### Introduction

Accurate S-wave velocity (*V*s) structure of sedimentary layers and the uppermost crust in the landward slope of a subduction zone provide important information about the rheology of the overriding plate. Knowledge of the *V*s structures in a shallow part should enhance the accuracy of a wide range of studies, such as hypocenter determination and consideration of rock properties. Since the P-wave velocity (*V*p) structures can generally have high resolution, a reliable estimation of the S-wave velocities would provide an accurate estimation of the ratio of P- and S-wave velocities (*V*p*/V*s), which is useful information for the estimation of rock properties (Ayres and Theilen 1999) and fluid pressure properties (Kodaira et al. 2004).

In the subduction zone of the Japan trench, the recent acquisition of marine multichannel seismic surveys has provided high spatial resolution *V*p structures of the sediment and upper crust (Miura et al. 2003; Takahashi et al. 2004). By contrast, the estimation of *V*s structures has been limited. Consequently, seismic surveys in the Japan trench could provide estimations of the *V*s structures based on the *P* to *S* conversion waves observed in data from Ocean Bottom Seismometers (OBSs) (Fujie et al. 2018). Although both seismic interferometry and receiver function methods are useful for estimating *V*s with OBS records (Yao et al. 2011; Akuhara et al. 2020; Yamaya et al. 2021), obtaining spatially high-resolution *V*s models has been difficult because the distance between OBS locations is typically greater than 6 km along active source seismic profiles and greater than 20 km apart for passive surveys.

In recent years, distributed acoustic sensing (DAS) measurements have been applied to seismic observations (e.g., Zhan 2020). DAS system enables us to measure strain or strain rate with very high spatial resolution over a long distance. The DAS measurement was performed on both the land and seafloor. For example, Dou et al. (2017) estimated shallow shear wave velocity structures by applying the ambient noise method to DAS records on land. In a marine area, Spica et al. (2020) estimated the *V*s structure of shallow sediments (to 3 km depth) by applying a frequency–wavenumber (FK) analysis to the seafloor DAS measurements obtained off the Sanriku coast of Japan. However, their estimation of *V*s structures was limited to a shallow sediments (< 3.0 km depth). Obtaining the *V*s structure throughout the crust is important for deriving the *V*p/*V*s and the forearc rock properties across subduction zones. The phase velocity of the overtone or long period Rayleigh waves is useful for estimating *V*s upper crustal layers, since the phase velocities of these waves are sensitive to *V*s.

In the present study, we applied a seismic interferometry method to seafloor DAS records to extract the overtone or long-period (> 5 s) Rayleigh waves. We also proposed a practical method using FK filtering to extract surface waves in the low-frequency range from short-time records with FK filtering. We then estimated the phase velocities from the extracted Rayleigh waves. Finally, we obtained a 2-D *V*s structural model of the sedimentary layers and the upper crust for the offshore area of the Sanriku coast of northeastern Japan.

### Data and methods

#### Data

In 1996, the Earthquake Research Institute of the University of Tokyo installed a seafloor seismic tsunami observation system that uses an optical fiber cable in the offshore area of Sanriku (Fig. 1). The length of the buried cable is approximately 120 km, and it extends between 0 and 47.7 km from the coast and covers an area with a sea depth from 0 to 2750 m (Shinohara et al. 2022). The system contains six spare fibers; therefore, DAS measurements have been conducted using a DAS interrogator unit from AP Sensing GmBH (Cedilnik et al. 2019) since 14 February 2019. In the present study, the DAS system recorded strain. The data were recorded at a temporal sampling frequency of 500 Hz, and the duration of the observations in this study was selected to be approximately 13 h. The sensing range, spatial resolution, and gauge length adopted in this study were 100 km, 5.1 m, and 40.79 m, respectively. Shinohara et al. (2019, 2022) and Spica et al. (2020) have described the cable setup and measurement quality of this DAS system.

#### Calculation of CCFs with an FK filter

Before calculating and stacking the cross-correlation functions (CCFs) of the DAS background noise records, we divided the entire dataset of the DAS array into 10 km-subarrays with a moving window of 75% overlap (Fig. 1). We decimated the original records by first reducing the sampling frequency from 500 to 2 Hz and then stacking 10 adjacent data to enhance the signal-to-noise ratios (SNRs) after filtering the anti-spatial aliasing. The resulting spatial interval of the data was then 51 m.

