Volume 123, Issue 9 p. 8125-8142
Research Article
Open Access

Slip-Deficit Rate Distribution Along the Nankai Trough, Southwest Japan, With Elastic Lithosphere and Viscoelastic Asthenosphere

Akemi Noda

Corresponding Author

Akemi Noda

National Research Institute for Earth Science and Disaster Resilience, Tsukuba, Japan

Correspondence to: A. Noda,

[email protected]

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Tatsuhiko Saito

Tatsuhiko Saito

National Research Institute for Earth Science and Disaster Resilience, Tsukuba, Japan

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Eiichi Fukuyama

Eiichi Fukuyama

National Research Institute for Earth Science and Disaster Resilience, Tsukuba, Japan

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First published: 29 August 2018
Citations: 42

Abstract

In southwest Japan, great earthquakes have occurred on the plate interface along the Nankai trough with a recurrence time of approximately 100 years. Most previous studies estimated slip deficits on the seismogenic zone from interseismic Global Navigation Satellite System (GNSS) velocity data assuming slip-response functions for an elastic medium. The observed surface velocities, however, include effects of viscoelastic relaxation in the asthenosphere caused by slip motion associated with seismic cycles. If the elastic responses cannot adequately approximate the deformation of the viscoelastic medium, the results of an inversion analysis could be biased. If the recurrence interval is greater than the effective relaxation time in the elastic-viscoelastic layered structure, we were able to formulate an inverse problem for the estimation of slip-deficit rates from GNSS velocity data with completely relaxed slip-response functions for a later stage of the seismic cycle. Analyzing surface velocity data from GNSS daily coordinate data between March 2005 and February 2011 together with seafloor geodetic data, we estimated the slip-deficit rate distribution by the strain data inversion method. There were significant differences between the results using elastic and completely relaxed responses. Although the result with elastic responses shows a high coupling zone in the coastal region, it was located trenchward when completely relaxed responses were used. We found that the peak slip-deficit rate increases as the thickness of the lithosphere becomes thinner. Moreover, we succeeded in appropriately separating elastic-viscoelastic deformation due to plate coupling from rigid block motion.

Key Points

  • Slip-deficit rates along the Nankai trough were estimated from GNSS data and seafloor geodetic data, using elastic-viscoelastic structures
  • When viscoelastic relaxation is included, a high slip-deficit rate zone was located more trenchward than those in pure elastic response
  • Our strain data inversion succeeded in decomposing observed displacement into the contributions from plate coupling and rigid block motions

1 Introduction

In southwest Japan, the oceanic Philippine Sea (PHS) plate is descending beneath the continental Eurasian (EU) or Amur (AM) plate along the Nankai trough (Figure 1). In Figure 1, arrows show the relative velocities of the PHS plate to the EU/AM plate calculated from global plate motion models: NUVEL-1A (DeMets et al., 1994), MORVEL (DeMets et al., 2010), and GSRM v.2.1 (Kreemer et al., 2014). The NUVEL-1A model, estimated from earthquake slip directions along subduction zones of the circum-PHS plate, is slightly smaller than the other models (approximately 4.5 cm/year). The MORVEL and GSRM v.2.1 models are based on Global Positioning System (GPS) measurements and are almost the same (approximately 6.0 cm/year). Figure 1 also shows the 3-D geometry of the upper boundary of the descending PHS plate, which was determined from the topography of ocean floors and the hypocenter distributions of earthquakes (Hashimoto et al., 2004).

Details are in the caption following the image
Tectonic setting in southwest Japan. The iso-depth contours (10-km interval) represent 3-D geometry of the plate interface between the Eurasian plate and the Philippine Sea plate. Arrows indicate the convergence velocity vectors of the PHS plate to the EU/AM plate in three plate motion models: NUVEL-1A, GSRM v2.1, and MORVEL. Muroto Promontory, Cape Shiono, and Lake Hamana are indicated by short black arrows. EU = Eurasian; AM = Amur; PHS = Philippine Sea.

At the plate interface, great thrust-type earthquakes have repeatedly occurred with a recurrence time of approximately 100 years (Ando, 1975). These large earthquakes excited strong ground motion and tsunamis, which caused serious damage around this region (e.g., Furumura & Saito, 2009; Kim et al., 2016). The latest M > 8 events are known as the 1944 Tonankai earthquake and the 1946 Nankai earthquake. The occurrence of interplate earthquakes corresponds to the release of accumulated stress, which is caused by an interseismic plate coupling (slip deficit, Savage & Prescott, 1978; Spence & Turcotte, 1979).

A nationwide dense Global Navigation Satellite System (GNSS) network in Japan, GNSS Earth Observation Network (GEONET), has been operated by the Geospatial Information Authority of Japan since 1996, and we can now monitor the crustal movements of the Japan Islands due to increase in the slip deficit at the plate interface. To reveal the process of stress accumulation leading to the great earthquakes along the Nankai trough, there have been a large number of studies to estimate precise interseismic slip-deficit rates at the plate interface using GNSS displacement rate (velocity) data. Most previous studies used displacement response functions for an elastic half-space model to invert the slip-deficit rate distribution from interseismic GNSS velocity data (e.g., Loveless & Meade, 2016; Yokota et al., 2016; Yoshioka & Matsuoka, 2013). However, the interseismic GNSS data actually include not only instantaneous elastic responses due to the slip deficit but also the effect of viscoelastic relaxation in the asthenosphere caused by the slip history of seismic cycles on the plate interface. The slip history is composed of coseismic slip motion in interplate earthquakes and aseismic slip motion during interseismic periods. If the elastic responses cannot adequately approximate the deformation of the viscoelastic medium due to the slip motions, the results of the inversion analysis could be biased.

Savage and Prescott (1978) kinematically simulated the surface deformation produced by seismic cycles on a transform fault in an elastic lithosphere overlying a viscoelastic asthenosphere, and they compared it with that in an elastic half-space model. The elastic-viscoelastic layered structure can be approximated by the elastic half-space model if the recurrence interval of earthquakes T is less than the characteristic time τM(=2η/μ). The parameters μ and η are the rigidity and the viscosity of the asthenosphere, respectively. However, the difference between the two models becomes evident for T > 5τM.

