### 2.1 Defects of existing methods

Typically, governing equations are solved to model a thermal structure in the deep crust by assuming factors such as the heat conductivity of rocks, the radioactive heat production, fluid convection, and partial melting, using heat flow data obtained at the ground surface (Jaupart and Mareschal, 2010). However, available heat flow data are not so dense. Although thermal gradient data in the inland areas of the Japanese Islands were considerably increased by Tanaka et al. (2004), they are still insufficient for our purpose, as described in Section 1, from the viewpoint that a statistical approach is required to analyze the data contaminated by shallow noises such as groundwater flow (e.g., Okubo et al., 2005) (Fig. 3). Further problems include bias due to frequent sampling from a hydrothermal field (Tanaka et al., 2004). Another problem is that the solidus temperature of the crustal materials depends on the chemical composition, pressure, and the quantity of H_{2}O and CO_{2} (Ashworth and Brown, 1990), and cannot be definitely given. It was also revealed in recent years that the heat conductivity of crustal materials is much lower than previously thought under high-temperature conditions, and, hence, the crust may be partially melted owing to heat generation by crustal deformations such as orogenic movements (Whittington et al., 2009).

Hasegawa et al. (2000) suggested another method to model the thermal structure of the crust: they estimated the thermal structure of the upper crust in the Tohoku district, Japan, from *P*-wave perturbations obtained by travel time tomography. It would be difficult, however, to apply their method to the lower crust because of the lack of information on the mineral distribution. Therefore, the problems in modeling the thermal structure of the deep crust are due to limitations in the information available on the distribution of material properties.

### 2.2 A method proposed in this study

We first estimate the temperature both immediately above and beneath the target depth of the lower crust, and then use the model under the assumption that the lower crust temperature is intermediate between those temperatures (Fig. 2). More specifically, the temperature at the bottom of the seis-mogenic layer is first estimated. Next, the temperature at a depth of more than 45 km (upper mantle) is inferred. Finally, temperatures between the two depth points are linearly interpolated at each latitude and longitude. A three-dimensional thermal structure model is, thus, constructed all over the Japanese islands through this process. Meanwhile, in areas where the upper boundary of the Philippine Sea Plate (PSP) is at a depth of 45 km or less, we made use of existing research data to estimate the temperature distribution along the upper boundary of the PSP. Then, we linearly interpolate the temperatures in the regions between the upper boundary of the PSP and the bottom of the seismogenic layer. Temperatures at greater depths than the upper boundary of the PSP fall outside the realm of our research. Details for each step are described below.

#### 2.2.1 The temperature of the bottom of the seismogenic layer

According to Turcotte and Schubert (2002), the strength of rocks in a brittle region is a function of depth *z*:

where Δ*σ*_{
b
} is a tectonic deviatoric stress, *μ* is the friction coefficient, *ρ* is density, *g* is the gravitational acceleration, and *p* is pore pressure. The upper and lower signs apply to reverse and normal faults, respectively. On the other hand, rocks flow at a depth deeper than the brittle-ductile transition (BDT) depth, and the flow strength Δ*σ*_{
f
} is expressed as the constitutive law of dislocation creep (e.g., Bürgmann and Dresen, 2008):

where *A* is the material constant, *Q* is the activation energy, *P* is pressure, *V* is the activation volume, and *n* and *r* are the stress and fugacity exponents, respectively, all of which are determined experimentally for each mineral. is the viscous strain rate, *f*h_{2}o is the fugacity of water, *R* is the gas constant, and *T* is the absolute temperature. At the depths where brittle to ductile transition occurs, the following equation holds:

Thus, the strength curve is described as the well-known ‘Christmas-tree’ model (Fig. 4).

A strong correlation is widely known to exist between the bottom depth of the seismogenic layer and the BDT depth (e.g., Sibson, 1982; Ito, 1990). This study sets appropriate values to the parameters other than the temperature *T* at the bottom of the seismogenic layer for Eqs. (1) and (2), and solves for *T* assuming that Eq. (3) holds at that depth.

This study uses D90 as the representative value for the bottom depth of the seismogenic layer, which is the depth at which more than 90% of earthquakes occur at a shallower depth (e.g., Tanaka, 2004). We obtained D90 through the following steps. First, we extracted all hypocenters (315,131 events), other than low-frequency events and artificial events, that occurred in the period from October 2001 to December 2008 from a unified hypocenter catalog of the Japan Meteorological Agency, with standard errors in the latitudinal and longitudinal directions within 0.05°, with standard errors in focal depth within 3 km, and with magnitudes larger than 1. Note that we did not exclude aftershock sequences of any earthquakes. Grid points were placed at 0.2° latitudinal and longitudinal increments within the range of 124.7°E–147.1°E and 25.5°N–46.5°N, enough to cover the target area depicted in Fig. 1. At each grid point, events were extracted in which the epicenters lie within a 10-km radius from the grid point. Finally, the events were sorted in order of depth, and D90 was defined from the upper 90% of the events counting from the shallowest earthquake. Further, earthquake counts were limited to those having a depth shallower than 40 km, or shallower than the Pacific plate and PSP, or shallower than the Moho discontinuity by Katsumata (2010). Some grid points were re-processed with a radius of 20 km whenever the number of events was less than ten. The distribution pattern of D90 obtained in this way (Fig. 5) is very similar to those of previous studies (e.g., Tanaka, 2004; Omuralieva et al., 2012).