We computed the CCFs by first dividing continuous records into 10 min time window segments. We allowed for a 50% overlap of each segment to improve SNR. We computed the weighted average of the cross spectra between two channels, following Takagi et al. (2021) and Takeo et al. (2013, 2014). This processing was conducted in the frequency domain. We avoided any earthquake signals by calculating the mean power between 0.025 and 0.2 Hz in the frequency domain for each time window and discarding any data segments whose amplitude exceeded 10 times the amplitude of the previous time window. Takeo et al. (2013) provide more detailed explanations for the calculation of CCFs.

In general, the DAS instrumental noise has a coherent phase at the same time for all stations (Tribaldos and Ajo-Franklin 2021), and this source of noise particularly affects seismic interferometry studies. In the present study, we applied an FK filter to the DAS data to remove this zero-lag noise and to enhance the SNR before calculating and stacking the CCFs. The high spatial density of the DAS data enables the application of spatial Fourier transform; therefore, we performed a 2-D Fourier transform in the FK domain (Hudson et al. 2021; Atterholt et al. 2022). The zero-lag noise, which has an infinite phase velocity, is mapped to the zero wavenumber in the FK domain. We designed an FK filter \(w\left(f,k\right)\) as:

$$\begin{array}{c}w\left(f,k\right)=g\left(f,k,{c}_{\text{min}}\right)*\left(1-s\left(f,k,{c}_{\text{max}}\right)\right), \end{array}$$

(1)

where \(g\left(f,k,{c}_{\text{min}}\right)\) is a 10% Tukey window used to eliminate signals slower than the cut-off minimum phase velocity \({c}_{\text{min}}\). \(g\left(f,k,{c}_{\text{min}}\right)\) is defined as:

$$g\left( {k,f,c_{{\min }} } \right) = \left\{ \begin{array} {ll} 1,& {\text{for}}\, - 0.4k_{{\max }} < k < 0.4k_{{\max }} \\ 0, & {\text{for}}\,k < - 0.5k_{{\max }} \;{\text{or}}\,k > 0.5k_{{\max }} \\ \frac{1}{2}\left[ {1 + \cos \left\{ {\pi \left( {1 - \frac{k}{{0.1k_{{\max }} }}} \right)} \right\}} \right], &{\text{otherwise}} \\ \end{array} \right.,$$

(2)

where \({k}_{\text{max}}=4\pi f/{c}_{\text{min}}\). Furthermore, \(s(f,k,{c}_{\text{max}})\) is an 8-th order Kaiser filter that removes the signal faster than the maximum phase velocity \({c}_{\text{max}}\), including the zero-lag noise. In this study, we enhanced the surface wave signals by setting \({c}_{\text{min}}\) and \({c}_{\text{max}}\) to 0.35 km/s and 4.00 km/s, respectively.

#### Phase velocity estimation

In this section (Phase velocity estimation), we obtained the phase-velocity dispersion curves for the Rayleigh waves using the CCFs of the ambient noises recorded by the DAS observation. At the same time, we determined the 1-D *V*s structures and errors of the phase velocities. In the next section (Inversion of the S-wave velocity structure), we provide more accurate estimates of the 1-D *V*s structures using only the reliable frequency range of the dispersion curves (i.e., those with small errors) obtained in this section.

We adopted the spatial auto-correlation (SPAC) method to estimate the phase velocity (Aki 1957; Okada 2006; Nishida et al. 2008b), because the SPAC method represents the observed cross-spectra assuming a laterally homogeneous structure and a homogenous source distribution. According to Nakahara et al. (2021), the synthetic cross-spectrum for radial strain records (e.g., DAS) is defined as:

$$\begin{array}{c}{S}_{ij}^{\text{syn.}}\left(\upomega ,{c}_{\text{R}},{c}_{\text{L}}\right) = {A}_{\text{R}}\left(\upomega \right)\left[3{J}_{0}\left(\frac{\upomega }{{c}_{\text{R}}}{d}_{ij}\right)-4{J}_{2}\left(\frac{\upomega }{{c}_{\text{R}}}{d}_{ij}\right)+{J}_{4}\left(\frac{\upomega }{{c}_{\text{R}}}{d}_{ij}\right)\right]+{B}_{\text{L}}\left(\omega \right)\left[{J}_{0}\left(\frac{\upomega }{{c}_{\text{L}}}{d}_{ij}\right)-{J}_{4}\left(\frac{\upomega }{{c}_{\text{L}}}{d}_{ij}\right)\right], \end{array}$$