In the PHS subduction zone around Japan, by modeling postseismic deformation after large earthquakes, the viscosity of the asthenosphere is estimated to be relatively small (approximately 5 × 1018 Pa s; e.g., Matsu'ura & Iwasaki, 1983; Suito, 2017) compared to the global average of 4 × 1019 Pa s (Cathles, 1975). Assuming the recurrence interval of great interplate earthquakes along the Nankai trough to be 100 years and the rigidity in the asthenosphere to be 60 GPa, the ratio T/τM is approximately 19. This suggests that the stress relaxation in the asthenosphere strongly affects interseismic deformation around the Nankai trough, although the mechanism is thrusting, and we need to take the viscoelastic response into account in modeling the surface deformation.

The most orthodox way to estimate the slip-deficit rate distribution with consideration of the effects of viscoelastic relaxation is to model all slip histories on the plate interface that affect the observed geodetic data. Fukahata et al. (2004) modeled two recent seismic cycles of Nankai earthquakes, assuming the completely same slip history in each cycle. They estimated the temporal changes of a one-dimensional slip distribution with respect to the distance from the trough during one seismic cycle from geodetic data using viscoelastic slip-response functions. By using a method similar to that described by Fukahata et al. (2004), Ito and Hashimoto (2004) estimated the two-dimensional slip history during one seismic cycle in the Nankai region. However, these inversion methods have some problems: the time series data of geodetic measurements covering one seismic cycle (generally for an order of 100 years) are needed, and the recurrence periods are actually not the same.

In contrast, Noda et al. (2013) proposed a method to include the effect of a viscoelastic asthenosphere by using completely relaxed slip-response functions. Using a similar method, Hashimoto et al. (2009) analyzed GNSS data for an interseismic period of 1996–2000 to estimate the interseismic slip-deficit rate distribution along the Nankai trough. However, since the GNSS measurements are restricted to the land surface, the spatial resolution was low in the offshore region. We can now use seafloor geodetic data obtained with GPS-Acoustic technique to improve the resolution in the offshore region, although the accuracy of positioning is lower than that of GNSS (e.g., Yokota et al., 2016). Additionally, the density and accuracy of GEONET were improved thereafter (e.g., The GEONET Group, Geographical Survey Institute, 2004).

Here we estimated the precise slip-deficit rate distribution on the plate interface along the Nankai trough based on Noda et al. (2013). We took into account the viscoelastic relaxation in the asthenosphere and investigated how the lithosphere-asthenosphere structure model affects the estimation of a slip-deficit rate distribution. First, we obtained GNSS velocity data from time series analysis of GNSS daily coordinate data for four periods, by which the postseismic stress relaxation after the 1944 Tonankai and 1946 Nankai earthquakes decays sufficiently. We then analyzed the highly accurate GNSS velocity data observed on land together with the newly obtained seafloor geodetic data in order to estimate an interseismic slip-deficit rate distribution with completely relaxed responses. Finally, we compared the estimated slip-deficit rate distribution with those obtained with different lithosphere-asthenosphere structure models.

2 The Formulation of an Inversion Analysis Taking Viscoelastic Relaxation in the Asthenosphere Into Account

Based on Noda et al. (2013), we derived the following expressions for the interseismic surface velocities that result from the increase of the slip deficit at a plate interface. We defined a coordinate on the plate interface by ξ = (ξ1, ξ2). The whole plate interface Σ is divided into a seismogenic region Σs and the remaining steady slip region Σ − Σs. We represent the slip motion w(ξ, t) at a point ξ and time t(>0) by the superposition of steady plate subduction at the slip rate of the relative plate motion Vpl(ξ), the steady slip-deficit at a constant rate urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0001, and the coseismic slip wk(ξ):
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0002(1)
Here H(t) denotes the Heaviside function, and wk(ξ) and Tk represent the coseismic slip distribution and the occurrence time of the kth earthquake, respectively. The dot in urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0003 indicates differentiation with respect to time. Assuming the recurrence interval to be longer than the effective relaxation time (τe) of the elastic-viscoelastic layered structure, we obtained the surface velocities urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0004 at a point x and time t due to the slip deficit at the plate interface:
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0005(2)
where Ui(x, t; ξ, t) denotes the ith component of the quasi-static displacement response functions at a point x and time t due to unit slip at a point ξ and time t. The response functions Ui(x, t; ξ, t) approach certain constants urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0006 as stress relaxation in a viscoelastic asthenosphere proceeds. We refer to urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0007 as a completely relaxed response hereafter. Although we consider a variable recurrence time ΔTk = Tk − Tk − 1 in contrast to Noda et al. (2013), equation 2 is still valid. The first and second terms on the right side of equation 2 are surface velocities caused by a steady slip all over the plate interface and a steady slip deficit in the seismogenic region, respectively. As shown by Matsu'ura and Sato (1989), these surface velocities due to steady slip motions become constant rates in time, and they are equal to the completely relaxed responses to the slip rates (see Appendix A). The third term represents the surface velocities caused by postseismic stress relaxation in the asthenosphere and varies with time.
In the case of convergent plate boundaries, the first term necessarily causes intrinsic crustal deformation (Hashimoto et al., 2004; Matsu'ura & Sato, 1989; Sato & Matsu'ura, 1992). When all observation stations are on the same plate, however, the deformation due to steady subduction is smaller than that due to steady slip deficits by an order of magnitude (Noda et al., 2013). We hence ignore the first term in the present study. The third term due to postseismic stress relaxation on the asthenosphere exponentially decays with time after the earthquake. Finally, if we can approximate the third term to be zero, we can formulate a linear inverse problem of estimating unknown steady slip-deficit rates from the observed surface velocities with the completely relaxed responses:
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0008(3)

The representation of interseismic velocities in equation 3 is almost the same form as that used in most studies for the estimation of a slip-deficit distribution except that the completely relaxed response functions are used instead of elastic response functions.

