Introduction

Fault frictional strength, i.e., the stress on the fault at failure, dictates when a fault that is loaded fails. If frictional strength controls slip nucleation, it implies that stronger faults can sustain higher levels of stress before failing in an earthquake relative to weaker faults. Direct measures of absolute stress and fault strength near where earthquakes nucleate are elusive and often not possible, which hampers our detailed understanding of the mechanics and physics of earthquake nucleation1,2,3. Earthquake stress drop (Δσ), i.e., the amount of shear stress released during an earthquake, is a parameter that can be estimated from seismological observations and is commonly inferred to be related to fault strength and the rupture process4,5,6,7,8,9,10. In practice, Δσ is measured from the corner frequency of the earthquake source spectrum and is a function of the total slip over the rupture area3,11,12,13. It is often intuitively inferred that high Δσ is indicative of stronger faults4,5,6,7,9. However, counterexamples exist where weak faults generate high slip per unit area during an earthquake and have correspondingly high measurable Δσ, calling such assumptions into question14,15.

A common approach for testing the dependence of Δσ on fault strength is to evaluate whether Δσ correlates with depth, or overburden16. The rationale is that fault strength is expected to increase with depth in the Earth’s brittle lithosphere17, therefore, if the Δσ depends on fault strength, it should also increase with depth. However, large uncertainties and scatter common to Δσ estimates3 make identifying robust correlations between Δσ and depth challenging. Studies in various tectonic settings and at a range of spatial scales report both an increase of Δσ with depth4,18,19 and a lack thereof20,21, leading to contradictory interpretations22. Part of the contradictory interpretations may rest in the fact that the dependence of fault strength on depth is not well understood quantitatively. Many estimates are based on Byerlee’s friction law23 and assumptions of the fluid pressure, and do not determine the forces that condition the stress field. Furthermore, most of the work that investigates the depth dependence of Δσ is limited to depth ranges of 10-15 km18,21,22. As a result, correlations between Δσ and depth may be difficult to tease out, especially given the uncertainties in Δσ measurements3,22.

In this study, we investigate how measured Δσ values correlate with depth and maximum shear stress (τmax) in the brittle Japanese forearc lithosphere, above the plate interface. We focus on Δσ values obtained for earthquakes in the northeastern Japanese forearc for the following reasons: 1) it hosted intense and widespread seismicity at depths of up to ~60 km following the Mw 9.0 Tohoku-Oki megathrust earthquake on 11 March 2011 (ref. 24) (Fig. 1). 2) The aftershock sequence was recorded by a dense network of borehole seismometers25 that enables detailed analysis of Δσ values over a 60+ km depth range in the area surrounding two forearc transects, namely Iwaki and Sendai (Fig. 1). 3) The forearc seismicity included many normal-faulting events that showed that large parts of the forearc must have experienced deviatoric tension (where vertical stress exceeds horizontal stress) after the Tohoku-Oki earthquake24,26,27. Deviatoric tension results from the gradient in potential energy imposed by the continental-margin relief28, which can be used to constrain the total stress in a forearc by means of force-balance models29,30. 4) Dielforder et al.29 found that τmax estimates (half the differential stress) obtained from force-balance models for the Japanese forearc show a similar increase with depth as median earthquake Δσ values. However, detailed quantitative analysis of earthquake Δσ and its correlation with depth and τmax, was beyond the scope of their study. In this study, we expand the temporal interval and methodological approaches presented in ref. 29 to provide new quantitative insights on the relationship between Δσ, depth, and fault strength.

Fig. 1: Forearc seismicity and stress drop (Δσ) estimates in the study region following the 11 March 2011 Mw 9.0 Tohoku-Oki megathrust earthquake.
figure 1

Seismicity and Δσ estimates for two different periods: 11 March 2011–10 March 2012 and 11 March 2012–31 December 2021. a Seismotectonic setting of the study region. Filled circles, squares, and diamonds represent single-spectrum (no attenuation correction), spectral-ratio, and attenuation-corrected (single-spectrum) Δσ estimates, respectively (see Methods). Fig. S1 details the distribution of Δσ estimates from the methods used in this paper. Gray dots indicate earthquakes without Δσ estimates. Coseismic slip contour lines are from ref. 31. Dashed black lines in map view and cross-sections are slab isodepths from ref. 32. Dashed blue rectangles indicate the surface trace of 200-km-wide swaths bracketing profiles (solid blue lines) across the Iwaki (Iw) and Sendai (Se) transects of the Japanese forearc. b Seismicity within the 200-km wide swaths is projected onto the central plane of cross-section (solid blue line in a) along Iw, (bottom row) and Se (top row) transects. Cross-sections are identical for finite-element force-balance models in Fig. 2.

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We start by first evaluating the depth dependence of Δσ for the two forearc transects near the cities of Iwaki and Sendai in two time periods, including the first year following the Tohoku-Oki earthquake (11 March 2011–10 March 2012) and the following decade (11 March 2012–31 December 2021) (Fig. 1). We then evaluate the Δσ dependence on τmax in the brittle lithosphere by comparing Δσ values with results from 2D finite-element, force-balance models29. The models yield representative estimates of τmax in the Japanese forearc along the Iwaki and Sendai transects in the first postseismic year and hence allow a comparison with median Δσ values from March 2011 to March 2012. We examine the correlation between Δσ and τmax for the first year after the Tohoku-Oki earthquake and the Δσ values in the following decade to understand the implications for failure conditions in the forearc region.

Results

Stress-drop catalog and finite-element models

We obtain an initial 18,016 Δσ estimates (Fig. S2a) from earthquakes with Japan Meteorological Agency magnitudes MJMA ≥ 2.5 (moment magnitudes Mw ≥ 2.1 estimated here, see Methods) by fitting spectral corner frequency (fc) of individual uncorrected (Fig. S3) and attenuation-corrected earthquake spectra (Figs. S4 and S5), and event-pair spectral-ratios3,33,34,35 (Fig. S6). The attenuation-corrected single-spectrum fc estimates apply a method that corrects individual spectra for attenuation in a confined volume using fc values obtained from fitting neighboring spectral-ratio event pairs34,35 (see Methods, Figs. S4, S5, and S7). The correction of earthquake spectra for propagation effects leads to fc estimates comparable to those from a spectral-ratio approach35. We refer to the resulting observations as attenuation-corrected single-spectrum fc and Δσ estimates to differentiate them from fc and Δσ estimates derived from spectral fitting, in which attenuation and non-source-related terms are not removed. We estimate Δσ from S-waves and account for a 1D depth-dependent shear-wave velocity (see Methods). For events where corner frequency estimates are derived from multiple methods, we assign a Δσ value based on the following priority: first, the target event spectral-ratio estimate; second, the empirical Green’s function (eGf) event spectral-ratio estimate; third, the attenuation-corrected single-spectrum estimate; and finally, the uncorrected single-spectrum estimate.

