On the correlations between the largest foreshocks and mainshocks of earthquake sequences in Taiwan

We collected a data set of mainshocks and their respective largest foreshocks of 38 earthquake sequences in Taiwan. The plot of local magnitude, M_L, of a mainshock (denoted by M_Lm) versus M_L of its largest foreshock (denoted by M_Lf) shows an increase in M_Lm with M_Lf. This indicates that for Taiwan’s earthquakes the bigger the largest foreshock is, the larger the mainshock is. The plot of the epicentral distance, Δ (in km), from the largest foreshock to the mainshock versus M_L of the mainshock exhibits a weak increase in Δ with M_Lm as Δ<10 km. The plot of the focal depth of the largest foreshock and that of the mainshock shows a linear increase in the former along with the latter for most event-pairs. Let T be the interval between the occurrence time of the largest foreshock and the mainshock. The plot of T versus M_Lm exhibits that the mainshock will occur within 5 days, with the highest probability of 1 day, after the occurrence of the largest foreshock. Let H be the hypocentral distance between the largest foreshock and the mainshock. The plot of T versus H reveals a slight increase in T with H when T>1 day.

1 Introduction

Prior to earthquake rupture, there is a complex nucleation stage where physical and chemical phenomena might happen. Foreshock activities are a type of preseismic phenomena involved in numerous forthcoming earthquakes (e.g., Papadopoulos et al., 2018; and cited references therein) and thus may be considered as a tool to aid in the prediction of earthquakes (e.g., Papazachos, 1975; Jones and Molnar, 1979; Knopoff et al., 1982; Jones et al., 1982; Jones, 1984; Jones, 1985; Agnew and Jones, 1991; Molchan et al., 1999; McGuire et al., 2005; Ogata, 2011; Ogata and Katsura, 2012; Ogata and Katsura, 2014; Wang et al., 2015; Wang et al., 2016; Wang, 2016; Papadopoulos and Minadakisf, 2016; Riga and Balocchi, 2017; Papadopoulos et al., 2018; Seit et al., 2019; Wang, 2021a; Wang, 2021b). Foreshocks may accurately pinpoint the time and location of the mainshock. For example, foreshocks were an important factor in the successful prediction of the 4 February 1975, M7.3 Haicheng, PRC, earthquake (e.g., Wu et al., 1976; Jones et al., 1982; Xu et al., 1982; Wang et al., 2006). Of course, in addition, numerous authors (e.g., Dodge et al., 1996; Brodsky and Lay, 2014; Kato et al., 2016) have shown some evidence of foreshocks migrating towards the location of the mainshock.

We assume that Taiwan is a good region for studying the above-mentioned problem due to the following reasons: 1) seismic observations have been carried out there for a long time, 2) there are excellent seismic observation networks, and 3) there are abundant earthquake data. Several authors (e.g., Hsu, 1971; Tsai et al., 1977; Wu, 1978; Tsai, 1986) proposed that Taiwan is located at an oblique collision zone between the Eurasian plate (EP) and the Philippine Sea plate (PSP). The collision boundary between the two plates is almost along the Longitudinal Valley (LV), marked with ‘LV’ in Figure 1. The PSP has been moving northwestward at a speed of ∼80 mm/year (Yu et al., 1997) to collide with the EP. In northern Taiwan, the subduction zone of the PSP is beneath the EP. In southern Taiwan, the EP moves from west to east and the subduction zone of the EP is beneath the PSP. Active orogeny due to the collision of these two plates causes complex tectonics and geological features in the region (119.5–122.5 °E and 21.5–25.5 °N). The complex tectonics have resulted in a non-uniform spatial earthquake distribution (Hsu, 1961; Hsu, 1966; Hsu, 1971; Wang, 1988; 1998). High and heterogeneous seismicity means Taiwan is one of the best natural laboratories for earthquakes. Seismological studies have been conducted in Taiwan for more than a century (Wang, 1988; 1998; Wang et al., 1994).

FIGURE 1

FIGURE 1. Epicentral distribution of 38 event-pairs of the mainshocks (open circle) and the largest foreshocks (solid circle). ‘LV’ denotes the Longitudinal Valley in eastern Taiwan.