3 Observation Data

We used the daily coordinates of GNSS stations (GEONET F3 Solution) in southwest Japan from March 1996 to February 2011. We excluded the daily coordinate data after May 2011 to avoid the coseismic and postseismic deformation caused by the Mw 9.0 Tohoku-Oki earthquake. We extracted long-term crustal movements from the GNSS time series data for four overlapping periods of 6 years each (I, from March 1996 to February 2002; II, from March 1999 to February 2005; III, from March 2002 to February 2008; and IV, from March 2005 to February 2011). For details, see Appendix B.

The white vectors in Figure 2a show the surface horizontal velocity data in period IV, calculated from only the second term in equation B1. Fixing the location of Fukue station (an open triangle in Figure 2), we obtained significant northwestern movements around the Pacific coast, which suggests an accumulation of elastic strains caused by a strong locking at the EU-PHS plate interface. The maximum horizontal velocity reaches 49 mm/year.

Details are in the caption following the image
Time changes of interseismic horizontal velocities obtained from GNSS and GPS-Acoustic observations in southwest Japan. (a) Surface velocities in period IV (from March 2005 to February 2011). (b–d) Residual velocities obtained by subtracting surface velocities in period IV (Figure 2a) from that in period I (from March 1996 to February 2002), that in period II (from March 1999 to February 2005), and that in period III (from March 2002 to February 2008). White arrows: The relative velocity vectors at GNSS stations to the reference point Fukue station, which is indicated by an open triangle in each panel. Black arrows: The seafloor velocities with respect to the Amur plate (Yokota et al., 2016). Thick black line: The Nankai trough. GNSS = Global Navigation Satellite System; GPS = Global Positioning System.

According to equation 2, while the surface velocities caused by the steady subduction and the steady slip deficit are constant, the velocities caused by postseismic stress relaxation in the asthenosphere vary with time. To compare the horizontal velocities in other time periods with those in Figure 2a, we show the residual horizontal velocities obtained by subtracting the horizontal velocities in Figure 2a, as reference velocity data, from horizontal velocities in periods I, II, and III in Figures 2b–2d, respectively. We could not detect any systematic temporal changes in the horizontal velocities in Figures 2b–2d, except for crustal movements due to the Tokai slow slip (from October 2000 to July 2005, Suito & Ozawa, 2009) around Lake Hamana (Figure 2c), which almost ended before period IV. Figure 3a shows the surface vertical velocity data in period IV. Figures 3b-3d show the residual vertical velocities obtained by subtracting the vertical velocities in period IV from the vertical velocities in periods I, II, and III, respectively. We could not identify any systematic temporal changes in the vertical velocities, although the differences in Figures 3b–3d were greater than those in the horizontal velocities shown in Figures 2b–2d.

Details are in the caption following the image
Time changes of interseismic vertical velocities obtained from GNSS observations in southwest Japan. (a) Surface velocities obtained in period IV (from March 2005 to February 2011). (b–d) Residual velocities obtained by subtracting surface velocities in period IV (Figure 3a) from that in period I (March 1996 to February 2002), that in period II (from March 1999 to February 2005), and that in period III (from March 2002 to February 2008). White and gray bars: The uplift and subsidence rates relative to the average value in the period, respectively. GNSS = Global Navigation Satellite System.

In addition, assuming the viscosity and the rigidity of the asthenosphere to be 5 × 1018 Pa s and 60 GPa, respectively, the characteristic relaxation time in the asthenosphere (η/μ) becomes approximately 2.6 years, and the effective relaxation time of the lithosphere-asthenosphere system is approximately 30 years. Hence, the postseismic stress relaxation after the 1944 Tonankai and 1946 Nankai earthquakes decayed sufficiently by 1996. For reference, we numerically evaluated the postseimic deformation assuming a simple source model in supporting information Text S1 and Figures S1 and S2.

In the present study, we analyzed the horizontal and vertical velocity data in period IV with the completely relaxed responses following equation 3, and we estimated the interseismic slip-deficit rate distribution along the Nankai trough. In addition to GNSS velocity data, we used horizontal velocity data on the seafloor (the black vectors in Figure 2a) reported by Yokota et al. (2016). The seafloor velocity data were obtained from GPS-Acoustic observations, which started after 2006.

The interseimic horizontal velocity data (Figure 2) include not only crustal deformation due to slip deficits at the plate interface but also some rigid translation and rotation motions of crustal blocks. Crustal movements due to rigid block translation and rotation cannot be theoretically explained by dislocation sources at the upper surface of the subducting PHS plate. The movements thus behave as coherent noise in inversion analysis and will seriously bias the inversion result. In this study we employed a strain data inversion method (Noda et al., 2013), in which we removed the block translation and rotation motions from the observed velocity data by the transformation of horizontal displacement vectors into horizontal strain tensors to estimate unbiased inversion result. Following Noda et al. (2013), we generated the optimum triangular mesh of the GNSS stations with the Delaunay triangulation method and transformed the 620 horizontal velocities into the average horizontal strain rates of 1,114 triangular elements composed of three adjacent GNSS stations as shown in Figures 4a–4c. We show invariants of horizontal strain tensors in Figures 4a–4c, although we actually used three horizontal strain components ε11, ε12, ε22(1, East; 2, North) for the inversion analysis. To analyze the vertical velocities in the same framework as the horizontal velocities, we also transformed the 605 vertical velocities urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0009 (J = A, B) into 1,703 tilt rates dAB between two adjacent GNSS stations (i.e., two vertices of a triangular element), PA and PB as
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0010(4)
Details are in the caption following the image
The horizontal strain rates and tilt rates transformed from GNSS velocity data from March 2005 to February 2011. (a) Maximum shear strain rate. (b) Dilatation rate. (c) Direction of maximum horizontal contraction. (d) Tilt rate. The arrows indicate downward tilt direction. The EW and NS components of tilt rate are shown in supporting information Figure S3. GNSS = Global Navigation Satellite System.

where lAB denotes the baseline length between the stations. We show the tilt rates in Figure 4d.