We impose an initial quality-control step that removes 2704 attenuation-corrected Δσ estimates and 4843 single-spectrum Δσ estimates with poorly constrained corner frequency (fc) values, leaving a total of 10,469 estimates (Fig. S2a). The less well-constrained values result from one or both of the following: (1) an insufficient number of station observations or (2) an estimated fc that lies outside the observational bandwidth. Of the 10,469 estimates, there are 510 target events and 156 eGf from spectral-ratio fitting, 7193 events from individual attenuation-corrected spectral fitting, and 2610 events from individual uncorrected spectral fitting (Fig. S2b). The Methods section details the event selection, Δσ calculation, and quality-control criteria used for spectral estimation and fitting. Individual Δσ estimates span four orders of magnitude ( ~ 0.1–461 MPa), and are comparable to other studies3,36. However, we note that 90% of Δσ estimates (9421 out of 10,469) range between 0.6 and 11.4 MPa (5th and 95th percentiles, respectively). Median Δσ values show little variation along the strike of the margin and more pronounced variations along the dip direction of the subducting slab (Fig. S9). The quality-control procedures help minimize uncertainties and ensure that the relative variation in Δσ reflects local stress heterogeneity or different physical processes in the earthquake source2,3,10. Because we are interested in the primary trend of Δσ with respect to depth and bulk stress conditions, we focus on variations in median Δσ values calculated for bins with equal numbers of events.

We determine the τmax in the Japanese forearc in the first year after the Tohoku-Oki earthquake along the Iwaki and Sendai transects (Fig. 1) using new 2D finite-element models of force-balance that are slightly modified from ref. 29 (Fig. S10, Methods). The models compute the total stress in the forearc resulting from the superposition of topographic and tectonic stresses. Topographic stress results from the gradient in potential energy that arises in the Earth’s gravitational field between areas of lower and higher elevation37. The large elevation difference between the oceanic trench and mountains and volcanoes in the upper plate of a subduction zone causes a strong gradient in potential energy that reduces the horizontal stress in the forearc relative to the vertical stress28. Tectonic stress in the forearc results mainly from the shear stress on the megathrust. The shear stress increases the horizontal stress and compresses the upper plate, as well as counteracts the effect of topography. In cases where tectonic stress exceeds topographic stress, the forearc will be under deviatoric compression and may fail by reverse faulting. Conversely, if topographic stress predominates, it will create deviatoric tension, and the forearc may fail by normal faulting. Therefore, depending on the local topographic relief and megathrust shear stress, the forearc stress state may alternate between deviatoric compression and deviatoric tension.

The finite-element models determine the magnitude of the megathrust shear stress that is both low enough to permit deviatoric tension in forearc areas that failed by normal faulting after the Tohoku-Oki earthquake, and high enough to cause deviatoric compression in forearc areas that failed by thrust faulting after the mainshock. As discussed in previous studies29,30, only a narrow range of megathrust shear stress values fulfill the boundary conditions, such that the total stress state can be constrained within reasonable uncertainties. The models presented here go a step further to account for viscoelastic stress relaxation in the first year after the Tohoku-Oki earthquake38. We restrict the models to the first postseismic year because the stress field at later periods may be increasingly affected by the relocking of the megathrust, which is quantitatively poorly understood.

Here, we report total stress in terms of maximum shear stress, τmax = (σ1–σ3)/2, where σ1 and σ3 are the maximum and minimum principal stresses, respectively (Fig. 2). The maximum shear stress values allow a direct comparison of model results with Δσ values. Note that the models do not compute the failure stress on discrete faults. The model results show τmax values on the order of 10 s of MPa, with maximum values of ~45 MPa and ~70 MPa for Iwaki and Sendai, respectively (Fig. 2). The values are in agreement with previous estimates for the Japanese forearc30,39 but are an order of magnitude lower than suggested by classic lithospheric strength models inferred from Byerlee’s law17,23. The low stress magnitudes mainly result from the following: subduction megathrusts have effective friction coefficients that are an order of magnitude lower40,41,42 than the static friction coefficient of rocks determined by laboratory experiments23. The low effective friction leads to low megathrust shear stress that causes nominal compression in the forearc. The shear stress along the megathrust decreases even more during a megathrust earthquake, which can locally reduce forearc compression and switch the stress field to deviatoric tension, as observed for the Tohoku-Oki earthquake24,26,30. If the forearc is under deviatoric tension and topographic stress dominates, τmax is limited by the gradient in potential energy, for which existing elevation differences on Earth limit it to ~100 MPa or less.

Fig. 2: Maximum shear stress (τmax) from finite-element models for the Iwaki (left) and Sendai (right) transects.
figure 2

Modeled τmax 1 day (top row), 6 months (middle row) and 12 months (bottom row) after the Tohoku-Oki earthquake. Viscoelastic stress relaxation leads to temporal changes in τmax. The dashed line indicates the crust-mantle boundary. Note that the stress contrast along the Sendai transect at ~250 km from the trench results from a change in material parameters in the model that would likely be more gradual in nature. See Fig. 1 for location of transects.

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The differences in τmax between the two models mainly result from differences in the spatial distribution of normal faulting and thrust faulting events, as well as the general aftershock distribution. The strong aftershock seismicity and normal-faulting activity at >160 km from the trench along the Iwaki transect requires very low megathrust shear stresses and conditions the comparatively low stress in the forearc. By comparison, deviatoric tension and normal faulting were restricted to <160 km from the trench along the Sendai transect27,30, which requires slightly higher megathrust shear stresses than along the Iwaki transect and causes slightly higher τmax values29,30. The stress pattern in the forearc is further affected by differences in forearc topography and the slab curvature between the two transects29,30. Regardless of the differences between the Iwaki and Sendai transects, τmax tends to increase with depth along the megathrust and to decrease with distance from it, such that τmax varies both with depth and laterally. The postseismic relaxation reduces τmax in the mantle wedge by 5–15 MPa. τmax changes are more pronounced in the Sendai model, where rapid stress relaxation in the viscoelastic mantle results in a stress contrast at the contact with the elastic mantle wedge at ~250 km from the trench (Fig. 2). The stress contrast results from the change in material parameters in the model and would likely be more gradual in nature.

