Mitsui et al.
Progress in Earth and Planetary Science(2024) 11:33
https://doi.org/10.1186/s40645-024-00638-7

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open Access

Quantifying the expansion rates
of aftershock zones for magnitude‑7 class
earthquakes around the Japanese archipelago
Yuta Mitsui1* , Yuya Utagawa2 and Ayaka Miyamoto3

Abstract
Earthquakes (mainshocks) trigger sequences of aftershocks, the frequency of which diminishes following a power-law
decay, while the spatial domain of these aftershocks extends logarithmically over time. The delineation of the aftershock zone can be modulated by variables beyond the magnitude of the mainshock, encompassing the locationof the fault (whether the fault is at a plate boundary), the depth at which the event occurs, and the prevailing local
stress conditions. Here, we evaluate the expansion rate of aftershock zones by analyzing earthquakes of magnitude-7
class in the vicinity of the Japanese archipelago. Prior studies have offered approximate assessments of expansion
rates; however, our approach involves the utilization of a straightforward algorithm for the automated estimation of this metric, facilitating the compilation of a catalog. Across the dataset, no pronounced correlations werediscerned between the expansion rate and other examined parameters. Yet, an inverse relationship is identified
between the expansion rate and the b value of aftershocks for mainshocks occurring at plate boundaries. This observation suggests that the expansion rate of aftershock zones predominantly mirrors the stress field following the mainshock. Such a pattern is not detected in mainshocks occurring within the plate’s interior. While the expansion rateof aftershock zones is likely influenced by various factors, aftershock zones may expand more rapidly with higher
differential stress in areas surrounding hypocenters of major interplate earthquakes of magnitude 8 or 9.
Keywords Aftershocks, Expansion rates, Seismic b value, Interplate earthquakes, Intraplate earthquakes, Stresscondition
1 Introduction
When an earthquake (mainshock) transpires, significant forces are exerted on the adjacent regions, leading
to the occurrence of aftershocks. Initially, the force emanating from the mainshock is disseminated to the surrounding area through both dynamic and static stress*Correspondence:
Yuta Mitsui
mit@shizuoka.ac.jp
1
Department of Geosciences, Shizuoka University, Shizuoka 422‑8529,
Japan
2
Formerly at Department of Geosciences, Shizuoka University,
Shizuoka 422‑8529, Japan
3
Formerly at Graduate School of Science and Technology, Shizuoka
University, Shizuoka 422‑8529, Japanchanges (King et al. 1994; Felzer and Brodsky 2006; Toda
et al. 2012). It is well established that the magnitude of
the mainshock is directly proportional to the extent
of the aftershock region (Tajima and Kanamori 1985;
Yamanaka and Shimazaki 1990). Subsequently, processes
that are delayed in time prompt alterations in the locations of aftershocks. Instances of such processes include
postseismic deformation subsequent to the mainshock(Matsuzawa et al. 2004; Perfettini and Avouac 2004; Hsu
et al. 2006) or the localized diminution of rock strength
facilitated by pore fluids (Nur and Booker 1972; Yukutake
and Iio 2017; Miller 2020), both of which can engender
aftershocks. The spatial domain of aftershocks undergoes
expansion over time (Chatelain et al. 1983; Henry and© The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
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Mitsui et al. Progress in Earth and Planetary Science(2024) 11:33

