Introduction

Slow earthquakes are typically observed in the transition zones between locked and creeping zones on plate boundaries (Obara and Kato 2016). The Nankai Trough plate interface, off the coast of and beneath southwest Japan (Fig. 1), is host to several types of slow earthquakes. Long-term slow slip events (SSEs), with durations of months or years, have been observed by using data from the global navigation satellite system (GNSS) in the Bungo Channel (Hirose et al. 1999), the Kii Channel (Kobayashi 2014), and the Tokai region (Ozawa et al. 2002) (see Fig. 1 for locations). Short-term SSEs, with durations measured in days (Obara et al. 2004), occur along with low-frequency tremor in a belt along the downdip side of the locked zone. Very low-frequency earthquakes, with predominant periods of tens of seconds, have been documented on both downdip and updip sides of the locked zone (Ishihara 2003; Obara and Ito 2005). Low-frequency tremor also occurs on the updip side of the locked zone (Sakai et al. 2007; Obana and Kodaira 2009). Among these slow earthquakes, long-term SSEs are the largest events, and they are located near the locked zone along the Nankai Trough, where future great megathrust earthquakes are postulated. Therefore, long-term SSEs may have a large influence on physical conditions around the locked zone.

Fig. 1
figure 1

Map of the study area in southwest Japan. a Location and plate tectonic setting. Heavy curves here and on the main map indicate plate boundaries. b Locations in the Nankai Trough study area. The dashed rectangle encloses the fixed region. Large numbered dots indicate GNSS stations. Heavy lines connecting two stations represent baselines for measurement of length changes. Small dots numbered 1–60 show the approximate location of the 25 km depth contour on the Nankai Trough subduction interface. The oblique rectangle in western Shikoku indicates an example of 100 km × 50 km area centered on dot 10

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In addition to the long-term SSEs already mentioned, small long-term SSEs have also been observed in and near the Nankai Trough (Kobayashi 2010; Takagi et al. 2016; Ozawa 2017). Kobayashi (2017) used GNSS observations to objectively detect long-term SSEs along the Nankai Trough and reported that they are not distributed homogeneously along the strike direction. In particular, no long-term SSE was detected during the first two decades of GNSS observations on the Kii Peninsula, including its easternmost tip, the Shima Peninsula (Fig. 1b). This paper reports a small long-term SSE that occurred beneath the Shima Peninsula during 2017 and 2018, the first detection of such an event from GNSS data.

Data and methods

Detection of long-term SSEs by GNSS

The GNSS Earth Observation Network System (GEONET) is operated in Japan by the Geospatial Information Authority of Japan (GSI). We used the GSI’s F3 daily coordinate datasets (Nakagawa et al. 2009) from March 1996 to February 2019 for this study. In the GEONET F3 strategy, global positioning system (GPS) data are analyzed with Bernese GPS Software version 5.0, and the final orbit and Earth rotation parameters of the International GNSS Service (IGS) are used to estimate static daily coordinates (Ozawa 2017). Details can be found in Nakagawa et al. (2009) and Munekane (2010).

We applied the objective detection method described by Kobayashi (2017), extending the GNSS record in that paper from June 2017 to February 2019. The method is described briefly here. We first removed the steady-state or background surface displacement velocity, annual and semiannual variations, and offsets due to antenna/receiver replacement and earthquakes. Offsets due to antenna replacements were subtracted by using an offset dataset from GSI (http://mekira.gsi.go.jp/JAPANESE/corrf3o.dat). Coseismic offsets of large earthquakes were removed by calculating the difference in the 10-day means of the daily coordinates before and after each earthquake; however, no earthquake occurred that required the removal of coseismic offsets from the added months of data after the record analyzed by Kobayashi (2017). We then subtracted the common mode error (Wdowinski et al. 1997) of the fixed region (dashed rectangle in Fig. 1b) from the coordinates of all stations, in effect treating the entire rectangle region as fixed. The residual displacements are thus referred to as unsteady displacements. For each station, we calculated the component of motion in the direction S55°E, which is the direction of the upper plate with respect to the subducting Philippine Sea plate (N55°W; Miyazaki and Heki 2001).

The long-term SSEs along the Nankai Trough are centered at 25–30 km depth, on the upper surface of the subducting plate (e.g., Ozawa et al. 2002, 2013; Kobayashi 2014). We determined the positions of 60 points above the 25 km depth contour of the subducting plate (plate configuration of Hirose et al. 2008) at intervals of 0.1° from longitude 132.1°E to 138.0°E (numbered points in Figs. 1b and 2a) and centered a 100 km × 50 km rectangle on each point with its long axis oriented in the direction of plate subduction. For each rectangle, the component of motion in the direction S55°E was calculated from the mean of the horizontal components of GNSS stations within the rectangle.