Depth variation of stress-drop values

We investigate the depth dependence of Δσ using computed values from two 200-km-wide swaths along the Iwaki and Sendai transects (dashed boxes in Fig. 1) and evaluate them in two time periods: 11 March 2011–10 March 2012 (1 year), and 11 March 2012–31 December 2021 (~decade) (Fig. 1). Median Δσ values exhibit an overall increase with depth along both transects and time intervals (~0.8 MPa every 10 km, Fig. 3, Fig. 1 and Fig. 2S). Because the Δσ estimation incorporates a 1D, depth-dependent shear-wave velocity model, we can rule out that the observed increase of Δσ with depth is an artifact of a constant shear velocity assumption19,21,22 (see Methods). We compute the coefficient of correlation values (Pearson’s r value) between median Δσ and depth over the two time periods for both forearc regions (Fig. 3a–d). The r value is larger for the Iwaki transect (r ~ 0.99) than for the Sendai transect (r ~ 0.86), where the hypocenters in the latter span a smaller depth range (Fig. 3). We note that the r-values are nearly independent of the number of events used to calculate the median Δσ values (Fig. S11), and that the overall increase of median Δσ with depth persists for all methods employed here to estimate Δσ (Figs. S12 and S13). We also note that the respective methods sample different depth ranges, and the combination of Δσ estimates from all methods is necessary to maintain continuous depth coverage (Figs. S12 and S13). The overall trend of increasing median Δσ with depth is interrupted by intervals of constant median Δσ values, which is most pronounced for the Iwaki transect at depths of ~15–30 km (Fig. 3a, b, Figs. S12b and S13b).

Fig. 3: Stress drop (Δσ) values with respect to depth of earthquakes within the 200-km-wide swath profiles for the Iwaki (Iw), and Sendai (Se) transects.
figure 3

Results for two temporal intervals: a, c 11 March 2011–10 March 2012, and b, d 11 March 2012–31 December 2021. Median Δσ values are calculated for bins of 200 events with 50 % overlap. The reported coefficients of correlation (r-values) refer to linear fitting of median Δσ values obtained from all Δσ estimates (white squares). See Fig. 1 for the location of swath profiles. Figs. S12 and S13 show Δσ estimates from attenuation-corrected and uncorrected single-spectrum estimates separately. Horizontal and vertical lines associated to mean and median values indicate the 5th and 95th percentiles. Legend for all panels is reported in panel d.

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We also verify whether the increase in earthquake Δσ with depth is robust and not an artifact of poorly resolved hypocentral depths outside the seismic network (offshore) by testing whether a similar depth dependence is also evident in a cluster of onshore earthquakes (Fig. S14). We focus on a cluster near the city of Iwaki owing to the excellent station coverage, where several stations are located nearly coincident with earthquake epicenters, allowing for well-constrained hypocentral depths (Fig. S14). The increasing median Δσ with depth also persists for the onshore cluster and is consistent with the observations from wider forearc areas in both transects.

The median Δσ values estimated exclusively from spectral-ratio fitting and the median values estimated using all methods (spectral-ratio fitting, individual spectra fitting from attenuation-corrected and uncorrected spectra) have similar trends, although the median spectral-ratio Δσ values are slightly larger. The larger spectral-ratio Δσ estimates agree with previous studies that commonly find larger spectral-ratio Δσ estimates relative to single-spectrum fitting34,43 (See also detailed discussion in the Methods and Figs. S12 and 13). Finally, we find that the Δσ-depth dependence is similar for the first year after the Tohoku-Oki earthquake to the following decade (Fig. 3). Earlier work using alternate methods to estimate Δσ values in a different time interval (2004–2011) finds consistent trends with depth for earthquakes shallower than 80 km in the same region19. In addition to upper-plate events, the estimates in ref. 19 also includes interplate events, with 1191 out of 1563 events occurring within 10 km of the plate interface. Moreover, Δσ estimates in ref. 19 do not include earthquakes at depths <10 km, which in our study, are located mostly near the city of Iwaki (Fig. 1, Iwaki transect) and were triggered by the Tohoku-Oki earthquake29. The consistency of our results, independent of method and with other studies, suggests that the relation between Δσ and depth is a robust feature that is independent of the number of individual Δσ estimates and of the time interval considered.

Correlation between stress drop values and maximum shear stress

The results in Fig. 3 suggest that Δσ correlates with depth within the brittle forearc lithosphere. The most plausible physical explanation for the depth dependence might be increasing fault strength with lithostatic stress. If the explanation is valid, it then follows that Δσ should correlate more generally with τmax, which may also vary laterally (Fig. 2). We test the more general correlation of Δσ with τmax using the independent modeling constraints in both the Iwaki and Sendai transects. For both transects, we project the earthquake hypocenters within a 200-km-wide swath onto the central plane of cross-section (dashed boxes and solid lines Fig. 1). We then determine the nearest respective τmax values in the 2D finite-element model for the projected hypocenter locations and associate them with their respective earthquake Δσ estimates (see Methods). We use the same Δσ estimates as those for the depth correlation, but limit the analysis to the period covered by the finite-element models, corresponding to the first year after the Tohoku-Oki earthquake (March 11, 2011–March 10, 2012). We consider the Iwaki and Sendai forearc transects separately and expect that the differences in τmax along-strike of the margin within each 200-km swath are smaller than the difference between the two models for the respective transects. The stress difference between the models is due to characteristic differences in forearc topography, slab curvature, aftershock distribution and fault kinematics between the swaths; however, these parameters vary only slightly within the swaths29.

We find generally larger median Δσ values in regions with larger modeled τmax (Fig. 4). There is a strong correlation between median Δσ and modeled τmax values along both transects, with a slightly higher correlation for the Iwaki transect (r ~ 0.96) relative to the Sendai transect (r ~ 0.88). We note that the Sendai seismicity occurs over a smaller depth interval that spans a smaller range of modeled τmax values. The strong correlation between Δσ values and τmax in the Iwaki transect is also evident when considering only spectral-ratio Δσ estimates (r ~ 0.98, Fig. 4c). The lower number of spectral-ratio Δσ estimates in Sendai enables calculating only one median Δσ value (Fig. 4d), yet the individual value is in agreement with the trend defined by all Δσ values (Fig. 4b). The calculated r-values are nearly insensitive to the number of events used to calculate median Δσ values (Fig. S11). To further assess the robustness of the observed correlation between Δσ values and τmax, we investigate a subset of earthquakes clustered near the city of Iwaki separately (Fig. S14). The median Δσ values near Iwaki also show a clear increase with τmax (Fig. S14), which corroborates the findings obtained from the entire datasets for both transects.

