# A possible scenario for earlier occurrence of the next Nankai earthquake due to triggering by an earthquake at Hyuga-nada, off southwest Japan

- Mamoru Hyodo
^{1}Email author, - Takane Hori
^{1}and - Yoshiyuki Kaneda
^{2}

**Received: **29 September 2015

**Accepted: **7 January 2016

**Published: **19 January 2016

## Abstract

Several recent large-scale earthquakes including the 2011 Tohoku earthquake (*M*
_{w} 9.0) in northeastern Japan and the 2014 Iquique earthquake (*M*
_{w} 8.1) in northern Chile were associated with foreshock activities (*M*
_{w} > 6). The detailed mechanisms between these large earthquakes and the preceding smaller earthquakes are still unknown; however, to plan for disaster mitigation against the anticipated great Nankai Trough earthquakes, in this study, possible scenarios after *M*
_{w} 7-class earthquakes that frequently occur near the focal region of the Nankai Trough are examined through quasi-dynamic modeling of seismic cycles. By assuming that simulated Nankai Trough earthquakes recur as two alternative earthquakes with variations in magnitudes (*M*
_{w} 8.7–8.4) and recurrence intervals (178–143 years), we systematically examine the effect of the occurrence timing of the *M*
_{w} 7 Hyuga-nada earthquake on the western extension of the source region of Nankai Trough earthquakes on the assumed Nankai Trough seismic cycles. We find that in the latter half of a seismic cycle preceding a large Nankai Trough earthquake, an immature Nankai earthquake tends to be triggered within several years after the occurrence of a Hyuga-nada earthquake, then Tokai (Tonankai) earthquakes occur with maximum time lags of several years. The combined magnitudes of the triggered Nankai and subsequent Tokai (Tonankai) earthquakes become gradually larger with later occurrence of the Hyuga-nada earthquake, while the rupture timings between the Nankai and Tokai (Tonankai) earthquakes become smaller. The triggered occurrence of an immature Nankai Trough earthquake could delay the expected larger Nankai Trough earthquake to the next seismic cycle. Our results indicate that triggering can explain the variety and complexity of historical Nankai Trough earthquakes. Moreover, for the next anticipated event, countermeasures should include the possibility of a triggered occurrence of a Nankai Trough earthquake by an *M*
_{w} 7 Hyuga-nada earthquake.

## Keywords

## Introduction

*M*

_{w}8 earthquakes have been recurring at the Nankai Trough since 684 CE, causing severe damage to the Japanese Islands. As shown in Fig. 1, approximately nine successive Nankai Trough earthquakes (in 1946 and 1944, 1854, 1707, 1605, 1498, 1361, 1099 and 1096, 887, and 684) have been documented in the historical record over the past 1300 years (Ishibashi 2004). As summarized in Ishibashi (2004), occurrences of these historical Nankai Trough earthquakes have been found in historical documents around Kyoto, the Japanese capital from the end of the eighth century to the beginning of the seventeenth century. Since the region around Kyoto is strongly impacted by not only the Nankai Trough earthquakes but also by nearby inland earthquakes, the occurrences of historical Nankai Trough earthquakes have been chosen by carefully compiling reliable historical descriptions or records that detail the typical interplate event along the Nankai Trough; these include strong ground motion, felt aftershocks in Kyoto, large tsunami damages along the Pacific coast, coseismic vertical deformation at coastal regions (uplift of the capes Muroto and Oma’ezaki, subsidence of the Kochi plain), and temporal inactivities of specific hot springs. By comparing the timing of a possible Nankai Trough earthquake with contemporary geological evidences of the occurrence of strong shaking (traces of liquefaction at ruins) or large tsunamis (tsunami deposits in coastal lakes), the reliability of the possible Nankai Trough events has been checked from another viewpoint (e.g., Sangawa 2009; Shishikura et al. 2011). These contemporary geological evidences especially help historical seismologists constrain the spatial extent or magnitude of the historical Nankai Trough earthquakes. In Fig. 1, the horizontal thick black solid lines denote certain rupture segments in historical Nankai Trough earthquakes. Judging from the completeness of reliable historical materials, thick and thin black broken lines are probable and possible rupture segments in historical Nankai Trough earthquakes, respectively. The thin dotted lines denote unknown rupture segments during the historical Nankai Trough earthquakes. White blanks indicate seismic gaps. The gray broken line in 1605 denotes a tsunami earthquake with no evidence of severe seismic hazard. As shown in Fig. 1, a recurrence interval between the 1099 Kowa Nankai earthquake and 1361 Ko’an Nankai Trough earthquake is 262 years, which is almost twice as long as the average interval after the 1361 Nankai earthquakes. Therefore, historical seismologists consider there may be a hidden Nankai Trough earthquake during this period. However, a new reliable earthquake around the thirteenth century has not been confirmed from historical materials yet there exist contemporary traces of liquefaction that are not inconsistent with the damages of the Nankai Trough earthquake in Osaka and the southern tip of the Kii peninsula (Sangawa 2011).