4 The Method of Inversion Analysis

4.1 Model Setting

In Figure 5, the gray iso-depth contours show the 3-D geometry of the EU-PHS plate interface (Hashimoto et al., 2004). The interseismic slip-deficit rate distribution urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0011 on the EU-PHS plate interface Σ(ξ) is represented by the superposition of known basis functions Skl(ξ). Specifically, for each orthogonal component of the slip-deficit vector, we represent the distribution by the superposition of 249 normalized bicubic B splines, Skl(ξ) = Nk(ξ1)Nl(ξ2), with a local support Δξ= 32 km in both the directions ξ1 (east) and ξ2 (north), as shown in Figure 5a. The cubic spline function is also plotted in Figure 5b. The slip deficit rate is given by
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0012(5)
Details are in the caption following the image
Plate interface in southwest Japan and model setting. (a) The gray isodepth contours (10-km intervals) represent the Eurasian-Philippine Sea plate interface. Open circles: The central points of bicubic B-splines for inversion analysis. The light gray lines connecting circles show the orthogonal directions of k and l. (b) Graphic representation of cubic B-spline functions Nk (k = 1, …,5). The black lines show basis functions. We can represent slip deficits near the updip edge of the plate interface by using the basis functions overlapping the trough, in which the slip deficit on the seaward side of the trough is set at zero (broken lines). The sum of basis functions, shown by the gray bold line, represents a smooth slip-deficit rate distribution on the landward side of the trough and a discontinuous change to zero at the trough.

where urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0013 is the unknown coefficient, and K and L represent the numbers of functions, Nk(ξ1) and Nl(ξ2). The superscripts P and C denote the primary and complementary components, respectively. The primary component is parallel to the direction of the steady slip, and the complementary component is perpendicular to it. To permit slip deficits at the trough, the basis functions are distributed not only landward but also seaward beyond the trough, as shown in Figure 5b.

The crust-mantle rheological structure is modeled by a horizontally layered structure composed of an elastic surface layer (the lithosphere) and a Maxwell-type viscoelastic substratum (the asthenosphere), as shown in Table 1. The rigidity and bulk modulus are compatible with P wave and S wave velocities beneath southwestern Japan (e.g., Matsubara et al., 2009). We assumed the thickness of the lithosphere to be 60 km. We computed completely relaxed responses by using a semianalytical solution for linear viscoelastic medium under gravity derived by Fukahata and Matsu'ura (2006). After stress relaxation is completed in the asthenosphere, the lithosphere behaves like an elastic plate floating on water, independent of the viscosity of the asthenosphere.

Table 1. The Structural Parameters of the Lithosphere-Asthenosphere Model
Thickness (km) Density (kg/m3) Rigidity (GPa) Bulk modulus (GPa) Viscosity (Pa s)
Lithosphere 60 3000 40.0 66.7
Asthenosphere 3400 60.0 130.0 5 × 1018

4.2 Strain Data Inversion

Following the method of strain data inversion (Noda et al., 2013), the observation equation can be represented in the following vector form:
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0014(6)
with
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0015(7)
where dh is an Nh × 1 dimensional (Nh= 3 × the number of triangular elements) horizontal strain rate data vector; dv is an Nv × 1 dimensional (Nv= the number of pairs of adjacent GNSS stations) tilt rate data vector; a is an M × 1 dimensional (M = 2KL) model parameter vector composed of aP with the elements urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0016 and aC with the elements urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0017 (k = 1, …, K; l = 1, …, L); Hh and Hv are Nh × M and Nv × M dimensional coefficient matrices, respectively; and e is a (Nh + Nv) × 1 dimensional error vector. The data error e obeys Gaussian distribution, e~N(0, E). We solved equation 6 for the model parameter a from the horizontal strain rate data and tilt rate data shown in Figure 4.
The covariance matrix E of data error e is expressed in the following form:
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0018(8)
with
urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0019(9)
where E is a covariance matrix of observation errors of GNSS velocity data, R is a matrix to transform horizontal velocity vectors and vertical velocities into average strain rate tensors for individual triangle elements and tilt rates for individual pairs of adjacent GNSS stations, respectively, and D is a diagonal matrix representing the relative weight of individual data, which consists of a diagonal matrix Dh for strain rate tensors and a diagonal matrix Dv for tilt rates.

The first and second terms on the right side of equation 8 express random and systematic errors, respectively, and the parameter c2 controls the relative weight of systematic errors to random errors. The random errors come mainly from observation errors, and the systematic errors come mainly from discrepancy between the theoretical models used for inversion analysis and the real Earth. For the covariance matrix of observation error E, we assumed the variances of the horizontal velocities at GNSS stations, the vertical velocities at the GNSS stations, and the horizontal velocities on the seafloor to be σ2, 32σ2, and 102σ2 (mm/year)2, respectively, and the nondiagonal elements to be zero. Here σ2 is an unknown scale factor to be determined from observed data through the inversion analysis. For the velocity data at the GNSS station, the ratio of the variances to σ2 is based on Nishimura et al. (2004). Although that study investigated GNSS data in northeastern Japan, the ratio of observation errors would be applicable to the current study because the data are from the same observation network. For horizontal velocities on the seafloor, the ratio of the variances to σ2 is based on Yokota et al. (2016).

For the matrix D, which regulates the structure of the covariance matrix of systematic errors, we assumed the matrix Dh to be σ2I (I is an identity matrix). This means that the systematic errors included in the horizontal strain rates are constant; that is, the systematic errors in horizontal velocities increase with the distance from a dislocation source at the plate interface. As demonstrated by Muto et al. (2016), the vertical displacements are much more sensitive to the difference of structure models compared to the horizontal displacements. We thus assumed the variance of the systematic errors of tilt rates to be 22 times larger than that of horizontal strain rate tensors, Dv = 22σ2I. During the inversion analysis, we determined the appropriate value of c2 so that the optimum value of σ2 becomes consistent with the variance of observation errors of GNSS horizontal velocities on the order of 10−2 (mm/yr)2 (e.g., Nishimura et al., 2004).