In Taiwan, numerous earthquakes, including the 20 April 1935, M_s7.2 Hsinchu- Taichung earthquake (e.g., Hsu, 1971; Miyamura, 1985) and the 20 September 1999 M_L7.3 (or M_w7.6) Chi-Chi earthquake (e.g., Ma et al., 1999; Chen et al., 2001; Chen et al., 2002; Shin and Teng, 2001; Wang et al., 2005; Wang, 2019), caused severe damage in the region. To reduce seismic hazards, the research on forecasting forthcoming earthquakes is hence quite important in the region for both academic interest and public need. The possibility to use foreshocks as a precursor in Taiwan has long been considered (cf. Wang, 2021a; Wang, 2021b; and cited references therein). Since foreshocks are usually identified after the mainshock occurs, it is not easy to use them as a useful precursor to predict the forthcoming mainshock. Nevertheless, seismologists still assess the occurrence of a forthcoming mainshock from a group of events that could be foreshocks and occur alongside other possible precursors. In order to investigate the possible role of foreshocks in the assessment of a forthcoming mainshock, it is necessary to study the correlations between the mainshock and the largest foreshock that is a significant representative of foreshocks. In this study, we consider five sorts of correlations between the mainshock and its largest foreshock: 1) the correlation between the magnitude of the former and that of the latter, 2) the correlation between the epicentral distance from the former to the latter and the mainshock magnitude, 3) the correlation between the focal depth of the largest foreshock and that of the related mainshock, 4) the correlation between the time interval that is measured from the occurrence time of the former to that of the latter and the mainshock magnitude, and 5) the correlation between the time interval and the hypocentral distance from the former to the latter.

A detailed description of seismological observations, especially the Taiwan Telemetered Seismographic Network (TTSN) operated by the Institute of Earth Sciences, Academia Sinica, before 1991 can be seen in Wang (1989), Wang (1998) and Yeh et al. (1989). Since 1991, a new seismic network, named the Taiwan Seismic Network (TSN), operated by the Central Weather Bureau (CWB), has been upgraded from the old CWB seismic network. In 1992, the TTSN was merged into the TSN. Since then, numerous new stations have been constructed. A detailed description of the TSN can be seen in Shin (1992) and Shin and Chang (2005). Currently, the TSN is composed of 72 stations, each equipped with three-component, digital velocity seismometers. The seismograms are recorded in both high- and low-gain forms. This network provides high-quality digital earthquake data.

Several magnitude scales have been taken to quantify the earthquakes in Taiwan (e.g., Hsu, 1971; Wang and Miyamura, 1990; Shin, 1992; Wang, 1992; Wang, 1998; Chen et al., 2007). The magnitude scales are Hsu’s magnitude (M_H), duration magnitude (M_D), surface-wave magnitude (M_s), moment magnitude (M_w), and local magnitude (M_L) which were used in different periods of seismological observations. In the following, the magnitudes of earthquakes in use are unified to be the local magnitude, M_L, determined by the CWB (Shin, 1992). Chen et al. (2007) have inferred the correlations between M_L and other magnitude scales.

The selection of a foreshock area that is based on the value of epicentral distance, Δ, between the mainshock and the farthest foreshock in consideration is a not yet clearly defined problem. Different authors often use different upper-bound values of Δ. Several examples are 30 km for M≥7.0 mainshocks by Jones and Molnar (1979), 70 km for M≥3.0 mainshocks by Knopoff et al. (1982), 15 km for M≥5 mainshocks by Lin (2009), 200 km for M≥8.1 mainshocks by Papadopoulos and Minadakisf (2016), 1,000 km for M≥5.8 mainshocks by Riga and Balocchi (2017), 10 km by Trugman and Ross (2019), 2 km by Wu and McLaskey (2022), and 3 km by Peng and Mori, (2022) and Wetzler et al. (2023). Clearly, the spectrum is quite wide. In addition, numerous authors studied the correlation between foreshocks and mainshocks and the criteria for selecting the foreshocks and their mainshocks (e.g., Rhoades and Evison, 2004; De Santis et al., 2015; Gulia and Wiemer, 2019; Cianchini et al., 2020; Console et al., 2020). Gulia and Wiemer (2019) studied the possible discrimination between foreshocks and mainshocks. De Santis et al. (2015) and Cianchini et al. (2020) applied the revised accelerated moment release to foreshocks revealing an acceleration pointing to the mainshock. Console et al. (2020) found that ten out of 14 earthquake sequences in Italy were characterized by multiple mainshocks of a similar magnitude. Their studies will be helpful for understanding the correlation between foreshocks and mainshocks and for the criteria of selecting the foreshocks and related mainshocks.