(3)

where \(\omega\) is the angular frequency, \({A}_{\text{R}}\) is the power spectrum for the Rayleigh wave, \({B}_{\text{L}}\) is the power spectrum for the Love wave, \({J}_{n}\) is the nth-order Bessel function of the first kind, \({d}_{ij}\) is the distance between *i*th and *j*th channels, \({c}_{\text{R}}\) is the phase velocity of the Rayleigh wave, and \({c}_{\text{L}}\) is the phase velocity of the Love wave. In Eq. 3, the first and second terms represent the Rayleigh and Love waves, respectively. In general, the Love waves have less energy than the Rayleigh waves at a period range below 0.1 Hz (Nishida et al. 2008a). Furthermore, as mentioned by Nakahara et al. (2021), Rayleigh waves predominate in the far field. The envelope of the Rayleigh wave term decays with an order of \({d}^{-1/2}\), while the Love wave term decays with an order of \({d}^{-3/2}\) in the far field. For these reasons, and given the small estimated contribution of the Love wave terms, we obtained our misfit function \(E\) as:

$$\begin{array}{c}L2\left(\omega ,{c}_{\text{R}}\right)=\sum_{ij}{\left(Re\left[{S}_{ij}^{\text{obs}}\left(\omega \right)\right]-{S}_{ij}^{\text{syn.}}\left(\omega ,{c}_{\text{R}}\right)\right)}^{2}, \end{array}$$

(4)

$$\begin{array}{c}E\left({\beta }_{l}, {h}_{l}\right)=\frac{1}{\omega }\int L2\left(\omega ,{c}_{R}\left(\omega ;{\beta }_{l};{h}_{l}\right)\right)d\omega , \end{array}$$

(5)

where \({S}_{ij}^{\text{obs}}(\omega )\) and \({S}_{ij}^{\text{syn.}}\left(\omega \right)\) are the observed and synthetic cross-spectra between *i*th and *j*th channels calculated from Eq. 3, respectively. \({\beta }_{l}\) and \({h}_{l}\) are the *V*s and thickness at the *l*th layer, respectively. We followed the methods described by Takeo et al. (2022) and Yoshizawa & Kennett (2002) to estimate the phase velocity model. \({\beta }_{l}\) and \({h}_{l}\) are model parameters for minimizing \(E\left({\beta }_{l}, {h}_{l}\right)\) in Eq. 5. In this study, we adopted the simulated annealing algorithm of Goffe et al. (1994) as a global optimizer in our search for model parameters. We avoided numerical instability by constraining the *V*s at each layer at greater than 80% of the value of the layer directly above it.

In practice, we estimated the phase velocity and evaluated the errors in the estimated phase velocity. First, we estimated the dispersion curves of the phase velocity from the cross-spectra for all pairs of channels for each subarray. We then obtained a 1-D model with six layers, where the deepest layer had an infinite thickness for each subarray. We adopted the scaling relationship between *V*p, *V*s, and density described by Brocher (2005). Next, searching by trial and error, we divided the subarrays into two groups according to the distances from the coast and frequency range. The dispersion curves of the phase velocity were estimated using the only fundamental mode of the Rayleigh wave at the frequency range between 0.08 and 0.50 Hz for one group that had distances ranging from 28 to 52 km from the coast. For the other group with distances ranging from 52 to 75 km, the fundamental mode (0.08–0.50 Hz) and the first higher mode (0.25–0.50 Hz or 0.08–0.50 Hz) of Rayleigh wave were used to obtain a dispersion curve. Due to large incoherent noise near the coast and offshore areas, the spatial range of the present data processing was limited from 28 to 75 km from the coast. Synthetic phase velocities were calculated using DISPER80 (Saito 1988).

We also used the bootstrap method (Efron 1992) to estimate errors. We aggregated station pairs randomly selected from all station pairs, allowing for overlap for a bootstrap sample. We then calculated the standard deviation of 100 dispersion curves estimated from each of the 100 bootstrap samples. The *V*s structures estimated from the data of all pairs of channels for each subarray were used as the reference model.