Das 2001), accompanied by a power-law decay in the frequency of aftershocks (Omori 1894; Utsu 1961).
The expansion rate of the aftershock zone is subject toinfluence by various factors, including the location of thefault (whether the fault is at a plate boundary), ambient
temperature and pressure conditions, and local stressconditions. Nonetheless, a consensus has emerged inrecent years around the general characteristic that expansion is proportional to the logarithm of time (Peng andZhao 2009; Kato and Obara 2014; Hainzl et al. 2016;
Ross et al. 2017). An underlying physical mechanism thatshows promise is the concept that the aseismic afterslip following the mainshock serves as an effective trigger for aftershocks (Wesson 1987; Kato 2007; Ariyoshi
et al. 2019). Beyond the afterslip, various transient (viscous) phenomena, such as power-law creep, are posited
to play a critical role immediately following megathrustearthquakes (Agata et al. 2019; Morikami and Mitsui2020). Independent of the presence of afterslip or othertransient mechanisms, the aftershock zone is anticipatedto undergo expansion to a certain degree as a result of asequential occurrence of aftershocks (Helmstetter andSornette 2002; Ozawa and Ando 2021). As mentioned
above, a variety of cases have been reported and models
have been proposed. However, there remains a notablescarcity of research that examines and compares actual
seismicity in a comprehensive manner.
In this study, we examine the expansion rate of theaftershock zone and its determinants by analyzing magnitude-7 class earthquakes occurring in the vicinity of
the Japanese archipelago, an area characterized by its
advanced seismic observation networks. Initially, we
perform an objective estimation of the aftershock zone
expansion rate and compile a comprehensive catalog.
Building on this groundwork, we proceed to investigate
the controlling factors driving aftershock generation. Our
exploration extends beyond the established characteristics of aftershocks, such as power-law decay and logarithmic spatial expansion, aiming to uncover new insights
into the complex natural systems that govern seismic
events.

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 comprises
earthquakes of JMA magnitude 2.0 or greater, occurring within the timeframe of 2000–2021. During thisperiod, major interplate earthquakes of magnitude 9(the Tohoku earthquake in 2011) and magnitude 8 (theTokachi-oki earthquake in 2003) occurred; however,
due to the vastly different scale of aftershock activity

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associated with these events compared to smaller earthquakes, they are not categorized as “mainshocks” inthis study. Additionally, aftershocks occurring within
one day of these events are also excluded from beingclassified as “mainshocks.” Among the remaining earthquakes, the largest recorded a magnitude of 7.4.
Earthquakes with JMA magnitudes ranging from
7.4 to 6.9, occurring around the Japanese archipelago
between 2000 and 2021, are identified as potential“mainshocks” for the analysis of the aftershock zone
expansion rate. From the perspective of balancing sufficient signal strength and a substantial population size,
this study focuses on earthquakes down to a magnitudeof 6.9. In this context, earthquakes in the vicinity of theJapanese archipelago can be categorized into subduction interplate earthquakes, which exhibit low-angle
reverse faulting akin to the angle of the subducting oceanic plate, and intraplate earthquakes, including inlandand intraslab events, as delineated by the centroid
moment tensors (refer to Supplementary Table S1). In
our categorization, we also refer to isobaths of the subducting plate and earthquake reports from the Headquarters for Earthquake Research Promotion. Due tothe inferior quality (significant estimation errors) of the
hypocenter catalog data, outer-rise earthquakes, which
occur at a considerable distance from the seismic networks, are excluded from our study.
Consequently, we designate the 25 earthquakes
depicted in Fig. 1 and Table 1 as “mainshocks.” Amongthese, the magnitude 7.3 earthquake in 2011, cataloged
as mainshock number 14 in Table 1, demands particular examination. Occurring approximately 2 days before
the magnitude 9.0 Tohoku earthquake, this event is
posited as a potential foreshock to the latter, more significant seismic occurrence (Hirose et al. 2011). In lightof this, our analysis is confined to the examination ofaftershocks that occurred prior to the Tohoku earthquake, with the aim of avoiding the influence of aftershock activity associated with the magnitude 9 event. In
Table 1, mainshock numbers 3 and 4, with magnitudesof 7.1 and 7.4 respectively, occurred in close proximity
to each other within 1 day, and their aftershock zones
overlap (Park and Mori 2005). In our analysis for thisstudy, both events are utilized; however, readers conducting their own analyses may consider excluding oneof the two to avoid potential confounding effects dueto the overlapping aftershock zones. In the case of the
Kumamoto earthquake (magnitude 7.3) in 2016 (identified as mainshock number 22 in Table 1), prior indepth studies (e.g., Uchide et al. 2016; Yagi et al. 2016)
have documented the active dynamic triggering of distant earthquakes at locales unassociated with the faultof the mainshock. This paper treats it similarly to otherMitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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time since the mainshock and the three-dimensional(hypocentral) distance from the mainshock hypocenter.
The definition of the aftershock zone is conductedin accordance with the well-established methodology
in previous studies (Gardner and Knopoff 1974; van
Stiphout et al. 2012). Specifically, the temporal window of
the range is defined by the elapsed time t [days] since themainshock, and the spatial window is defined by the distance d [km] from the hypocenter of the mainshock, as
delineated by the following equation:

Fig. 1 Magnitude-7 class “mainshocks” around the Japanesearchipelago. The beachballs display the double couple partof the centroid moment tensors estimated by JMA. Interplateand intraplate earthquakes, which can be determinedfrom the centroid moment tensors (Supplementary Table S1),
are shown in red and blue. The vague gray beachballs show
the outer-rise earthquakes, which were excluded from our study. The
light pink lines depict the trench and the isobaths of the subducting
Pacific Plate, plotted at 20-km intervals up to a depth of 80 km(Iwasaki et al. 2015; Lindquist et al. 2004). The violet lines represent
those of the subducting Philippine Sea Platemainshocks, yet a cautious approach may be warranted
when discussing it in detail.
We investigate the spatiotemporal dynamics of aftershocks, defined as earthquakes occurring in proximity
to a “mainshock.” Despite the spatial diversity observedin aftershock distributions, we utilize the three-dimensional distance from the “mainshock” hypocenter as an
isotropic spatial coordinate to enable a cohesive analysis. This approach facilitates a standardized discussion across different seismic events. The spatiotemporaldefinition of aftershocks is not straightforward. In thisstudy, we initially conduct our analysis using definitioncommonly employed in existing research. Subsequently,
we also evaluate the implications of slightly altering thedefinition.

3 Methods
3.1 Estimation of the expansion rate of aftershock zone

The expansion rate of the aftershock zone has traditionally been deduced from hypocenter data through visualinspection. However, in the present study, this rate isquantitatively estimated through a regression analysis,
incorporating two variables: the logarithm of the elapsed

t = 100.032M+2.7389(1)

d = 100.1238M+0.983(2)where M is the magnitude of the mainshock. In defining
the aftershock zones for the 25 mainshocks examinedin this study, the temporal parameter t is set between
911.4 and 945.6 days, and the spatial parameter d ranges
from 68.7 to 79.3 km. Moreover, in light of the proximity
between certain mainshocks (Table 1), the implications ofutilizing shorter temporal interval (t ) will be examined.
Figure 2 presents an illustration of the expansion of
the aftershock zone and the calculation of the expansion
rate for the mainshock numbered 10 in Table 1, namely
the 2008 off Ibaraki earthquake (Kubo and Nishikawa
2020). Within Fig. 2a–d, it is evident that the aftershockzone expands over time. To quantify this expansion rate,a scatterplot depicting the logarithm of elapsed time versus the hypocentral distance (Fig. 2e) is utilized, followed
by a simple linear regression analysis of the points along
the outer boundary. The regression analysis focuses onaftershocks at the greatest hypocentral distances as time
progresses. It is important to note that the points defining the outer boundary are determined based on the spatiotemporal distribution of the aftershock hypocenters,
without the imposition of any artificial time window for
their selection. We present more information for all the
25 mainshocks, in Supplementary Data S1.
3.2 Estimation of Gutenberg–Richter b value

We focus on the Gutenberg–Richter b value (Gutenberg
and Richter 1944) as a potential indicator of local stressassociated with the expansion of the aftershock zone.
The Gutenberg-Richter law posits that, for any given
magnitude M , the number of earthquakes exceeding
this magnitude, denoted as N , adheres to the expression
log10 N ∝ −bM . This b value demonstrates a tendency tonegatively correlate with differential stress (Scholz 1968;
Wyss 1973), and has been extensively employed in the
research about the stress conditions on faults (Schorlemmer et al. 2005; Nanjo et al. 2012; Spada et al. 2013; Chiba
2020).

Mitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Table 1 Magnitude-7 class “mainshocks” around the Japanese archipelago in this study
Date (JST)Longitude, latitude [deg.]

