Abstract
The deformation transient following large subduction zone earthquakes is thought to originate from the interaction of viscoelastic flow in the asthenospheric mantle and slip on the megathrust that are both accelerated by the sudden coseismic stress change. Here, we show that combining insight from laboratory solidstate creep and friction experiments can successfully explain the spatial distribution of surface deformation in the first few years after the 2011 M_{w} 9.0 TohokuOki earthquake. The transient reduction of effective viscosity resulting from dislocation creep in the asthenosphere explains the peculiar retrograde displacement revealed by seafloor geodesy, while the slip acceleration on the megathrust accounts for surface displacements on land and offshore outside the rupture area. Our results suggest that a rapid mantle flow takes place in the asthenosphere with temporarily decreased viscosity in response to large coseismic stress, presumably due to the activation of powerlaw creep during the postearthquake period.
Introduction
Postearthquake deformation can be interpreted as a process of relaxing the stress perturbation caused by the earthquake rupture. It generally consists of the deformation due to continued, mostly aseismic slip on the megathrust (afterslip)^{1} and viscoelastic relaxation in the asthenosphere^{2}. Afterslip relaxes the stress perturbation by localized deformation in the region of the fault plane that surrounds the earthquake rupture. Viscoelastic flow relaxes the coseismic stress change by distributed, plastic deformation in the surrounding mantle^{3,4}. The postearthquake deformation of the 2011 M_{w} 9.0 TohokuOki earthquake was captured by a wide array of landbased^{5,6} and seafloor^{7,8,9} instruments. This widespread observation network captured a complex postearthquake deformation field. Some neartrench seafloor stations moved seaward, in the opposite direction to the longterm subduction motion, while others moved landward (Fig. 1a). The postearthquake vertical motion was also complex, with many seafloor stations moving in opposing directions than that on land. Several studies^{7,8,10,11,12} claim that viscoelastic relaxation largely contributed to these patterns.
The 2011 M_{w} 9.0 TohokuOki earthquake induced a large stress perturbation in the surrounding lithosphere that accelerated the flow in the oceanic asthenosphere and in the mantle wedge. It is natural to expect that viscoelastic relaxation during the postearthquake period can be described by the constitutive properties of peridotite, a rock assemblage of mostly pyroxene and olivine, under high temperature and pressure conditions^{13}. Likewise, afterslip may be controlled by the frictional properties of the megathrust. Laboratory experiments suggest that the plastic deformation of mantle rocks is accommodated by a thermally activated flow that obeys a powerlaw relation between stress and strain rate^{14,15}. The friction between the subducting slab and the upper plate is governed by a laboratoryderived kinematic friction law^{16,17} that predicts the velocity of afterslip based on the stress evolution. Incorporating the laboratoryderived constitutive properties for viscoelastic flow and afterslip successfully explained the deformation that followed the 2012 M_{w} 8.6 Indian Ocean earthquake^{4}, for which the surrounding rheological structure is rather simple. In contrast, most studies of the TohokuOki earthquake employed simplified rheological models with linear viscoelastic flow in the mantle and kinematic afterslip^{8,11,12,18,19,20,21}, or explored more realistic rock properties in twodimensional models^{10,22}. This limitation of approach is due in part to the difficulty in dealing with the combination of the geometrical complexity and the nonlinear governing equations. Several of the linear viscoelastic models inferred from the TohokuOki earthquake include a thin lowviscosity (weak) layer along the lithosphere–asthenosphere boundary (LAB) in the upper mantle^{8,11,21}. A sharp decrease of seismic velocity at LAB^{23,24} has been attributed to the presence of water or partial melts, which upholds the existence of a lowviscosity layer as a permanent rheology structure^{8}. This interpretation remains controversial, as these findings require explanations consistent with mineral physics^{14,15}.