Fig. 2
figure 2

Detection of long-term SSEs. a Map showing locations (dots 1–60) at ~ 25 km depth on the Nankai Trough subduction interface. b Time-longitude (dot number) plot showing displacements during long-term SSEs within 100 km × 50 km rectangles centered on dots 1–60. Colored regions have correlation coefficients greater than 0.6 and 2-year displacements greater than 2 mm. See the text for details. T: Tokai long-term SSE (Ozawa et al. 2016), S: Shima Peninsula (this study), K: Kii Channel (Kobayashi 2017), WS: western Shikoku (Kobayashi 2010), B1, B2: Bungo Channel (Ozawa et al. 2007; Ozawa et al. 2013). Labels (1), (2) and (3) indicate migrations discussed in the text. The data record ends in February 2019; note that results for latest 1.5 years may change as later data are included

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Displacement components obtained by this procedure included the postseismic deformations associated with the March 11, 2011, Tohoku earthquake (M 9.0) and the September 5, 2004, off-Kii Peninsula earthquake (M 7.4). The postseismic component of the Tohoku earthquake was estimated by a logarithmic function using a mean time series of the stations, which had no unsteady displacements other than the postseismic deformation of the Tohoku earthquake. Assuming that the same time constant can be applied to the time series of displacement in the direction S55°E, the same logarithmic function was subtracted, fitting only the amplitude to the time series. The correction procedure was similar for the 2004 event, but the postseismic deformation overlapped the 2000–2005 long-term SSE in the Tokai region (Suito and Ozawa 2009). Therefore, assuming that the postseismic displacement was proportional to the coseismic displacement, the amplitude coefficient was estimated from the coseismic displacement of the stations near the epicenter.

At this point in the procedure, we evaluated the cross-correlation between the displacement time series and a 3-year ramp function, which consisted of a linear slope over a 1-year period connecting flat slopes in the previous year and the following year. For more details, see Kobayashi (2017).

Estimation of slip distribution

The slip vector on the plate interface was estimated from the surface unsteady north–south and east–west horizontal displacements using the inversion technique of Yabuki and Matsu’ura (1992), with some modifications based on the formulation of Okada (1992). This inversion technique represents the slip distribution as a superposition of B-spline functions. In this study, slip vectors were represented by linear combinations of boxcar functions, and the values of coefficients were estimated by using a Bayesian model with a smoothness constraint for regularization. Scale factors were chosen for the covariance and smoothness parameters that minimized Akaike’s Bayesian information criterion (ABIC, Akaike 1980). Slip vectors were calculated for a 12 × 12 grid of rectangular sources on the plate interface, using the plate interface configuration of Hirose et al. (2008). The grid intervals and rectangle dimensions were about 10.1 km along the strike of the Nankai Trough and about 9.8 km along the dip direction. The slip was set to zero outside the grid. Dip angles were derived from the plate configuration by using continuous curvature splines in tension (Smith and Wessel 1990). The slip direction was constrained within 45° from the direction of plate subduction, using the nonnegative least squares method (Lawson and Hanson 1974).

Results and discussion

The spatiotemporal distribution of the correlation between displacement in the S55°E direction and the ramp function from March 1996 to February 2019 is shown in Fig. 2b, which displays periods when the correlation coefficient was greater than 0.6 and the 2-year displacement was greater than 2 mm. Note that the results for latest 1.5 years may change as later data are included. After the Tokai long-term SSE (labeled T in Fig. 2b) subsided in 2016, displacement with high correlation migrated to the vicinity of the Shima Peninsula (labeled S) and continued there until July 2018. Note that the displacement with high correlation shown in Fig. 2b are not directly related to slip on the plate boundary beneath there. For example, high correlation related to 2000 Tokai long-term SSE extended to the Shima Peninsula. However, no slip was estimated in the Shima Peninsula (Ozawa et al. 2002). Displacements in the direction S55°E and correlation coefficients between the displacement and the ramp function at a point near the Shima Peninsula are shown in Additional file 1: Figure S1.

Figure 3a shows the horizontal unsteady displacement on the Shima Peninsula from April 2017 to October 2018 with respect to the fixed region. In the procedure of the previous section, the postseismic deformation of the 2011 Tohoku earthquake is subtracted from the mean displacement time series for each grid, but it is not subtracted from the displacement of each station. To eliminate the postseismic deformation from the earthquake, displacement of each station from January 2016 to April 2017 was subtracted. According to the approximate function of postseismic deformation used by Kobayashi (2017), the postseismic displacement rates during these two periods differed by less than 1%. The unsteady displacement consisted of southeastward horizontal movements of 7–8 mm; there was no systematic displacement in the vertical component. Baseline lengths between two stations on the Shima Peninsula and two stations to the northwest showed slow extension of 7–8 mm from spring 2017 to autumn 2018 (Fig. 4). Because the unsteady displacement continued for more than a year and no other long-term slow slips occurred around this area during this period, it appears to correspond to a single long-term SSE.