Depth [km]Magnitude

Type

1

October 6, 2000

2

May 26, 2003

133.349, 35.274

8.96

7.3

Intra

141.651, 38.821

72.03

7.1

3

Intra

September 5, 2004

136.798, 33.033

37.58

7.1

Intra

4

September 5, 2004

137.141, 33.138

43.54

7.4

Intra

5

November 29, 2004

145.276, 42.946

48.17

7.1

Inter

6

December 6, 2004

145.343, 42.848

45.84

6.9

Inter

7

March 20, 2005

130.176, 33.739

9.24

7.0

Intra

8

August 16, 2005

142.278, 38.150

42.04

7.2

Inter

9

March 25, 2007

136.686, 37.221

10.70

6.9

Intra

10

May 8, 2008

141.608, 36.228

50.57

7.0

Inter

11

June 14, 2008

140.881, 39.030

7.77

7.2

Intra

12

July 19, 2008142.265, 37.521

31.55

6.9

Inter

13

September 11, 2008

144.152, 41.776

30.86

7.1

Inter

14

March 9, 2011143.280, 38.329

8.28

7.3

Inter

15

April 7, 2011141.920, 38.204

65.89

7.2

Intra

16

April 11, 2011

140.673, 36.946

6.42

7.0

Intra

17

June 23, 2011142.591, 39.948

36.40

6.9

Inter

18

July 10, 2011

143.507, 38.032

34.00

7.3

Intra

19

December 7, 2012

143.867, 38.020

49.00

7.3

Intra

20

July 12, 2014142.321, 37.050

33.00

7.0

Intra

21

February 17, 2015

143.193, 39.872

12.72

6.9

Inter

22

April 16, 2016

130.763, 32.755

12.45

7.3

Intra

23

November 22, 2016141.604, 37.355

24.50

7.4

Intra

24

February 13, 2021

141.699, 37.729

55.38

7.3

Intra

25

March 20, 2021

141.628, 38.468

59.46

6.9

Inter

The longitude, latitude, and depth of hypocenters and the magnitude are based on the earthquake catalog provided by JMA. The type of earthquakes, interplate(“inter”) or intraplate (“intra”), is determined from the centroid moment tensors estimated by JMA

Addressing the magnitude completeness problem
in earthquake catalogs (e.g., Ogata and Katsura 1993;
Nanjo et al. 2010; Mignan 2012), the recent development of the b-positive method (van der Elst 2021)
marks a significant advancement in accurately estimating b values. The original Gutenberg–Richter law canbe expressed as a probability density function for earthquakes exceeding a threshold magnitude Mc , which is
influenced by the conditions of seismic observation, as
follows:f (x) = βexp(−βx)

for x ≥ 0

(3)

where x = M − Mc and β = bln(10). Mc is generally not
uniform across space and time; for instance, early aftershocks tend to be obscured by overlapping seismic waves(Kato et al. 2012; Omi et al. 2013; Sawazaki and Enescu
2014). The b-positive method (van der Elst 2021) utilizes the difference in magnitude, denoted as M ′ , rather
than the absolute magnitudes. Namely, it employs themagnitude differences ( M ′ ≥ 0.1) between successive

earthquakes where the magnitude of the second earthquake exceeds that of the first. This approach culminatesin the following relationship:f M ′ = βexp(−βM ′ )(4)

The b value for each aftershock sequence can be estimated based on the maximum likelihood method (Aki1965), yielding the estimated b value as follows:b=

log10 (e)M ′ − 0.05

(5)where M ′ is the mean magnitude difference and 0.05
represents the resolution of the catalog (Guo and Ogata
1997). We present histograms of the magnitude differences for all the 25 mainshocks, in Supplementary Data
S2. The 90% confidence interval is obtained through theanalysis of 500 bootstrapping samples.

Mitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Fig. 2 Expansion of the aftershock zone and estimation of the expansion rate for the 2008 off Ibaraki interplate earthquake (magnitude 7.0). Panels
a–d show the spatiotemporal distribution of the aftershocks by elapsed time from the mainshock. Panel e exhibits the scatterplot of the logarithmof the elapsed time versus the hypocentral distance. The pink circles denote the temporal progression of the locations of aftershocks at the greatest
hypocentral distances (outer boundary). The slope of the dashed red line represents the estimated expansion rate of the aftershock zone, whereas
the range of error is indicative of the 90% confidence interval