Fig. 4: Stress drop (Δσ) values as a function of modeled maximum shear stress (τmax) for events within the 200-km-wide swath profiles along the Iwaki (Iw) and Sendai (Se) transects.
figure 4

τmax values are from the 2D finite-element models shown in Fig. 2. Stress-drop values are from the year following the Tohoku-Oki earthquake (11 March 2011–10 March 2012; see text for additional information). a, b Individual and median Δσ values (colored and white squares, respectively) obtained from attenuation-corrected and uncorrected spectra, and spectral-ratio corner-frequency (fc) fitting. c, d Individual and median Δσ values obtained from spectral-ratio fc fitting. Median Δσ values are calculated for bins of 200 events with 50% overlap. Individual Δσ values are color-coded according to depth. Oblique dashed lines denote isolines of Δσ/τmax ratios. Reported r-values are from linear fitting of median Δσ values of all events in the respective panel (white squares). See Fig. 1 for the location of swaths bracketing the profiles. Horizontal and vertical lines associated to mean and median values indicate the 5th and 95th percentiles.

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Figure 4 further illustrates individual Δσ values color-coded by depth, together with the median Δσ values. The individual Δσ values also reflect a dependence on depth, with larger Δσ values being associated with greater depth. However, the data also reveal that depth is not the only controlling factor, because similar τmax values can occur at variable depth (Fig. 2). Stated in another way, events at similar hypocentral depth, i.e., with a common color in Fig. 4, can be associated with a broader range of τmax values (e.g., 2–5 MPa in Fig. 4a, or 20–40 MPa in Fig. 4b).

Median Δσ values are generally smaller than modeled τmax values, which is consistent with a partial stress release during earthquakes13. The ratio between median Δσ from all observations and τmax varies between 0.1 and 0.3 (dashed diagonal lines in Fig. 4), suggesting that earthquakes within the study region release, on average, 10 – 30% of τmax. Similarly, when we examine the Δσ values from spectral ratio fitting only, we find a similar range of Δσ:τmax ratios of 0.1–0.5 (Fig. 4b, d). In detail, we find that the Δσ–τmax ratios are slightly lower along the Sendai transect (Fig. 4b, d). In addition, the Iwaki transect shows a transition from relatively higher to lower Δσ-τmax ratios with increasing τmax (Fig. 4a). The lower Δσ-τmax ratios along the Iwaki transect (Fig. 4a) are primarily caused by events occurring within the mantle wedge (Fig. 1b). They are similar to the ratios obtained for Sendai, where a large portion of the seismicity occurs at depths >30 km within the mantle wedge (Figs. 1b and  4b). We therefore test whether the lower ratios are an artifact of the model setup, either due to the viscoelastic stress relaxation or due to the lower rigidity (Young’s modulus E) of the crust (Ec = 60 GPa) compared to the mantle (Em = 150 GPa). Both disabling viscoelasticity and removing the rigidity contrast in the models (Ec, Em = 60 GPa) slightly alter the τmax values, but still yield lower Δσ–τmax ratios of 0.1–0.2 in the mantle wedge (Fig. S15). This indicates that the lower ratios are not an artifact of these model parameters.

The inferred average stress release during earthquakes that ranges from 10–50%, or slightly lower when considering Δσ estimates from all methods (10–30%, Fig. 4a, b), is consistent with the upper end of the range observed in laboratory experiments44,45,46. Refs. 45,46 report Δσ values that are 10–12% or less of τmax, while ref. 44 report a wider range of 4-49%. However, many studies document that the uncertainties in absolute values of Δσ estimates are high, while the relative differences in Δσ values, given consistent data and methodological approach, may be well-constrained3. Therefore, we focus on our interpretation of relative changes rather than on absolute values. In the context of absolute uncertainty, we observe some individual Δσ estimates that are larger than τmax values (data points above the 1:1 line in Fig. 4). Ratios larger than one may reflect either Δσ estimate uncertainties or local stress anomalies arising from geological complexity that are not captured in the finite-element models. A less likely possibility is that they may result from rupture overshoot, where a rupture releases more than the total available shear stress47. It is beyond the scope of this study to determine the causes of individual Δσ values that exceed τmax values.

We next examine the stress ratios in 3D grids along each transect to look for spatial variability. Each grid has a spacing of 0.15° × 0.15° × 20 km in latitude, longitude, and depth, respectively, with grid nodes orthogonal to the trace of the Iwaki and Sendai transects. We associate events within a 15 km radius of each grid node and calculate the ratio between the Δσ and average τmax values. We report ratios at grid nodes with a minimum of five associated Δσ values (Fig. 5a–c). Overall, the Δσ/τmax ratios span a narrow range of values that are comparable with the results in Fig. 4, although the Iwaki area contains slightly larger Δσ-τmax ratios up to ~0.6. The ratios for the Iwaki transect exhibit a median value of 0.17 with 5th and 95th percentile values of 0.06 and 0.41, respectively (Fig. 5d). The ratios for the Sendai transect exhibit a median value of 0.11 with 5th and 95th percentile values of 0.06 and 0.23, respectively (Fig. 5d).

Fig. 5: Spatial variability of median stress drop to mean modeled maximum shear stress (Δσ/τmax) ratios.
figure 5

Ratios were calculated over a 3D grid with dimensions 0.15° × 0.15° × 20 km in latitude, longitude, and depth, respectively. Reported values are for grid nodes with a minimum of five Δσ estimates within a 15 km radius. Results at (a) 10 km, b 30 km, and c 50 km depth. Median Δσ values include estimates from single-spectra (with and without attenuation correction) and spectral-ratio corner-frequency fitting. d Histogram of all values displayed in (a–c).

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Discussion

Our study investigates the correlation of earthquake Δσ with depth and modeled lithospheric τmax in the brittle northeastern Japanese forearc following the 11 March 2011 Tohoku-Oki megathrust earthquake. We find that median Δσ values positively correlate with depth in the seismically active forearc lithosphere (Fig. 3). Several studies have investigated the relation between Δσ and depth4,18,20,21, however, whether or not Δσ correlates with depth remains unresolved22. Difficulties in documenting the depth dependence of Δσ may result from large uncertainties and the scatter in Δσ estimates, which commonly range over 3–4 orders of magnitude3. Our quality-control criteria ensure that relative changes of median Δσ values are well-constrained and robust. In particular, the incorporation of a depth-dependent shear-wave velocity model and strict spectral-ratio event-pair selection has been shown to effectively reduce non-source-related effects on Δσ values. While we vary the rupture velocity using the shear-wave velocity at the source depths, it is important to note that other variability in rupture velocity will be mapped into the estimated Δσ values. One common source of uncertainty is insufficient depth-dependent attenuation corrections22. However, we apply a depth-dependent attenuation correction that has been shown to effectively remove attenuation from individual spectra and provide Δσ estimates that are similar to spectral ratio estimates (see Methods and ref. 35). We further discuss the depth-dependent attenuation correction below.