Here, we consider the threedimensional response of the lithosphere–asthenosphere system following the 2011 M_{w} 9.0 TohokuOki earthquake with powerlaw viscoelastic flow in the mantle and afterslip on the megathrust, incorporating a realistic velocity structure for the Japanese margin, Earth’s sphericity and laboratoryderived, nonlinear rock constitutive properties. We assume that the viscoelastic flow of the upper mantle is accommodated by steadystate dislocation creep, with the following stress–strain rate relationship^{14}
where ε_{M} is the norm of the strain in the Maxwell element in a Burgers material (see Methods), A_{M} is a preexponential factor, C_{OH} and r are the water concentration and its exponent, σ is the norm of deviatoric stress tensor, n is the stress exponent, H = Q + pΩ is the activation enthalpy, R is the universal gas constant, and T is the temperature. The enthalpy incorporates the activation energy Q and the activation volume Ω and depends on the confining pressure p. In addition, we incorporate the transient creep that is thought to take place during the early stage of postearthquake transients^{4,25}. We use a model that includes the transient effect of dislocation creep^{4}, as
where ε_{K} is the norm of the strain in the Kelvin element in a Burgers material, A_{K} is a preexponential factor and G_{K} is a work hardening coefficient. Here we use the same parameters as in Eq. (1) with A_{K} = A_{M} and G_{K} = G, where G is rigidity. We combine dislocation creep with diffusion creep, but the latter does not play a significant role in our shortterm simulations (see Methods). For the same reason, we did not include the transient effect of diffusion creep. We assume that the velocity of afterslip on the megathrust is governed by the ratedependent and statedependent friction, given by the constitutive law,
combined with the aging law^{17},
where V is slip velocity, V_{*} is the reference velocity, τ is the shear traction, τ_{s*} is the steadystate frictional resistance, and Δτ_{s} is a state variable analogous to the “strength as a threshold”^{26}. A is a parameter that controls the fracture energy consumed during fault slip, the frictional parameter B controls strength recovery, and L controls the slip weakening distance. Simulating the dynamics of this nonlinear system in threedimensions with realistic elastic, frictional, and viscoelastic properties requires stateoftheart modeling strategies^{27,28} (see Methods). Following this approach, we show the postearthquake deformation in Tohoku to be caused by rapid flow in the asthenosphere, due to temporarily decreased viscosity because of coseismic stress.
Results
Cumulative 2.8 year postearthquake displacement
The temperature profile used in Eqs. (1) and (2) is based on a twodimensional model for the Tohoku region^{29}, which we expanded along strike with a mantle temperature of 1380 °C (Fig. 1b), compatible with another study^{4}. We converted the background shortening rate of 10^{−8} yr^{−1} to determine the background stress based on the rheological law^{30}. For the initial condition of the simulation, we borrow the coseismic slip (Fig. 1a) and the fault constitutive properties (i.e., V, τ, Δτ_{s}, A, B and L) (Figs. 1, 2) from a simulation of giant earthquakes in the Tohoku region^{31} (see Methods for details). We divide the region into three plates: a continental plate that includes the NorthAmerican and Eurasian plates and two oceanic plates, the Pacific and the Philippine Sea plates. Each tectonic plate consists of an elastic layer near the surface (the crust and the lithospheric mantle) and a viscoelastic mantle layer below (Figs. 1, 3). The elastic and viscoelastic layers in the three plates share the same elastic properties (Fig. 1c).
Our simulated deformation shows similar patterns to the observation data for the cumulative 2.8 year postearthquake displacement in the horizontal direction (Fig. 4a) when we choose the following rock properties K = 10^{0.56} MPa^{−n} s^{−1}, C_{OH} = 1000 ppm H Si^{−1}, Q = 430 kJ mol^{−1}, r = 1.2, Ω = 13.5 cm^{3} mol^{−1}, and n = 3 (see Methods). For simplicity, we assumed a similar average water content in the oceanic asthenosphere and in the mantle wedge, even though water concentration may be larger in the mantle wedge corner due to slab dehydration^{32}. The values adopted for the activation energy and the activation volume fall well within the uncertainties constrained by laboratory experiments^{15}, i.e., Q = 410 ± 50 kJ mol^{−1} and Ω = 11 ± 3 cm^{3} mol^{−1} for olivine, despite the required extrapolation to different temperature and pressure conditions. This indicates that the laboratoryderived rheological and frictional models with the proper in situ conditions allow us to make firstorder predictions about how the lithosphere–asthenosphere system will deform in response to a large earthquake.