We then constructed a Bayesian model with direct and indirect prior information. The direct prior information bounds the complementary components of slip-deficit rate, perpendicular to the direction of plate subduction (Matsu'ura et al., 2007), and the indirect prior information constrains the roughness of slip-deficit rate distribution (Yabuki & Matsu'ura, 1992). We used the NUVEL-1A model for the direction of plate subduction. We objectively determined the relative weights of the direct and indirect prior information to the observed data with Akaike's Bayesian Information Criterion (Akaike, 1980).

5 Slip-Deficit Rate Distribution

Applying the inversion method described in section 4.2 to the horizontal strain rate data and tilt rate data, we estimated the optimum slip-deficit rate distribution along the Nankai trough. The optimum value of σ2 is estimated to be 4.2 × 10−2 (mm/yr)2 for c2 = 10−7 (m−2). Table 2 shows the averages of the random errors and systematic errors calculated from the optimum values of σ2 and c2. The systematic errors are greater than the random errors in the horizontal strain rates and tilt rates from the GNSS data, whereas the systematic errors and the random errors of the horizontal strain rates on the seafloor are comparable.

Table 2. The Averages of Random Errors, Systematic Errors, and Total Errors Calculated From the Inversion Result of σ2 and c2
Random errors (year−1) Systematic errors (year−1) Total errors (year−1)
Horizontal strain rates from the GNSS data 1.6 × 10−8 6.5 × 10−8 8.1 × 10−8
Tilt rates from the GNSS data 4.7 × 10−8 1.3 × 10−7 1.8 × 10−7
Horizontal strain rates on the seafloor 7.8 × 10−8 6.5 × 10−8 1.4 × 10−7
  • Note. GNSS = Global Navigation Satellite System.

We show the slip-deficit rate distribution on the EU-PHS plate interface in Figure 6a, together with the estimation errors in Figure 6b. In Figure 6a, the high slip-deficit rate zone (>40 mm/year) extends broadly from the Tokai region to the Nankai region along the plate boundary. The depth range of the high slip-deficit rate zone is approximately 0–30 km. Figure 6a shows two peaks of slip-deficit rate, the 10-km-deep peak off the Muroto Promontory (Figure 1) and the 5-km-deep peak off the Kii Peninsula (Figure 1). The slip-deficit rate in the Tokai region is somewhat smaller than that in the Nankai and the Tonankai regions.

Details are in the caption following the image
The interseismic slip-deficit rate distribution and corresponding estimation errors. (a) The slip-deficit rate distribution estimated from strain rate data and tilt rate data. White arrows: The slip-rate vectors. The blue and red color scales: Slip-deficit and slip-excess rates, respectively. The contours are at intervals of 20 mm/year. (b) Estimation errors (1σ) of the slip-deficit rate distribution in Figure 6a. The contours are shown at intervals of 5 mm/year. The outside of the model region is shown in gray.

The estimation error increases with the distance from the observation stations on land and reaches approximately 25 mm/year near the trough. The local minimums of the estimation error offshore correspond to the locations of seafloor stations (Figure 2a). The results in Figures 6a and 6b suggest that the high slip-deficit rate zone from the Tokai region to the Nankai region is reliable. The slip-excess and slip-deficit rates southeast from Kyushu, however, might not be reliable because of the large estimation errors. The slip-excess rates southeast of Kyushu might be attributed to expansion around the Sakurajima volcano, which cannot be properly modeled by the slip model used in the present analysis. The plate convergence rate is approximately 60 mm/year in the MORVEL and GSRM v2.1 models (Figure 1). The peak value of the slip-deficit rates reaches approximately 80 mm/year in the Nankai region, which is consistent with the convergence rate if the 2σ error is taken into account. This suggests a fully locked state of the plate interface.

Coseismic slip distributions in the 1944 Tonankai and 1946 Nankai earthquakes were estimated from geodetic data (e.g., Sagiya & Thatcher, 1999; Yabuki & Matsu'ura, 1992), tsunami data (e.g., Baba & Cummins, 2005), seismic data (Ichinose et al., 2003), and both of geodetic and seismic data (Murotani et al., 2015). The coseismic slip distributions estimated from geodetic data, which are a superposition of coseismic slip from both the 1944 and the 1946 earthquakes, show two high-slip zones off Shikoku region and off Kii Peninsula. The tsunami and seismic data analyses, which can resolve the two events, show that the demarcation point between source regions of the two events is off Cape Shiono (Figure 1). The two peaks of the slip-deficit rate off Muroto Promontory and off Kii Peninsula (Figure 6a) correspond to the high-slip zones estimated from geodetic data, while the estimated high slip-deficit rate zones are located more trenchward than the coseismic high-slip zones. The peaks of the slip-deficit rates close to the trench are rather compatible with the inversion results of coseismic slip distribution using seismic data. The source region in the 1944 Tonankai earthquake does not reach the high slip-deficit zone in the Tokai region, but the 1854 Ansei-Tokai earthquake is considered to have occurred in the Tokai region (Seno, 2012).

We did not observe any discontinuity of the slip-deficit rate off Cape Shiono (Figure 6a). However, the shear stress changes calculated from the slip-deficit rate distribution show a bimodal pattern off Kii Peninsula (supporting information Figure S4). In simulations of spontaneous rupture, frictional heterogeneities off Kii Peninsula are introduced to reproduce the rupture patterns observed at the Nankai trough (e.g., Hok et al., 2011; Hyodo et al., 2016). The combination of heterogeneities in accumulated shear stress and frictional properties off Kii Peninsula might have behaved as a rupture propagation barrier between the Tonankai and Nankai regions.

