Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes

Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes Wei Liu, Xueting Wei, Xiaofei Chen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7074847/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Slow earthquakes represent a distinct class of fault slip phenomena characterized by prolonged source duration and reduced seismic radiation compared to regular earthquakes. While essential for understanding seismic cycles and megathrust earthquakes, the mechanisms governing their rupture processes remain elusive. Here we present the finite-fault inversions of very low-frequency earthquakes (VLFs) in the Nankai Trough, a subtype of slow earthquakes producing detectable seismic waves while maintaining resolvable rupture dimensions. Comparison with two nearby regular earthquakes reveals that VLFs exhibit a distinct two-stage process: (1) rapidly accelerating slip confined to hypocentral regions, and (2) progressive deceleration and spontaneous termination within the initial slip zone, without lateral propagation. Source dynamic simulations reconcile these observations through a slip-weakening friction model, suggesting that a low slip-weakening rate facilitates slow self-arresting ruptures consistent with VLFs. Our findings provide observational insights into VLFs rupture dynamics, suggesting that slow earthquakes may experience limited propagation and self-arresting evolutions. Earth and environmental sciences/Solid Earth sciences/Seismology Earth and environmental sciences/Solid Earth sciences/Geophysics Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction The Nankai subduction zone, a locus of recurring megathrust earthquakes 1 – 3 , also hosts a diversity of slow earthquakes 4 – 11 . These slow events, observed in the shallow transitional zone between locked and stably sliding regions 12 , 13 , span a broad spectrum from seconds-scale elastodynamic rupture to multi-year aseismic slip. Unlike regular earthquakes, slow earthquakes, while capable of radiating detectable seismic waves, typically exhibit prolonged source durations, reduced seismic radiation, and orders-of-magnitude slower slip rates 14 , 15 . Compelling evidence links these slow events to shear slip mechanisms similar to regular earthquakes, with emerging implications for their potential precursory roles in megathrust cycles 16 – 18 . However, a critical unresolved question remains: Are slow earthquakes a scaled-down version of regular seismic processes with reduced stress release, or do they embody a fundamentally distinct rupture process with unique characteristics? While laboratory experiments 19 – 22 and numerical simulations 23 – 28 have captured essential features of slow events, direct observations of the rupture processes of slow earthquakes remain limited, posing challenges to achieving a comprehensive understanding of their controlling mechanisms and potential interactions with megathrust seismicity. Among the diverse family of slow earthquakes, VLFs, characterized by their dominant frequencies falling between 0.01 and 0.1 Hz 10 , 11 , stand out as promising candidates for elucidating the source processes of slow earthquakes. VLFs exhibit their slow source processes through rupture radii of 5–10 km, source durations of 10–100 seconds, and stress drops of ~ kPa, which are two to three orders of magnitude lower than those of regular earthquakes with comparable seismic moments 29 , 30 . However, unlike other slow earthquakes whose rupture scales are either too small for seismic resolution or too prolonged for waveform-based analysis, VLFs generate discernible seismic waves with resolvable rupture dimensions, making them particularly well-suited for investigating rupture processes. Given the frequent co-occurrence of VLFs and other slow earthquakes 11 , 31 – 38 , a shared source mechanism is likely, thus offering critical insight into the physics of slow earthquakes. But, as of now, the rupture process and properties of VLFs remain unresolved. Here, we investigate three VLFs along with two regular earthquakes occurring in the Nankai region using broadband seismic records from the NIED F-net network. We reconstruct the spatiotemporal rupture processes of these events by performing finite fault inversions. Our results demonstrate that the rupture region of each VLF is limited to an approximate 5 km radius around the hypocenter. Contrary to the two regular earthquakes, which exhibit clear rupture propagation, these VLFs indicate spatially confined rupture development. Instead, slip initiates and slowly self-arrests near the centroid, with peak slip rates consistently localized there throughout the rupture process. Additionally, source dynamics simulations based on the boundary integral equation method suggest that a low slip-weakening rate leads to a similar rupture pattern of slow self-arresting rupture, characterized by initiation and automatic termination within the nucleation patch. Our study, to the best of our knowledge, presents the first direct inversion of slow earthquake rupture processes, suggesting that the slow self-arresting ruptures, confined to the nucleation zone without lateral propagation may govern the source mechanisms of VLFs. Results and discussion Finite-Fault Inversion of VLFs and regular earthquakes Shallow VLFs are frequently observed offshore along the Nankai Trough, often occurring in conjunction with other slow earthquakes at the shallow plate interface 33 , 37 , 38 . These VLFs exhibit a spatially discrete distribution within the Nankai subduction zone, clustering in distinct patches 33 , 39 – 42 , which suggests lateral variations in lithology and frictional properties 43 – 46 . Although previous study 29 provides constraints on the source characteristics of shallow VLFs, including source durations of 10–20 seconds, rupture radii of 5–10 km, and moment magnitudes (Mw) of 3–4, the spatiotemporal evolution of VLF ruptures remains poorly resolved due to their low dominant frequencies and limited signal-to-noise ratios. To bridge this gap, we perform rupture process inversions for multiple spatially distributed VLFs in the Nankai region. Specifically, our analysis focuses on three VLFs selected from the catalog of Takemura et al. 12 , based on variance reductions (VR) exceeding 70%. These events include VLF1 (Mw 3.8, depth 8.02 km), which occurs on 13 March 2018 12:28:09 (134.7°E, 32.7°N); VLF2 (Mw 4.14, depth 6.74 km) on 14 May 2018 17:02:27 (135.3°E, 32.7°N); and VLF3 (Mw 3.8, depth 5.97 km) on 28 April 2009 08:12:25 (137.3°E, 33.4°N). The estimated source durations for these events are 16 s, 27 s, and 17 s, respectively 12 , which are sufficient to resolve the detailed rupture processes through inversion analysis. Despite their spatial separation, the three VLFs exhibit a consistent focal mechanism, characterized by low-angle thrust faulting with strike angles parallel to the trench axis, suggesting slip on the plate boundary interface 12 . Critically, to establish a comparative framework for rupture processes and enhance the robustness of our results, we extend our analysis to regular earthquakes with rupture dimensions comparable to VLFs near their clusters. Based on the effective rupture radius of approximately 5 km for shallow VLFs 29 , which corresponds to a magnitude range of 5–6 for regular earthquakes, we investigate two Mw 5–6 earthquakes that exhibit source mechanisms consistent with those of the VLFs. According to the U.S. Geological Survey (USGS) earthquake catalog, the first event (EQ1) occurs in 1 April 2016 02:39:08 at (136.4°E, 33.4°N) with a magnitude of Mw 5.9, while the second event (EQ2) takes place on 3 December 2021 00:28:28, at (135.1°E, 33.8°N) with a magnitude of Mw 5.2. Following the finite-fault inversion method detailed in Methods, we analyze VLF1, VLF2 and VLF3 using broadband velocity records from 13 F-net stations (Fig. 1 ), applying 0.02–0.05 Hz bandpass filtering to optimize signal-to-noise ratios. To facilitate comparison with the rupture characteristics of regular earthquakes, we perform inversions for both EQ1 and EQ2. For consistency with the frequency band used in the VLF inversions, the waveforms of EQ1 and EQ2 are also filtered to 0.02–0.05 Hz. The inversion results (Fig. 2 ) suggest that the source durations of VLFs1-3 are significantly longer than those of EQ1-2. The evolution of moment rate is similar initially for all three VLF events, with VLF2 and VLF3 showing a subsequent decrease followed by a renewed increase (Fig. 2 b). The peak values of the moment rate functions for VLF1-3 are 3–4 orders of magnitude smaller than those of EQ1-2, consistent with the characteristics of VLFs, indicating that VLFs rupture more slowly and release less stress. The rupture radii of VLFs 1–3 are approximately 3–5 km, consistent with previous estimates of the effective rupture area for VLFs 29 (Fig. 2 c). Peak slip for each VLF event is concentrated near its centroid, and VLF3 attains a maximum slip of 0.95 cm. The synthetic seismograms demonstrate a reasonable agreement with the observed waveforms across multiple stations for all three VLFs (Fig. 2 d). Waveform discrepancies likely arise from deviations between the reference 1-D layered velocity model and the actual velocity structure, especially in the horizontal components. Furthermore, all three VLFs occur within the sedimentary layers overlain by seawater, adding to the complexity of the