For the current sequence of reliable Nankai Trough earthquakes, there have been large variations in the magnitudes (*M*
_{w} = 8.0–8.7) and recurrence intervals (90–262 years) of successive Nankai Trough earthquakes, with different rupture extents along the Nankai Trough. Additionally, in some successive earthquakes, the eastern and western segments of the Nankai Trough have ruptured separately, with large variations in time lags: about 3 years between the 1099 and 1096 events, 2 years between 1946 and 1944, several days for the 1361 events, and 30 h in the 1854 events. In most of the events with such rupture separation, the eastern segment ruptured first, followed by the western segment. For the recent 1946 and 1944 Nankai Trough earthquakes, which were observed by modern observation instruments such as seismograms and tide gauges, the hypocenter locations were estimated from the observed data (Kanamori 1972): the 1944 Tonankai earthquake initiated at Kumano-nada, on the eastern side of the Kii Peninsula, while the 1946 Nankaido earthquake initiated in the eastern part of the Kii channel, on the western side of the southern tip of the Kii Peninsula (see solid and open stars in Fig. 1). Only for the 1498 Meio event is there a possibility that rupture on the western segment preceded that on the eastern segment, based on combined interpretation of Japanese and Chinese historical documents (Tsuji 1999).

The recent successive earthquakes in 1946 and 1944 were the smallest events among the historical Nankai Trough events. Seventy years have passed since those events, approaching the minimum recurrence interval (90 years between the 1944 and 1854 events) between successive Nankai Trough earthquakes. Accordingly, the application of the time-predictable model (Shimazaki and Nakata 1980) to the recent Nankai Trough earthquake sequence predicts a short recurrence interval to the next earthquake, considering the small magnitudes of the 1946 and 1944 events. Hence, the occurrence of the next Nankai Trough earthquake has been anticipated for several decades (Headquarters for Earthquake Research Promotion 2001).

Recently, the *M*
_{w} 9 Tohoku earthquake occurred unexpectedly in 2011 and generated a very large tsunami. Following this unexpected great earthquake, the damage expectation and the occurrence probability for the next Nankai Trough earthquake have been revised by government agencies (e.g., Cabinet Office 2012; Headquarters for Earthquake Research Promotion 2013). However, the published damage expectation was based on a kinematic model for a much larger earthquake scenario (*M*
_{w} ~ 9) than the historical earthquakes (*M*
_{w} ~ 8). It is not necessary that the damage expectation due to kinematic scenarios of *M*
_{w} 9 earthquakes is greater than the damages due to possible *M*
_{w} 8 earthquakes with various rupture patterns, and the damage estimate based only on one large *M*
_{w} 9 earthquake might be inadequate. Therefore, it is indispensable to examine not only an *M*
_{w} 9 scenario but also various possible *M*
_{w} 8 scenarios.

Many complex physical processes that contribute to earthquake generation, such as fault friction and fault interactions, are not well known. Although we cannot model all of the processes, we should examine the possible and more physically plausible scenarios predicted by various models. Recently, numerical simulations of earthquake occurrences, than those obtained via kinematic earthquake models, have been executed for the purpose of obtaining various earthquake scenarios that are more physically plausible, assuming a fault constitutive law and fully elastodynamic interactions during earthquake ruptures (e.g., Hok et al. 2011; Noda and Lapusta 2013). In Japan, for interpreting previous seismic cycles and examining possible future scenarios, quasi-dynamic seismic cycle simulations have been widely conducted based on the rate- and state-dependent friction law and quasi-static slip response functions. For the Japan Trench in the northeastern part of Japan Islands, quasi-dynamic earthquake cycle modeling with rate- and state-dependent friction and stress accumulation due to the subducting Pacific plate have been implemented, with the aim of examining possible seismic cycles including multi-scale asperities with the patch size of 10^{0}–10^{2} km (e.g., Shibazaki et al. 2011; Kato and Yoshida 2011; Ohtani et al. 2014; Ariyoshi et al. 2014). Additionally, along the Nankai Trough, similar quasi-dynamic seismic cycle simulations of *M*
_{w} 8-class earthquakes due to the subduction of the Philippine Sea plate have been conducted (e.g., Hori et al. 2004; Hori 2006). Moreover, aseismic events recently observed in the Nankai Trough region have also been successfully modeled in the framework of quasi-dynamic earthquake cycle model. Long-term slow slip event (SSE) beneath Tokai, eastern part of the Nankai trough region, was reproduced with *M*
_{w} 8-class Nankai Trough earthquakes (Hirose and Maeda 2013). As for the western side of the Nankai Trough, Shibazaki et al. (2010) and Matsuzawa et al. (2010) modeled SSEs, occurring in the deeper part of the seismogenic zone of the *M*
_{w} 8-class earthquake beneath Shikoku, by using a modified version of the rate- and state-dependent friction laws. Subsequently, it has been discussed that the stress buildup process in an *M*
_{w} 8 earthquake cycle affects the recurrence behavior of SSEs (Matsuzawa et al. 2010; Ariyoshi et al. 2012)

In order to deduce the possible mechanism for the variation in magnitude and recurrence intervals in the three recent Nankai Trough seismic cycles (1944–1946, 1854, and 1707), Hori (2006) modeled the three seismic cycles by focusing on the relationship between the variation of recurrence intervals and the time gaps between the ruptures of the Tokai (eastern) and Nankai (western) earthquakes. Thus, by assuming frictional heterogeneity beneath the Kii Peninsula, which behaves as a barrier to rupture propagation, he found that Nankai Trough earthquakes tend to initiate off Kumano-nada, where the oceanic plate is subducting at a steep dip angle, and the increase in time intervals between rupture on the eastern and western segments with seismic cycles (from concurrent rupture to several years of time lag) occurs simultaneously with shortening of the recurrence intervals (111–95 years). Although a quantitative comparison of the recurrence intervals shows large gaps between simulated and historical values, this is qualitatively consistent with the increasing time intervals between rupture on the eastern and western segments in the three recent seismic cycles (e.g., simultaneous in 1707, 30 h in 1854, and 2 years in 1944 and 1946) and the shortening of the recurrence intervals (209, 147, and 90 years). Furthermore, Hori et al. (2009) suggested that seismic cycle simulations with nesting asperities (hierarchical asperity) with different fracture energies explained much larger variation of earthquake magnitudes and recurrence intervals because of the remarkable change in the stressing rate near the earthquake nucleation point (off Kumano-nada for Nankai Trough earthquakes) between seismic cycles. By applying the hierarchical asperity model to the Nankai Trough seismic cycles, Hyodo and Hori (2013) reproduced the large variations in magnitude (*M*
_{w} = 8.58–8.97) and recurrence intervals (168–204 years) for the Nankai Trough earthquakes. The difference in the recurrence intervals is consistent with that of the historical sequence, except for the short recurrences of the Showa events. As mentioned above, some characteristics of the Nankai Trough earthquake sequence have been explained as variations of the seismic cycles. Thus, these quasi-quantitative reproductions of previous patterns of Nankai Trough earthquakes suggest that we can use such earthquake cycle models to examine possible Nankai Trough earthquake scenarios.