Around the Bungo Channel (Figure 1), aseismic interplate slip events (which are called long-term slow slip events, or L-SSEs) occurred repeatedly at the interval of approximately 6 years. The moment magnitude and the maximum slip of the L-SSEs in 1997, 2003, and 2010 events range from Mw 6.8 to 6.9 and from 15 to 19 cm, respectively (Yoshioka et al., 2015). The cumulative slip deficit for 6 years (20–40 mm/year × 6 year = 12–24 cm) calculated from the slip-deficit rate beneath the Bungo Channel in Figure 6a is consistent with the maximum slip of the L-SSEs, which suggests that most of the cumulative slip deficits beneath the Bungo Channel are released by aseismic slip in the L-SSEs.

6 Discussion

6.1 Random and Systematic Errors

The horizontal strain rates and tilt rates computed from the optimum slip-deficit rate distribution are shown in Figure 7. From the comparison between Figures 4a–4c and 7a–7c, we can see that the estimated slip-deficit rate distribution explains the observed strain rate data well. For tilt rates, southward ground down in the coastal regions jutting out into ocean (around Lake Hamana, Cape Shiono, and Muroto Promontory) and northward ground down in the inland area (around the Seto Inland Sea and base of the Kii Peninsula) in Figure S3b are explained by the computed tilt rates in Figure S5b.

Details are in the caption following the image
The computed strain rates and tilt rates from the interseismic slip-deficit rate distribution shown in Figure 6a. (a) Maximum shear strain rate. (b) Dilatation rate. (c) Directions of maximum horizontal contraction. (d) Tilt rate. Arrows: The downward tilt direction. The EW and NS components of the tilt rate are shown in supporting information Figure S5.

Figure 8 shows the residual strain rates and tilt rates obtained by subtracting the theoretical strain rates and tilt rates in Figure 7 from the respective observed rates shown in Figure 4. The root-mean-squares (RMSs) of the residual strain rates at the GNSS stations, the residual tilt rates at the GNSS stations, and the residual strain rates on the seafloor are 8.2 × 10−8 (year−1), 1.1 × 10−7 (year−1), and 8.1 × 10−8 (year−1), respectively. The RMSs of the residual strain rates and tilt rates at the GNSS stations are obviously greater than the random errors estimated from the observation errors of the GNSS data, while they are basically consistent with the total errors shown in Table 2. This suggests the importance of systematic errors in inversion analyses of GNSS data. Although the RMS of the residual strain rates on the seafloor seems comparable to the averages of random errors and systematic errors in Table 2, this might not be statistically significant because of the small data set (only 14 triangle elements) on the seafloor.

Details are in the caption following the image
The residual strain rates and tilt rates obtained by subtracting the calculated strain rates and tilt rates from the observations. (a) Maximum shear strain rate. (b) Dilatation rate. (c) Directions of maximum horizontal contraction. (d) Tilt rate. Arrows: The downward tilt direction. The EW and NS components of the tilt rate are shown in supporting information Figure S6.

Some localized high shear strain rate zones can be identified in Figure 8. The localized shear zones would be caused by intraplate inelastic deformation, which is not modeled in the inversion for interplate slip-deficit distribution. We therefore treated the localized high shear strain rates as systematic errors due to the imperfection of the theoretical models. They are likely related to the process of stress accumulation leading to the occurrence of intraplate earthquakes, but we consider this issue to be beyond the scope of the present study.

6.2 Effects of Structure Model

In the present study, we estimated the slip-deficit rate distribution by using the completely relaxed responses. As mentioned above in section 4.1, the viscosity of the asthenosphere does not affect the completely relaxed responses. Assuming the horizontally layered structure of the lithosphere and the asthenosphere, the parameter that mainly controls the completely relaxed response is the thickness of the lithosphere. To investigate the effects of the structure model on the estimation of the slip-deficit rate distribution, we estimated the slip-deficit rate distributions by using the completely relaxed responses with various lithosphere thicknesses HL = ∞ (elastic half-space), 80, 70, 60 (the same as in Figure 6), 50, and 40 km as shown in Figure 9.

Details are in the caption following the image
Comparison of the interseismic slip-deficit rate distributions estimated from strain rates and tilt rates using six values of lithosphere thickness (HL): (a) Elastic half-space, (b) 80, (c) 70, (d) 60, (e) 50, and (f) 40 km.

The profiles of the estimated slip-deficit rate distributions along the lines A–B (the Nankai region), C–D (the Tonankai region), and E–F (the Tokai region) are shown in Figure 10. We can see that the high slip-deficit rate zones move trenchward with the decreasing thickness of the lithosphere, whereas these zones are located near coastal regions when the elastic half-space responses are used. In addition, the maximum slip-deficit rate increased with the decreasing thickness of the lithosphere. The change in the location of the high slip-deficit rate zone was significant in the Nankai and the Tonankai regions, and it was not significant in the Tokai region.

Details are in the caption following the image
Profiles of the interseismic slip-deficit rate distributions. (a) Locations of profiles. (b–d) Slip-deficit rate distributions along the lines A–B, C–D, and E–F, respectively.

To interpret the changes in the estimated slip-deficit rate distribution in Figures 9 and 10, we compare a completely relaxed response with an elastic response when the same slip-deficit rate distribution is given (supporting information Text S2 and Figure S7). In and around the source region, the deformation in the elastic model is greater than that in the model with completely relaxed asthenosphere (completely relaxed model). However, far from the source region, the completely relaxed response is greater than the elastic response. In other words, the elastic response is localized in a narrower zone than the completely relaxed response. This resulted in the high slip-deficit rate zones located closer to the shore (deeper) when viscoelastic relaxation in the asthenosphere was not considered (e.g., Wang et al., 2012). Additionally, the GNSS and GPS-Acoustic observation network are almost in the region where the elastic response is greater than the completely relaxed response. For this reason, the completely relaxed model required more slip-deficit rates than the elastic model to reproduce the horizontal strain and tilt rate data.

