time window compared to those obtained through the traditional approach. Specifically, a correlation coefficient of 0.95 was noted for the b-positive method, as opposed to 0.69 for the traditional approach. Only for the b-values calculated using the traditional approach, there is a tendency for the b-values associated with 10-day aftershocks to be slightly lower than those for 1000-day aftershocks. When testing the difference in mean b-values using the Wilcoxon test, the null hypothesis that both are equal is rejected for the traditional approach, with a p-value of less than 2%.

Fig. 4

Comparison of the b-values between the 10-day aftershocks and the 1000-day aftershocks. The blue dashed line represents $y=x$, a straight line with a slope of 1. a For the traditional approach. b For the b-positive method

5 Discussion

The temporal variation of b-values, which is more readily observed using the traditional approach as compared to the b-positive method, may be misinterpreted as temporal variations in differential stress that do not actually exist.

For the sake of simplification, discussions will henceforth consider the average of all 25 aftershock sequences. In the traditional approach, the average b-value for aftershocks over 10 days is 0.67, while that for 1000 days is 0.73. Substituting these into the empirical relationship between b-value and differential stress, $b=1.23-0.0012({\sigma }_{1}-{\sigma }_{3})$, proposed by a previous study (Scholz 2015), the differential stresses are calculated to be 467 MPa and 417 MPa, respectively. This differential stress variation of 50 MPa is likely apparent, because the average b-value based on the b-positive method, for both 1000-day and 10-day aftershocks, remains almost unchanged at 0.78.

In research that discusses the detailed temporal changes in stress due to variations in the b-value (e.g., Nanjo et al. 2012; Tormann et al. 2015), robust estimation of the b-value is fundamentally essential. For this purpose, estimations based on the more stable b-positive method, over the traditional approach, could serve as one viable option.

While the b-positive method leads to more stable maximum likelihood estimates compared to the traditional approach, it is important to note that the b-positive method is not superior in every aspect. For example, as shown in Table 2, the confidence interval for the b-positive method tends to be slightly larger than that for the traditional approach. This is because the b-positive method only uses positive magnitude differences, reducing the number of data points. The reason for not using negative magnitude differences is that they are more likely to occur after a large earthquake in a seismic sequence and are therefore susceptible to the decreased detection capability for smaller earthquakes (van der Elst 2021). This point may have a non-negligible impact in certain cases. Furthermore, in addition to the goodness-of-fit test performed with certain thresholds set in this study, other approaches for estimating ${M}_{c}$ have been proposed by previous studies. For example, the modeling of detection capability (Ogata and Katsura 1993) represents an essentially superior attempt for estimating ${M}_{c}$. A comparison between these approaches and the b-positive method should be conducted in the future.

6 Conclusion

We conducted the comparative analysis of the estimated b-values derived from the traditional approach and the b-positive method. We found that the b-positive method tends to yield more stable b-values across different aftershock time windows than the traditional approach. Variations in b-values, as interpreted through the traditional approach, may be mistakenly attributed to temporal changes in differential stress that, in fact, do not exist. To circumvent such misinterpretations, the b-positive method emerges as a critical alternative.