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).

In Taiwan, numerous earthquakes, including the 20 April 1935, M _s 7.2 Hsinchu- Taichung earthquake (e.g., Hsu, 1971; Miyamura, 1985) and the 20 September 1999 M _L 7.3 (or M _w 7.6) 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.

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 _L 6.4 (or M _D 5.7) 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 _L 5.2 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.

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.

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.

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.