The thickness of the lithosphere, or the bottom depth of the lithosphere, is considered to be related to temperature and varies widely with the age of the plate. Around the Nankai trough, to explain surface displacements during one seismic cycle of the Nankai earthquake, Ito and Hashimoto (2004) and Fukahata et al. (2004) assumed the thickness of lithosphere to be 35 and 30 km, respectively. Off the Kii peninsula (Figure 1), Suito (2017) assumed the thickness to be 40 km to explain postseismic displacements following the 2004 earthquake off the Kii Peninsula (Mw 7.3) that occurred inside the PHS plate. In the Kanto region (Figure 1), which is east of the model region in the present study, the 1923 Kanto earthquake occurred on the upper boundary of the PHS plate. Matsu'ura and Iwasaki (1983) interpreted the postseismic movements after the earthquake using a structure model with a 60-km-thick lithosphere. On the other hand, the seismic lithosphere-asthenosphere boundary of the subducting PHS plate was detected at the depth of 50–70 km as discontinuities of wave velocity (Kumar & Kawakatsu, 2011; Tonegawa & Helffrich, 2012). Among the lithosphere thickness of 40–60 km, we selected the result with the 60-km-thick lithosphere as an optimum model, because the results with the 40- to 50-km-thick lithosphere showed slip-deficit rates that were much larger than the convergence velocity at the plate interface.

In this study, we ignored heterogeneity in the structure, such as a subducting slab and different thicknesses between the continental and oceanic lithosphere. Comparison of horizontal layered and heterogeneous models performed by Yoshioka and Suzuki (1999) helps us consider the difference between two models in viscoelastic relaxation, although they did not investigate the completely relaxed responses directly. From their simulations of postseismic surface deformations due to viscoelastic relaxation caused by the 1946 Nankai earthquake, we can see that the horizontal displacements in a horizontally layered structure with a 50-km-thick lithosphere are similar to that in a heterogeneous structure with the subducting slab. This is because surface deformation due to local viscoelastic flow in the wedge mantle is limited owing to the shallow dip angle of the slab in the young subduction zone, and the deformation due to viscoelastic flow beneath the oceanic lithosphere is dominant. Hence, we could interpret the 50-km-thick lithosphere in Yoshioka and Suzuki (1999) and the 60-km-thick lithosphere in our model as the bottom depth of the elastic slab beneath the seismogenic zone.

On the other hand, the vertical displacements are relatively sensitive to heterogeneity in the structure. In the present analyses, we set systematic errors of tilt rate greater than that of horizontal strains to avoid a seriously biased inversion result. However, as Yoshioka and Suzuki (1999) pointed out, we need a realistic 3-D heterogeneous viscoelastic structure to substantially decrease systematic errors. This remains a challenge for the future.

6.3 Rigid Body Rotation and Translation

We used the horizontal strain rates, instead of the horizontal velocities that are commonly used, in order to avoid biases in slip-deficit rate distributions caused by unexplainable coherent noise due to rigid block translation and rotation. Strictly speaking, if the observation network of the GNSS stations covers a block boundary, the strain rate data could still contain localized high strain rates on the block boundary caused by displacement discontinuity between two adjacent crustal blocks. On the other hand, the slip-response functions in this study reproduce deformation in a relatively broad zone, because of the distance of the observation network from the plate boundary and the relatively coarse spatial resolution of slip-deficit rate distribution (open circles in Figure 5a). Additionally, deformation of the lithosphere in completely relaxed responses occurs in a broader zone than that in elastic responses. Thus, the localized high strain rates could not be reproduced by the model and would hardly bias the inversion results. Actually, we can see that localized high shear strain rates in the observed strain rate data (Figure 4a) remain in the residual strain rates (Figure 8a).

For reference, we performed test analyses to verify the inversion result in Figure 6a, as follows. We divided the strain and tilt rate data on land into five equally sized subsamples. We omitted a single subsample from the data and performed the same analysis as in section 4.2 using the remaining four subsamples. We obtained five inversion results from the data set from which each subsample was removed (supporting information Text S3 and Figure S8). These results were robust, which suggests that the gross feature of the slip-deficit rate distribution does not depend on a specific outlier on the block boundaries.

Figure 11 shows the residual horizontal velocities obtained by subtracting the horizontal velocities calculated from the optimum slip-deficit rate distribution in Figure 6a from the observed velocities in Figure 2a. The residual horizontal velocities should be composed mainly of observation errors of the GNSS data, systematic errors due to imperfection in the structure model, and some rigid translation and rotation of crustal blocks. From Figure 11, we can see steep gradients of horizontal velocities across major tectonic zones in southwest Japan: right-lateral shear deformation along the Median Tectonic Line (Ikeda et al., 2009; Tsutsumi & Okada, 1996), right-lateral shear deformation along the Beppu-Shimabara graben (Matsumoto, 1979), and right-lateral shear and compressional deformation along the Niigata-Kobe Tectonic Zone (Sagiya et al., 2000).

Details are in the caption following the image
The residual horizontal velocities obtained by subtracting the calculated velocities from the observed velocities. White arrows: The relative velocity vectors to the reference point Fukue, which is indicated by an open triangle. The thick gray lines show representative tectonic zones in southwest Japan: NKTZ, Niigata-Kobe Tectonic Zone; MTL, Median Tectonic Line; BSG, Beppu-Shimabara Graben.

Showing relative velocities to the reference point Fukue located in the backarc region of the southwestern Japan arc, we can recognize a westward block movement of the forearc sliver on the southern side of the major tectonic zones, which could presumably be caused by oblique plate convergence along the Nankai trough. We can also see the westward translation motion even on the seafloor, although the seafloor data have large observation errors. However, a single block motion could not explain all of the westward movements in forearc region from 130°E to 138°E. It might be possible for the forearc region to be redivided into some subblocks as proposed by Wallace et al. (2009).