Recently, through similar numerical simulations of interplate seismic cycles along only the western half of the Nankai Trough from Kyushu to Shikoku, Nakata et al. (2014) found that there exists a possible earthquake scenario in which the Nankai earthquake rarely initiates off the western edge of Shikoku Island about 5 years after the occurrence of the Hyuga-nada earthquake, although most Nankai and Hyuga-nada earthquakes occur independently, and most Nankai earthquakes initiate at the Kii Peninsula, on the eastern edge of their model region. Since such a rare Nankai earthquake is triggered by a Hyuga-nada earthquake, the recurrence interval from the previous Nankai earthquake is considerably shortened, thus its magnitude is smaller than those with non-triggered ruptures. *M*
_{w} 7 Hyuga-nada earthquakes have historically occurred at the western extension of the source area of Nankai Trough earthquakes, and in the 1498 Meio event, one of the Hyuga-nada earthquakes may have enhanced the initiation of the Nankai earthquake at the western edge of Shikoku Island. Hence, we propose to evaluate the possible effect of realistic conditions with an *M*
_{w} 7-class asperity for the entire Nankai Trough, including the eastern segments.

In this paper, by extending the seismic cycle model of Nakata et al. (2014) toward the eastern part of the Kii Peninsula, we first examine the conditions under which the westward triggering of Nankai earthquakes occurs. Then, we systematically examine the possible effect of trigger timing on the recurrence intervals of Nankai Trough earthquakes and rupture timing between the eastern and western segments. Finally, we discuss the possibility of the triggered Nankai Trough earthquake scenario as the next earthquake.

## Methods

Equation 1 is an approximated expression of fault shear stress *τ* in the subduction direction using the static slip response to unit forward slip *G*, slip deficit on the plate interface *V*
_{
p
}
*t − u*, and fault slip rate *V*. The first term on the right-hand side of this equation represents a quasi-static fault shear stress as the spatial integration of the product of *G* and slip deficit over the target plate interface *Ω*, where *u* and *V*
_{
p
} are the slip and prescribed back-slip rate on the plate interface, respectively. *V* and *u* have the relation of *du/dt* = *V*. The second term represents the effect of radiation damping introduced by Rice (1993), the coefficient *η* controls the degree of the effect of the seismic radiation, and *η = μ/*2*β* is assumed following Rice (1993), where *μ* and *β* are the rigidity and shear wave velocity of the material surrounding the source region, respectively. In the following, we set *μ =* 30 GPa and *β* = 3.27 km/s.

Equation 2 represents a rate- and state-dependent friction, where *V*
_{
*
}, *τ*
^{
*
}
_{
s
}, and *Δτ*
_{
s
} are an arbitrary reference slip velocity on the frictional surface, the strength of a fault sliding with a velocity of *V*
_{
*
}, and the differential fault strength measured from *τ*
^{
*
}
_{
s
}, respectively. Because Eq. 2 implies the slip rate *V* is accelerated or decelerated from *V*
_{
*
} depending on the difference between the fault shear stress *τ* and the variable *τ*
_{
s
} (=*τ*
^{
*
}
_{
s
} + *Δτ*
_{
s
}), the variable *τ*
_{
s
} is regarded as a threshold to acceleration of the fault with respect to the fault shear stress *τ*. Hence, we call the variable *τ*
_{
s
} as the “strength” of the fault. A frictional parameter *A* (=*aσ*) controls the variation of *V* with respect to the certain difference of stress and strength, where parameter *a* (and *b*, as will be described below) is a frictional coefficient, and *σ* is the effective normal stress. The differential strength *Δτ*
_{
s
} here is defined as *Δτ*
_{
s
} = *B* ln (*V*
_{
*
}
*θ*/*L*), where *θ* is the state variable representing a contact state of the interface and *B* (=*bσ*) and *L* are also frictional parameters controlling strength recovery and slip weakening. It should be noted that we can obtain the ordinary rate- and state-dependent friction equation by substituting the definition of *Δτ*
_{
s
} in Eq. 2. Equation 3 is an equivalent differential equation to the composite equation describing the evolution of the state variable *θ* (Kato and Tullis 2001), although the integrand is *Δτ*
_{
s
} instead of *θ*. The composite equation deduced by Kato and Tullis (2001) was made by slightly modifying existing rate- and state-dependent friction laws (slip law and slowness law) originally developed by Dieterich (1979) and Ruina (1983) so that a wide range of experimental observations are fitted. The first term on the right-hand side of Eq. 3 expresses the time-dependent increase of *Δτ*
_{
s
} (healing effect) if the slip rate *V* is small enough (*V~V*
_{
c
}), while the second term represents the slip-dependent decrease of *Δτ*
_{
s
} (slip-weakening effect). Thus, *V*
_{
c
} switches the behavior of Eq. 3, depending on the slip rate *V*, and is called the cutoff velocity. Following Kato and Tullis (2001), we assume *V*
_{
c
} = 10^{−8} m/s in Eq. 3 to fit the theoretical frictional property to the experimental data of friction on granite surfaces obtained by Blanpied et al. (1998). For *V* >> *V*
_{
c
}, the steady-state shear stress *τ*
^{ss} is defined for *dΔτ*
_{
s
}
*/dt* = 0, and *τ*
^{ss} = *τ*
^{
*
}
_{
s
} + (*A* − *B*) ln (*V*/*V*
_{
*
}). When *A* − *B* < 0, the steady-state stress decreases with an increase in slip velocity (velocity weakening), possibly leading to unstable slip (earthquake). In the case of *A* − *B* > 0, stress increase with slip velocity (velocity strengthening) and steady slip or aseismic slip can occur. Accordingly, *A* (=*aσ*), *B* (=*bσ*), and *L* in Eqs. 2 and 3 largely affect the seismic cycles on the plate interface. Though *A* (=*aσ*) and *B* (=*bσ*) depend on the normal stress *σ* which could vary with seismic cycles, we neglect the temporal variation of *σ* for simplification.