The correlation between Δσ and depth is supported by the consistent results in our analysis of a sub-cluster of earthquakes with robust depth estimates near the city of Iwaki (Fig. S14). We therefore interpret the correlation between Δσ and depth in the two forearc transects across different time periods (Fig. 3) to represent physical differences. We also note that the depth dependence of Δσ would potentially be obscured if Δσ values were evaluated for a restricted depth range. The positive correlation between Δσ and depth emerges in our dataset expressly because of the wide hypocentral depth range considered. In fact, on average, for every 10 km increase in depth we observe an ~0.8 MPa increase in Δσ (Fig. 3). For example, if the available depth range of Δσ values were to be restricted to 15–30 km, the correlation between Δσ and depth would be difficult to discern, as Δσ would only be expected to increase by ~1.2 MPa (e.g., Fig. 3a, c). Further, our results show good agreement with a previous study using an alternative approach in a different time period in the same region19. ref. 19 estimated Δσ values for earthquakes occurring at depths shallower than 80 km, with 1191 out of 1563 events located within 10 km of the plate interface. The authors also observed an increase in Δσ values with depth, with the exception interval between ~10–30 km, where Δσ values were roughly constant. Ref. 48 observed a decrease in source duration with depth during the aftershock sequence of the 2010 Mw 8.8 Maule megathrust earthquake, which would also translate to an increase in Δσ values. (Given that source duration is inversely related to corner frequency, and therefore to Δσ, a decrease in source duration reflects an increase in Δσ, consistent with our findings). Although their analysis is not limited to earthquakes above the plate interface, it includes a large number of upper plate, forearc events in the widespread seismicity following the Maule earthquake49. We note that there are also studies that do not observe increasing Δσ with depth in a subduction forearc. A recent study in northern Chile reports increasing Δσ values of upper plate forearc seismicity to ~20 km depth, followed by a decrease at greater depths50. Although the differing depth trend may stem from methodological factors involved in Δσ estimation, such as applied quality-control criteria and data availability (e.g., ref. 50 has roughly a factor of 10 fewer stations available). One possible reason for the different trends might be that ref. 50 aggregates data over a structurally complex, 700-km-wide forearc region using a catalog that spans a longer time period and includes several large megathrust earthquakes (Mw 7.6, 7.7, 8.1). The dataset may therefore introduce additional variability related to factors controlled by the earthquake cycle stage. In the context of the complexities mentioned before, the observed decrease in Δσ values within the lower forearc crust may reflect a stress contrast between the upper and lower crust in the overthickened Chilean forearc crust51.

Previous studies have shown that a correlation between Δσ and depth can arise if the variation in attenuation (or Q-value, which is inversely related to attenuation) is not properly accounted for22. As described in the Methods, we validate the reliability of the correlation between Δσ and depth by examining the variation of attenuation with depth. The attenuation curves for a sample volume in the forearc show a general trend of constant or slightly increasing attenuation with depth that is visible when comparing deeper ( > 40 km) and shallower events with similar epicentral locations (Fig. S7). We note that most of the deeper events ( > 40 km) occur close to the top of the slab to the south of the city of Iwaki (Fig. 1, Iwaki forearc transect). Subduction forearcs exhibit a complicated attenuation structure that would render a laterally constant attenuation structure with a homogeneous reduction with depth unlikely52. In fact, dehydration of the subducting slab, sediment underplating, mantle wedge serpentinization, and possible additional subduction-related processes can locally increase attenuation at larger depths near the top of the slab and lead to a spatially heterogenous structure52. For instance, many Δσ studies in different settings assume a homogeneous reduction of attenuation (or increase of Q-value) with depth22. The fact that our attenuation curves suggest systematic, gradual changes, as opposed to a sporadic spatial variation, suggests that the spatial changes are indeed related to the physical conditions of the volume in which they were derived. Thus, we infer that the correlation between Δσ and depth represents real physical variations with depth.

We observe that the increase of median Δσ values with depth is concomitant with an increase in τmax. As the τmax is indicative of fault strength, the increase in Δσ likely reflects a dependence on fault strength that is consistent with mechanical models and laboratory results44,53,54. It should be noted, however, that the modeled τmax relates to fault strength only in seismically active areas. The modeled stress values are not indicative of fault strength in areas that show no seismicity and where rocks may deform aseismically. Moreover, the homogenous material properties in the models (see Methods) do not capture local stress heterogeneities that may arise from geological complexity55. The dependence of Δσ on fault strength is therefore most robust for median Δσ values and descriptive of bulk behavior. Individual Δσ values may still reflect other factors, such as variation in rupture velocity8, fault slip rate10, fault kinematics56, lithology10, or variation in the amount of slip per unit area14,15.

The comparison of Δσ values with τmax also explains the comparatively constant Δσ values at 15–30 km depth along the Iwaki transect (Fig. 3a). The respective seismicity occurs in a forearc region at ~70–180 km from the trench, which shows only minor variations of τmax with depth of a few MPa (Figs. 1–2). Accordingly, the fault strength should be nearly constant over that same depth range. This outcome corroborates the hypothesis of near-constant crustal strength, as inferred from deep borehole data indicating an increasing pore fluid pressure with depth57. Finally, we find that the ratio between median Δσ values and τmax show only slight variations across the forearc, but the ratios may be somewhat lower in the mantle relative to the crust (Figs. 4–5).

Taken together, our results indicate that the Δσ in earthquakes tends to increase with τmax and fault strength, but the fractional Δσ relative to crustal τmax is similar for all faults, independent of their strength, at least on average. Notably, the median Δσ values from March 2012 to December 2021 are almost identical to the values from the first postseismic year (Fig. 3), except for a slight increase in Δσ at 50-60 km depth along the Iwaki transect (Fig. 3b). On closer inspection, the multiyear Δσ dataset shows that the elevated Δσ values are associated with seismicity near the megathrust in the southernmost part of the Iwaki transect that initiated between 2013 and 2015 (Fig. S16h). Otherwise, the median Δσ values are nearly constant through time along Iwaki and Sendai transects (Figs. 3 and S16). Likewise, the spatial distribution of earthquake hypocenters is similar between the first postseismic year and the following decade (Fig. 1). We interpret the spatio-temporal consistency in the seismicity distribution and Δσ values to indicate that the strength of active faults has not changed significantly since the Tohoku-Oki earthquake, while postseismic processes such as viscoelastic relaxation38 and reloading of the upper plate58 drive the faults repeatedly to failure.