Upon evaluating Eq. (1), the densities were calculated by converting the *P*-wave velocities (Matsubara et al., 2008) using an empirical relation by Christensen and Mooney (1995). The friction coefficient was set to 0.6 (Byerlee, 1978), and the pore pressure was calculated under the assumption of hydrostatic pressure. The fault type was set to be reverse faults in northeastern Japan and Hokkaido, as strike-slip faults in southwestern Japan, and as normal faults in a part of Kyushu, according to the Earthquake Research Committee (2001). The value of the tectonic stress Δ*σ*_{
b
} for a strike-slip fault was set as the average of those for reverse and normal faults.

The horizontal strain rate in Eq. (2) was set using the analysis results of the Global-Positioning-System (GPS) data of Sagiya et al. (2000). Experimental results for quartz (Rutter and Brodie, 2004) were used for the parameters of Eq. (2) as the weakest mineral that composes granite. The fugacity of water was calculated by multiplying the litho-static pressure by the fugacity coefficient of Holland and Powell (1998). Finally, Eq. (3) was solved for the unknown value *T*.

#### 2.2.2 Thermal structure below 45 km

The thermal structure at a depth of more than 45 km was modeled using the method reported by Nakajima and Hasegawa (2003). First, seismic attenuation tomography data of the mantle are converted to a three-dimensional, relative-thermal-structure model by using equation (16) of Karato (1998), in which the quality factor has a proportional relation to the exponential of the inverse of absolute temperature. Next, by taking into account the temperatures indicated by mantle xenoliths at a certain reference point, the relative-thermal-structure model is converted to an absolute-thermal-structure model.

For attenuation data, we used Nakamura (2009)’s result of the three-dimensional attenuation tomography of *S* waves in the crust and the upper mantle across the Japanese Islands. The intervals between the grid points for tomography were 0.2° both in the latitudinal and longitudinal directions, and a 30-km interval was used for depth. Only the tomography data with sufficient resolution from a checkerboard test (Nakamura, 2009) were utilized in this study.

The points in which mantle xenoliths were found are located in the west of southwestern Japan and in the north of northeastern Japan (Fig. 1), and the equilibrium temperatures of the samples between those areas are known to be systematically different (Arai et al., 2007). We set Ichi-nomegata and Aratoyama as respective reference points for northeastern and southwestern Japan, and set the equilibrium temperatures to 950°C and 1050°C, respectively, using the research result of Arai et al. (2007) as a reference. In most cases, the mantle xenoliths obtained in the Japanese Islands were spinel peridotite (Abe and Arai, 2005), which stably exists in the depth range shallower than about 60 km according to a *P-T* diagram (e.g., Gill, 2010). Therefore, we set the representative depth, at which the mantle xenoliths in the Japanese Islands had existed, to 45 km as the intermediate depth between the Moho boundary (about 30 km) and 60 km.

Two absolute-thermal-structure models were thus created using the temperatures at the respective reference points, based on a relative-thermal structure model obtained using the attenuation tomography data of the mantle. These two models were integrated into a single model by taking a weighted average of the two models, and by gradually varying the weight of values in central Japan.

#### 2.2.3 Temperature of the upper boundary of the Philippine Sea Plate (PSP)

The temperature of the upper boundary of the PSP was given by assuming a thermal gradient for each region. According to Yoshioka and Murakami (2007), the temperature of the upper boundary of the PSP at a 30-km depth, where it subducts along the Nankai Trough, ranges between 350 and 450°C. Based on the fact that the maximum focal depths of small earthquakes in the Wadati-Benioff zone become significantly deeper in the Kyushu and Tokai-Kanto regions, Hasegawa et al. (2010) speculates that the thermal gradients are lower in these areas. Considering these previous studies, the thermal gradient on the upper boundary of the PSP was assumed to be 9°C/km in Kyushu, 15°C/km in Shikoku to central Japan, 11°C/km in southern Kanto, and 7.5°C/km in northern Kanto. A model by Nakajima and Hasegawa (2007), and Nakajima et al. (2009), was used for the depth of the upper boundary of the PSP.