Kato and Yoshida (2011) conducted a numerical simulation for understanding the mechanics of the 2011 Tohoku-oki earthquake. In this model, a strong asperity with higher effective normal stress was assumed at a shallower part of the plate interface. This shallow strong asperity controls the occurrence cycle of great earthquakes. The present paper discusses pore pressure distribution along the interface at the Pacific plate subducting beneath northern Honshu (Tohoku), Japan. Assuming that the permeability is exponentially dependent on effective stress and that the proportional constant drops to a lower value at a critical effective stress due to pore collapse, we obtained nearly hydrostatic pore pressure at a shallow part and nearly lithostatic pressure at a deep part of the plate interface. Assuming a different permeability model, in which increase of confining pressure has a larger effect than decrease of pore pressure, we obtained similar results. Such pore pressure distributions provide a possible generation mechanism for a shallow strong asperity.
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 The Mw = 9.0 great Tohoku-oki earthquake of March 11, 2011, broke the entire seismogenic depth of the plate interface between the subducting Pacific plate and the overriding plate at northern Honshu, Japan. Finite fault models estimated from seismic waveforms, tsunami, Global Positioning System (GPS), and ocean bottom displacement data indicate that the seismic slip extended about 500 km along the Japan trench and about 200 km along the dip direction and the largest seismic slip greater than 30 m occurred at a shallower part of the fault off Miyagi prefecture and seismic slip tended to decrease with increasing depth [e.g., Fujii et al., 2011]. Large M7 earthquakes repeatedly occurred at the plate interface off Miyagi and the summed moment release by these M7 interplate earthquakes was significantly smaller than that expected from the relative plate motion [Yamanaka and Kikuchi, 2004]. GPS observations indicate that the plate interface off Miyagi was firmly locked and the slip deficit rate was close to the relative plate rate [e.g., Nishimura et al., 2004]. Historical documents and tsunami deposits indicate that a great earthquake of M > 8 took place off Miyagi in 869 [e.g., Minoura et al., 2001].
Kato and Yoshida  (hereinafter referred to as paper 1) conducted a numerical simulation of the earthquake cycle at a subduction zone using a 2D model to understand the mechanics of the 2011 great Tohoku-oki earthquake, assuming the rate- and state-dependent friction law. Figure 1 shows the assumed depth dependence of the effective normal stress σneff. In this simulation, the strong shallow asperity is realized by higher values of effective normal stress σneff and/or large values of characteristic slip distance L at a shallow part of the plate interface. Because σneff is given by the difference between lithostatic pressure and hydrostatic pore pressure, the large contrast of σneff results from higher pore pressure at deeper parts of the plate interface. The simulation results successfully explained the following important features of observations: (1) Great earthquakes that break the entire seismogenic plate interface repeatedly occur at recurrence intervals longer than several hundred years. (2) The largest seismic slip of a great earthquake occurs at the shallow part of the plate interface. (3) Slip deficit is accumulated at the deeper part of the plate interface where M7 earthquakes repeatedly occur during the interseismic period of the great earthquakes. The simulated depth distribution of the seismic slip showed a large slip at a shallow part, and this distribution agrees well with the result obtained by Tanioka et al. (http://cais.gsi.go.jp/YOCHIREN/activity/191/image191/017-018.pdf) from joint inversion analysis of tsunami, GPS, and seafloor geodetic data.
 Because the pore fluid pressure distribution along the plate interface is very important to understand the occurrence of interplate earthquakes, the issue has been discussed in a number of studies (see introduction of Seno ). Extremely high pore pressure within fault zones is considered to be one of the possible causes of weak plate boundaries such as the San Andreas fault [e.g., Rice, 1992]. However, drilling into the San Andreas fault zone showed no evidence of anomalously high pore pressure within the fault [Zoback et al., 2010]. The pore pressure distribution has not been constrained well so far. In the present paper, we address this issue and discuss a possible mechanical origin for the shallow asperity.
2. Nearly Hydrostatic Pore Pressure at a Shallow Part
Figure 1 shows depth dependence the effective normal stress σneff assumed in the model of paper 1. σneff is given by (ρ − ρw)gy for y ≤ 20 km, where y is depth, ρ = 2.8 × 103 kg/m3, ρw = 1.0 × 103 kg/m3, and g = 9.8 m/s2. σneff is assumed to be 88.2 MPa for y > 20 km. Although the expression of (ρ − ρw)gy contains errors due to ignoring the deviatoric stress component in the fault zone, the errors are not significant.