Effective viscosity and time series of displacement
The temporal and spatial evolution of effective viscosity after the giant earthquake naturally results from the nonlinear constitutive relations (1) and (2) and plays an important role in the rapid and complex deformation that occurs during the postearthquake period^{33}. In response to the large (above 1 MPa) stress perturbation in the upper mantle, the effective viscosity (see Methods for the definition) was largely reduced shortly after the earthquake in the depth of 100–200 km in the oceanic mantle and 80–180 km in the mantle wedge (Fig. 5). Temporal increase of effective viscosity explains well the time series of horizontal displacement in the station MYGI and some land stations that are aligned in the trench normal direction from the epicenter (Fig. 6). The misfit in the station MYGW is likely due to the dominance of the elastic response due afterslip there, which we discuss in the Discussion section.
Decomposition of source mechanisms and induced deformation
In our model, the postearthquake displacements result from mechanically coupled afterslip and ductile flow. Both mechanisms are initially driven by the coseismic stress change, but they subsequently influence each other. Despite that coupling, the kinematics of deformation can be uniquely attributed to one source mechanism or the other: the displacements are a linear function of the slip and viscous strain distribution^{34,35,36,37}. We exploit these relationships (see Methods) to unravel the relative contributions of afterslip and viscoelastic flow within the subduction zone (Figs. 4, 7). The flow of lowviscosity mantle material below the trench axis drives westward motion around the trench, explaining the continued displacement of the seafloor stations located above the coseismic rupture (MYGI, KAMS, and KAMN, Fig. 4b). The accelerated flow in the mantle wedge contributes to the eastward displacement of GPS stations on land. Afterslip on the megathrust is essential to explaining the deformation on land, but also the spatial pattern of displacement of the seafloor stations, such as eastward displacement seen in the stations FUKU and MYGW (Fig. 4b). Both these stations are in locations where viscoelastic flow produces little horizontal displacement, making the postearthquake response due to the afterslip dominant there (Fig. 7).
Discussion
Remarkably, the spatial distribution of effective viscosity derived from laboratory data and coseismic stress change is similar to those inferred from optimization of simplified linear viscoelastic models^{8,11,21}. The effective viscosity shortly after the earthquake is around 2 × 10^{17} Pa s at the minimum both in the mantle wedge and the oceanic mantle. This is equivalent to the viscosity in a linear transient creep model that fits observed postearthquake deformation during the early stage^{8}. The LAB, originally identified as a lowseismicvelocity layer^{23,24}, has also been associated with a permanent lowviscosity structure. However, our result suggests that the LAB hosts a rapid mantle flow with temporarily decreased viscosity in response to large coseismic stress, rather than a permanent lowviscosity layer. A recent experimental study suggests that the presence of water, which has been invoked to explain a permanent lowviscosity structure at the LAB, is not compatible with the low seismic velocity^{38}. Further studies are required to unravel the nature of the LAB.
Despite the excellent fit at numerous stations in the farfield, there remain a few discrepancies with the nearfield data, presumably because our model does not include some fine details of the coseismic rupture offshore. For example, the simulated horizontal displacement at the station FUKU is nearly half of the measured one, despite a good agreement in the azimuthal direction. A peak of the amplitude of afterslip in the dashed rectangle in Fig. 4b should be slightly closer to station FUKU to better fit the data, perhaps indicating that the coseismic slip was overestimated in this region. Such afterslip distribution should also fit better the horizontal displacements in the southern part of the land area (the dashed rectangle in Fig. 4a). In addition, the displacement time series in the station MYGW (Fig. 6) shows larger displacements in the plate convergence direction compared to the observed one. Figure 4b suggests that this is because the azimuthal direction of the elastic response due to the afterslip is almost parallel to the plate convergence direction, while the observation presents a displacement in the southeast direction. Smaller afterslip at the south of Onagawa (the dotdashed rectangle in Fig. 4b), which is more consistent to the estimated afterslip distributions in previous studies^{8,11}, is likely to produce a displacement with a similar azimuthal direction to the observation. In the vertical displacement, significant uplift is observed in the forearc (the purple circles in Fig. 7).
In the trenchnormal profile of the stations MYGI and MYGW, although viscoelastic flow in the simulation produces uplift in this region, subsidence due to afterslip cancels it out (the green circles in Fig. 7). A significant portion of this uplift in viscoelastic flow is due to stress change associated with afterslip, which we inferred from simulations of viscoelastic flow that exclude afterslip (the green circles in Fig. 8a). Without the interaction between afterslip and viscoelastic flow, the computed 2.8year horizontal displacements are reduced by more than 10% in some of the land stations, and the vertical ones change by more than 30% in many stations in both the land and the seafloor (Fig. 8b). As afterslip in the near field can be highly sensitive to the details of the coseismic rupture, these residuals may be caused by still unresolved slip patterns of the mainshock. Nevertheless, our results highlight significant nonlinear interactions among coseismic slip, afterslip, and viscoelastic flow.