Note that we abbreviate the functional expressions as variables in Eqs. 2 and 3: the variables *V*
_{
*
}, *τ*
_{
*s
}, *A*, *B*, and *L* are functions with respect to only the spatial coordinate (i.e., constant in time), while *V*
_{
c
} (=10^{−8} m/s) is constant with respect to time and space. *V*, *τ*, and *Δτ*
_{
s
} are unknown variables with respect to time and space.

*Ω*is divided into a set of

*N*small finite sub-faults. Then, simultaneous differential equations with respect to slip rate

*V*

_{ i }and differential fault strength

*Δτ*

_{ s,i }are deduced, where the subscript

*i*indicates an index for the sub-faults (1 <

*i*<

*N*). The differential equation with respect to

*Δτ*

_{ s,i }is directly obtained from Eq. 3, and it is obvious that the evaluation of

*dΔτ*

_{ s,i }

*/dt*on each sub-fault

*i*can be executed by using the variables on its own sub-fault only. Meanwhile, the concrete expression of the evolution of slip rate

*V*

_{ i }is deduced from Eqs. 1 and 2 as follows:

Here, subscripts *i* and *j* indicate indexes for the sub-faults (1 < *i, j* < *N*). Thus, it should be noted that the direct evaluation of *dV*
_{
i
}
*/dt* for all *i* requires *O*(*N*
^{2}) operation, since *G*
_{
ij
}, which is the slip response on sub-fault *i* due to unit slip on sub-fault *j*, becomes an *N × N* dense matrix. Hence, in the quasi-dynamic earthquake cycle simulation with a large sub-fault number *N*, direct evaluation of Eq. 4 requires a large memory for *G*
_{
ij
}, and it becomes the most time-consuming process.

For efficient calculation of seismic cycles, many attempts to reduce memory sizes and computation time associated with Eq. 4 have been examined (Kato 2008; Tullis et al. 1999; Ohtani et al. 2011; etc.). Among these, we adopted the procedure used in Ohtani et al. (2011), which applied the method of hierarchical matrices (H-matrices) to multiplicative computations of *N* × *N* slip response function matrix and the slip deficit rate vector. In contrast to other attempts, the quasi-dynamic simulation using H-matrices is easily applicable to a seismic cycle on a 3-D curved interface, since H-matrices are just mathematical approximations as low-rank compressed representations of the original dense matrix *G*
_{
ij
}. Thus, independent of the target fault configuration or earthquake slip direction, H-matrix approximation is applicable to quasi-dynamic cycle simulations and is suitable to our target problem. Following Ohtani et al. (2011), the original computation time of arithmetic operation with *O*(*N*
^{2}) will be efficiently reduced to *O*(*N* log*N*) for *N* which we treat here (10^{5}–10^{6}), while memory size will be reduced to about *O*(*N*).

For more efficient computation of seismic cycles, we execute simulations with H-matrices on massively parallel computers. In the parallelized simulation code, the total sub-faults are divided into groups depending on the number of usable CPUs. Then, the velocities *V*
_{
i
} and differential strengths *Δτ*
_{
s,i
} in each group are assigned on a CPU in charge of each group. Also, the CPU stores only a subset of H-matrices that are used for the evaluation of velocity rates *dV*
_{
i
}
*/dt* on each group. Though each CPU contains only velocities of a group in charge of each CPU, current velocities on all sub-faults are necessary for evaluating Eq. 4. Hence, before evaluating Eq. 4 on each CPU, velocities on all sub-faults are collected through the message passing interface (MPI) communication; then, the evaluations of Eq. 4 are calculated on each CPU. Calculated slip velocity rates and strength rates are used for the time integration by the adaptive-stepping fifth-order Runge-Kutta method (Press et al. 1996).

*Y*-

*Z*plane form homogeneous rectangular grid cells (see Fig. 3a). For these sub-faults, we prepare the slip response

*G*

_{ ij }in Eq. 4 as the stress change enhances the slip of the sub-fault

*i*due to unit slip of sub-fault

*j*. Since sub-faults generally form non-planar configurations, we approximate a sub-fault with a set of three triangles, as shown in Fig. 3b. Then,

*G*

_{ ij }is evaluated at the center of the sub-fault

*i*(crosses in Fig. 3b) as the combined stress change due to unit slips on a set of three triangles representing sub-fault

*j*. The direction of a unit slip on each triangle in a sub-fault is determined from the direction of subduction and the configuration of the triangle (i.e., strike and dip) as follows. First, we divide the horizontal unit vector with the direction of subduction direction into two components: the strike direction of the triangle and the direction perpendicular to it. Then, the latter component of the unit vector is rotated to the direction of the dip of this triangle, and this rotated vector is assumed to become the dip component of the unit slip. Finally, we define the tangential slip which can occur on the triangle as the dislocation along the unit vector consisting of the strike and dip components as mentioned above.