4 Results
Figure 3 and Table 2 present the estimated expansion
rates of the aftershock zones for the 25 magnitude-7 classmainshocks, as depicted in Fig. 1 and Table 1. The spatial distribution and the triangle diagram of focal mechanisms (Frohlich 1992), as illustrated in Fig. 3, do notreveal any discernible patterns in the expansion rates ofaftershock zones. Nonetheless, from a physical perspective, it is plausible to conjecture that the occurrence ofaftershocks may be influenced by factors such as the location of the fault (whether the fault is at a plate boundary),
the ambient temperature and pressure conditions, or the
local stress conditions. Consequently, we undertake a statistical examination of these aspects.
Initially, we examine the presence of significant disparities in the expansion rates of aftershock zones
between interplate and intraplate earthquakes, asdelineated in Table 1. The average expansion rate forinterplate earthquakes is determined to be 12.46 [km/decade], in contrast to 12.79 [km/decade] for intraplate
earthquakes, with “decade” in this context referring toa single digit in the base-10 logarithm. No significant
difference exists between the two types of earthquakes.
Subsequently, we shift our attention to the focal
depth of the mainshock as a parameter approximately
indicative of the ambient temperature and pressure
conditions, and investigate its correlation with the
expansion rate of the aftershock zones. The correlation
coefficient is calculated to be − 0.28, implying that thedepth at which the event occurs does not exert a controlling influence on the expansion of aftershock zones.
Finally, we establish a correlation between the seismic b value of the aftershocks (Table 2), as an indicator of local differential stress (Scholz 1968, 2015), andthe expansion rate of the aftershock zones, depicted inMitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Fig. 3 Estimated aftershock zone expansion rates for the magnitude-7 class mainshocks (Fig. 1). The left figure represents the spatial distribution,
while the right figure shows the triangle diagram of focal mechanisms (Frohlich 1992), to demonstrate the estimated expansion rates

Fig. 4. The correlation coefficient is -0.19, indicating no
significant correlation.
However, in Fig. 4, when distinguishing between mainshocks of interplate earthquakes and those of intraplate
earthquakes, a certain characteristic becomes apparent.
Namely, when isolating only the interplate earthquakes,
a clear negative correlation is observed between the
expansion rate of the aftershock zones and the b value.
The correlation coefficient was found to be − 0.75, with ap value of 0.01. Mainshocks No. 13 and No. 14 hold significant importance. This negative correlation implies apositive correlation between the aftershock expansion
rate and local stress after the mainshock, because theb value is negatively correlated with local differential
stress (Schorlemmer et al. 2005; Nishikawa and Ide 2014;
Scholz 2015). It would be mechanically plausible when
considering the simpler aftershock propagation along theplane-like mature faults at plate boundaries. Additionally,
aseismic afterslip or other transient mechanisms mentioned in the introduction section might contribute tothis simpler propagation pattern.
In the relationship between the aftershock expansion rate and the b value for interplate earthquakes(Fig. 4), mainshocks No. 13 and No. 14 are characterized
by their high expansion rates and low b values. Notably, the hypocenters of these mainshocks are located inclose proximity to the larger magnitude 8 and 9 earthquakes, compared to other mainshocks, suggesting
significant implications. For example, the 2008 event(No. 13) at longitude 144.152°, latitude 41.776°, depth
30.86 km, was near the magnitude 8.0 2003 Tokachi-oki
earthquake (longitude 144.079°, latitude 41.779°, depth

45.07 km), and the 2011 event (No. 14), at longitude
143.280°, latitude 38.329°, depth 8.28 km, occurred near
the magnitude 9.0 2011 Tohoku-oki earthquake (longitude 142.861°, latitude 38.104°, depth 23.74 km), potentially serving as a foreshock, as previously mentioned.
This implies that the areas surrounding the hypocentersof major earthquakes of magnitude 8 or 9 tend to havehigher differential stress compared to other areas. Suchan effect is less apparent in the intraplate mainshocks,
where their focal mechanisms and the plate boundary
slip substantially differ.
The persistence of high stress in the area surrounding the hypocenter 5 years after the magnitude 8.0 2003
Tokachi-oki earthquake aligns with a previous study, suggesting that the stress release of this earthquake did not
significantly alter the stress field (Tormann et al. 2015).
Although numerous cases have been documented whereb values increase in surrounding areas following large
earthquakes (Gulia and Wiemer 2019), the 2003 Tokachioki earthquake appears not to conform to this pattern. To
further explore this issue, additional investigation might
be warranted using statistical models such as the epidemic-type aftershock sequence (ETAS) model, properly
defining background seismicity distinct from aftershocks(Ueda and Kato 2023).