velocity medium. Detailed distributions of slip, slip duration, rupture velocity for VLFs1-3 and EQ1-2 are presented in Supplementary Figs. 1–2, while Supplementary Figs. 3–7 provides detailed waveform comparisons of VLF1–3 and EQ1–2 across all used stations. Shallow VLFs rupture characteristics A central question remains regarding the spatiotemporal rupture processes of VLFs: do they represent low-stress-drop analogs of regular earthquakes, or do they exhibit fundamentally distinct rupture characteristics from regular seismic events? To address this, we further analyze finite fault inversion results to reconstruct the spatiotemporal rupture processes of VLF1, VLF2, and VLF3, as well as EQ1 and EQ2 (Fig. 3 a). Supplementary Figs. 8–12 provide detailed rupture processes of VLFs1-3 and EQ1-2. The rupture processes of VLFs1-3 and EQ1-2 exhibit clear and distinguishable differences. For EQ1 and EQ2, rupture initiates at the hypocenter, followed by outward propagation. EQ1 exhibits an average rupture velocity of 1.40 km/s predominantly toward the northwest, while EQ2 propagates at 1.60 km/s with a primarily eastward direction. Both events exhibit clear rupture fronts, with peak slip rates concentrated at their leading edges, as the slip region dynamically propagates with the front. Simultaneously, slip rates gradually decrease near the model centroid as the rupture progresses. Ultimately, slip terminates near the leading edge of the rupture front. In contrast, all three VLF events originate within the hypocentral region, with slip predominantly concentrated near the center of the fault, and without rupture propagation. At different moments during the rupture process, the regions of slip on the fault remain nearly unchanged. Throughout the rupture process, the maximum slip rate for all three VLFs remains localized at the center of the modeled fault area, persisting there until it decays to zero. Moreover, the hypocenter appears to be the last region to arrests, a behavior that contrasts with the characteristics of EQ1 and EQ2, where slip arrests near the rupture front. These distinct features indicate the fundamentally different rupture dynamics of VLFs when compared to regular earthquakes. The slip rates during VLF1 appears to be more concentrated in the central region, whereas the rupture processes of VLF2 and VLF3, although both originating in the hypocentral area, exhibit a slightly more distributed slip rate. This may be attributed to limitations in the resolution and accuracy of the inversions. Alternatively, this pattern could arise from the presence of multiple closely spaced nucleation zones, potentially leading to the initiation of several sequentially small events. In addition, the three VLFs exhibit notably longer source durations compared to EQ1 and EQ2. VLF1 reaches its peak slip rate at ~ 2.8 s, followed by a gradual decay over ~ 24.8 s until rupture termination. Similarly, VLF2 and VLF3 attain peak slip rates at ~ 2.4 s, with decay phases lasting ~ 43.2 s and ~ 28.8 s, respectively, before slip arrest. In contrast, EQ1 reaches its peak slip rate at ~ 2.0 s and terminates within ~ 1.6 s, while EQ2 peaks at ~ 1.2 s and terminates after ~ 1.0 s. Although the slip rates of the three VLFs increase rapidly, their slip rate decay phases are significantly prolonged, indicating that the rupture termination mechanisms of VLFs may differ from those of regular earthquakes. The consistent rupture characteristics observed in the three VLFs across distinct locations and different occurrence times suggest a shared rupture process underlying VLFs, thereby enhancing the credibility of the inversion results. Moreover, as identical frequency band filtering and inversion procedures are applied during the inversion of both EQ1-2 and VLFs1-3, the observed differences in rupture characteristics appear unlikely to result from inversion artifacts and may instead reflect intrinsic distinctions in their source dynamics. These distinctive spatiotemporal rupture characteristics imply that VLFs may arise from different rupture processes and underlying mechanisms compared to regular earthquakes. The distinct rupture processes identified here, characterized by the propagation-limited nature of VLFs within confined zones and prolonged source durations, necessitate physical mechanisms beyond conventional rupture dynamic models. To reconcile these observations with fault mechanics, we explore source dynamics frameworks capable of generating such spatially confined ruptures. Notably, previous source dynamics simulations based on the slip-weakening friction law 51 – 53 indicate the presence of a distinct type of rupture when the dimensionless critical slip distance is large (suggesting a low weakening rate). In these cases, slip is initiated within the nucleation zone (the localized patch where rupture begins in the simulations) but fails to extend beyond this region. Unlike typical rupture propagation, the slip does not spread outward; instead, it gradually decelerates and ultimately self-arrests within the nucleation zone, forming a slow self-arresting rupture (SSAR) 27 , 49 , 50 . Additionally, SSARs exhibit longer durations and stress drops that are two orders of magnitude smaller than those of sub-Rayleigh ruptures/regular earthquakes of the same magnitude 50 . Thus, to investigate this further, we employ the slip-weakening frictional law 51 – 53 and the boundary integral equation method (BIEM) 54 – 56 to generate a SSAR and a sub-Rayleigh rupture for comparison. We consider a dip-slip fault in elastic media, featuring a slip-weakening homogeneous patch enclosed by an unbreakable boundary (Fig. 3 b). The simulation details and parameters are provided in the Methods and Supplementary Table 1. The SSAR attains its peak slip rate approximately 3.4 s after initiation, followed by a decay until slip stop around 9.9 s (Fig. 3 c). Despite using a slip-weakening frictional model, where frictional stress decreases linearly with slip, the rupture does not propagate outward but instead self-arrests within the nucleation patch, without external influence. Furthermore, the maximum slip rate is consistently concentrated at the center of the simulated domain, which also serves as the final point of arrest for the rupture. These rupture evolution characteristics, including slip rates around \(\:{10}^{-5}\) m/s, are highly similar to those observed in VLF1, VLF2, and VLF3. In contrast, sub-Rayleigh ruptures or regular earthquakes, once triggered within the nucleation zone, rapidly expand beyond its boundaries and continue to propagate outward (Fig. 3 d). The peak slip rate consistently located at the rupture front throughout the rupture process. These rupture characteristics are consistent with those observed in EQ1-2. Consequently, VLFs and SSARs exhibit fundamentally similar rupture characteristics. Integrated with prior studies reporting comparable source parameters and waveforms 49 , 50 , these observations indicate that SSARs may share essential source dynamics with VLFs. Discussions and conclusions Finite-fault inversions reveal spatially confined ruptures in VLFs with source durations substantially exceeding those of regular earthquakes, providing new insights into slow earthquake processes. Two possible mechanisms may explain these observations. First, VLFs could be regular earthquakes but encounter immediate strength barriers arresting propagation. While this explains the absence of rupture expansion, it conflicts with source parameters scaling of VLFs. Regular earthquakes with comparable rupture dimensions exhibit source durations lasting only a few seconds, whereas VLFs have source durations extending to 10–50 s. Moreover, regular earthquakes exhibit 2–3 orders of magnitude higher average stress drops, slip, and slip rates than VLFs. Additionally, such barriers would need to fortuitously surround every VLF patch. Alternatively, VLFs may represent SSARs that spontaneously arrest and remain confined to nucleation regions. Assuming the second mechanism to be more accurate, it implies that the stress and frictional conditions governing VLFs and SSARs may share similar characteristics. Previous dynamic simulations based on the slip-weakening law indicate that, variations in frictional properties and stress conditions give rise to distinct rupture types, including supershear rupture, sub-Rayleigh rupture, self-arresting rupture, and SSAR 27 , 49 , 50 (Supplementary Fig. 13). Supershear ruptures and sub-Rayleigh ruptures can propagate continuously until encountering a barrier that forces their slip to stop, whereas self-arresting ruptures and SSARs spontaneously terminate even in the absence of barriers. SSARs are particularly unique in that they initiate and terminate entirely within the nucleation zone without propagating. They are further characterized by prolonged source durations, relatively small stress drops, reduced slip, and lower average slip rates. In the framework of the slip-weakening law, the difference between the peak shear strength and residual stress is referred to as the breakdown stress drop \(\:{T}_{\text{u}}\) , while the slip required for the reduction from peak strength to residual stress is characterized as the critical slip distance \(\:{D}_{\text{c}}\) ​ (Fig. 4 a). The ratio of \(\:{T}_{\text{u}}\) to \(\:{D}_{\text{c}}\) ​ (absolute value) represents the weakening rate, which governs the frictional weakening process during fault slip: $$\:\varvec{W}=\left|{\varvec{T}}_{\mathbf{u}}/{\varvec{D}}_{\mathbf{c}}\right|$$ 1 . At higher weakening rates \(\:W\) , frictional strength decreases rapidly for a given amount of slip, promoting further slip and allowing the rupture to propagate outward until it encounters stress barriers that impede its expansion (Fig. 4 a). Rupture behavior under varying initial stress conditions may manifest as self-arresting rupture, sub-Rayleigh rupture, or supershear rupture. However, as the weakening rate \(\:W\) decreases, the frictional strength decreases slowly, resulting in slow slip with insufficient strength drop to drive rupture propagation (Fig. 4 b). When the weakening rate falls below a critical threshold, rupture transitions to SSAR under nearly all initial stress conditions (Supplementary Fig. 13). Assuming a nucleation patch radius of 2 km, an S-wave velocity of 3.464 km/s, a P-wave velocity of 6 km/s, and a density of 2670 kg/m³, the critical weakening rate for the occurrence of SSARs is estimated to be 18.2 MPa/m. This indicates that SSARs are possible only when the weakening rate falls below this critical threshold. Building on this, we propose a conceptual framework to elucidate the occurrence and slip dynamics of VLFs (Fig. 4 c). Along the subduction plate interface near the trench axis, a localized region with low weakening rates is likely to exist within a horizontal range of approximately 20 km and a depth of about 5 km. As depth increases, the weakening rate gradually grows larger. The critical weakening rate for the occurrence of SSARs is situated along the plate interface, resulting in SSARs being confined to regions on the plate interface, while only regular earthquakes are possible in areas below the interface. Furthermore, the presence of significant anisotropy and heterogeneity within the sedimentary layers of subduction zones 43 – 46 likely contributes to the gradual increase in weakening rate in the horizontal direction, moving away from the trench axis. This spatial variation in weakening rate results in the preferential occurrence of VLFs within regions closer to the trench axis. Assuming that the initial shear stress on faults hosting SSARs and sub-Rayleigh ruptures is identical at 1 MPa, the significantly lower shear strength of SSARs (1.02 MPa) compared to regular earthquakes (1.62 MPa) suggests that SSARs can be triggered by stress perturbations on the order of ~ kPa. Such perturbations may arise from slow plate motion, tidal forces, or seismic waves. In contrast, the initiation of regular earthquakes requires substantially higher stress increments, which may explain the frequent occurrence of VLFs in this region compared to the relative scarcity of regular earthquakes exceeding Mw 5. Previously, Fukuyama and Madariaga 57 demonstrated that large critical slip distances could lead to immediate rupture termination outside of an asperity. Also, numerical simulations 58 reveal that under certain slip-weakening parameters, ruptures could spontaneously arrest on the fault, failing to reach dynamic instability. Furthermore, Campillo and Ionescu 59 , in their investigation of dynamic instability initiation on a slip-weakening fault, found that ruptures remain stable when the slip-weakening rate falls below a certain threshold. While these analyses and simulations suggest the existence of self-arresting earthquakes, they may have often been overlooked or dismissed, and excluded from the conventional definition of 'earthquakes'. Our current study bridges this gap between theoretical predictions and observational evidence by presenting the observationally evidence of such ruptures through finite-fault inversions of VLFs. Notably, previous studies have shown that stick-slip models under ~ kPa effective normal stress can also generate VLFs, but with rupture propagation similar to that of regular earthquakes 28 . In contrast, our inversion results suggest that the SSAR, which exhibits fundamentally different rupture dynamics from regular earthquakes, may serve as a plausible source model for shallow VLFs. This finding also provides a consistent framework for understanding VLFs and LFEs, where the source parameters and waveform characteristics are well explained by SSARs 27 , 49 , 50 . We present the spatiotemporal rupture processes of shallow VLFs in the Nankai region. Using finite-fault inversion, we resolve the rupture processes of three shallow VLFs in the Nankai region and two nearby regular earthquakes. Our results imply that shallow VLFs exhibit spatially confined slip, limited rupture propagation, and spontaneous termination, in contrast to the sustained rupture propagation characteristic of nearby regular earthquakes. Comparisons with dynamic rupture models based on slip-weakening laws suggest that VLFs may represent SSARs that originate within the nucleation zone and spontaneously terminate there without further propagation. Furthermore, the frictional conditions of SSARs imply that VLFs may develop under relatively low weakening rates. Our results reveal that the Nankai shallow VLFs release stress through non-propagating rupture processes and provide potential evidence for SSARs occurring along the plate interface. Methods Finite fault inversions Here we adopt a wavelet-domain method first proposed by Ji, et al. 60 to perform finite fault inversions. In finite fault inversions, the fault plane can be considered as a series of subfaults, and the observed seismic waveform can be described as (e.g., Hartzell and Heaton 61 ; Ide 62 ; Ji, et al. 63 ): $$\:u\left(t\right)=\sum\:_{m=1}^{M}\sum\:_{n=1}^{N}{\mu\:}_{mn}{A}_{mn}{D}_{mn}{\dot{S}}_{mn}\left(t\right)*\delta\:\left(t-{\varDelta\:t}_{mn}\right)*\left[{G}_{mn}^{1}\left(t\right)\text{cos}{\theta\:}_{mn}+{G}_{mn}^{2}\left(t\right)\text{sin}{\theta\:}_{mn}\right]$$ 2 here \(\:M\) and \(\:N\) indicate the along-strike and down-dip directions, \(\:{\mu\:}_{mn}\) , \(\:{A}_{mn}\) , \(\:{D}_{mn}\) , and \(\:{\theta\:}_{mn}\) are the shear modulus, area, slip amount, and rake angle of the subfault \(\:mn\) , respectively. \(\:{G}_{mn}^{1}\left(t\right)\) and \(\:{G}_{mn}^{2}\left(t\right)\) are Green’s functions in the along-strike and down-dip directions, respectively. \(\:{\dot{S}}_{mn}\left(t\right)\) is the normalized slip-rate function, which can be simplified as a symmetric cosine function to simulate the slip history by one parameter, the slip duration \(\:{t}_{mn}^{s}\) (e.g., Ji, et al. 63 ). \(\:{\varDelta\:t}_{mn}\) is the time difference between the real arrival time of the rupture front and the assumed one on the subfault, which can be represented by a rupture velocity \(\:{V}_{mn}\) . Hence, the slip distribution of one earthquake can be described by \(\:{D}_{mn}\) , \(\:{\theta\:}_{mn}\) , \(\:{V}_{mn}\) , and \(\:{t}_{mn}^{s}\) . In our wavelet-domain method, the observed data \(\:{u}^{obs}\) and synthetic waveforms \(\:{u}^{syn}\) are transferred to wavelet coefficients \(\:{\alpha\:}_{j}^{obs}\) and \(\:{\alpha\:}_{j}^{syn}\) ( \(\:j\) means the \(\:j\) th scale in the wavelet domain), and the inversion is performed to find the best solution of the objective function \(\:\phi\:\left(\varvec{m}\right)\) : $$\:\phi\:\left(\varvec{m}\right)=\sum\:_{j={j}_{min}}^{{j}_{max}}{W}_{j}\times\:{‖{\alpha\:}_{j}^{obs}-{\alpha\:}_{j}^{syn}‖}_{L1+L2}+Constraints$$ 3 here \(\:{W}_{j}\) is the weight of the \(\:j\) th scale, \(\:{j}_{min}\) and \(\:{j}_{max}\) are the minimum and the maximum scales used. Following Ji, et al. 63 , the combination of the \(\:L1\) and \(\:L2\) norms is used to measure the misfit ( \(\:{\alpha\:}_{j}^{obs}-{\alpha\:}_{j}^{syn}\) ). We adopt the Simulated Annealing algorithm (Rothman 64 ; Sen and Stoffa 65 ; Tarantola 66 ) to search for the best solution. In this study, velocity waveforms recorded in NIED F-net are used to recover the rupture processes of the 3 VLFs and 2 EQs. We select stations with relatively high signal-to-noise ratios and epicentral distances of less than 500 km for the inversion analysis. Based on these criteria, 13 F-net stations are selected for VLFs1-3, 15 stations for EQ1, and 20 stations for EQ2. All the waveforms are filtered to 0.02–0.05 Hz and resampled with a 0.4-s sampling interval. In the wavelet domain, we choose wavelet scales 6–7 (corresponding to a frequency band of ~ 0.02–0.05 Hz) to perform inversions. We use the FK method 67 to calculate the theoretical Green’s functions based on the IASP91 model 68 . For each VLF, we consider the lower-dipping nodal plane determined by Takemura, et al. 12 as the causative fault plane (Supplementary Table 2). The fault plane is then divided into 25 x 25 subfaults with dimensions of 0.5 km along strike and 0.5 km along dip. During the inversion, for each subfault, the slip amplitude is allowed to vary from 0 to 3 cm, and the rake angle is from 120° to 180°. The slip duration is allowed to vary from 2.0 to 20.0 at a 2.0-s interval, and the rupture velocity of each subfault can vary from 0.1 to 3.0 km/s. As for EQs, the lower-dipping nodal plane from NIED catalog are considered (Supplementary Table 2). The fault planes are also divided into 25 x 25 subfaults with dimensions of 0.5 km x 0.5 km, the same as VLFs. For EQs, the slip amplitude is allowed to vary from 0 to 2 m, and the rake angle is from 60° to 120°. The slip duration is allowed to vary from 1.6 to 16.0 at a 1.6-s interval, and the rupture velocity of each subfault can vary from 0.1 to 3.0 km/s. We