By setting *ΔY* = 1 km and *ΔZ* = 0.1 km, as in Fig. 3a, and discretizing the model plate interface into sub-faults less than 1 km^{2} in size, the total number of sub-faults treated here is 308,736. For these discretized sub-faults, the set of slip responses *G*
_{
ij
} in Eq. 4 is evaluated in a homogeneous elastic half space, and they are compressed as H-matrices as mentioned above. Then, the vertical position of the free surface of the elastic half space is set at 5 km depth of the interface geometry of Baba et al. (2002). This means that the upper limit of the model interface does not reach the seafloor (i.e., free surface). When it reaches the seafloor, the shallow slip becomes too large, and the rupture becomes difficult to stop, as shown in Hyodo and Hori (2013).

We project the same distribution of *A*, *B*, *L,* and *V*
_{
p
} as in Nakata et al. (2014) onto the plate interface from Kyushu to Shikoku; then, for the frictional parameters *A*, *B*, and *L*, we extrapolate the depth profiles for eastern Shikoku further along the eastern plate interface (Fig. 2b, c). For back-slip rate *V*
_{
p
}, the direction is assumed to be N 55° W similar to Nakata et al. (2014). In size of back-slip rate, a gradual decrease is assumed from the southern tip of the Kii Peninsula to the Izu Peninsula (Fig. 2a). Such eastward decrease of back-slip rate is based on analysis of the crustal deformation velocity (Heki and Miyazaki 2001). Here, the decrease in the back-slip rate is given perpendicular to the direction of back-slip (see Fig. 2a). Note that the eastern and western edges of the model plate interface are cut along the direction of N 55° W to avoid boundary effects.

Additionally, in extending the model plate interface eastward, we incorporate the frictional heterogeneity around the Kii Peninsula (Fig. 2b, c) assumed by Kodaira et al. (2006) and Hori (2006). The heterogeneity beneath the Kii Peninsula is based on structural analyses, and it acts as the segment boundary between the Nankai and Tokai earthquakes in numerical simulations (Kodaira et al. 2006). As shown in Kodaira et al. (2006), at the segment boundary between Tokai and Nankai earthquakes, two kinds of remarkable structures are found from seismic imaging: one is a high-density and high-velocity dome body beneath Cape Shionomisaki. The other is a highly fractured oceanic crust caused by the strike slip fault system located at the shallower side of the dome structure. Kodaira et al. (2006) considered that the high-density and high-velocity dome on the plate interface increases the normal stress on the plate interface and is strengthened against the rupture. Hence, in the position of the dome, they assigned large *B* − *A* following the assumption of high *σ*, and also large *L* for large fracture energy. For fractured oceanic crust, they considered the fracture zone must be weak and assigned *A* − *B* > 0 to avoid stress accumulation. Using these ideas of Kodaira et al. (2006), in this study, we follow their model setting around the segment boundary.

*Δτ*

_{ H }(x) and no associated strength change due to a simulated

*M*

_{w}7.5 Hyuga-nada earthquake (see Fig. 5f), dividing Eq. 2 for after the Hyuga-nada earthquake (

*t*=

*T*

_{ H }

^{ + }) by that for before the Hyuga-nada earthquake (

*t*=

*T*

_{ H }

^{ − }) leads to the following:

where *V*(x,*T*
_{
H
}
^{
+
}) is a modified slip velocity at *x* due to the occurrence of the Hyuga-nada earthquake and is accelerated (for *Δτ*
_{
H
} > 0) or decelerated (for *Δτ*
_{
H
} < 0). Accordingly, by calculating the seismic cycles with the modified velocity and pre-strength distribution as initial conditions, we can evaluate the effect of the Hyuga-nada earthquake on the basic Nankai Trough earthquake cycle for arbitrary times.

## Results

*M*

_{w}7.5), it is regarded as a triggered event caused by coseismic slip or after-slip following the Hyuga-nada earthquake. However, the occurrence frequency of triggered Nankai earthquakes is too low (only several times during the 2000-year simulation) to investigate the conditions under which Nankai earthquakes are triggered by a Hyuga-nada earthquake based on observation data. Hence, we executed numerical experiments that enable us to evaluate the effect of the Hyuga-nada earthquake on the arbitrary timing of the representative Nankai Trough cycles.

We utilize Eq. 5 to evaluate the effect of the occurrence timing of an *M*
_{w} 7.5 Hyuga-nada earthquake on Nankai Trough seismic cycles. Then, we explain the representative behaviors of the Nankai Trough seismic cycles by removing the *M*
_{w} 7 patches in the Hyuga-nada region from the above seismic cycle model.

*M*

_{w}> 8.5) and white (

*M*

_{w}< 8.5) stars in Fig. 6. As indicated in Figs. 6 and 7, the time intervals preceding the larger Nankai Trough earthquakes are shorter than those preceding the smaller earthquakes, and the variation of recurrence intervals is about 150–180 years, which is less than the variation of historical earthquakes (90–262 years). Two types of earthquakes are mainly caused by the large gap in

*L,*or fracture energy, in the seismogenic zone, as shown in Fig. 2b, similar to Hyodo and Hori (2013). At Hyuga-nada, about 5 m of after-slips occur only after the larger Nankai Trough earthquakes. However, in the inter-seismic periods of the Nankai Trough earthquake, a regime of stable sliding is dominant (blue line in Fig. 6).