To our knowledge, this study provides some of the first quantitative evidence of a correlation between median earthquake Δσ values and lithospheric stress in nature. Our finding that stress released in earthquakes is proportional to stress at failure suggests that seismic observations can be used to infer relative strength differences in the Earth’s brittle lithosphere (Fig. 6).

Fig. 6: Conceptual model summarizing the main results of this study.
figure 6

Earthquake stress drop (Δσ) correlates with maximum shear stress (τmax) and fault strength. Seismic failure occurs and stress is released when stress values reach the fault strength. On average, the stress released during earthquakes is related to stress, suggesting that Δσ can be used as a marker for relative crustal strength. Fault strength exhibits lateral spatial variability and not just variation with depth.

Full size image

Data and methods

Earthquake catalog and waveforms

Hypocentral solutions and P- and S-phase arrivals originate from the Japan Meteorological Agency (JMA) (https://www.data.jma.go.jp/svd/eqev/data/bulletin/hypo_e.html). We restrict the list to forearc events by considering events located from 1 km above the top of the slab32 to the surface, and excluding events in close proximity to the volcanic arc (~west of the 80 km slab isodepth, Fig. 1). We impose quality-control criteria that limit analysis to earthquakes located using a minimum of 10 P and/or S phase picks from high-sensitivity borehole stations (Hi-net) with vertical errors ≤5 km, latitude and longitude errors ≤0.025°, and origin times between 11 March 2011 (after the Tohoku mainshock) and 31 December 2021 with magnitude MJMA ≥ 2.5. The initial list of events considered for stress drop calculations consisted of 39,295 events with an average latitude error of 0.006 ± 0.003° (~0.7 ± 0.3 km), average longitude error of 0.01 ± 0.005° (~0.9 ± 0.4 km), and average vertical error of 1.6 ± 0.9 km.

We downloaded event waveforms from Hi-net borehole stations, which are available from the National Research Institute for Earth Science and Disaster Resilience59 for each of the earthquakes considered. We use waveforms from a total of 259 borehole stations (Fig. S14a) and deconvolve them by their instrument response. For earthquakes with MJMA ≥ 3.5, we apply a time domain recursive filter that simulates seismic waves recorded with broadband seismometers, as described in ref. 60. The latter step ensures robust estimates of corner frequencies of M3.5+ earthquakes as they approach and become lower than the short-period instrument corner frequency.

Stress-drop estimates

We employ three different methods to estimate fc of earthquakes in the Japanese forearc: 1) fitting of individual earthquake spectra that have not been corrected for attenuation, 2) spectral-ratio fitting of co-located event pairs, 3) single-spectrum fitting of attenuation-corrected spectra. We then use fc obtained from the different methods to calculate a Δσ value. For all the methods, the first step consists of estimating the displacement spectral amplitude using Thomson’s multi-taper method61. We use S-wave windows starting 0.2 seconds before the phase arrival that contain 90%, 80%, and 70% of the energy at stations within hypocentral distances of 25 km, 25-50 km, and ≥50 km, respectively62. This energy-based approach ensures that the window captures the main signal while minimizing noise and contamination from later phases. We obtain a median S-wave signal length of 3.1 seconds, with the 5th and 95th percentiles at 1.3 and 10.8 seconds, respectively. We ensure that each spectrum has a signal-to-noise ratio (SNR) ≥ 3 in a magnitude-dependent frequency band for a given event that we determine by calculating corner frequencies corresponding to theoretical stress drop values of 0.05 and 500 MPa, to define lower and upper-frequency limits, respectively.

Method: single-spectrum fitting of uncorrected spectra

We fit individual spectra using a trust-region-reflective minimization algorithm63:

$${\varOmega }_{t}\left(f\right)=\frac{{\varOmega }_{0}{e}^{-\left(\pi {ft}/Q\right)}}{{\left[1+{\left(f/{f}_{c}\right)}^{\gamma n}\right]}^{1/\gamma }}$$
(1)

where Ω0 is the long-period spectral amplitude, f is the spectral frequency, t is the travel time, Q is the quality factor, fc is the corner frequency, and n is the high-frequency falloff rate13,64. We use a spectral shape constant γ of 2 (Boatwright model)64, because it provides the lowest residuals across all spectra. We then calculate the seismic moment (M0) using the fitted Ω0 values as:

$${M}_{0}=\frac{4\pi \rho {c}^{3}R{\varOmega }_{0}}{{U}_{\phi \theta }}$$
(2)

where ρ is the density, c is the S-wave velocity at the depth of the hypocenter, R is the station-event hypocentral distance, and Uϕθ is the mean radiation pattern for S-waves65. We use S-wave velocities at the hypocentral depth taken from an existing, local 1D velocity model from the JMA (vjma2001 available from: https://www.data.jma.go.jp/svd/eqev/data/bulletin/catalog/appendix/trtime/trt_e.html, last accessed March 2024). We then calculate mean values and 95% confidence intervals for each event using a delete-one jackknife-mean66 where S-wave estimates from at least five stations are available. Inherent tradeoffs between fc, n, and Q occur during the individual spectral fitting process and lead to less stable estimates of fc. To reduce the tradeoff between fitting parameters of the uncorrected spectra and increase the stability of fc, as well as initial M0 estimates, we hold n fixed to 2.5 in the final fitting procedure. We obtain the best value by which to fix n by initially allowing Q and n to vary between 500–3000 and 2–4, respectively, and retaining the averaged values that lead to the lowest residuals. The initial calculation produced average values of 1121 for Q and 2.5 for n. We then fix n to 2.5 and let Q vary between 500–3000 for the final fc estimates for single-spectrum fitting of uncorrected spectra (Fig. S3).