 Following Rice's  model as applied to the San Andreas fault, we model a dipping fault zone along the plate interface as a channel of uniform width d assuming that a fault zone is more permeable than the adjoining rock of the plates, and that fluid is provided from a source at the bottom of the model region (Figure 2). We ignore fluid loss from the fault zone to the adjoining rock, and also ignore variations in the fluid mass density. Then we assume that the upward volumetric flow rate q along the dipping fault is independent of depth. Letting K be permeability at y, and Pp be pore pressure, the steady state Darcy's flow is
where y′ is distance along the fault, η is viscosity, and θ is a dip angle of the plate interface.
 Permeability K of various rocks is expressed as a function of effective pressure [e.g., David et al., 1994]
where σ is confining pressure, σ* is a constant. When pore pressure is equal to lithostatic pressure, Pp = σ = ρgy, the permeability becomes independent of depth, and is given by K = K0. Under this condition, from equation (1) we obtain
 Assuming that the pore pressure is not higher than the lithostatic pressure along the fault zone, equation (3) gives the lower limit kmin of K0 for a given fluid influx q, for the cases with uniform K0. When K0 has a heterogeneous distribution, K0 can locally be smaller than kmin.
 From equations (1) and (2), we calculate the pore pressure distribution for various values of σ* and K0 (Figure 3). Values of η and θ were assumed to be 0.001 Pa s and 20°, respectively, and ρ, ρw, and g were assumed to be the same as paper 1. Note that the results are independent of values of q. Compiled published data of σ* for different rocks by David et al.  shows that σ* is 50 to 700 MPa for sandstones. Figure 3 indicates that when σ* = 300 MPa and K0 > 10 kmin, pore pressure at depths <20 km is near hydrostatic pressure. When σ* = 100 MPa, pore pressure for K0 > 100 kmin is nearly hydrostatic, and pore pressure for K0 > 10 kmin is closer to hydrostatic than lithostatic. Nearly hydrostatic pore pressure at a shallow part is a key condition for generating a strong shallow asperity. These results suggest that the strong shallow asperity is realized by a uniform K0 model. However, this simple model cannot reproduce both hydrostatic pressure at a shallow part and nearly lithostatic pressure at a deep part.
3. Nearly Lithostatic Pore Pressure at a Deep Part
David et al.  investigated permeability dependence on effective pressure and found that, for sandstone, a sudden decrease in permeability is induced by pore collapse beyond a critical pressure. Sometimes, there was a sharp drop in permeability by about three orders of magnitude immediately following the onset of pore collapse and grain crushing at the critical pressure. We take this pore collapse into account when calculating pore pressure distributions. For simplification, we assume that K0 = K1 at a shallow part while beyond a certain depth at which the effective pressure reaches a critical value, K0 drops to K2.
Paper 1 assumed that at a deep part, the effective normal stress is independent of depth (Figure 1) as Rice  assumed. In such situations, because permeability is a function of effective stress, we obtain the relation between fluid flux and the effective stress σ2eff at the deep part as
 We then numerically calculate pore pressure and permeability using equations (1) and (2) from the shallower depth under boundary condition of Pp = 0 at y′ = 0, for K0 = K1 = 56 kmin = 10−15m2, σ* = 100 MPa. When the effective pressure reaches the critical pressure at a certain depth, we set K0 = K2 for the deeper part taking K2/K1 = 0.03, 0.04. We assume a critical pressure of 290 MPa, which is the average of four sandstone samples measured by David et al.  (excluding one tested sample with extremely low value). Figure 4 shows that a sufficiently deep part has depth independent flow in which equations (4) and (5) are satisfied. At the transient part (19 < y < 30 km), pore pressure decreases with upward flow, resulting in higher effective stress that leads to lower permeability, and therefore a large pore pressure gradient. Figure 4c shows distribution of effective stress, which is similar to the assumed stress distribution of paper 1; at a shallow part the pore pressure is nearly hydrostatic, and at a deep part the pressure is nearly lithostatic. The value and depth of the effective stress peak are determined by the critical pressure of pore collapse. The effective stress at a deep part is determined by equation (5), or degree of pore collapse represented by K2/K1. If we take into account depth dependence of η, distribution of K0/η controls the pressure distribution. As η decreases with depth, higher degree of pore collapse is needed to obtain the similar pressure distribution.