#### Inversion of the S-wave velocity structure

The phase velocity of the Rayleigh wave estimated by seismic interferometry was used to obtain a 1 – D *V*s isotropic model. The 1-D *V*s structure was determined by minimizing the misfit function *E,* as defined by:

$$\begin{array}{c}E = \sqrt {\frac{1}{N}\sum {\left[ {\frac{{c_{{obs}} \left( \omega \right) - c_{{syn}} \left( {\omega ;\beta _{l} ;h_{l} } \right)}}{{c_{{err}} \left( \omega \right)}}} \right]^{2} ,} } \end{array}$$

(6)

where *N* is the number of estimated phase velocities, and \({c}_{obs}\), \({c}_{syn}\) and \({c}_{err}\) indicate the observed, synthesized, and uncertainty of the phase velocity, respectively. We obtained a stable result by selecting only a frequency band with phase velocities with an error lower than 0.1 km/s for the inversion analysis. We again set the *V*s (\({\beta }_{l})\) and thickness (\({h}_{l})\) as the model parameters to minimize the value obtained from Eq. 6. We then used the simulated annealing algorithm method to obtain the optimal model parameters. The bootstrap average models estimated from the phase velocity estimation were used as the reference model. We estimated errors using the bootstrap method once again. Here, we aggregated a final dispersion curve randomly selected from the 100 dispersion curves estimated in the previous section (Takeo et al. 2013).

The 1-D *V*s structures were estimated on the assumption that we could ignore the effect of lateral heterogeneity for profiles perpendicular to the seafloor cable. Previous seismic reflection and refraction surveys support this assumption since profiles perpendicular to the seafloor cable show little lateral heterogeneity to a depth of 10 km (e. g. Takahashi et al. 2004).

### Results and discussion

We computed the CCFs between all possible pairs of each subarray, and the resulting CCFs clearly showed a surface wave (Fig. 2). The group velocities of about 0.5 km/s were much slower than the surface waves from land observations in the same period range. The surface wave with low velocity corresponded to a special type of Rayleigh wave or Scholte wave (Stokoe et al. 1991; Bagheri et al. 2015) because the Love wave had less energy than the Rayleigh wave in the period range below 0.1 Hz. The SNR was much higher for the CCFs with the FK filter than without the FK filter. We used the method of Bensen et al. (2007) to estimate the SNR.

We selected the dispersion curve using the CCFs (Figs. 3-A and 3-E). Extracting the Rayleigh waves by interferometry enabled a stable estimation of the dispersion curves with errors less than 0.1 km/s by bootstrapping. Near the coast (Fig. 3A), the fundamental Rayleigh wave was dominant in the frequency range between 0.2 and 0.5 Hz. By contrast, in the case of an offshore subarray (Fig. 3E), both the fundamental (0.5–0.1 Hz) and the 1st higher mode (0.5–0.2 Hz) Rayleigh waves predominated. This result clearly indicates that the thickness of the low-*V*s layer differs between the near-coast and offshore areas. We determined the *V*s structure using only the phase velocities of the fundamental mode of Rayleigh waves in the near-coast area at distances between 28 and 52 km. In the area where the distance was greater than 52 km from the coast, both the fundamental and first higher modes of the Rayleigh waves were used for phase velocity estimation.

We inverted the measured phase velocities into 1-D *V*s structures (Fig. 3) below each subarray and calculated the 1-D *V*s structures (Fig. 3B and F) and normalized *V*s sensitivity kernels (Fig. 3C, G, and I) of the Rayleigh waves. Errors were calculated using the bootstrap method. A small error was estimated in the depth range where *V*s values were slower than 2.0 km/s. The synthetic phase velocities calculated by DISPER80 using the 1-D *V*s inversion results were consistent with the observations (Figs. 3D and H). Given that the sensitivity kernel had high sensitivity in the depth range of *V*s slower than 2.0 km/s, our inversion provided a *V*s structure with small errors at that depth range of *V*s. By contrast, the layers of *V*s greater than 2.0 km/s were not estimated well. However, the *V*s sharply increases to those greater than 2.0 km/s at the boundary between the shallow slow layers and the layers below, and the mutual velocity ranges, with errors, are clearly separated. Therefore, the depth was reliably estimated for the discontinuity between the shallow slow layers and the layers below.