Method: spectral-ratio fitting

Non-source related terms, such as site and path effects, can bias fc estimates and propagate to the final stress drop estimate. Where data permit, we implement a spectral-ratio approach to deconvolve all non-source-related terms from the spectral signal to ensure accurate estimates of fc (Fig. S6). The ratio between two co-located event spectra at a specific station cancels the influence of site and path effects, leaving the ratio between the two earthquake source spectra. The approach offers an effective way to remove path effects, including attenuation, to provide more robust estimates of the earthquake source fc relative to individual, uncorrected spectral estimates. The frequency bandwidth of sufficient SNR dictates whether fc estimates from one or both events in the pair are recoverable67 (Fig. S6). We require viable event pairs to have cross-correlation values ≥ 0.7 on three components of full waveforms of events within 5 km hypocentral distance and magnitude differences ≥ 0.5. A magnitude difference ≥ 0.5 ensures a minimum standard for the selection of event pairs with fc values that are resolvable in the spectral-ratio fitting. The displacement spectral ratio Ωr(f) between two event spectra can be written as:

$${\varOmega }_{t}\left(f\right)={\varOmega }_{0r}{\left[\frac{{1+\left(f/{f}_{c2}\right)}^{\gamma n}}{{1+\left(f/{f}_{c1}\right)}^{\gamma n}}\right]}^{1/\gamma }$$
(3)

where fc1 and fc2 are the corner frequencies of the larger-magnitude target event and the smaller-magnitude empirical Green’s function event (eGf), respectively. The spectral shape constant γ is set to 2, consistent with our single-spectrum estimates, while n is set to 2 (Boatwright-model64). We require at least five S-wave station ratios for individual event pairs, and manually review the spectral ratio fits to ensure high quality and check whether fc1 and fc2, or only fc1 values, are resolvable. We note that in cases where the fc of the target or the eGf events are resolvable for multiple ratios (i.e., a single event can be included in multiple pairs), the final fc results from a delete-one jackknife-mean using all spectral ratio estimates.

Method: Single-spectrum fitting of attenuation-corrected spectra

Estimating fc from spectral ratios imposes strict conditions on the data that are commonly met by only a small subset of events. However, it is possible to make use of the spectral-ratio estimates to compute a spatially confined attenuation correction for individual event spectra that occur in the crustal volume near spectral-ratio pairs. Applying such a correction significantly increases the number of attenuation-corrected source spectra that can be used to estimate fc. Including this approach enables us to robustly correct 75% of the stress drop estimates for attenuation through either the spectral-ratio approach or the single-spectrum attenuation-corrected. We refer to the fc and subsequent Δσ estimates obtained from the attenuation-correction method as single-spectrum attenuation-corrected fc and Δσ estimates to differentiate them from standard single-spectrum fc and Δσ estimates that are not corrected for attenuation.

Specifically, we use the method of refs. 34,35 to correct individual spectra for local attenuation. The method uses the spectral-ratio fc estimates for target-eGf pairs (spectral-ratio method, described above) and estimates attenuation curves for the neighboring volume. Following ref. 34, the effect of the propagation path, site, and geometrical spreading on the spectra (attenuation curves) is derived by comparing a theoretical source model to the observed spectrum (Fig. S4). As in ref. 35, we use the omega-squared moment spectrum \({\widetilde{M}}_{0}\) (f) of ref. 64 for the theoretical source model as the Boatwright-model produces the lowest residuals for uncorrected single-spectra fitting:

$${\widetilde{M}}_{0}(f)={M}_{0}/{[1+(f/{f}_{c})^{4}]}^{1/2}$$
(4)

For each event with a spectral-ratio fc estimate, we use M0 from single-spectrum fitting to calculate a theoretical source spectrum (Eq. 4). To derive attenuation curves for each station-event combination, we take the ratio between the observed spectrum at each component and the theoretical spectrum34 (Fig. S4). We require each theoretical spectrum to be derived using S-wave fc spectral-ratio estimates constrained from a minimum of five stations. The spectrum of a given nearby earthquake (which is then assumed to have occurred in a confined region with common attenuation properties) is then corrected using mean attenuation curves (minimum 10) for each station and component from events within a 5-km radius (Figs. S5 and S7). We only use curves from close neighbors to ensure resolvability of variable attenuation paths for distinct source-station pairs. The attenuation corrections derived from our data in the Japanese forearc suggest that attenuation may increase slightly with depth and is dictated by the specific source-station travel paths (Fig. S7). The difference in attenuation is most visible when comparing deeper ( > 40 km) and shallower events (Fig. S7). Forearcs exhibit a complicated attenuation structure that does not decrease uniformly with depth as might be expected for a simple 1D velocity model, likely the result of subducting slab dehydration, sediment underplating, mantle wedge serpentinization, and possible additional subduction-related processes that may locally increase attenuation at larger depths52. An individual attenuation-corrected spectrum is the ratio between the observed spectrum and the associated averaged-attenuation curve (Fig. S7). We average attenuation-corrected spectra across channels and fit Equation 4 to retrieve an attenuation-corrected fc estimate for an individual event-station combination. We fit the attenuation-corrected spectra by keeping n fixed to 2. In contrast to individual spectra that are not corrected for attenuation, individual attenuation-corrected spectra are not fit for Q. We estimate mean values and confidence intervals as is done for fitting of the individual spectra without an attenuation correction.

All methods

The subsequent step uses M0 values, and moment magnitudes (Mw)68, from uncorrected single-spectrum fitting together with fc values, either from single-spectrum fitting (with or without attenuation corrections), or spectral-ratio fitting methods, to calculate Δσ values assuming a circular crack model with radius r = kβ/fc as ref. 50:

$$\Delta \sigma =\frac{7}{16}\frac{{M}_{0}}{{r}^{3}}$$
(5)

where β is the shear-wave velocity at the hypocentral depth and k is a constant that is set to 0.26, assuming a symmetrical circular model with a rupture velocity of 0.8 β12.

We refine the initial list of Δσ estimates obtained using fc estimates from single-spectrum and attenuation-corrected spectra by comparing their fc estimates with those obtained from spectral-ratio fitting for events where both estimates are available (Fig. S8). We use the comparison between single-spectrum uncorrected fc, single-spectrum attenuation-corrected fc, and spectral-ratio fc to discern reliable fc ranges and a minimum number of observations for robust fc estimates (Fig. S8). We use uncorrected single-spectrum Δσ estimates with fc ≥ 3 only if they have at least 20 or more observations. In addition, we reject uncorrected single-spectrum Δσ estimates with Mw ≤ 3. We observe that the fitting of attenuation-corrected spectra more reliably resolves the fc of events with larger fc (i.e., smaller magnitudes) relative to the uncorrected single-spectrum-fitting method. In the case of attenuation-corrected single-spectrum fc estimates, we require at least 10 individual fc observations to use Δσ estimates for events with fc ≥ 7 and/or Mw ≤ 2.5. The quality-control steps described above reduce the trend of decreasing Δσ with magnitude (or larger fc). The trend is commonly attributed to decreasing SNR values at lower magnitudes that cause observational bandwidth limitations near the corner frequency of smaller earthquakes34,69.