also adopt some model constraints to stabilize the inversions, including slip smoothing (minimizing the difference between the slip in adjacent subfaults), temporal smoothing (compressing the irresolvable roughness in rupture fronts), and seismic moment fitness (minimizing the difference between the total moment obtained and a priori one). We set the weight between the constraint and waveform fitness to 0.1 (e.g., Ji, et al. 60 ; Shao, et al. 69 ; Liu and Yao 70 ). Boundary integral equation method We adopt the boundary integral equation method (BIEM) to numerically simulate fault slip evolution, including nucleation, propagation and termination. The extended BIEM of Zhang & Chen 54 , 55 comes from the representation theorem, $$\:{u}_{n}\left(x,t\right)={\int\:}_{-\infty\:}^{\infty\:}d\tau\:{\iint\:}_{\varSigma\:}\left[{u}_{i}\left(\xi\:,\tau\:\right)\right]{c}_{ujpq}{\nu\:}_{j}\partial\:{G}_{np}\left(x,t-\tau\:;\xi\:,0\right)/\partial\:{\xi\:}_{q}d\varSigma\:$$ 4 . Substituting Eq. ( 4 ) into the isotropic and homogeneous constitutive relationship \(\:{\tau\:}_{ij}=\lambda\:{\delta\:}_{ij}{\partial\:}_{m}{u}_{m}+\mu\:\left({\partial\:}_{j}{u}_{i}+{\partial\:}_{i}{u}_{j}\right)\) and through regularization and discretization, we obtain the following discrete boundary integral equation for updating the shear stress field on a planar fault: $$\:{\varvec{\tau\:}}^{\varvec{i}\varvec{j}\varvec{k}}={\varvec{\tau\:}}_{0}+{\sum\:}_{\varvec{l},\varvec{m},\varvec{n}}{\varvec{C}}^{\varvec{i}\varvec{j}\varvec{k},\varvec{l}\varvec{m}\varvec{n}}{\varvec{V}}^{\varvec{l}\varvec{m}\varvec{n}}$$ 5 , where \(\:{\tau\:}^{ijk}\) , \(\:{\tau\:}_{0}\) and \(\:{V}^{lmn}\) are the stress, initial stress and slip velocity, respectively. The superscripts ‘ ijk ’ denote the spatial grids of the fault plane and the temporal steps of the field points, while ‘ lmn ’ denote those of the source points. \(\:{C}^{ijk,lmn}\) is the kernel that links slip velocity to stress. The detailed expression of \(\:{C}^{ijk,lmn}\) is presented by Zhang & Chen 55 . For our full-space model, we rewrite Eq. ( 5 ) as, $$\:{\varvec{\tau\:}}^{\varvec{i}\varvec{j}\varvec{k}}={\varvec{T}}_{\varvec{e}}+{\varvec{\sigma\:}}_{\varvec{r}}+{\sum\:}_{\varvec{l},\varvec{m},\varvec{n}}{\varvec{C}}^{\varvec{i}\varvec{j}\varvec{k},\varvec{l}\varvec{m}\varvec{n}}{\varvec{V}}^{\varvec{l}\varvec{m}\varvec{n}}$$ 6 , where \(\:{T}_{e}={\tau\:}_{0}-{\sigma\:}_{r}\) . In our simulations, we apply the slip-weakening friction law 51 – 53 . The slip-weakening friction law is as follows: $$\:\varvec{T}\left(\varvec{D}\right)=\left\{\begin{array}{c}\left(1-\frac{\varvec{D}}{{\varvec{D}}_{\mathbf{c}}}\right)({\varvec{\sigma\:}}_{\mathbf{u}}-{\varvec{\sigma\:}}_{\mathbf{r}})+{\varvec{\sigma\:}}_{\mathbf{r}},\:\:D<{\varvec{D}}_{\mathbf{c}}\:\\\:{\varvec{\sigma\:}}_{\mathbf{r}},\:\:D\ge\:{\varvec{D}}_{\mathbf{c}}\end{array}\right.$$ 7 , where \(\:{\sigma\:}_{\text{u}}\) is the peak shear strength, \(\:{\sigma\:}_{\text{r}}\) is the residual stress, \(\:D\) is slip and \(\:{D}_{\text{c}}\) is the critical slip distance. Once the rupture is triggered, the patch strength drops from peak strength \(\:{\sigma\:}_{\text{u}}\) to a small residual value, denoted \(\:{\sigma\:}_{\text{r}}\) . We denote the breakdown stress drop \(\:({\sigma\:}_{\text{u}}-{\sigma\:}_{\text{r}})\) as \(\:{T}_{\text{u}}\) and the dynamic stress drop \(\:({\sigma\:}_{0}-{\sigma\:}_{\text{r}})\) as \(\:{T}_{\text{e}}\) . Using Eq. ( 6 ) and the slip-weakening frictional law (7), the slip rate and stress at each simulation point at each time step can be obtained. In the simulations of SSAR and sub-Rayleigh ruptures, we maintain the wave velocities and nucleation zone radius of 5 km for both rupture types. We use a P-wave velocity of 6 km/s, an S-wave velocity of 3.464 km/s, and a density \(\:\rho\:\) of 2670 kg/m³. By varying the \(\:{D}_{\text{c}}\) , \(\:{T}_{\text{e}}\) , and \(\:{T}_{\text{u}}\) parameters, we simulate a SSAR and a sub-Rayleigh rupture. Our previous work 27 outlines the parameter domains corresponding to each rupture type. Specific parameter settings are detailed in Supplementary Table 1. The rupture is initiated by increasing the initial stress in the nucleation zone just above the yield strength, with its evolution governed by the slip-weakening law. Declarations Competing interests The authors declare no competing interests. Author contributions Xueting Wei, Xiaofei Chen and Wei Liu proposed the idea, designed the experiments, analysed the data, and wrote the paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grants 42304064, 42304061, 92155307 and 41874054), the National Key Research and Development Program of China (No. 2021YFC3000702), Guangdong Provincial Key Laboratory of Geophysical High-resolution Imaging Technology (2022B1212010002), Shenzhen Science and Technology Program (KQTD20170810111725321) and Center for Computational Science and Engineering at Southern University of Science and Technology. Data availability The F-net waveforms used in this work are available via the website of the National Research Institute for Earth Science and Disaster Resilience (NIED), Japan (at the following URL: http://www.hinet.bosai.go.jp/ ). The VLFs catalogue are attributed to Takemura, et al. 12 . 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Supplementary Files supportinfo.docx Supporting Information for "Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes" Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7074847","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":485425706,"identity":"ec8cd478-0b25-40a8-9dcc-3682f03e3558","order_by":0,"name":"Wei Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYBACPmYg8aGCgYGNHSYkAcQ8eLSwAbUwzjgDYQCBARFagJiZsw1EEq2FncdMmnHeNnmgC5k/8/z5I8c/u4Hxwds2BnlznA4DaincdtuwjZmBTZq3zcBY4s4BZsO5bQyGOxtwa7k9c9ttRpAWZt4Gg8QNEgkgvQwJBgfwaOGdc9u+DeIwg3qgFvbfhLU03E4EamGQ5mEzSDAA2sKMXwtb+c8Zx24ntzEztknObTM2nHEjsVlyzjkJww04tPDzH95s8KHmtu389ubDH978kZPnn5F88MObMht5XLYgAcYGZIYEQfWjYBSMglEwCnADAH0uS7bziAL1AAAAAElFTkSuQmCC","orcid":"","institution":"Southern University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Wei","middleName":"","lastName":"Liu","suffix":""},{"id":485425707,"identity":"987fc0f9-e690-4d5e-86cd-7a9d4230a732","order_by":1,"name":"Xueting Wei","email":"","orcid":"https://orcid.org/0000-0003-3207-9000","institution":"Wuhan University","correspondingAuthor":false,"prefix":"","firstName":"Xueting","middleName":"","lastName":"Wei","suffix":""},{"id":485425708,"identity":"9fe6ecc7-c241-446c-aed1-310314dbf608","order_by":2,"name":"Xiaofei Chen","email":"","orcid":"https://orcid.org/0000-0002-0950-8121","institution":"Southern University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xiaofei","middleName":"","lastName":"Chen","suffix":""}],"badges":[],"createdAt":"2025-07-08 12:25:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7074847/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7074847/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87025505,"identity":"f79b63b9-150a-4915-8ae2-5aa0c3cfe2d1","added_by":"auto","created_at":"2025-07-18 11:59:18","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":5121131,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSeismotectonic context. \u003c/strong\u003eF-net stations are shown as white filled triangles. Pale yellow dots denote offshore and onshore low-frequency earthquakes (LFEs)\u003csup\u003e13,47\u003c/sup\u003e. VLFs, positioned by Asano, et al.\u003csup\u003e39\u003c/sup\u003e, Nakamura and Sunagawa\u003csup\u003e48\u003c/sup\u003e, Takemura, et al.\u003csup\u003e40\u003c/sup\u003e, Takemura, et al.\u003csup\u003e41\u003c/sup\u003e, are shown as blue open circles. Gray filled squares represent the VLFs located in Takemura, et al.\u003csup\u003e12\u003c/sup\u003e. The three VLFs analyzed in this study are marked with yellow stars, and the two regular earthquakes are shown with pink stars, both accompanied by their focal mechanisms. Focal mechanisms for VLFs derive from Takemura et al.\u003csup\u003e12\u003c/sup\u003e, and those for regular earthquakes from the National Research Institute for Earth Science and Disaster Resilience (NIED). The bottom panel presents the cross-section along AA', with the green line indicating the slab interface.\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/c523f841a20da6816fe6ab4b.png"},{"id":87024555,"identity":"e1b270b1-460c-4af3-9219-cf8486bc44a8","added_by":"auto","created_at":"2025-07-18 11:51:18","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":2467963,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFinite-fault inversion of F-net waveforms. a\u003c/strong\u003e, Map of source regions and locations of VLF1, VLF2, VLF3, EQ1, and EQ2.\u003cstrong\u003e b\u003c/strong\u003e, Total moment-rate functions for VLF1, VLF2, VLF3, EQ1, and EQ2.\u003cstrong\u003e c\u003c/strong\u003e, Spatial distribution of fault slip for VLF1, VLF2, VLF3, EQ1, and EQ2. Blue circles indicate areas with a 4 km radius around the epicenters, while green circles represent areas with a 5 km radius. \u003cstrong\u003ed\u003c/strong\u003e, Comparison of observed (red lines) and synthetic (blue lines) three-component (E, N, Z) seismograms for VLF1 and EQ1 at five F-net stations, obtained through finite-fault inversion. Observed and synthetic velocity waveforms are bandpass-filtered between 0.02 and 0.05 Hz.