Our study demonstrates that a rheological model of the plate boundary based on independent geological and geophysical data can make realistic, firstorder predictions of the transient response of the lithosphere following giant earthquakes. Complex postearthquake deformation of a large subduction zone earthquake can be well explained by taking into account the laboratoryderived friction and viscoelastic flow laws in a threedimensional structural model. The discrepancy between the simulation and the data, particularily in vertical motions and in some seafloor stations, should be reduced, in principle, by refined models of the coseismic rupture and the in situ conditions, such as initial stress, temperature, and confining pressure, properties that are usually only constrained for long time scales^{29,39}. The approach is generally applicable to other oceancontinent subduction zones, implying that our understanding of viscoelastic properties and rocks friction may be detailed enough to predict the slow deformation of the lithosphere during the postseismic and interseismic periods.
Methods
Rheology model for upper mantle
We use the Burgerstype rheology, where the strain due to steadystate creep and transient creep are in series:
where ε_{v} is the viscous strain. In the Maxwell element, the strain rates for dislocation creep and diffusion creep add up, as
where η_{l} is a constant viscosity for diffusion creep. The viscosity for diffusion creep is 10^{1−2} times larger than effective viscosity for dislocation creep shortly after earthquakes of M_{w} 8.2 and 8.6^{4}, so the influence of diffusion creep is not expected to be very large in the 2.8 years deformation after the 2011 M_{w} 9.0 TohokuOki earthquake. We use η_{l} = 1 × 10^{19} Pa s for the whole of the region, which is nearly the average value of the viscosity structure estimated for steadystate 2D model around the Japan Trench^{30}. In a tensor notation,
We define the effective viscosity \(\eta^{{\mathrm{eff}}} = \sigma /2\dot \varepsilon\), thus
where \(\eta _{\mathrm{M}}^{{\mathrm{eff}}}\) is effective viscosity in the Maxwell element and
In the same manner, we can write the transient dislocation creep (2) in the tensor notation as
where q_{ij} = σ_{ij}−2G_{K}(ε_{K})_{ij} and q = (q_{kl}q_{kl})^{1/2}. Then, the effective viscosity of the transient dislocation creep is
where \(\eta _{\mathrm{K}}^{{\mathrm{eff}}}\) is effective viscosity in the Kelvin element.
Our temperature pattern (Fig. 1b) in the elastic slab is significantly different from the reference thermal model^{29} in that it keeps a low temperature even in the depth deeper than 200 km. However, the absolute temperature does not affect the simulation results significantly because the high pressure at these depths hardens the material. In the simulation, we use the values proposed from laboratory experiments^{15} for K, r, and n, while Q and Ω were chosen within the error bar obtained in the same experiments, so that the computed displacement values are more consistent with the data. We set the C_{OH} value as an average in the upper mantle. Further study on more detailed variation of measured displacement should require considering heterogeneous distribution of water content^{4,40}.