Finally, a 2-D *V*s profile (Fig. 4) was obtained from the 1-D *V*s structures at each subarray. Although each 1-D *V*s structure at a given subarray was composed of six layers, we classified this 2-D profile into four units with a clear velocity discontinuity (Fig. 4). The velocities of some contiguous layers were coincident in terms of errors. In those cases, we unified the contiguous layers into one unit. The *V*s of Unit-1 at depths of 0.7–2.6 km and Unit-2 at the depth of 0.9–4.0 km were slower than 0.6 km/s and approximately 0.7 km/s, respectively. The *Vs* of Unit-3 at depths of 1.2–6.8 km and Unit-4 layers below the Unit-3 were 1.1–1.8 km/s and greater than 2.0 km/s, respectively.

We compared our 2-D *V*s structure (Fig. 4) with the *V*s and *V*p structures obtained in other studies to interpret these units. The *V*s values of Unit-1 and Unit-2 layers, at less than 0.6 km/s and about 0.7 km/s, respectively, were consistent with the results presented by Spica et al. (2020) with the same profile. Takahashi et al. (2004) provided the *V*p structure estimated by refraction surveys along a profile parallel to our DAS cable, positioned northward, with a distance of about 20 km from the seafloor cable (Fig. 1). The *V*p values of Unit-1 and Unit-2 in the present study were determined as 1.7 and 2.3 km/s, respectively, and these layers were interpreted as Neogene sediment by Spica et al. (2020). The thicknesses of both Unit-1 and Unit-2 increased seaward for a distance range from 28 to 47 km. This increase in thickness is also consistent with the results reported by previous studies. In distance range further from the coast between 47 and 75 km, however, the thicknesses of Unit-1 and Unit-2 showed very small variations.

The thickness of Unit-3 was 0.6 km at a distance of 28 km from the coast and increased with a horizontal distance. Unit-3 became the thickest (4.1 km) at a distance from 52 km from the coast. The thickness of Unit-3 decreased to 1.9 km at a distance of 65 km. At distances between 65 and 75 km, Unit-3 appeared to pinch out. These strong lateral heterogeneities in Unit-3 were also evident in the *V*p model by Takahashi et al. (2004). The discontinuity observed between Unit-3 and Unit-4 can be interpreted as the boundary between sediments and the island arc uppermost crust consistent with the interpretation reported by Takahashi et al. (2004). Along the same Japan trench offshore Ibaraki and the south of the study area, Yamaya et al. (2021) obtained *V*s values of the deepest sediment of approximately 2.1 km/s using records from pop-up ocean bottom seismometers. In contrast, Unit-3 in this study has smaller *V*s compared to the results off Ibaraki. We attribute the difference in *V*s values for the two regions to differences in porosities and lithologies in the lowermost sedimentary layers.

Although our *V*s profile and the *V*p profile of Takahashi et al. (2004) are 20 km apart (Fig. 1) and the spatial resolution of our results is much higher than that of Takahashi et al. (2004), we calculated the average value and errors of the *V*p*/V*s (Fig. 4) using the *V*p estimated by Takahashi et al. (2004) to support our interpretation of the results. The errors were defined as twice the standard deviation during calculation of the average. The average *V*p/*V*s values of Unit-1 and Unit-2 were 5.00 and 3.34, respectively. The *V*s in Unit-3 of 1.1–1.8 km/s was obtained from this study and the *V*p in Unit-3 was estimated to be 4.2 km/s by Takahashi et al. (2004). As a result, the average *V*p/*V*s value in Unit-3 become 3.12. On the other hand, the *V*p/*V*s of Unit-4 was 1.98 from a *V*s of > 2.0 km/s and a *V*p of 5.0 km/s.