From the measurements of the aftershock areas of seven of Taiwan’s earthquakes with 6≤M≤7.5, Hsu (1971) concluded that the dimension of the aftershock area somewhat increases with the magnitude of the mainshock, even though he did not infer a formula to correlate the two parameters. Since the aftershock areas measured by Hsu (1971) range from 9.9 × 10² to 8.9 × 10³ km², the linear dimension of the aftershock area varies from 30 km to 100 km, with an average of ∼65 km. In Taiwan, the foreshock area is usually smaller than the aftershock area (e.g., Chen and Wang, 1984; Chen et al., 1990; Chan et al., 2019). Hence, we consider the upper-bound value of Δ to be 15 km for Taiwan’s earthquakes in this study.

In this study, we will compile the data of M_L≥5 mainshocks and their respective largest foreshocks. The earthquake magnitudes of the mainshock and the largest foreshock are denoted by M_Lm and M_Lf, respectively, hereafter. The focal depth of an earthquake is denoted by D (in km). In order to distinguish the focal depths of the mainshock and the largest foreshock, they are denoted by D_m and D_f, respectively, hereafter. The time interval (in day) between the occurrence time of the largest foreshock and that of the mainshock is denoted as T. The epicentral distance (in km) between the epicenter of the largest foreshock and that of the mainshock is denoted as Δ. The hypocentral distance (in km) between the hypocenter of the largest foreshock and that of the mainshock is denoted as H. Therefore, this study will focus on five correlations as mentioned above: M_Lm versus M_Lf; Δ versus M_Lm; D_f versus D_m; T versus M_Lm; and T versus H. The five issues that have included the correlations in size, spatial distribution, and time window between the largest foreshocks and mainshocks will thus provide basic information for studying earthquake physics of earthquake sequences in advance. We hope our studies will be valuable for the studies about the foreshock-mainshock trigger, which is an important problem in earthquake physics, for example, the cascade model for foreshock generation (Ellsworth and Bulut, 2018; McLaskey, 2019; Yoon et al., 2019; Wu and McLaskey, 2022). Of course, we have to emphasize the point that the existence of relationships between foreshocks and mainshocks is meaningful only if the mainshock is preceded by a foreshock sequence.

2 Data

Some authors studied the foreshocks and mainshocks of several of Taiwan’s earthquake sequences. We first compiled the data of those studies from the literature. Chen and Wang (1984) and Chen et al. (1990) observed the occurrences of foreshocks before the 10 May 1983 M_L6.4 (or M_D5.7) Taipingshan earthquake. The foreshocks that first happened on May 7 and continued for about 2 days before the mainshock were located within the source area. The mainshock occurred in the southern part of the foreshock area. The largest foreshock with M_L=5.5 occurred about 3 days before the mainshock.

Lin (2009) studied the foreshock activities of 10 M_L≥5 earthquake sequences with significantly felt foreshocks having M_L≥4.0 in Taiwan during 1990–2004. The largest foreshock whose focal mechanism is similar to its mainshock occurred 5 days before and at a distance of 15 km from the mainshock. Lin (2010) studied the foreshock activities of the 4 March 2008, M_L5.2 Taoyuan earthquake in southern Taiwan. He found that the earthquake was preceded by two groups (A and B) of foreshocks that clustered along the major fault plane and dipped to the southeast. Group A, consisting of 29 micro-earthquakes with 0.6≤M_L≤2.2, occurred several hours before the mainshock. Group B, including 35 events with the largest one having M_L=4.0, started about 20 min (or 0.3 h) before the mainshock. These events are listed in Table 1 with event numbers from 01 to 09.

TABLE 1

TABLE 1. The event-pair number (No), date, longitude, latitude, focal depth (D, in km), and local magnitude (M_L) of 38 pairs of the largest foreshocks and mainshocks. In the last column, “F” and “M” denote the largest foreshock and mainshock, respectively.