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/1621637815ffc5c0370a2351.png"},{"id":87024559,"identity":"d36d4833-42e7-4721-9382-5e55890a0e97","added_by":"auto","created_at":"2025-07-18 11:51:18","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":3052788,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRupture process analysis. a\u003c/strong\u003e, Slip-rate evolution over time along the fault for VLF1, VLF2, VLF3, EQ1, and EQ2, derived from near-field waveform inversion. The orange circle outlines a circular area with a radius of 4.5 km centered on the epicenter. Time intervals are indicated in the top-left corner of each panel. \u003cstrong\u003eb\u003c/strong\u003e, Schematic diagram of the numerical model geometry and frictional properties. A circular nucleation patch (brown) is embedded in a slip-weakening region (yellow), surrounded by an unbreakable fault domain (grey). \u003cstrong\u003ec\u003c/strong\u003e, Slip-rate evolution of a simulated SSAR with a time interval of 1.3 s. \u003cstrong\u003ed\u003c/strong\u003e Slip-rate evolution of a sub-Rayleigh rupture with a time interval of 1.2 s.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/94ea33a1eaa875a2ba4e82af.png"},{"id":87024558,"identity":"da3f6c17-a8e9-4407-9785-78d0d0d2d109","added_by":"auto","created_at":"2025-07-18 11:51:18","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1038952,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eInterpretation of Nankai VLF rupture dynamics. a\u003c/strong\u003e, Illustration of a slip-weakening model for regular earthquake behavior. \u003cstrong\u003eb\u003c/strong\u003e, Proposed mechanism for stress release via slow, self-arresting rupture producing VLFs. \u003cstrong\u003ec\u003c/strong\u003e, Interpretive cross-section along line AA’. Red solid line outlines the plate boundary locations. Gray filled squares represent the VLFs identified by Takemura, et al.\u003csup\u003e12\u003c/sup\u003e. The colors in the figure represent different regions of slip-weakening rates inferred from the phase diagram of rupture dynamics. Low slip-weakening rates near the trench may facilitate VLFs/SSARs. As depth increases and distance from the trench grows, the weakening rate gradually rises, leading to the occurrence of regular earthquakes.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/d467674627d017ebbe8ca4ac.png"},{"id":95314186,"identity":"c12ead11-9c33-40a2-ada1-4d60087cbcb9","added_by":"auto","created_at":"2025-11-06 15:52:34","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":11784087,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/32b3db2b-a4fd-49c2-b471-c222bbed796a.pdf"},{"id":87024560,"identity":"fcaf1917-62ba-4533-ac04-3023e331f541","added_by":"auto","created_at":"2025-07-18 11:51:18","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":5071678,"visible":true,"origin":"","legend":"Supporting Information for \"Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes\"","description":"","filename":"supportinfo.docx","url":"https://assets-eu.researchsquare.com/files/rs-7074847/v1/6d1b5d4b9c5f97ebc0f2b183.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe Nankai subduction zone, a locus of recurring megathrust earthquakes\u003csup\u003e\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e, also hosts a diversity of slow earthquakes\u003csup\u003e\u003cspan additionalcitationids=\"CR5 CR6 CR7 CR8 CR9 CR10\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e. These slow events, observed in the shallow transitional zone between locked and stably sliding regions\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e,\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e, span a broad spectrum from seconds-scale elastodynamic rupture to multi-year aseismic slip. Unlike regular earthquakes, slow earthquakes, while capable of radiating detectable seismic waves, typically exhibit prolonged source durations, reduced seismic radiation, and orders-of-magnitude slower slip rates\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e. Compelling evidence links these slow events to shear slip mechanisms similar to regular earthquakes, with emerging implications for their potential precursory roles in megathrust cycles\u003csup\u003e\u003cspan additionalcitationids=\"CR17\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. However, a critical unresolved question remains: Are slow earthquakes a scaled-down version of regular seismic processes with reduced stress release, or do they embody a fundamentally distinct rupture process with unique characteristics? While laboratory experiments\u003csup\u003e\u003cspan additionalcitationids=\"CR20 CR21\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e and numerical simulations\u003csup\u003e\u003cspan additionalcitationids=\"CR24 CR25 CR26 CR27\" citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e have captured essential features of slow events, direct observations of the rupture processes of slow earthquakes remain limited, posing challenges to achieving a comprehensive understanding of their controlling mechanisms and potential interactions with megathrust seismicity.\u003c/p\u003e\u003cp\u003eAmong the diverse family of slow earthquakes, VLFs, characterized by their dominant frequencies falling between 0.01 and 0.1 Hz\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e,\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e, stand out as promising candidates for elucidating the source processes of slow earthquakes. VLFs exhibit their slow source processes through rupture radii of 5\u0026ndash;10 km, source durations of 10\u0026ndash;100 seconds, and stress drops of ~\u0026thinsp;kPa, which are two to three orders of magnitude lower than those of regular earthquakes with comparable seismic moments\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e,\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e. However, unlike other slow earthquakes whose rupture scales are either too small for seismic resolution or too prolonged for waveform-based analysis, VLFs generate discernible seismic waves with resolvable rupture dimensions, making them particularly well-suited for investigating rupture processes. Given the frequent co-occurrence of VLFs and other slow earthquakes\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e,\u003cspan additionalcitationids=\"CR32 CR33 CR34 CR35 CR36 CR37\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e, a shared source mechanism is likely, thus offering critical insight into the physics of slow earthquakes. But, as of now, the rupture process and properties of VLFs remain unresolved.\u003c/p\u003e\u003cp\u003eHere, we investigate three VLFs along with two regular earthquakes occurring in the Nankai region using broadband seismic records from the NIED F-net network. We reconstruct the spatiotemporal rupture processes of these events by performing finite fault inversions. Our results demonstrate that the rupture region of each VLF is limited to an approximate 5 km radius around the hypocenter. Contrary to the two regular earthquakes, which exhibit clear rupture propagation, these VLFs indicate spatially confined rupture development. Instead, slip initiates and slowly self-arrests near the centroid, with peak slip rates consistently localized there throughout the rupture process. Additionally, source dynamics simulations based on the boundary integral equation method suggest that a low slip-weakening rate leads to a similar rupture pattern of slow self-arresting rupture, characterized by initiation and automatic termination within the nucleation patch. Our study, to the best of our knowledge, presents the first direct inversion of slow earthquake rupture processes, suggesting that the slow self-arresting ruptures, confined to the nucleation zone without lateral propagation may govern the source mechanisms of VLFs.\u003c/p\u003e"},{"header":"Results and discussion","content":"\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eFinite-Fault Inversion of VLFs and regular earthquakes\u003c/span\u003e\u003c/p\u003e\u003cp\u003eShallow VLFs are frequently observed offshore along the Nankai Trough, often occurring in conjunction with other slow earthquakes at the shallow plate interface\u003csup\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e,\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e. These VLFs exhibit a spatially discrete distribution within the Nankai subduction zone, clustering in distinct patches\u003csup\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e,\u003cspan additionalcitationids=\"CR40 CR41\" citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e, which suggests lateral variations in lithology and frictional properties\u003csup\u003e\u003cspan additionalcitationids=\"CR44 CR45\" citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e. Although previous study\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e provides constraints on the source characteristics of shallow VLFs, including source durations of 10\u0026ndash;20 seconds, rupture radii of 5\u0026ndash;10 km, and moment magnitudes (Mw) of 3\u0026ndash;4, the spatiotemporal evolution of VLF ruptures remains poorly resolved due to their low dominant frequencies and limited signal-to-noise ratios. To bridge this gap, we perform rupture process inversions for multiple spatially distributed VLFs in the Nankai region. Specifically, our analysis focuses on three VLFs selected from the catalog of Takemura et al.