 Pore collapse is not a unique mechanism that generates the effective stress distribution assumed in paper 1. Bernabe  pointed out that permeability data measured in the laboratory is expressed as
α takes values near 1.0 when pore pressure was applied before confining pressure, while it was much lower (0.4 ∼ 0.6) in the other cases. This relation can be written as
This form indicates that effective K0 decreases with depth.
 If we use equation (8) for all the depth, effective K0 becomes lower than kmin. To avoid this, we assume that effective K0 is given by equation (8) at depths ≤30 km, and a constant value of K0eff(30 km) for depth >30 km. The value of 30 km was chosen so that the resultant pressure distributions become similar to paper 1. Figure 5 for K0 = 56 kmin = 10−15m2 and σ* = 100 MPa shows the pore pressure distribution, permeability, and effective stress. We also obtained nearly hydrostatic pressure at a shallow part, and nearly lithostatic pressure at a deep part although the results show gradual change compared with the pore collapse model. These two kinds of calculation demonstrate that a decrease of K0 with increasing depth leads to an effective stress distribution similar to that assumed in paper 1.
 We will now estimate the upper limit of the fluid influx along the plate boundary. Iwamori and Zhao  assumed a subducting oceanic crust of 7 km thickness containing initially 6 wt.% H2O to model transportation of H2O along with the subduction of the Pacific plate beneath the northeast Japan arc. Using the same assumption, the maximum provided fluids with unit width is estimated to be 100 m3/yr m for a relative plate rate of 85 mm/yr. To estimate the upper limit of the fluid influx, we assume that all the fluids provided by this maximum source flows upward along the plate interface. The upper limit of the fluid influx is estimated to be 1 m3/yr m2 for the thickness of the fault zone, d in Figure 2, of 100 m. When calculating the pore pressure distribution in Figure 4, we assumed K1 = 56 kmin. From equation (3) with θ = 20°, we find that fluid flux |q| is 0.003 m3/yr m2 for K1 = 10−15 m2, and |q| is 0.15 m3/yr m2 for K1 = 50 × 10−15 m2, which is applicable for sandstone. These are 0.3% and 15% of the upper limit of the fluid influx, respectively. Although there is large uncertainty in the fluid influx estimation, it is considered likely that sufficient fluid flow is supplied along the northern Honshu plate interface.
 The present paper focuses on the possibility of the shallow strong asperity on the Tohoku-oki plate interface from the viewpoint of pore pressure distribution. We conclude that a decrease of K0 with depth provides a possible generation mechanism for a shallow region of higher effective normal stress, resulting in a shallow asperity as demonstrated in paper 1. Hasegawa et al.  studied temporal change in the stress field after the 2011 Tohoku-oki earthquake by stress tensor inversions of the focal mechanisms of earthquakes near the source region. They found that the deviatoric stress that caused the Tohoku-oki earthquake was mostly released by the earthquake. This means that σ2eff is probably lower than 88.2 MPa, which was assumed in paper 1. Seno  estimated the shear stress averaged over the seismogenic interface off Miyagi, in the eastern region of northern Honshu, to be approximately 20 MPa. Lower σ2eff is realized by K2 closer to kmin.
 Effective stress distributions having a similar pattern to Figure 4 may be found for other plate boundaries. As described above, the value and depth of the peak effective stress depend on K1, K2, and the critical pressure. If the degree of pore collapse K2/K1 is not significant, peak may not appear. Therefore the shallow strong asperities do not universally exist at plate interfaces. It is possible that the shallow strong asperities control the coupling strength between downgoing and upper plates, in addition to other physical properties such as convergent rate, slab age, and fault temperature, which have been pointed out in previous studies [e.g., Ruff and Kanamori, 1980; McCaffrey, 2007].
 We are grateful to H. Noda and an anonymous reviewer for valuable comments, which improved the manuscript. Discussions with T. Seno, T. Koyaguchi, and T. Matsuwzawa were useful for clarifying our thought.
 The Editor thanks Hiroyuki Noda and an anonymous reviewer for their assistance in evaluating this paper.