In the inversion for the interplate slip-deficit distribution, we modeled crustal deformation caused by slip motion on the plate interface, but not the rigid block translation and rotation of a forearc sliver. Hence, the rigid block motions of a forearc sliver behave as unexplainable coherent noise and could seriously bias the inversion result. In previous studies using surface velocity data without taking into consideration the rigid block motions, the slip-deficit rates in the strike-parallel direction would be overestimated to reproduce the westward movements due to the rigid sliver motion. For example, previous studies estimated high slip-deficit rates in the Tokai region as large as those in the Nankai region (e.g., Yokota et al., 2016). The high slip-deficit rates in the Tokai region could be caused by large westward rigid movements in the Chubu region.

7 Conclusions

Most previous studies estimated slip deficits on the plate interface along the Nankai trough from interseismic GNSS velocity data assuming elastic slip-response functions, but if the elastic responses cannot adequately approximate the deformation of the viscoelastic medium, the results of an inversion analysis could be biased. From the viscosity in the asthenosphere obtained by previous studies, we found that (1) the stress relaxation in the asthenosphere strongly affects the interseismic deformation around the Nankai trough, and (2) we need to consider the viscoelastic response in modeling the surface deformation.

In this study, we (i) employed completely relaxed slip-response functions to invert the slip-deficit rate distribution from geodetic data taking an elastic-viscoelastic layered structure into account, (ii) analyzed seafloor geodetic data together with inland GNSS data to constrain the slip deficit in the offshore region, and (iii) used the strain/tilt rate to avoid the bias from the rigid block translation and rotation. First, we analyzed GNSS daily coordinate data and obtained the surface velocity data for overlapping periods of 6 years between 1996 and 2011. There is no significant temporal change in the velocity data, which suggests that postseismic stress relaxations after the 1944 Tonankai and 1946 Nankai earthquakes decayed sufficiently. Next, applying the strain data inversion method to the GNSS data from 2005 to 2011 together with the seafloor geodetic data, we estimated the slip-deficit rate distribution with completely relaxed slip responses. The optimum slip-deficit rate distribution using a structure model with a 60-km-thick lithosphere shows two peaks of slip-deficit rate off the Muroto Promontory and off the Kii Peninsula.

There is a significant difference between the results using elastic and completely relaxed responses. While the result using elastic responses shows a high slip-deficit rate zone in coastal regions, the high slip-deficit rate zones are located trenchward when completely relaxed responses were used. As the thickness of the lithosphere becomes thinner, the peak value of the slip-deficit rate increases. Finally, we obtained residual horizontal velocities by subtracting the calculated velocities due to the optimum slip-deficit rate distribution from the observed velocities. We can see significant crustal block movements of forearc slivers on the southern side of major tectonic zones. This suggests that we appropriately divided the elastic deformation due to slip deficits and rigid block motion that comes from inelastic crustal deformation.

Acknowledgments

We thank three anonymous reviewers for their useful comments and suggestions for improving our manuscript. A. N. also thanks Shunsuke Takemura for a helpful discussion on the structure of the PHS plate. This study was supported by the National Research Institute for Earth Science and Disaster Resilience (Research Project Large Earthquake Generation Process) and JSPS KAKENHI grant JP18K03809. We used the GEONET F3-Solution (daily coordinate data of GNSS stations) published by the Geospatial Information Authority of Japan (http://datahouse1.gsi.go.jp/terras/terras_english.html). We evaluated viscoelastic slip-response functions with the computer program developed by Fukahata and Matsu'ura (2006). For plotting we used the Generic Mapping Tools (Wessel & Smith, 1998).

    Appendix A: Expression of the Surface Velocity Field Caused by Steady Slip Motion

    We consider a system composed of an elastic lithosphere and a viscoelastic asthenosphere under gravity. It is not necessarily a horizontally layered model. Then, we can represent surface displacements ui(x, t) due to steady slip motion on the plate interface Σ at the rate urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0020 that started at the time t = 0 as
    urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0021(A1)
    where Ui(x, t; ξ, τ) denotes the ith component of the quasi-static displacement response functions in the lithosphere-asthenosphere structure at a point x and time t due to unit slip at a point ξ and time τ. Substituting t = t − τ, we obtain
    urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0022(A2)
    Equation A2 is equivalent to a time integration of surface displacements due to a unit step slip at a point ξ and time t = 0. As the time t increases, the quasi-static response functions Ui(x, t; ξ, 0) tend to approach certain constants, that is, the completely relaxed responses urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0023. After the effective relaxation time τe of the lithosphere-asthenosphere system, t ≥ τe, we can rewrite equation A2 as:
    urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0024(A3)

    If t ≥ τe, equation A3 hence becomes to increase in proportion to the time t at the rate of the completely relaxed responses.

    Differentiating both sides of equation A3 with respect to t, the expression of surface velocities caused by steady slip motion becomes constant in time:
    urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0025(A4)

    Appendix B: Time Series Analysis of GNSS Daily Coordinate Data

    In order to extract long-term crustal movements from the GNSS time series data, we fit a parametric model
    urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0026(B1)
    to the time series of each coordinate component (i= EW, NS, UD) at every GNSS station. Here the first term on the right side of equation B1 is an adjustable constant, the second term is a secular trend determined from interseismic GNSS data, the third and fourth terms are the seasonal variations with periods ≤1 year, the fifth term is coseismic steps due to earthquakes that occurred at t = Tr (r = 1, …, R). We estimated the optimum values of the adjustable parameters ai, bi, urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0027, urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0028, and urn:x-wiley:21699313:media:jgrb53001:jgrb53001-math-0029 for each coordinate component at every GNSS station for each period with the least squares method.

    Each period of the analysis (periods I, II, III, and IV) included one L-SSE, which repeatedly occurred in the Bungo Channel at an interval of 5–7 years (e.g., Yoshioka et al., 2015). We removed the total offset during the L-SSE in the same way as the coseismic steps (supporting information Figure S9). Here we chose the appropriate number of seasonal modes Q with Akaike's Information Criterion (Akaike, 1974). After the optimization of adjustable parameters, we picked out the second term in equation B1 as the surface velocity data.