Essentially, both types of Nankai Trough earthquakes are initiated at the plate interface beneath Kumano-nada. Then, the initiated ruptures propagate eastwards and rupture the eastern segments. At the same time, westward ruptures also begin to traverse the barrier (heterogeneity shown in Figs. 2b, c) at the Kii Peninsula, crossing the barrier at different time intervals after the eastward rupture, as shown in Fig. 7. However, the rupture in the third cycle (Fig. 7f) is initiated at the western end of Shikoku and propagates eastward. This type of rupture is rarely seen, as it occurs only once in a few thousand years. As a result, there are three types of seismic cycles, depending on the size and rupture patterns.

As shown above, the behaviors of the small Nankai Trough earthquake at the end of the third cycle in Figs. 4 and 5 are representative of the triggering effect of the *M*
_{w} 7.5 Hyuga-nada earthquake. To quantitatively evaluate how the triggering effect can modify the intrinsic behavior of the Nankai Trough seismic cycle, we executed numerical experiments by applying the velocity perturbation due to the *M*
_{w} 7.5 earthquake shown in Eq. 5 to the three seismic cycles in Figs. 6 and 7. For evaluating Eq. 5, we adopted a slip distribution of the Hyuga-nada earthquake, as shown in Fig. 5f. Although this earthquake triggered a Nankai earthquake as shown in the above simulation, its slip distribution with maximum slip of about 6 m is not so different to other Hyuga-nada earthquakes that occurred during the interseismic periods of the Nankai Trough earthquakes (see the steps of the blue line in Fig. 4a).

For cycle 1, as shown in Fig. 8a, the resultant interval of the seismic cycle varies sharply at 80 % of the time scale of the original cycle. For triggering before 80 % of cycle 1, both the rupture initiation position and timing of the next Nankai Trough earthquake are not affected, while for triggering timed after 80 %, Nankai earthquakes are initiated off Ashizuri with a delay of 3–4 years after the triggering, then the earthquake rupture propagates to the east. The maximum shortening of the seismic cycle is about 23 years, where the original interval of cycle 1 is 143 years. This is the case for the triggering at 81 % of the cycle.

Different to cycle 1, the triggered seismic cycle interval in the case of cycle 2 varies gradually (Fig. 8b), and the maximum shortening of the cycle is only 8 %, with a trigger timing of 86 %. As the original interval of cycle 2 is 178 years, the maximum shortening of the seismic cycle is about 14 years, and the interval between the trigger timing and the next Nankai Trough earthquake is about 11 years. For triggering before 60 %, rupture initiates at Kumano-nada, as shown in Fig. 7a, b, d, and e, while later triggering leads to the rupture initiation in western Shikoku.

For cycle 3 in Fig. 8c, the triggering effect varies sharply, similar to cycle 1, although the trigger timing, which minimizes the interval of the modified cycle, is much earlier than that of cycle 1: about 57 % of the original cycle for cycle 3 and 81 % for cycle 1. For triggering before 57 %, rupture initiates at Kumano-nada, but without the triggering effect, rupture initiates off Ashizuri in cycle 3. Meanwhile, for triggering after 57 %, the Nankai earthquakes are initiated off Ashizuri 2–8 years after the triggering. Among them, the maximum shortening of the seismic cycle is about 57 years for the original cycle of 149 years, corresponding to a shortening of about 40 % of the original cycle. This significant advance in time can only be seen for cycle 3. The triggered earthquakes initiated off Ashizuri tend to be arrested at the barrier beneath the Kii Peninsula, after which the eastern part of the Kii Peninsula ruptures with time lags as shown in the simulation results in Fig. 5g, h.

## Discussion

*M*

_{w}7 patches. The time series in Fig. 12 is evaluated offshore Ashizuri where many triggered earthquakes initiate. As shown in this figure, trigger start times at 80 and 55 % of cycles 1 and 3, respectively, correspond to a slip velocity level of around 2 × 10

^{−3}

*V*/

*V*

_{ p }. On the other hand, for cycle 2, the slip velocity reaches that level just before 100 %. Since a higher slip velocity means a higher stress level relative to the strength, the velocity level is a type of threshold for triggering in our model. Although it is difficult to resolve a slip velocity on the order of 10

^{−3}

*V*/

*V*

_{ p }from observation, the velocity reaches the order of 10

^{−1}

*V*/

*V*

_{ p }at 85 and 70 % for cycles 1 and 3, respectively. The latter level may be resolved from observation.

From the numerical experiment results, a large advance in time of small Nankai Trough earthquakes due to a Hyuga-nada earthquake occurs only when the interplate coupling at the western part of Shikoku Island is weakened, and the slip velocity reaches a certain level, as in cycle 3 (Fig. 12). Hence, the significance of the Hyuga-nada earthquakes for the next Nankai Trough earthquake depends strongly on the current state of its earthquake cycle.