Fig. 3
figure 3

Map of the Shima Peninsula showing horizontal unsteady displacement and estimated slip distribution. a Horizontal displacements for the period April 2017 to October 2018 used for estimating fault slip. Black arrows indicate observations, and white arrows indicate calculated values. Ellipses at the tips of the black arrows show standard deviations of the coordinates. b Estimated slip distribution on the plate interface. Arrows indicate the motion of the overriding plate with respect to the subducting Philippine Sea plate. Ellipses at the tips of the arrows indicate estimation errors. Solid squares are locations of GNSS stations. Dashed contours show the depth of the top of the subducting plate (Hirose et al. 2008)

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Fig. 4
figure 4

Baseline lengths (dots) between pairs of GNSS stations plotted in Fig. 1b. The upward direction indicates extension. The shaded period corresponds to the period of Fig. 3a

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The slip distribution on the plate interface, estimated from the unsteady displacements using the inversion technique, is shown in Fig. 3b. The maximum slip, about 3 cm, was located beneath the Shima Peninsula. The plate interface at that location is about 25–30 km deep, which is consistent with the depths of other long-term SSEs along the Nankai Trough. The estimated moment magnitude corresponding to the total slip was 6.5, assuming a rigidity of 40 GPa.

Long-term rate of moment release of the short-term SSEs along the Nankai Trough is roughly constant (Nishimura et al. 2013). However, recurrence interval of the short-term SSEs decreased during the long-term SSEs of the Bungo Channel and the Tokai region (Hirose and Obara 2005; Hiramatsu et al. 2008). Short-term SSEs occurred around the Shima Peninsula, during this long-term SSE period and the steady-state period. Therefore, we confirm whether the short-term SSE has been activated during the long-term SSE of the Shima Peninsula. Ochi et al. (2016a, b, 2017a, b, 2018a, b, 2019) estimated rectangle faults of short-term SSEs using the strain, tilt and groundwater level data. We calculated displacements by the formulation of Okada (1992) using these parameters of the rectangle faults. We adopted the same period length just before the long-term slow slip period as the steady-state period. Maximum horizontal displacements on the Shima Peninsula caused by the short-term SSEs for the period of the long-term SSE (from April 2017 to October 2018) and the steady-state period (from October 2015 to April 2017) were 7.0 mm and 4.5 mm, respectively. The fault parameters of the short-term SSEs for each period are shown in Additional file 2: Tables S1 and S2. The short-term SSEs around the Shima Peninsula became active during the period of the long-term SSE; therefore, we estimated the contribution of the short-term SSE activation included in the estimated slip of the long-term SSE. The displacement due to the short-term SSE activity of the steady-state period has already been subtracted as a part of displacement velocity. Then, the contribution of the activated short-term SSEs during the long-term SSE period was 2.5 mm. Therefore, remaining unsteady displacement is considered to be due to the long-term SSE. The estimated moment magnitude corresponding to the long-term SSE alone was 6.4. Thus, we report the first documentation of a long-term SSE on the Shima Peninsula.

This updated record of SSEs may have implications for the locked region of the plate interface. Figure 2b shows a smooth connection of grid cells with moderate correlation (near 0.6) (labeled (1) in Fig. 2b) between the 2017–2018 long-term SSE documented here (labeled S) and the Tokai long-term SSE from 2013 to 2016 (labeled T). There are some similar connections between SSEs in the figure. For example, it shows a connection (labeled (2)) between the 2003 long-term SSE in the Bungo Channel (labeled B1) and the 2005 western Shikoku long-term SSE (labeled WS) (Kobayashi 2010; Takagi et al. 2016). Similar connection (labeled (3)) is also seen between the 2010 long-term SSE in the Bungo Channel (labeled B2) and the 2014–2016 long-term SSE in the Kii Channel (labeled K). Note that during the latter event, the area of highest correlation appears to have migrated through central and eastern Shikoku, where no long-term SSE have been detected in GNSS. If the areas of moderate correlation imply small amounts of slip on the plate interface, it may be that small slow slips are being transmitted. Simulations of the seismic cycles along the Nankai Trough based on a rate- and state-dependent friction law indicate that locked regions become smaller with time during the interseismic period (Hirose and Maeda 2013; Nakata et al. 2014). The weakening of coupling in the locked region may appear as new areas of slow slip as well as changes of the steady state. Ozawa et al. (2012) revealed a preceding aseismic slip of the 2011 Tohoku earthquake. Koulali et al. (2017) pointed out an increasing amplitude of the SSE toward the 2016 Te Araroa earthquake, New Zealand. The expansion of the long-term SSE slip area and the increase in slip amount may cause stress changes in the adjacent rocked zone and promote the occurrence of a large earthquake. To explore this issue further, it is important to characterize the spatiotemporal transitions of long-term SSEs.

Conclusion

We used GNSS daily coordinate data to detect unsteady displacement on the Shima Peninsula, central Japan. This displacement represents the first long-term SSE to be observed on the Shima Peninsula from spring 2017 to autumn 2018. Horizontal unsteady displacement with respect to the fixed region showed southeastward movements of 7–8 mm on the Shima Peninsula during this SSE. Baseline lengths between stations on the Shima Peninsula and stations to the northwest indicated slow extension starting in 2017. The slip distribution on the plate interface, estimated using the inversion technique, indicates moment release beneath the Shima Peninsula equivalent to an earthquake with a moment magnitude of 6.4.