5 Discussion
We examine the definition of the aftershock zone thathas been established (Gardner and Knopoff 1974; van
Stiphout et al. 2012). Given the high background seismicity surrounding the Japanese archipelago, the current definition of the time window (ranging from t =Mitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Table 2 Estimated parameters for each mainshock (Table 1)
Expansion rate

b value[km/decade](aftershock)1

14.61 ± 2.08

0.954 ± 0.049

2

12.88 ± 1.86

0.797 ± 0.032

3a13.37 ± 2.84

0.827 ± 0.037

4a

13.14 ± 1.84

0.846 ± 0.040

513.37 ± 1.47

0.669 ± 0.042

6

13.04 ± 1.53

0.675 ± 0.050

7

12.14 ± 3.19

0.868 ± 0.052

88.65 ± 1.95

0.735 ± 0.046

99.06 ± 1.41

0.856 ± 0.038

10

10.09 ± 2.02

0.839 ± 0.042

11

10.39 ± 2.04

0.908 ± 0.037

12

9.17 ± 2.01

0.796 ± 0.044

1318.62 ± 1.56

0.527 ± 0.049

14b

20.07 ± 4.91

0.481 ± 0.055

15

9.61 ± 3.98

0.761 ± 0.014

1614.66 ± 2.55

0.808 ± 0.014

17

12.67 ± 2.40

0.658 ± 0.028

18

13.74 ± 1.73

0.829 ± 0.029

19

6.10 ± 2.55

0.875 ± 0.034

20

10.47 ± 2.25

0.853 ± 0.030

21

6.83 ± 1.82

0.630 ± 0.041

22c

26.81 ± 8.74

0.892 ± 0.027

23

12.23 ± 1.96

0.841 ± 0.020

24d12.61 ± 3.23

0.838 ± 0.027

d12.08 ± 3.18

0.746 ± 0.058

25a

In proximity to each other within one day with overlapped aftershock zonesbWith only aftershocks up to about two days before the magnitude 9 Tohoku
earthquake in 2011c

With active dynamic triggering of distant earthquakesdWith only aftershocks up to the end of 2021 (due to catalog limitation)

The expansion rate is derived from the regression analysis on the logarithmof the elapsed time and the hypocentral distance. The b value (aftershocks)
represents the b-positive estimation results for aftershocks. The error ranges
represent the 90% confidence intervals

911.4 to 945.6 days) from Eq. (1) does not preclude thepossibility of incorporating earthquakes unrelated toaftershock activity within this timeframe.
Taking into account the results obtained thus far(Supplementary Data S1), where the time parameter
ranged from t = 911.4 to 945.6 days, we modified thisparameter to t = 10 days in order to conduct an analysis
focused exclusively on the early aftershocks. The duration of 10 days corresponds to an order of magnitude
greater than the one-day period previously examined as
early aftershocks in previous studies (Omi et al. 2013,2014).
Figure 5 presents a comparison between the estimated results of aftershock zone expansion rates and b

Fig. 4 Scatterplot of the aftershock zone expansion rateversus the seismic b value of the aftershocks. The range of error
is indicative of the 90% confidence intervalvalues with the time parameter set to t = 10 days andthe results from the original time parameters. The correlation coefficients are sufficiently high at 0.71 and
0.95, respectively, indicating that the estimated values do not significantly change even when the timeparameter t is altered by two orders of magnitude. Supplementary Data S3 and S4 show results analogous tothose in Supplementary Data S1 and S2, with the timeparameter t adjusted to 10 days.
We examine the correlation with the various parameters upon adjusting the time parameter to t = 10 days.
Despite the two orders of magnitude change in the valueof t , there are no fundamental changes to the findings
discussed in the results section; no significant correlation
is observed between the aftershock zone expansion rateand either the depth of the mainshock or b value. Whenthe analysis is limited to the interplate earthquakes, thecorrelation coefficient between the aftershock zone
expansion rate and b value slightly decreases to -0.60(from -0.75), which is still meaningful.
Furthermore, we address the results obtained whenusing another scaling relations (Uhrhammer 1986) thatinvolve t [days] and d [km], which are different fromthose in Eqs. (1) and (2), in the definition of aftershocks.
The equations are given by:

t = exp(1.235M − 2.87)(6)d = exp(0.804M − 1.024)(7)