Since the number of events obtained by the above-mentioned three groups of researchers is too small to make a comprehensive study, we compiled a large data set from the CWB’s earthquake catalogs from 2005 to 2022 (CWB, 2022). The time interval between two mainshocks is longer than 5 days. This makes it easy to distinguish two mainshocks. Considering the previous studies, the criteria selection of the mainshock and its largest foreshock are Δ ≤15 km for epicentral distance and T≤5 days for the time difference between the largest foreshock and the mainshock. In order to avoid the possibility of multiple events, the event-pairs with δM_L=M_Lm-M_Lf<0.2 are not taken into account. In total, there are 38 mainshocks with 4.6≤M_Lm≤6.8 (during 1983–2022) together with their respective largest foreshocks with 4.0≤M_Lf≤6.4. The focal depths are 1.26 km≤D_m≤19.8 km for the mainshocks and 0.57 km≤D_f≤58.9 km for the foreshocks. From the Central Weather Bureau, Taiwan, the uncertainties in epicenter and focal depth are, respectively, about 2 km and 5 km. The earthquake data in use are listed in Table 1. The epicentral distribution is displayed in Figure 1 in which an open circle and a solid circle represent, respectively, the epicenter of a mainshock and that of its largest foreshock. Figure 2 displays the depth distribution of the mainshocks and that of the largest foreshocks. The number above each bar is the number of events in the relative depth range. Clearly, most of the event pairs are located in a focal depth range from 0 to 20 km.

FIGURE 2

FIGURE 2. Depth distribution of the number of mainshocks and the largest foreshocks in a depth unit of 5 km. The upper and lower numbers above each bar are the number of mainshocks and foreshocks, respectively, in the relative depth range.

Numerous authors (e.g., Rau and Wu, 1995; Ma et al., 1996; Kim et al., 2005) inversed the three-dimensional velocity structures in the Taiwan region from the earthquake data. They inferred that the crust-upper mantle boundary with v_p=7.5 km/s is almost in the range of 35–45 km. Hence, a depth of 40 km is here considered as a boundary to classify the events: the crustal events with D≤40 km and the upper-mantle or subduction-zone events with D>40 km. From Table 1 and Figure 2, it can be seen that all mainshocks are crustal events with D≤20 km. Two foreshocks are mantle events with D>40 km and the rest are crustal events with D≤40 km.

3 Results

3.1 The correlation between M_Lm versus M_Lf

In order to explore the correlation between the M_L of a mainshock (denoted by M_Lm) and M_L of its largest foreshock (denoted by M_Lf), the plot of M_Lm versus M_Lf for the 38 event-pairs is shown in Figure 3. Although the data points are somewhat scattered, results still show an increase in M_Lm with M_Lf and may be described by the following relationship:

$M_{Lm}=\left(1.59\pm 0.47\right)+\left(0.79\pm 0.10\right)M_{Lf}.(1)$

FIGURE 3

FIGURE 3. Plot of M_Lm (M_L for mainshocks) versus M_Lf (M_L for the largest foreshocks). The solid line represents the linear regression equation and the parameter γ is the correlation coefficient.

The correlation coefficient of the linear regression equation between M_Lm and M_Lf is evaluated to be 0.8. This reveals that the regression equation, Eq. 1, is good enough to represent the correlation between M_Lm and M_Lf. Equation 1 indicates that for Taiwan earthquake sequences the bigger the largest foreshock is, the larger the mainshock is.

3.2 The correlation of Δ versus M_Lm

In order to explore the correlation between the epicentral distance, Δ (in km), from the largest foreshock to the mainshock with M_Lm, the plot of Δ versus M_Lm for the 38 event-pairs is shown in Figure 4. Since the data points are quite scattered, we cannot recognize a relationship between Δ versus M_Lm. Nevertheless, considering only the data points with Δ<10 km, we can still see that Δ slightly increases with M_Lm.

FIGURE 4

FIGURE 4. Plot of the epicentral distance (in km), Δ, from the largest foreshock to the mainshock versus the mainshock magnitude, M_Lm.