At the Bungo channel, near the Hyuga-nada region, repeated slow slip events (SSEs) have been reported since the 1990s (e.g., Hirose et al. 1999). As discussed in Nakata et al. (2014), the occurrence of SSEs may indicate the weakening of interplate coupling near the source area of the SSE. Therefore, repeated SSEs in the Bungo channel may correspond to weak coupling around the western part of the source area of the Nankai earthquake, such as in cycle 1 or 3. The estimated slip deficit rate in the SSE source area for recent decades is 0–4 cm/year, which means the slip velocity is more than 40 % of the plate convergence rate (Yoshioka et al. 2015). Meanwhile, the last *M*
_{w} 7 Hyuga-nada earthquake occurred in 1968, about 47 years ago. The plate convergence rate (6.5 cm/year) and the assumption of strong coupling at the source area of the Hyuga-nada earthquake lead to a slip deficit accumulation of about 3 m. In addition, after several more decades, the slip deficit on the Hyuga-nada earthquake patch will reach 4–5 m, and this amount of slip deficit will be enough to cause an *M*
_{w} 7.5 earthquake.

This indicates that a triggered Nankai earthquake is likely to occur if an *M*
_{w} 7 Hyuga-nada earthquake occurs. Although such a triggered Nankai earthquake was not confirmed along the Nankai Trough (i.e., the 1498 Meio earthquake is just a candidate of triggered earthquakes), it is known that several recent large earthquakes (*M*
_{w} > 8) have occurred with foreshocks. The 2011 Tohoku earthquake (*M*
_{w} 9.0) along the Japan trench occurred after 2 days of *M*
_{w} 7.3 foreshock (e.g., Ando and Imanishi 2011); the 2014 Iquique earthquake (*M*
_{w} 8.1) occurred within 2 weeks of foreshock activities with *M*
_{w} > 6 (Ruiz et al. 2014). These foreshocks might trigger the following *M*
_{w} > 8 earthquakes, as discussed in this study. Additionally, if the Hyuga-nada earthquake occurs within several decades from now, the triggered Nankai earthquake becomes immature because the elapsed time is now only about 70 years since the previous Nankai earthquake of 1946 and 1944. Thus, rupture separation between the western and eastern segments could occur for an immature and small Nankai Trough earthquake, as shown in Figs. 9c and 10c for 55–80 % trigger timing.

In contrast, if an *M*
_{w} 7 Hyuga-nada earthquake does not occur in the next several decades, no matter where the next Nankai earthquake is initiated, the next Nankai Trough earthquake may be a short time interval rupture type of *M*
_{w} > 8.5, as shown in Figs. 9c and 10c for 80–100 %.

As a current countermeasure against the next Nankai Trough earthquake, a real-time earthquake and tsunami monitoring system called dense oceanfloor network system for earthquakes and tsunamis (DONET; Kaneda et al. 2011) 1 and 2 are deployed at Kumano-nada and in the Kii channel, respectively. These areas were chosen based on a typical scenario where a Tokai (or Tonankai) earthquake initiates at Kumano-nada, then at various time intervals, a Nankai earthquake is triggered at the Kii channel, similar to the 1944 and 1946 earthquakes. However, based on the scenario proposed here, the triggered Nankai earthquake might occur before Tokai (or Tonankai) earthquake within the following several decades. Therefore, we need the real-time monitoring system to begin observations from Hyuga-nada to the western Shikoku region, the anticipated possible hypocenter location, as soon as possible. In addition, for such a scenario, detailed estimates of possible seismic and tsunami damage should be calculated, similar to those for maximum-class earthquakes.

From the numerical experiment results, the 1498 Meio Nankai Trough earthquake could be interpreted as a triggered event due to the occurrence of Hyuga-nada earthquake. In 1498, a Hyuga-nada earthquake occurred on July 9, then the Meio Tokai earthquake on the eastern Nankai Trough segments occurred after two and a half months (September 20) as shown in Fig. 1. Although the rupture timing of the western Nankai Trough segments in the Meio events has been in dispute so far, our results here suggest that a scenario which a Nankai earthquake occurred between the occurrences of the Hyuga-nada and Tokai earthquakes is physically plausible. Then, this triggered scenario for the 1498 Meio earthquake corresponds to triggered event in cycle 2 of our numerical experiments, since the preceding 1361 Ko’an Nankai Trough event was the larger one among the successive Nankai Trough earthquakes. As mentioned in the “Results” section, the minimum interval between a Hyuga-nada and triggered Nankai earthquakes is 11 years for cycle 2. The discrepancy of the simulated and 1498 trigger timings may indicate that our assumption of the Hyuga-nada earthquake with a moment magnitude of 7.5 is too small for the 1498 Hyuga-nada earthquake, because the 1498 Hyuga-nada earthquake was large enough to cause the tsunami hazards at the Chinese continent (Tsuji 1999). Since we only examined the effect of the occurrence timing of the Hyuga-nada earthquake with a fixed magnitude and location on the Nankai Trough earthquake cycles, how the variations in the magnitude or the position of the trigger earthquake affects nearby seismic cycles will be briefly considered. In our trigger model obeying Eq. 5, the sub-fault *i* off Ashizuri is accelerated from *V*
_{
i
}
^{
−
} to *V*
_{
i
}
^{
+
} depending of the static stress change *Δτ*
_{
H,i
} due to the occurrence of a Hyuga-nada earthquake. Then, during the small period *Δt*, slip with the amount of *U*
_{
i
} = *V*
_{
i
}
^{
+
}
*Δt* is induced, and both shear stress and strength on the sub-fault *i* are decreased; the strength about ~*B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
} is decreased by the slip weakening. Shear stress change on sub-fault *i* becomes *G*
_{
ij
}
*U*
_{
j
}, since it is affected by slips of other sub-faults. Hereafter, we apply Einstein summation convention to only the index *j*. Then, according to Eq. 2, the quantity *G*
_{
ij
}
*U*
_{
j
} + *B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
} represents whether the sub-fault *i* is accelerated (if *G*
_{
ij
}
*U*
_{
j
} + *B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
} > 0) or decelerated (if *G*
_{
ij
}
*U*
_{
j
} + *B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
} < 0) due to triggering, and it exponentially acts for the acceleration or deceleration. Our simulation results show that *G*
_{
ij
}
*U*
_{
j
} + *B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
} for the sub-fault *i* off Ashizuri has a non-negligible positive value due to the *M*
_{w} 7.5 Hyuga-nada earthquake at a later stage in cycles 1 and 3, and it grows to trigger the Nankai earthquake several years after the Hyuga-nada earthquake. Therefore, as long as the spatial distribution pattern of induced slip *U*
_{
j
} (1 < *j* < *N*) is not so different to our simulation result, Hyuga-nada earthquakes with a greater magnitude or at a closer position may dramatically shorten the time lag between a Hyuga-nada earthquake and Nankai earthquake. From this point of view, the foreshocks for the 2011 Tohoku and the 2014 Iquique earthquakes might be interpreted as triggering sources (*M*
_{w} > 6) located close to the target earthquake. Let us consider the occurrence of a very small (*M*
_{w} << 6) local earthquake next to the sub-fault *i*. Then, the local earthquake may only increase the meaningful stress on the sub-fault *i*. In this situation, stress change on sub-fault *i* will become *G*
_{
ii
}
*U*
_{
i
} due to the isolation of the induced slip. Then, the quality for acceleration, *G*
_{
ii
}
*U*
_{
i
} + *B*
_{
i
}
*U*
_{
i
}/*L*
_{
i
}, tends to be negative, because *G*
_{
ii
} is generally negative with a large absolute value. Thus, it may be hard for a single small local earthquake to accelerate nearby asperity, even if the local small earthquake generates the same stress change in the next sub-fault *i* as that of a distant *M*
_{w} 7.5 large earthquake.