We initially obtain a total of 18,016 Δσ estimates, including: 510 estimates of target events and 156 estimates of eGf events in event pairs obtained from spectral-ratio fc, 9897 estimates obtained from attenuation-corrected single-spectrum fc, and 7453 estimates obtained from uncorrected single-spectrum fc (Fig. S2a). After implementing quality-control criteria, the restricted stress-drop catalog that we use for comparison with modeled maximum shear stress consists of 10,469 events including: 510 estimates for target events and 156 estimates for the eGf events obtained from spectral-ratio fc, 7193 estimates obtained from attenuation-corrected single-spectrum fc, and 2610 estimates obtained from uncorrected single-spectrum fc (Fig. S2b).

We observe slightly larger median Δσ values from spectral-ratio fc fitting relative to median Δσ values from individual, attenuation-corrected and uncorrected fc spectral fitting. The observed differences increase with depth and with increasing maximum shear stress (Figs. 2–3). Δσ values obtained from uncorrected single-spectrum fc fitting are susceptible to travel-path and site effects that are difficult to fully account for with individual spectral-fitting approaches. Spectral-ratio corrections remove the effects of travel-path and site for perfectly co-located events, and so spectral-ratio fitting is hampered by their effects to a lesser degree34,43. When we examine the attenuation-corrected single-spectrum fc, they tend to more closely match the spectral ratio fc values and show less deviation from a 1:1 line than the uncorrected single-spectrum fc across the whole dataset (Fig. S8a, c). Strict quality-control metrics are applied to all single-spectrum data, such that fc estimates used in the study closely match spectral ratio fc values across a wide range of event sizes; however, attenuation-corrected single-spectra fc estimates are biased slightly low (Fig. S8b, d). Thus, we interpret the deviation between attenuation-corrected and uncorrected single-spectrum and spectral-ratio fitting as being related to travel-path and site effects. Nevertheless, the good agreement between relative values obtained from the three methods used here allows for a robust interpretation of the relative changes in Δσ values obtained from single-spectrum uncorrected and single-spectrum attenuation-corrected fc estimates, particularly for the magnitude range considered.

Maximum shear stress from finite-element models

We use 2D finite-element models from ref. 29, to determine the maximum shear stress, τmax = (σ1–σ3)/2, in the Japanese forearc along the Iwaki and Sendai transects in the first year after the Tohoku-Oki earthquake. The finite-element models are based on the modeling approach of refs. 29,30 and calculate the total stress in the forearc resulting from gravity and shear stress on the megathrust (Fig. S10). The models are created with the commercial finite-element software ABAQUS70 and comprise a rigid lower plate in frictional contact with an elastic upper plate that is subdivided into continental crust and mantle. A lithostatic pressure and an elastic foundation are applied to the bottom of the model to implement isostasy (arrows and springs in Fig. S10). The left-hand side of the model (back side of the upper plate) is free to move vertically, but is fixed in the horizontal direction. All models are meshed with linear tetrahedral elements with an average element edge length of ~2 km. The margin topography is approximated by the mean elevation, which we calculate from the ETOPO1 global relief model71 using TopoToolbox for MATLAB72. We approximate the slab geometry by fitting an arc with constant curvature through the upper 80 km of the Slab2.0 model32.

The megathrust is implemented as a frictional contact between the upper and lower plates and extends from the trench down to a depth of 60 km. The shear stress τ on the megathrust obeys the friction law for a cohesionless fault, τ = μ‘bσn, where μ‘b is the effective coefficient of megathrust friction and σn is normal stress. The shear stress is generated by displacing the lower plate in the downdip direction tangential to the plate interface. The displacement ensures that the entire plate interface is at a state of failure28. The stress and strain in the upper plate are independent of the total displacement of the lower plate. The values of the effective coefficient of megathrust friction have been constrained in ref. 29 and vary between 0.005-0.025 for the Iwaki transect and between 0.005-0.034 for the Sendai transect in the post-Tohoku model step investigated in this study. The μ‘b values in the pre-Tohoku model step (not considered in this study) vary between 0.02 and 0.02529. The difference between the pre-Tohoku and post-Tohoku μ‘b values is, on average, about 0.01 and relates to the coseismic change in megathrust shear stress, i.e., the coseismic stress drop in the Tohoku-Oki earthquake. The μ‘b values as well as the difference between the pre- and post-Tohoku values agree within uncertainties with previous estimates derived from force-balance models28,30,41, heat-dissipation models40, and stress drop models28,30,73.

The models of ref. 29. consider only elastic deformation in the upper plate and constrain the forearc stresses immediately after the Tohoku-Oki earthquake. To account for possible stress changes in the postseismic period38, we slightly modify the models from ref. 29 and implement a bi-viscous Burgers rheology for the lower crust and mantle wedge of the upper plate (Fig. S8). We use transient (Kelvin) and steady-state (Maxwell) viscosities of 1018 Pa s and 1020 Pa s, respectively, which are compatible with previous models and the global mantle average74,75,76. For lower viscosities (1017–1018), the stresses in the mantle wedge approach zero within the first months after the earthquake, which we consider unlikely. The implementation of the bi-viscous rheology allows considering viscoelastic stress relaxation in the first year after the Tohoku-Oki earthquake. Over longer periods of time, the stress in the forearc may be influenced by the reloading of the subduction megathrust58, which is not included in the finite-element models. Therefore, we determine maximum shear stress in the forearc only for the first postseismic year (March 2011 to March 2012) (Fig. 2, Fig. S8) and compare it with Δσ estimates of earthquakes that occur during the same period. We note that the viscoelastic relaxation has only a small effect on the correlation between median Δσ values and τmax values, and that we obtain similar results for the original models of ref. 29 (Fig. S15).

Comparison of stress drop values with modeled maximum shear stress values

To compare the earthquake Δσ values with modeled maximum shear stress values, we first project earthquake hypocenters located within 200-km-wide swaths bracketing the profiles in Fig. 1 onto a vertical plane of cross-section and determine the x and y coordinates for each projected hypocenter, i.e., the distance from the trench and depth below sea level. The swaths that bracket the profiles are oriented normal to the plate margin and are aligned parallel to the Sendai and Iwaki transects represented by the finite-element models (Fig. 1), such that the coordinates of the projected hypocenters and models agree. We then associate all maximum shear stress values from the model results that fall within ±1 km to each earthquake hypocenter and calculate the mean modeled stress values. As the finite-element models are meshed with linear tetrahedral elements with an average element edge length of ~2 km, the final mean value that is associated with individual earthquake Δσ values typically originates from 2–3 single values. To account for the relative timing of earthquakes with a Δσ estimate with respect to the Tohoku-Oki earthquake, we read only the stress values from the model time step that overlap with the earthquake origin time. The duration of the model time steps varies between 7 and 15 days, such that the difference between the event time and the model time never exceeds 1 week.