Because we found no previous reports on the *V*p/*V*s of marine sedimentary layers in our study area, we compared *V*p/*V*s with the values obtained in other marine areas for the similar geological ages. *V*p/*V*s values of sedimentary layers are known to vary depending on porosity of rocks, constitutive materials, and tectonic settings from place to place (e.g., Kodaira et al. 1996). We chose the ages of sediments as an indicator of the comparison rather than the mineral components, because porosity of medium primarily affects *V*s of sediments and uppermost crust under marine environment. A comparison of the thickness of each Unit in our results to those of the layers obtained by Takahashi et al. (2004) indicates that Unit-1 and Unit-2 are Neogene sediments, and Unit-3 is Cretaceous sediments. The *V*p/*V*s of the layer deposited in Neogene was estimated at 5.10 in Norway (Kvarven et al. 2015) and 3.2–4.4 in the landward slope on the southern Japan trench (Yamaya et al. 2021). These values are comparable to our estimation of *V*p/*V*s in Unit-1 and Unit-2. Kodaira et al. (1996) reported that the *V*p/*V*s of the sedimentary layers in Norway ranged from 5 to 3, although their ages were unknown. By contrast, Kvarven et al. (2015) reported a *V*p/*V*s of 1.78 for the Cretaceous sediment layer in Norway. On the other hand, Yamaya et al. (2021) showed a *Vp/Vs* of 2.1 for the Cretaceous sediment layer beneath the landward slope of the southern Japan trench. Although our *V*p/*V*s values are believed to have some uncertainty due to lower spatial resolution of *V*p results from the airgun-OBS surveys, the *V*p/*V*s for the Cretaceous sediment unit in our study region can be concluded to have large value obviously from a point of view of the errors estimated in this study. We therefore infer that the Cretaceous sediments in this study area is unconsolidated.

Now let us consider our result on a deep part or the crust. Using stationary seismic networks in land and sea by body-wave travel-time tomography, Matsubara et al. (2019) reported *V*p/*V*s of 1.6–1.9 for the upper crust below the northeastern Japan island arc. In our study, the *V*p/*V*s for Unit-4 seems to correspond to the uppermost part of the island arc crust from the point of view of their *Vp/Vs* Furthermore, aftershocks of the 2011 Tohoku-oki earthquake, as estimated by Shinohara et al. (2012), were located only in Unit-4 (Fig. 4), whereas no seismicity was experienced in the other three units. In summary, Unit-1, Unit-2, and Unit-3 consist of sedimentary layers, whereas Unit-4 represents the uppermost part of the igneous crust of an island arc.

Previous studies that estimated *V*s structures using DAS records have been limited to elucidation of a shallow part of the sediment layers. For example, Spica et al. (2020) estimated the *V*s structure to 3 km in depth. The application of the seismic interferometry with FK filtering to the DAS data now enabled the determination of phase velocities of surface waves whose periods are longer than those of previous studies. Using DAS data with high spatial resolution we were able to obtain an S-wave structure of lowermost sediment > 3.0 km depth and extending to 6.8 km depth where sediments were thickest.

The SPAC method is useful to estimate velocity structure using data from a seismic network; however, spatial aliasing due to a sparse distribution of stations causes uncertainty for the SPAC analysis. Because DAS measurement has a high-spatial density, spatial aliasing can be avoided. For example, we could prevent misidentification of a mode branch. *V*s structures in marine area have conventionally been estimated by the observation of converted waves during seismic surveys or by seismic interferometry using spatially dense OBSs data. Our method offers a new approach for determining the *V*s of shallow structures with high spatial resolution in marine areas using short-term DAS data and a seafloor cable. A shallow *V*s structure in a marine area with high resolution by our method leads to a detailed distribution of *V*p/*V*s in a shallow region, which is useful for considering tectonics and rock properties.

### Conclusions

The information of *V*s is important for understanding the rock properties of the upper crust and sediments on the seafloor. We applied a seismic interferometry method to DAS records obtained during a recording period of 13 h by a seafloor cable installed off Sanriku, Japan. We found that applying the FK filter to the DAS data before calculating the CCFs effectively enhances the surface waves. The phase velocities of the Rayleigh waves were calculated by the SPAC method. Finally, we inverted the *V*s structure from the phase velocities of the Rayleigh waves obtained from seismic interferometry. The *V*s structures of the sediment layers and the upper crust were consistent with those determined by refraction surveys. The seismic interferometry method with DAS data collected through seafloor cables could estimate *V*s structures with high spatial resolution.

The *V*s structure of sediment layers has been widely explored using seismic interferometry and receiver function methods with OBS data. However, their spatial resolution is constrained by the spatial density of the installed OBSs. By contrast, a *V*s structure with high spatial resolution can be estimated from short-term DAS records by applying the seismic interferometry method. Our results identified a new approach for estimating the heterogeneous *V*s structure of sediment layers and the upper crust in subduction zones.