\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e, based on variance reductions (VR) exceeding 70%. These events include VLF1 (Mw 3.8, depth 8.02 km), which occurs on 13 March 2018 12:28:09 (134.7\u0026deg;E, 32.7\u0026deg;N); VLF2 (Mw 4.14, depth 6.74 km) on 14 May 2018 17:02:27 (135.3\u0026deg;E, 32.7\u0026deg;N); and VLF3 (Mw 3.8, depth 5.97 km) on 28 April 2009 08:12:25 (137.3\u0026deg;E, 33.4\u0026deg;N). The estimated source durations for these events are 16 s, 27 s, and 17 s, respectively\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e, which are sufficient to resolve the detailed rupture processes through inversion analysis. Despite their spatial separation, the three VLFs exhibit a consistent focal mechanism, characterized by low-angle thrust faulting with strike angles parallel to the trench axis, suggesting slip on the plate boundary interface\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e. Critically, to establish a comparative framework for rupture processes and enhance the robustness of our results, we extend our analysis to regular earthquakes with rupture dimensions comparable to VLFs near their clusters. Based on the effective rupture radius of approximately 5 km for shallow VLFs\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e, which corresponds to a magnitude range of 5\u0026ndash;6 for regular earthquakes, we investigate two Mw 5\u0026ndash;6 earthquakes that exhibit source mechanisms consistent with those of the VLFs. According to the U.S. Geological Survey (USGS) earthquake catalog, the first event (EQ1) occurs in 1 April 2016 02:39:08 at (136.4\u0026deg;E, 33.4\u0026deg;N) with a magnitude of Mw 5.9, while the second event (EQ2) takes place on 3 December 2021 00:28:28, at (135.1\u0026deg;E, 33.8\u0026deg;N) with a magnitude of Mw 5.2.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFollowing the finite-fault inversion method detailed in Methods, we analyze VLF1, VLF2 and VLF3 using broadband velocity records from 13 F-net stations (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), applying 0.02\u0026ndash;0.05 Hz bandpass filtering to optimize signal-to-noise ratios. To facilitate comparison with the rupture characteristics of regular earthquakes, we perform inversions for both EQ1 and EQ2. For consistency with the frequency band used in the VLF inversions, the waveforms of EQ1 and EQ2 are also filtered to 0.02\u0026ndash;0.05 Hz. The inversion results (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) suggest that the source durations of VLFs1-3 are significantly longer than those of EQ1-2. The evolution of moment rate is similar initially for all three VLF events, with VLF2 and VLF3 showing a subsequent decrease followed by a renewed increase (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb). The peak values of the moment rate functions for VLF1-3 are 3\u0026ndash;4 orders of magnitude smaller than those of EQ1-2, consistent with the characteristics of VLFs, indicating that VLFs rupture more slowly and release less stress. The rupture radii of VLFs 1\u0026ndash;3 are approximately 3\u0026ndash;5 km, consistent with previous estimates of the effective rupture area for VLFs\u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec). Peak slip for each VLF event is concentrated near its centroid, and VLF3 attains a maximum slip of 0.95 cm.\u003c/p\u003e\u003cp\u003eThe synthetic seismograms demonstrate a reasonable agreement with the observed waveforms across multiple stations for all three VLFs (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ed). Waveform discrepancies likely arise from deviations between the reference 1-D layered velocity model and the actual velocity structure, especially in the horizontal components. Furthermore, all three VLFs occur within the sedimentary layers overlain by seawater, adding to the complexity of the velocity medium. Detailed distributions of slip, slip duration, rupture velocity for VLFs1-3 and EQ1-2 are presented in Supplementary Figs.\u0026nbsp;1\u0026ndash;2, while Supplementary Figs.\u0026nbsp;3\u0026ndash;7 provides detailed waveform comparisons of VLF1\u0026ndash;3 and EQ1\u0026ndash;2 across all used stations.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eShallow VLFs rupture characteristics\u003c/span\u003e\u003c/p\u003e\u003cp\u003eA central question remains regarding the spatiotemporal rupture processes of VLFs: do they represent low-stress-drop analogs of regular earthquakes, or do they exhibit fundamentally distinct rupture characteristics from regular seismic events? To address this, we further analyze finite fault inversion results to reconstruct the spatiotemporal rupture processes of VLF1, VLF2, and VLF3, as well as EQ1 and EQ2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). Supplementary Figs.\u0026nbsp;8\u0026ndash;12 provide detailed rupture processes of VLFs1-3 and EQ1-2. The rupture processes of VLFs1-3 and EQ1-2 exhibit clear and distinguishable differences. For EQ1 and EQ2, rupture initiates at the hypocenter, followed by outward propagation. EQ1 exhibits an average rupture velocity of 1.40 km/s predominantly toward the northwest, while EQ2 propagates at 1.60 km/s with a primarily eastward direction. Both events exhibit clear rupture fronts, with peak slip rates concentrated at their leading edges, as the slip region dynamically propagates with the front. Simultaneously, slip rates gradually decrease near the model centroid as the rupture progresses. Ultimately, slip terminates near the leading edge of the rupture front. In contrast, all three VLF events originate within the hypocentral region, with slip predominantly concentrated near the center of the fault, and without rupture propagation. At different moments during the rupture process, the regions of slip on the fault remain nearly unchanged. Throughout the rupture process, the maximum slip rate for all three VLFs remains localized at the center of the modeled fault area, persisting there until it decays to zero. Moreover, the hypocenter appears to be the last region to arrests, a behavior that contrasts with the characteristics of EQ1 and EQ2, where slip arrests near the rupture front. These distinct features indicate the fundamentally different rupture dynamics of VLFs when compared to regular earthquakes. The slip rates during VLF1 appears to be more concentrated in the central region, whereas the rupture processes of VLF2 and VLF3, although both originating in the hypocentral area, exhibit a slightly more distributed slip rate. This may be attributed to limitations in the resolution and accuracy of the inversions. Alternatively, this pattern could arise from the presence of multiple closely spaced nucleation zones, potentially leading to the initiation of several sequentially small events.\u003c/p\u003e\u003cp\u003eIn addition, the three VLFs exhibit notably longer source durations compared to EQ1 and EQ2. VLF1 reaches its peak slip rate at ~\u0026thinsp;2.8 s, followed by a gradual decay over ~\u0026thinsp;24.8 s until rupture termination. Similarly, VLF2 and VLF3 attain peak slip rates at ~\u0026thinsp;2.4 s, with decay phases lasting\u0026thinsp;~\u0026thinsp;43.2 s and ~\u0026thinsp;28.8 s, respectively, before slip arrest. In contrast, EQ1 reaches its peak slip rate at ~\u0026thinsp;2.0 s and terminates within ~\u0026thinsp;1.6 s, while EQ2 peaks at ~\u0026thinsp;1.2 s and terminates after ~\u0026thinsp;1.0 s. Although the slip rates of the three VLFs increase rapidly, their slip rate decay phases are significantly prolonged, indicating that the rupture termination mechanisms of VLFs may differ from those of regular earthquakes. The consistent rupture characteristics observed in the three VLFs across distinct locations and different occurrence times suggest a shared rupture process underlying VLFs, thereby enhancing the credibility of the inversion results. Moreover, as identical frequency band filtering and inversion procedures are applied during the inversion of both EQ1-2 and VLFs1-3, the observed differences in rupture characteristics appear unlikely to result from inversion artifacts and may instead reflect intrinsic distinctions in their source dynamics. These distinctive spatiotemporal rupture characteristics imply that VLFs may arise from different rupture processes and underlying mechanisms compared to regular earthquakes.