3.3 The correlation of D_f versus D_m

The plot of the focal depth (in km) of the largest foreshock, i.e., D_f, versus that of the mainshock, i.e., D_m, for the 38 event-pairs is displayed in Figure 5. Except for four event-pairs, the data points are distributed almost around the bi-section line as shown in Figure 5 with a dashed line, thus indicating a linear increase in D_f with D_m.

FIGURE 5

FIGURE 5. Plot of D_f (the focal depth of the largest foreshock in km) versus D_m (the focal depth of the mainshock in km).

3.4 The correlation of T versus M_Lm

The time interval between the occurrence of the largest foreshock to that of the mainshock is denoted by T (in day). The plot of T versus M_Lm for the 38 event-pairs is displayed in Figure 6A. Since the data points are scattered, a relationship between T and M_Lm cannot be recognized. Figure 6B exhibits the time interval distribution of the number of mainshocks in a time unit of 1 day. The value of T varies from 0 to 5 days. The number of event-pairs for which the largest foreshocks occurred within 1 day before the mainshocks is 27, accounting for 71% of the total number of event-pairs. The number of event-pairs rapidly decreases with increasing T as T>1 day.

FIGURE 6

FIGURE 6. (A) Plot of T (the time interval between the occurrence time of a mainshock and that of its largest foreshock) versus M_Lm and (B) The time interval distribution of the number of mainshocks in a time unit of 1 day. The number above each bar is the number of events.

3.5 The correlation of T versus H

Figure 7 shows the plot of T versus H for the 38 event-pairs. Although the data points are quite scattered, we can still observe two interesting phenomena: 1) For T<1 day, there are three clusters of data points: a large one distributing from H=0 km to H=5 km, with an average of 2.5 km, a moderate one around an average hypocentral distance of 15 km, and a small one around an average hypocentral distance of 9 km; and 2) For T>1 day, T slightly increases with H.

FIGURE 7

FIGURE 7. The plot of T versus H (i.e., the hypocentral distance between the largest foreshock and the mainshock).

4 Discussion

4.1 The correlation between M_Lm versus M_Lf

Figure 3 shows an increase in M_Lm with M_Lf that may be described by Eq. 1, M_Lm=1.59+0.79M_Lf, as displayed by a solid line in Figure 3. This implies that M_Lm may be estimated with some uncertainty when M_Lf is known.

Papazachos (1974), Papazachos (1975) plotted M_sf and M_sm for Greek earthquakes and Wu et al. (1976) did the same for Chinese events. Although a high-degree scatter of data points is present in the plots, the two groups of authors still suggested that there is a linear relationship between the two magnitudes. On the other hand, Jones and Molnar (1979) also made the plot of M_sf and M_sm from a large data set of world-wide earthquakes. From the plot, they claimed that even allowing for an error of 0.5 units in magnitude, no relationship between the two magnitudes can be seen. They argued that the crude relationships proposed by Papazachos (1974), Papazachos (1975) and Wu et al. (1976) are due to the limited magnitude range of their respective data. Hence, they concluded that foreshocks cannot be used to reliably estimate the magnitude of an impending earthquake. The correlation between the two magnitudes does not exist is either because the area that slips or the amount of slip during a foreshock is not a constant fraction of the fault area or displacement caused by the mainshock. For numerous world-wide mainshock-foreshock sequences with M_wm≥7, Jones and Molnar (1979) recognize a straight line to exhibit the upper bound magnitudes of the largest foreshocks and such a line increases with the mainshock magnitude. But they could not find a positive correlation between M_wf and M_wm. For seven mainshock-foreshock sequences with M_wm≥5 during 1966–1980 in the San Andreas fault system, California, United States, Jones (1984) did not find a positive correlation between M_wf and M_wm. On the other hand, from 156 worldwide mainshocks with 3≤M_wm≤8 (M_w=the moment magnitude), Riga and Balocchi (2017) inferred the relationships between M_wf and M_wm. The relationships are M_wm=0.695+1.048M_wf for the worldwide earthquake and M_wm= 0.610+1.147M_wf for 32 Italian events. The present result shown in Figure 3 is similar to those obtained by Papazachos (1974), Papazachos (1975), Wu et al. (1976), and Riga and Balocchi (2017) and different from that done by Jones and Molnar (1979). Peng and Mori, (2022) found no clear trend between foreshock size and mainshock size. From laboratory data, numerous authors (Scholz, 2015; Riviere, 2018) found that the b-value decreases as failure approaches and scales inversely with the effective normal stress. The present result of the correlation between foreshock size and mainshock size is consistent with their experimental ones.