In this study, we propose a possible trigger scenario in addition to the conventional occurrence scenario in which the Nankai Trough earthquakes initiate off Kumano-nada, similar to the 1946 and 1944 events. From the numerical experiments based on the proposed scenario, in some conditions, large variations in the recurrence intervals and magnitudes of Nankai Trough earthquakes are generated owing to the triggering effect of Hyuga-nada earthquakes. This suggests that simulations with eastern and western possible hypocenters predict more complicated seismic cycles, with larger variations of recurrence intervals and magnitudes, than those with a single hypocenter at Kumano-nada. Therefore, for more consistent reproduction of the historical Nankai Trough earthquake sequence, we should also allow several variations of hypocenter locations on the eastern segments. A seismic cycle modeling of the Parkfield segment of San Andreas Fault shows that spontaneous complexity in rupture dynamics can naturally give rise to variability in hypocenter location (Barbot et al. 2012). Though this is based on the fully elastodynamic modeling of seismic cycles, similar variability might be introduced in our quasi-dynamic model of Nankai Trough based on the more complex distribution of frictional parameters near Kumano-nada.

Seno (2012) pointed out that, based on reconsideration of the historical earthquakes, there are two possible rupture patterns on the eastern segments: the “Ansei”-type earthquake, which ruptures the D and E segments in Fig. 1, and the “Hoei”-type earthquake, which ruptures the C and D segments. Each type of earthquake has a recurrence interval of 350–400 years, and the two types of earthquakes have occurred alternately. Although the hypocenters are not explicitly mentioned in Seno (2012), since the 1944 Tonankai earthquake is categorized in Hoei-type, the hypocenters should be located on segment C in Kumano-nada. Meanwhile, for Ansei-type earthquakes, the hypocenters must be located on segment D in Enshu-nada or segment E in Suruga Bay. Hence, for the purpose of a more quantitative representation of the historical Nankai Trough earthquake sequence, applying the hypothesis of alternative hypocenters on the eastern segments to the quasi-dynamic seismic cycle simulations, which have modeled only the hypocenters in Kumano-nada, is an urgent task.

## Concluding remarks

To explore possible scenarios for the next Nankai Trough earthquake, we focused on Nankai earthquakes triggered by a preceding Hyuga-nada earthquake and investigated the conditions in which such triggering occurs. As shown by the results of detailed numerical experiments, after the occurrence of smaller Nankai Trough earthquakes such as the 1946 and 1944 earthquakes, the outer extensions of the ruptured area are stressed and slide with a higher slip velocity than after the larger Nankai Trough earthquakes. Therefore, a Hyuga-nada earthquake (*M*
_{w} 7.5) on the western extension of the western edge of a smaller Nankai Trough earthquake tends to trigger the occurrence of a Nankai earthquake, especially in the latter half of the seismic cycle after the smaller Nankai Trough earthquake. Hence, if a Hyuga-nada earthquake occurs within several decades from now, a Nankai earthquake may be triggered. Then, rupture separation between the western and eastern segments may also occur. Although a triggered Nankai Trough earthquake may be immature and small (8 < *M*
_{w} < 8.5) relative to the larger class of Nankai Trough earthquakes (*M*
_{w} > 8.5), revisions of current countermeasures against the next Nankai Trough earthquake should be performed as soon as possible. In contrast, if an *M*
_{w} 7 Hyuga-nada earthquake does not occur in the next several decades, no matter where the next Nankai earthquake is initiated, it may be a large (*M*
_{w} > 8.5), short time-interval rupture type of earthquake.

## Declarations

### Acknowledgements

We thank Sylvain Barbot and an anonymous reviewer for their useful comments and suggestions. The Generic Mapping Tools (GMT) package (Wessel and Smith 1995) was used to produce some of the figures. Some of the results were obtained using the K computer at the RIKEN Advanced Institute for Computational Science (Proposal number hp150215).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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