For the 25 mainshocks in this study, the temporalparameter t is set between 284.7 and 527.9 days, and thespatial parameter d ranges from 92.2 km to 137.8 km.
Supplementary Table S3 compiles the results for theMitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Fig. 5 A comparison between the estimated results of aftershock zone expansion rates and b values with the time parameter set to t =10 days and the results from the original time parameters. The left figure displays the results for the aftershock zone expansion rates (correlation
coefficient = 0.71), while the right figure represents the outcomes for the b values (correlation coefficient = 0.95). The range of error is indicativeof the 90% confidence interval

aftershock zone expansion rates and the b values. The
values of the aftershock zone expansion rates tend toincrease compared to the original results (Table 2), yetthere are no fundamental changes to the findings. Whenfocusing solely on interplate earthquakes, the correlation
coefficient between the aftershock zone expansion rate
and the b value stands at − 0.79.
Finally, we check the impact of the method used to estimate the b value. We have estimated the b value using
Eq. (5), which means that it has been estimated within
the range of M′ ≥ 0.1. The previous study (van der Elst
2021) recommended verifying a minimum magnitudedifference greater than 0.1; therefore, we attempt to estimate the b value at M′ ≥ 0.2. As a result, the estimated b
values differed by a maximum of 0.04 and less than 0.01
in more than half of the mainshocks compared to theoriginal values (Table 2). Therefore, this point does notaffect the main conclusions of this paper.
While there is ongoing potential for refinement in our
methodology for individual mainshocks, we proposethat we have effectively discerned shared characteristics
among aftershocks through the utilization of a straightforward approach.

6 Conclusions
We conducted a systematic investigation into the
expansion rates, measured in logarithmic time, of aftershock zones associated with 25 mainshocks of magnitude-7 class (Mj 7.4–6.9) occurring in the vicinity of
the Japanese archipelago. When limited to interplate

earthquakes, a negative correlation was observed
between the expansion rate of the aftershock zones andthe b value of the aftershocks. This reflects a tendency
for aftershock zones to expand more rapidly in areas
with higher differential stress, such as areas surrounding the hypocenters of major interplate earthquakes ofmagnitude 8 or 9.
Abbreviation
JMA	Japan meteorological agency

Supplementary Information
The online version contains supplementary material available at https://​doi.​
org/​10.​1186/​s40645-​024-​00638-7.
Additional file 1
Additional file 2
Additional file 3
Additional file 4
Additional file 5
Additional file 6
Additional file 7
Acknowledgements
We thank the Japan Meteorological Agency for providing the earthquake data.
The plate models by Iwasaki et al. (2015) were constructed from topography
and bathymetry data by Geospatial Information Authority of Japan (250-m
digital map), Japan Oceanographic Data Center (500m mesh bathymetry data,
J-EGG500, http://​www.​jodc.​go.​jp/​jodcw​eb/​JDOSS/​infoJ​EGG_j.​html) and Geographic Information Network of Alaska, University of Alaska (Lindquist et al.,2004). We are grateful to Taku Ueda, Futoshi Yamashita, and Aitaro Kato forfruitful comments and discussions. We appreciate two anonymous reviewers
for their insightful comments and suggestions, which have greatly improved
the manuscript.

Mitsui et al. Progress in Earth and Planetary Science(2024) 11:33

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Author contributions
YM conceived the study, verified the whole results, and drafted the manuscript. YU analyzed the data. AM developed the computer programs. All
authors contributed to the manuscript.

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Funding
This study is funded by Japan Society for the Promotion of Science KAKENHI
Grant 22H00176.
Availability of data and material
The datasets supporting the conclusions of this article are included within thearticle and its additional files. The hypocenter catalog and centroid moment
tensor catalog used in this study are available from JMA [http://​www.​data.​
jma.​go.​jp/​svd/​eqev/​data/​bulle​tin/​index_e.​html]. The earthquake reports canbe accessed through the Headquarters for Earthquake Research Promotion[https://​www.​jishin.​go.​jp/​main/​chousa/​chousa-​e.​htm].

Declarations
Competing interests
The authors declare that they have no competing interest.
Received: 29 February 2024 Accepted: 7 June 2024

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