Richter (1942, 1956) established the energy-magnitude law of earthquakes as follows:

$\log \left(E_s\right)=11.8+1.5M_s(2)$

in which E_s and M_s are the seismic-wave energy (in ergs) and the surface-wave magnitude, respectively. In Eq. 1, we use the local magnitude, M_L. Hence, we must transfer M_L to M_s. Chen et al. (2007) inferred the following relationship, M_s=−(0.53 ± 0.36)+ (1.03 ± 0.06)M_L, for Taiwan’s earthquakes. Inserting this relationship into Eq. 1 leads to

$M_{sm}=1.52+0.79M_{sf}.(3)$

From Eqs 2, 3, we have

$\log \left(E_{sf}/E_{sm}\right)=-2.28+0.32M_{sf}(4)$

where E_sf/E_sm is the ratio of seismic-wave energy of the largest foreshock to that of the mainshock. Equation 4 exhibits an increase in E_sf/E_sm with the magnitude of the largest foreshock. Table 1 and Figure 3 show that the maximum value of M_Lf in use is 6.4. This yields the maximum values of M_sf and E_sf/E_sm are 6.1 and 0.72, respectively.

4.2 The correlation of Δ versus M_Lm

Figure 4 exhibits the data points of Δ versus M_Lm. Since the data points are in general quite scattered, we cannot recognize a relationship between Δ versus M_Lm.

4.3 The correlation of D_f versus D_m

The plot of the focal depth of the largest foreshock, D_f, versus that of the mainshock, D_m, for the 38 event-pairs is displayed in Figure 5. This figure exhibits that, except for four event-pairs, the former almost linearly increases with the latter because the data points distribute almost around the bi-section line. This means that for most event-pairs in use, the largest foreshock and the mainshock occur almost at the same depth. This implies that we may estimate the focal depth of the mainshock with a high possibility after the largest foreshock occurred. Note that although this correlation exists for Taiwan’s earthquakes, we are not sure if it works for the events in other regions.

As mentioned above, an average depth of 40 km is taken as a boundary to classify the events: a crustal event with D≤40 km and an upper-mantle or subduction-zone event with D>40 km. Obviously, all mainshocks are crustal events with D≤20 km. On the other hand, two foreshocks that are the mantle events with D>40 km are followed by their respective mainshocks that occurred in the crust. Consequently, most of the event-pairs in the study occurred in the crust, especially in the upper crust.

4.4 The correlation of T versus M_Lm

Figure 6A reveals the plot of T versus M_Lm for 38 event-pairs in the study. The data points are somewhat scattered, and thus we cannot infer the relationship between T and M_Lm. This phenomenon also occurs with earthquakes in other regions. For seven mainshock-foreshock sequences with M_wm≥5 that occurred in the San Andreas fault system, California, United States, during 1966–1980, Jones (1984) could not find a positive correlation between log(T) versus M_Lm. Nevertheless, Figure 6A exhibits that the mainshock will occur within 5 days after the largest foreshock happened. Riga and Balocchi (2017) obtained T=0–3,000 days for 128 world-wide earthquakes with 3≤M_wm≤8 and T=0–1,400 days for 16 Italy events. They did not infer any relationship between T and the foreshock magnitude.

Figure 6B exhibits the time interval distribution of the number of mainshocks in a time unit of 1 day. Obviously, about 71 percent of mainshocks occurred within 1 day after their respective largest foreshocks happened.

4.5 The correlation of T versus H

Figure 7 shows the plot of T versus H for the 38 event-pairs. As mentioned above, for T<1 day, the data points form two clusters: the large one with H ranging from 0 km to 5 km, with an average of 2.5 km, and the small one with H ranging from 15 km to 17 km with an average of 16 km. On the other hand, for T>1, day T slightly increases with H. Results suggest the two possibilities for assessing the forthcoming mainshock: 1) If the mainshock occurs within 1 day after the occurrence of the largest foreshock, the most possible hypocenter of the former would have a hypocentral distance of 2.5 km from the latter or the possible one has a hypocentral distance of 15 km from the latter; 2) If the mainshock does not occur within 1 day after the occurrence of the largest foreshock, the hypocentral distance from the former to the latter will increase with the hypocentral distance.

