# Constraints on the three-dimensional thermal structure of the lower crust in the Japanese Islands

- Ikuo Cho
^{1}Email author and - Yasuto Kuwahara
^{1}

**65**:650080855

https://doi.org/10.5047/eps.2013.01.005

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013

**Received: **5 September 2012

**Accepted: **28 January 2013

**Published: **17 September 2013

## Abstract

We propose a method for modeling a three-dimensional thermal structure with a particular focus on the lower crust. Our method enables high-resolution modeling without heat flow data, but with earthquake hypocenter data in the crust and seismic attenuation data in the upper mantle. In our method, the temperature at the bottom of the seismogenic layer is estimated under the assumption that the bottom depth corresponds to a brittle-ductile transition zone. Next, the temperature beneath the Moho discontinuity is estimated using seismic attenuation data of the mantle and a few point temperature data inferred from mantle xenoliths. Finally, the temperatures between the two depths are linearly interpolated. Attempts to construct an actual model for the Japanese Islands are also described.

### Key words

Thermal structure lower crust modeling numerical simulation forecast## 1. Introduction

With the above as a background, this paper reports on the modeling of a three-dimensionally nonuniform thermal structure beneath the Japanese Islands focusing on the lower crust.

## 2. Method

### 2.1 Defects of existing methods

_{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

*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.

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.

## 3. Results

In the thermal distribution at 45-km depth in Fig. 6(b), ranging between 800 and 1200°C, the temperatures in northeastern Japan were generally lower than in southwestern Japan owing to differences in the reference temperature of the mantle xenoliths. Not shown here, the temperatures at the depths of the Conrad and Moho discontinuities (Katsumata, 2010) ranged between 400 and 700°C and between 600 and 900°C, respectively.

## 4. Discussion and Concluding Remarks

The model obtained in this study is consistent with previous studies. For example, Hasegawa et al. (2000) compared the spatial changes in the isothermal line at 300°C, 400°C, and 500°C, from *P*-wave perturbations and the cutoff depth of shallow seismicity, and showed that the cut-off depth generally matches the isothermal line at 400°C. Strict comparisons are difficult because their “cut-off depth” is not clearly defined, but their result does not contradict our model of the thermal distribution for D90 which has an average value of 385°C with a standard deviation of 24°C. Also, Tanaka (2004) estimated that the temperature at D90 in the Japanese Islands ranges between 250°C and 450°C on the basis of the thermal gradient data. In addition, our model corresponds well with the thermal distribution that is collaterally calculated in the simulation of the transportation of fluid within the crust and mantle (Iwamori, 2007).

*μ*is set to a value smaller than 0.6. For example, the D90 temperature at 10-km depth is 374°C when

*μ*is set to 0.6 and the other parameters (

*ρ*,

*f*H

_{2}O, ) to (2.8 g/m

^{3}, 65 MPa, 10

^{−15}strain/s), while it becomes 400°C when

*μ*is set to 0.4. Furthermore, even if

*μ*is fixed to 0.6, if the pore pressure is abnormally high such as 50% and 80% of the lithostatic pressure, the temperature can still take higher values of 385°C and 429°C, respectively.

We assumed D90 as the representative depth of BDT for simplicity. It should be noted, however, that more recent studies have suggested that the zone over which earthquakes may nucleate is actually a subset of the range over which brittle deformation occurs, with the seismogenic zone having a velocity-weakening frictional property (e.g., Scholz, 1998). Therefore, D90 possibly represents the depth at which velocity-strengthening starts to dominate, and the BDT lies in the deeper regions. In that case, our assumption that D90 coincides with the BDT may underestimate the depth of the BDT. But, on the other hand, it is speculated that the frictional-stability transition and the BDT are closely related to each other (e.g., Scholz, 2002), and, in fact, there is an observation that co-seismic frictional slip and aseismic plastic flaw coexist at the bottom of the seismogenic zone (e.g., Takagi et al., 2000). To further complicate matters, the BDT may not represent a sharp transition from brittle to ductile behavior, but rather a more gradual zone of semi-brittle-behavior (e.g., Kohlstedt et al., 1995). Given the lack of constraints on the frictional-stability and BDT, we retain the simplest assumption that D90 represents the BDT depth. Further refinement of this model may be possible with better information about the transitional properties at the bottom of the seismogenic zone in the future.

It is expected from Eqs. (1)–(3) that the BDT temperature becomes lower as the BDT depth becomes deeper. For example, while we can calculate the BDT temperature at 10-km depth as 374°C assuming *μ* is 0.6 and *p* the hydrostatic pressure, the temperature becomes 358°C when the depth is set to 15 km. In fact, a strong correlation was found between the temperature and the logarithmic depth of D90 of our model, with a correlation coefficient of −0.85. By comparison, the correlation coefficient between the temperature and the logarithmic strain rate was 0.44, much lower than that for D90.

It should be noted that the following problems arise upon using the GPS data as the strain rate for Eq. (2): (i) in Eq. (2) represents the viscous strain rate, but the GPS data are also affected by the elastic strain rate that cyclically accumulate during interplate earthquakes with a recurrence time interval about 100 years; (ii) differential stress that generates reverse and normal faults is the absolute value of the difference between principal stresses in the horizontal and vertical directions (Anderson’s Law), but GPS data analyses lean towards the evaluation of plane strain rate using horizontal components only. The vertical component has been recognized to have a poorer quality and has seldom been subjected to detailed examinations. This is true also for data provided by Sagiya et al. (2000), which we utilized in this study. Therefore, we used the horizontal maximum shear strain rate as the representative scalar data.

The heat flow data, used in the conventional method to model a thermal structure on the basis of a partial differential equation, are obtainable only at the ground surface, which is generally more than 10 km above the lower crust. Heat flow data are also much sparser than the hypocenter distribution data (Fig. 3). Furthermore, one should accurately know the mineral distribution and the environment in the lower crust to solve the partial differential equation. Therefore, our method can be expected to have a higher resolution and to be more robust than the conventional method to model the thermal structure of the lower crust for forecasting simulations of crustal deformations, despite the aforementioned flaws. The applicability of our method is owed to the high seismicity beneath Japan. Thus, our method is possibly applicable to regions other than Japan if they have high seismicity. We are planning to validate the usability of our method by constructing a viscoelas-tic crustal structure model beneath Japan and executing a crustal deformation simulation, based on the thermal structure model obtained in this study.

## Declarations

### Acknowledgments

Data on the seismic attenuation tomography were provided by Dr. Ryoichi Nakamura. Data on the Philippine Sea Plate depth were provided by Dr. Junichi Nakajima. Data on the Conrad and Moho discontinuity depths were provided by Dr. Akio Katsumata. Constructive comments by Dr. Susan Ellis and an anonymous reviewer significantly improved the manuscript. Data on hypocenters, used to determine D90, was processed through a joint effort between the Japan Meteorological Agency and the Ministry of Education, Culture, Sports, Science and Technology.

## Authors’ Affiliations

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