\u003c/p\u003e\u003cp\u003eThe distinct rupture processes identified here, characterized by the propagation-limited nature of VLFs within confined zones and prolonged source durations, necessitate physical mechanisms beyond conventional rupture dynamic models. To reconcile these observations with fault mechanics, we explore source dynamics frameworks capable of generating such spatially confined ruptures. Notably, previous source dynamics simulations based on the slip-weakening friction law\u003csup\u003e\u003cspan additionalcitationids=\"CR52\" citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/sup\u003e indicate the presence of a distinct type of rupture when the dimensionless critical slip distance is large (suggesting a low weakening rate). In these cases, slip is initiated within the nucleation zone (the localized patch where rupture begins in the simulations) but fails to extend beyond this region. Unlike typical rupture propagation, the slip does not spread outward; instead, it gradually decelerates and ultimately self-arrests within the nucleation zone, forming a slow self-arresting rupture (SSAR)\u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e,\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e. Additionally, SSARs exhibit longer durations and stress drops that are two orders of magnitude smaller than those of sub-Rayleigh ruptures/regular earthquakes of the same magnitude\u003csup\u003e\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e. Thus, to investigate this further, we employ the slip-weakening frictional law\u003csup\u003e\u003cspan additionalcitationids=\"CR52\" citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/sup\u003e and the boundary integral equation method (BIEM)\u003csup\u003e\u003cspan additionalcitationids=\"CR55\" citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e\u003c/sup\u003e to generate a SSAR and a sub-Rayleigh rupture for comparison. We consider a dip-slip fault in elastic media, featuring a slip-weakening homogeneous patch enclosed by an unbreakable boundary (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb). The simulation details and parameters are provided in the Methods and Supplementary Table\u0026nbsp;1. The SSAR attains its peak slip rate approximately 3.4 s after initiation, followed by a decay until slip stop around 9.9 s (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec). Despite using a slip-weakening frictional model, where frictional stress decreases linearly with slip, the rupture does not propagate outward but instead self-arrests within the nucleation patch, without external influence. Furthermore, the maximum slip rate is consistently concentrated at the center of the simulated domain, which also serves as the final point of arrest for the rupture. These rupture evolution characteristics, including slip rates around \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{10}^{-5}\\)\u003c/span\u003e\u003c/span\u003e m/s, are highly similar to those observed in VLF1, VLF2, and VLF3. In contrast, sub-Rayleigh ruptures or regular earthquakes, once triggered within the nucleation zone, rapidly expand beyond its boundaries and continue to propagate outward (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ed). The peak slip rate consistently located at the rupture front throughout the rupture process. These rupture characteristics are consistent with those observed in EQ1-2. Consequently, VLFs and SSARs exhibit fundamentally similar rupture characteristics. Integrated with prior studies reporting comparable source parameters and waveforms\u003csup\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e, these observations indicate that SSARs may share essential source dynamics with VLFs.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eDiscussions and conclusions\u003c/span\u003e\u003c/p\u003e\u003cp\u003eFinite-fault inversions reveal spatially confined ruptures in VLFs with source durations substantially exceeding those of regular earthquakes, providing new insights into slow earthquake processes. Two possible mechanisms may explain these observations. First, VLFs could be regular earthquakes but encounter immediate strength barriers arresting propagation. While this explains the absence of rupture expansion, it conflicts with source parameters scaling of VLFs. Regular earthquakes with comparable rupture dimensions exhibit source durations lasting only a few seconds, whereas VLFs have source durations extending to 10\u0026ndash;50 s. Moreover, regular earthquakes exhibit 2\u0026ndash;3 orders of magnitude higher average stress drops, slip, and slip rates than VLFs. Additionally, such barriers would need to fortuitously surround every VLF patch. Alternatively, VLFs may represent SSARs that spontaneously arrest and remain confined to nucleation regions. Assuming the second mechanism to be more accurate, it implies that the stress and frictional conditions governing VLFs and SSARs may share similar characteristics.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003ePrevious dynamic simulations based on the slip-weakening law indicate that, variations in frictional properties and stress conditions give rise to distinct rupture types, including supershear rupture, sub-Rayleigh rupture, self-arresting rupture, and SSAR\u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e,\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e (Supplementary Fig.\u0026nbsp;13). Supershear ruptures and sub-Rayleigh ruptures can propagate continuously until encountering a barrier that forces their slip to stop, whereas self-arresting ruptures and SSARs spontaneously terminate even in the absence of barriers. SSARs are particularly unique in that they initiate and terminate entirely within the nucleation zone without propagating. They are further characterized by prolonged source durations, relatively small stress drops, reduced slip, and lower average slip rates. In the framework of the slip-weakening law, the difference between the peak shear strength and residual stress is referred to as the breakdown stress drop \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e, while the slip required for the reduction from peak strength to residual stress is characterized as the critical slip distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e​ (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). The ratio of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e​ (absolute value) represents the weakening rate, which governs the frictional weakening process during fault slip:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\varvec{W}=\\left|{\\varvec{T}}_{\\mathbf{u}}/{\\varvec{D}}_{\\mathbf{c}}\\right|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e.\u003c/p\u003e\u003cp\u003eAt higher weakening rates \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:W\\)\u003c/span\u003e\u003c/span\u003e, frictional strength decreases rapidly for a given amount of slip, promoting further slip and allowing the rupture to propagate outward until it encounters stress barriers that impede its expansion (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). Rupture behavior under varying initial stress conditions may manifest as self-arresting rupture, sub-Rayleigh rupture, or supershear rupture. However, as the weakening rate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:W\\)\u003c/span\u003e\u003c/span\u003e decreases, the frictional strength decreases slowly, resulting in slow slip with insufficient strength drop to drive rupture propagation (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb). When the weakening rate falls below a critical threshold, rupture transitions to SSAR under nearly all initial stress conditions (Supplementary Fig.\u0026nbsp;13). Assuming a nucleation patch radius of 2 km, an S-wave velocity of 3.464 km/s, a P-wave velocity of 6 km/s, and a density of 2670 kg/m\u0026sup3;, the critical weakening rate for the occurrence of SSARs is estimated to be 18.2 MPa/m. This indicates that SSARs are possible only when the weakening rate falls below this critical threshold.\u003c/p\u003e\u003cp\u003eBuilding on this, we propose a conceptual framework to elucidate the occurrence and slip dynamics of VLFs (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ec). Along the subduction plate interface near the trench axis, a localized region with low weakening rates is likely to exist within a horizontal range of approximately 20 km and a depth of about 5 km. As depth increases, the weakening rate gradually grows larger. The critical weakening rate for the occurrence of SSARs is situated along the plate interface, resulting in SSARs being confined to regions on the plate interface, while only regular earthquakes are possible in areas below the interface. Furthermore, the presence of significant anisotropy and heterogeneity within the sedimentary layers of subduction zones\u003csup\u003e\u003cspan additionalcitationids=\"CR44 CR45\" citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e likely contributes to the gradual increase in weakening rate in the horizontal direction, moving away from the trench axis. This spatial variation in weakening rate results in the preferential occurrence of VLFs within regions closer to the trench axis. Assuming that the initial shear stress on faults hosting SSARs and sub-Rayleigh ruptures is identical at 1 MPa, the significantly lower shear strength of SSARs (1.02 MPa) compared to regular earthquakes (1.62 MPa) suggests that SSARs can be triggered by stress perturbations on the order of ~\u0026thinsp;kPa. Such perturbations may arise from slow plate motion, tidal forces, or seismic waves. In contrast, the initiation of regular earthquakes requires substantially higher stress increments, which may explain the frequent occurrence of VLFs in this region compared to the relative scarcity of regular earthquakes exceeding Mw 5.