Coseismic slip and fault friction setting
To compute the postseismic deformation, we borrow the frictional properties assumed in the simulations of Nakata and colleagues^{31}. The top of the subducting slab is modeled as a frictional interface loaded by the same tectonic forces that drive subduction. We assume the force balance
where τ_{i} and V_{i} are shear stress and slip velocity on the ith FEM node on the fault. V_{i} is in the direction opposite to the convergence rate (Fig. 1). v and v_{pl} are vectors whose components are V_{i} and (V_{pl})_{i}, the plate convergence rate. Here, the difference between v and v_{pl} is the source of deformation based on the back slip model^{41}, which assumes that the steadystate subduction does not contribute to the deformation at the free surface in the hanging wall. It means that the calculated displacement at the foot wall does not include the contribution from the subduction motion either. V_{pl} = 8.4 cm yr^{−1} is used for the whole region in this study. The second term introduces the effect of the seismic radiation damping^{42}. We use γ = 0.3G/2c, which is used in Nakata et al.^{31} to reproduce a shorter duration during the 2011 TohokuOki earthquake^{43}, where G is the rigidity and c is the shear wave velocity. In many previous studies, the simulations have been carried out assuming an elastic homogeneous halfspace, where \(\dot \varepsilon _{\mathrm{v}} = 0\). This makes F_{i} a linear function of v and enable F_{i} to be discretized by the boundary integral equation method (BIEM). In this study, we evaluate F_{i} directly by using the finite element method (see the next section), in which F_{i} can be a function of both v and \(\dot \varepsilon _{\mathrm{v}}\), and arbitrary geometry and material heterogeneity can be considered. We carry out time integration of Eq. (12) and the equations for the ratedependent and statedependent friction law (3) and (4) using an adaptive time step fifthorder Runge–Kutta algorithm^{44}. In our simulation, initial value of τ_{i} and Δτ_{si} is extracted from a time step right after the earthquake in the simulation of Nakata et al.^{31} (Fig. 2a), multiplied by 0.7 to bestfit the geodetic data (Fig. 9). The initial value of V_{i} is calculated with Eq. (3). Frictional parameters are also the same as in Nakata et al.^{31}, excluding that small patches for M7 earthquakes are removed (Fig. 2b). A and B values in Eqs. (3) and (4) are known to be normalstress dependent: A = aσ_{n} and B = bσ_{n}, where σ_{n} is the normal stress. See Nakata et al. for the normal stress distribution. V_{*} is set to be identical to V_{pl}.
Figure 9 shows the coseismic slip, the same as in Fig. 1, which we extracted from the cycle simulation results, and comparison between computed and observed coseismic displacement. Although this slip model is not inferred from observation data, it fits the horizontal component of coseismic crustal deformation data well when multiplied by 0.7. The stress distribution computed in response to this coseismic slip is used as the stress perturbation to compute powerlaw viscoelastic flow and afterslip evolution.
Finiteelement modeling
In the finiteelement modeling, we discretize the equations for viscoelastic deformation and fault friction using the mesh shown in Fig. 3. The mesh was constructed using an updated version of a meshing technique for quadratic tetrahedral elements based on a background structured grid^{28}. In the method, at first a uniform background cell covering entire targeted domain was used, and it defined the resolution of the layer interfaces as ds. The geometries of the ground surface and interfaces were simplified slightly to maintain good element quality. At the same time, unnecessary elements were merged to generate larger elements elsewhere. This method enables automated and robust construction of highresolution tetrahedral mesh directly from digital elevation model (DEM) data of crustal structure without creating a computeraided design (CAD) model. The updated version of the meshing algorithm carries out an additional post process to minimize the simplification of the geometry in the ground surface and interfaces as much as possible. Input elevation data sets are based on 900 m resolution topography data (JTOPO30), the CAMP model^{45} and a velocity data set for the Japanese Island^{46}. From these data sets we constructed a finite element model in which the geometry of layer boundaries is in 2km resolution (ds = 2 km) with slight modification. Using this finite element model, shear stress distribution on the fault, which is essential for computing stressdriven afterslip, is evaluated accurately in the target problem. The finite element mesh has 1,402,810,116 degreeoffreedom (DOF) and 346,885,129 tetrahedral elements. In viscoleastic material and elastic material, rigidity is G_{v} = 65 GPa and G_{e} = 45 GPa, respectively. Poisson’s ratio is ν = 0.25 and density is ρ = 3300 kg m^{−3} everywhere, which setting follows Sun et al. ^{8}. Confining pressure is calculated as p = ρgz, where g is the gravitational acceleration and z is depth.
To evaluate F_{i} in Eq. (12), we applied an algorithm based on a viscoelastic finite element formulation^{47,48}, which we modified to consider nonlinear viscoelasticity. Slip velocity v is input to the finiteelement model using the split node technique^{49} to evaluate response displacement rate. We consider the effect of gravity using surface gravity approximation^{50}. Since no inertia term is included in the equations, the problem is quasistatic, which ends up with solving an elliptic problem in every time step. It means we need to solve the system which has billions of DOF. We introduced a modified version^{51} of a massively parallel FEM solver for computing crustal deformation^{28} based on “GAMERA”^{27} (a physicsbased seismic wave amplification simulator, enhanced by a multiGrid method, Adaptive conjugate gradient method, Mixed precision arithmetic, Elementbyelement method, and pRedictor by Adams–Bashforth method).