Seif et al. (2019) constructed the plot of T versus Δ for the earthquakes occurring in California, United States. They showed that Δ varies from 0 to 80 km for all events in use, from 0 to 50 km for most of the events and from 50 km to 80 km for three events; while T is in the range from 0 to 10 days for all events. From their plot of T versus Δ, no positive correlation between T versus Δ can be recognized.

4.6 Implications of these correlations

The results from this study are positive. Nevertheless, foreshocks are not always robust features of all mainshocks, as pointed out by Wetzler et al. (2023). This means that it is not easy to assess a forthcoming mainshock just based on the largest foreshock. The present results will potentially help Taiwan’s seismologists estimate the magnitude, focal depth, epicentral distance from the largest foreshock, and occurrence time of the forthcoming mainshock after the largest foreshock of a group of events accompanied by other reliable precursors occurred. From Figure 3 for the correlation of M_Lm versus M_Lf, the mainshock magnitude may be estimated from M_Lm=1.59+0.79M_Lf. From Figure 4 for the correlation of Δ versus M_Lm, the epicentral distance of the mainshock from the largest foreshock is in general shorter than 15 km. From Figure 5 for the correlation of D_f versus D_m, the focal depth of the forthcoming mainshock is almost the same as that of the foreshocks for most of the mainshocks. From Figures 6A, B for the correlation of T versus M_Lm, the forthcoming mainshock will occur within 5 days, with the highest probability of 71% for 1 day, after the largest foreshock happened. Of course, there is the issue that this kind of assessment cannot work for numerous mainshocks, such as the 1999 M_L7.3 Chi-Chi earthquake, because no foreshock happened before them.

5 Summary

In this study, we collected mainshocks and their respective largest foreshocks of 38 earthquake sequences in Taiwan from the CWB database. The plot of local magnitude, M_L, of a mainshock (denoted by M_Lm) versus M_L of its largest foreshock (denoted by M_Lf) for the 38 event-pairs shows an increase M_Lm with M_Lf following Eq. 1. This indicates that the bigger the largest foreshock is, the larger the mainshock is. The plot of the epicentral distance, Δ (in km), from the largest foreshock to the mainshock versus M_Lm exhibits that, although the data points are in general quite scattered, there is a weak increase in Δ with M_Lm as considering only the data points with Δ<5 km. The plot of the focal depth of the largest foreshock versus that of the mainshock shows an increase in the former with the latter, with a linear correlation for most of the event-pairs in the study. The plot of T versus M_Lm exhibits that the data points are somewhat scattered. Nevertheless, T slightly increases with M_Lm. The plot of T versus H shows that the data points are somewhat scattered. For T>1 day, the plot of T versus H reveals a slight increase in T with H.

After the occurrence of the largest foreshock, we may estimate the values of four earthquake parameters, with some uncertainties, of the forthcoming mainshock: 1) the mainshock magnitude can be evaluated from Eq. 1, M_Lm=1.59+0.79M_Lf; 2) the epicentral distance of the mainshock from the largest foreshock is shorter than 15 km; 3) the focal depth of the mainshock is almost the same as that of the foreshock; and 4) the mainshock will occur within 5 days, with the highest probability of 1 day, after the respective largest foreshock happened. Note that, of course, these results cannot work for mainshocks without foreshocks.

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: CWB (2022). Earthquake Information, https://scweb.cwb.gov.tw/en-us/earthquake/data.

Author contributions

K-CC carried out the calculations and drafted the manuscript. J-HW improved the manuscript and interpretation. K-HK discussed the physical meaning of the results and corrected numerous typo errors. All authors contributed to the article and approved the submitted version.

Funding

This study was supported by the Institute of Earth Sciences, Academia Sinica, Taiwan, and partially by the Korean Meteorological Administration Research Development Program (KMI 2022-00610).

Acknowledgments

The authors would like to express their gratitude to the three reviewers and the editor for their valuable comments and suggestions for the substantial improvement of the article. We also thank the Central Weather Bureau for providing the earthquake data.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

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