We run the calculation using 2048 computer nodes (16,384 computer cores) of the K computer at RIKEN Center for Computational Science^{52}, each computer node of which has one CPU (Fujitsu SPARC64 VIIIfx 8 core 2.0 GHz) and 16 GB of memory, for nearly 10 h to obtain the postearthquake deformation for 2.8 years shown in Fig. 4.
Geodetic data
All the cumulative geodetic displacements plotted in the figures in this paper are adjusted to values relative to the stable part of the North American plate, on the basis of ITRF2005 model^{53}.
Viscoelastic and afterslip contributions
Figure 4b and the figures in the right in Fig. 6 present breakdown of computed displacement into contribution from elastic deformation due to afterslip and viscoelastic flow. In principle, calculated postearthquake deformation in this study can be decomposed into elastic response due to cummulative afterslip and viscoelastic strain (e.g. refs. ^{34,35,36,37}). For example, in the case of the Maxwelltype rheology model for simplicity, u_{original}, cumulative displacement vector at the GPS stations (corresponding to red arrows Fig. 4a), can be written as
where Δd and Δε_{v} are vectors for cumulative afterslip (corresponding to the black contour lines in Fig. 4b) and viscoelastic strain change, and G_{d} and G_{ε} are matrices for elastic Green’s functions to map afterslip and viscoelastic strain change to displacement at the GPS stations. u_{afterslip} = G_{d}Δd and u_{viscoelastic} = G_{ε}Δε_{v} correspond to the blue and red arrows in Fig. 4b, respectively. The second term of the righthand side is more complex in the case of the Burgerstype rheology model, but the discussion here still applies. Note that Δd includes slip driven by coseismic stress, stress due to viscoelastic deformation and stress due to afterslip itself. In the same manner, Δε_{v} includes strain change driven by coseismic stress, stress due to afterslip and stress due to viscoelastic relaxation itself. The contribution from each factor is nonlinearly coupled and cannot be decomposed from each other. u_{afterslip} and u_{viscoelastic} are calculated in the following three steps: 1. Extract accumulated 2.8 year afterslip distribution Δd that is computed based on the nonlinear interaction of the ratedependent and statedependent friction law and the nonlinear rock constitutive properties in the original simulation. 2. Compute elastic response displacement due to the cumulative after slip as u_{afterslip} = G_{d}Δd using the same finiteelement model. 3. Compute the difference u_{viscoelastic} = u_{original} − u_{afterslip} to recover the contribution from viscoelastic flow.
We also present a result postearthquake deformation simulation with “no interaction” between viscoelastic flow and afterslip (Fig. 8). In this simulation, we computed viscoelastic flow without the friction law (the red arrows in Fig. 8a), while computing afterslip without the nonlinear rock constitutive properties, only with pure elasticity. We finally summed up these to compute total deformation without their interaction (the red arrows in Fig. 8b).
Code availability
Computer codes for calculating viscoelastic relaxation and afterslip are available from the authors upon reasonable request.
Data availability
GPS data are available from the Geospatial Information Authority of Japan (http://terras.gsi.go.jp/). Other relevant data in this work are available from the authors upon reasonable request.
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Acknowledgements
We used GPS data provided by the Geospatial Information Authority of Japan. This study was supported by Post K computer project (Priority issue 3: Development of Integrated Simulation Systems for Hazard and Disaster Induced by Earthquake and Tsunami) and JSPS Fellowship (268867). We obtained the results using the K computer at the RIKEN Center for Computational Science (Proposal number hp160221, hp170249, and hp180207). S.B. was supported by the National Research Foundation (NRF) of Singapore under the NRF Fellowship scheme (National Research Fellow Awards Number NRFNRFF201304), the Singapore Ministry of Education (AcRF Tier 1 grant RG181/16), and by the Earth Observatory of Singapore, under the Research Centres of Excellence initiative. Some figures were produced using GMT software, developed by P. Wessel and W.H.F. Smith.
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R.A. and T.H. designed and conducted the study. R.A, S.D.B. and T.H. wrote the manuscript. R.A., K.F. and T.Ichimura wrote the simulation code. S.D.B. and M.H. contributed to refining the simulation algorithm. S.D.B. and T.Iinuma contributed to the modeling. R.N. prepared the data used in the afterslip calculation.
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Correspondence to Ryoichiro Agata.
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