\u003c/p\u003e\u003cp\u003ePreviously, Fukuyama and Madariaga\u003csup\u003e\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/sup\u003e demonstrated that large critical slip distances could lead to immediate rupture termination outside of an asperity. Also, numerical simulations\u003csup\u003e\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e\u003c/sup\u003e reveal that under certain slip-weakening parameters, ruptures could spontaneously arrest on the fault, failing to reach dynamic instability. Furthermore, Campillo and Ionescu\u003csup\u003e\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e\u003c/sup\u003e, in their investigation of dynamic instability initiation on a slip-weakening fault, found that ruptures remain stable when the slip-weakening rate falls below a certain threshold. While these analyses and simulations suggest the existence of self-arresting earthquakes, they may have often been overlooked or dismissed, and excluded from the conventional definition of 'earthquakes'. Our current study bridges this gap between theoretical predictions and observational evidence by presenting the observationally evidence of such ruptures through finite-fault inversions of VLFs. Notably, previous studies have shown that stick-slip models under ~\u0026thinsp;kPa effective normal stress can also generate VLFs, but with rupture propagation similar to that of regular earthquakes\u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e. In contrast, our inversion results suggest that the SSAR, which exhibits fundamentally different rupture dynamics from regular earthquakes, may serve as a plausible source model for shallow VLFs. This finding also provides a consistent framework for understanding VLFs and LFEs, where the source parameters and waveform characteristics are well explained by SSARs\u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e,\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e,\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eWe present the spatiotemporal rupture processes of shallow VLFs in the Nankai region. Using finite-fault inversion, we resolve the rupture processes of three shallow VLFs in the Nankai region and two nearby regular earthquakes. Our results imply that shallow VLFs exhibit spatially confined slip, limited rupture propagation, and spontaneous termination, in contrast to the sustained rupture propagation characteristic of nearby regular earthquakes. Comparisons with dynamic rupture models based on slip-weakening laws suggest that VLFs may represent SSARs that originate within the nucleation zone and spontaneously terminate there without further propagation. Furthermore, the frictional conditions of SSARs imply that VLFs may develop under relatively low weakening rates. Our results reveal that the Nankai shallow VLFs release stress through non-propagating rupture processes and provide potential evidence for SSARs occurring along the plate interface.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eFinite fault inversions\u003c/span\u003e\u003c/p\u003e\u003cp\u003eHere we adopt a wavelet-domain method first proposed by Ji, et al.\u003csup\u003e\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e\u003c/sup\u003e to perform finite fault inversions. In finite fault inversions, the fault plane can be considered as a series of subfaults, and the observed seismic waveform can be described as (e.g., Hartzell and Heaton\u003csup\u003e\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e\u003c/sup\u003e; Ide\u003csup\u003e\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e\u003c/sup\u003e; Ji, et al.\u003csup\u003e\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e):\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:u\\left(t\\right)=\\sum\\:_{m=1}^{M}\\sum\\:_{n=1}^{N}{\\mu\\:}_{mn}{A}_{mn}{D}_{mn}{\\dot{S}}_{mn}\\left(t\\right)*\\delta\\:\\left(t-{\\varDelta\\:t}_{mn}\\right)*\\left[{G}_{mn}^{1}\\left(t\\right)\\text{cos}{\\theta\\:}_{mn}+{G}_{mn}^{2}\\left(t\\right)\\text{sin}{\\theta\\:}_{mn}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ehere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:M\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:N\\)\u003c/span\u003e\u003c/span\u003e indicate the along-strike and down-dip directions, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mu\\:}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{mn}\\)\u003c/span\u003e\u003c/span\u003e are the shear modulus, area, slip amount, and rake angle of the subfault \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:mn\\)\u003c/span\u003e\u003c/span\u003e, respectively. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{G}_{mn}^{1}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{G}_{mn}^{2}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e are Green\u0026rsquo;s functions in the along-strike and down-dip directions, respectively. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\dot{S}}_{mn}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e is the normalized slip-rate function, which can be simplified as a symmetric cosine function to simulate the slip history by one parameter, the slip duration \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{t}_{mn}^{s}\\)\u003c/span\u003e\u003c/span\u003e (e.g., Ji, et al.\u003csup\u003e\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e). \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varDelta\\:t}_{mn}\\)\u003c/span\u003e\u003c/span\u003e is the time difference between the real arrival time of the rupture front and the assumed one on the subfault, which can be represented by a rupture velocity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}_{mn}\\)\u003c/span\u003e\u003c/span\u003e. Hence, the slip distribution of one earthquake can be described by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}_{mn}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{t}_{mn}^{s}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eIn our wavelet-domain method, the observed data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}^{obs}\\)\u003c/span\u003e\u003c/span\u003e and synthetic waveforms \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}^{syn}\\)\u003c/span\u003e\u003c/span\u003e are transferred to wavelet coefficients \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{j}^{obs}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{j}^{syn}\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:j\\)\u003c/span\u003e\u003c/span\u003e means the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:j\\)\u003c/span\u003e\u003c/span\u003eth scale in the wavelet domain), and the inversion is performed to find the best solution of the objective function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\phi\\:\\left(\\varvec{m}\\right)\\)\u003c/span\u003e\u003c/span\u003e:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\phi\\:\\left(\\varvec{m}\\right)=\\sum\\:_{j={j}_{min}}^{{j}_{max}}{W}_{j}\\times\\:{‖{\\alpha\\:}_{j}^{obs}-{\\alpha\\:}_{j}^{syn}‖}_{L1+L2}+Constraints$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ehere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{j}\\)\u003c/span\u003e\u003c/span\u003e is the weight of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:j\\)\u003c/span\u003e\u003c/span\u003eth scale, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{j}_{min}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{j}_{max}\\)\u003c/span\u003e\u003c/span\u003e are the minimum and the maximum scales used. Following Ji, et al.\u003csup\u003e\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e, the combination of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:L1\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:L2\\)\u003c/span\u003e\u003c/span\u003e norms is used to measure the misfit (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{j}^{obs}-{\\alpha\\:}_{j}^{syn}\\)\u003c/span\u003e\u003c/span\u003e). We adopt the Simulated Annealing algorithm (Rothman\u003csup\u003e\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e\u003c/sup\u003e; Sen and Stoffa\u003csup\u003e\u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e\u003c/sup\u003e; Tarantola\u003csup\u003e\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e\u003c/sup\u003e) to search for the best solution.\u003c/p\u003e\u003cp\u003eIn this study, velocity waveforms recorded in NIED F-net are used to recover the rupture processes of the 3 VLFs and 2 EQs. We select stations with relatively high signal-to-noise ratios and epicentral distances of less than 500 km for the inversion analysis. Based on these criteria, 13 F-net stations are selected for VLFs1-3, 15 stations for EQ1, and 20 stations for EQ2. All the waveforms are filtered to 0.02\u0026ndash;0.05 Hz and resampled with a 0.4-s sampling interval. In the wavelet domain, we choose wavelet scales 6\u0026ndash;7 (corresponding to a frequency band of ~\u0026thinsp;0.02\u0026ndash;0.05 Hz) to perform inversions. We use the FK method\u003csup\u003e\u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e\u003c/sup\u003e to calculate the theoretical Green\u0026rsquo;s functions based on the IASP91 model\u003csup\u003e\u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e68\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eFor each VLF, we consider the lower-dipping nodal plane determined by Takemura, et al.\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e as the causative fault plane (Supplementary Table\u0026nbsp;2). The fault plane is then divided into 25 x 25 subfaults with dimensions of 0.5 km along strike and 0.5 km along dip. During the inversion, for each subfault, the slip amplitude is allowed to vary from 0 to 3 cm, and the rake angle is from 120\u0026deg; to 180\u0026deg;. The slip duration is allowed to vary from 2.0 to 20.0 at a 2.0-s interval, and the rupture velocity of each subfault can vary from 0.1 to 3.0 km/s.\u003c/p\u003e\u003cp\u003eAs for EQs, the lower-dipping nodal plane from NIED catalog are considered (Supplementary Table\u0026nbsp;2). The fault planes are also divided into 25 x 25 subfaults with dimensions of 0.5 km x 0.5 km, the same as VLFs. For EQs, the slip amplitude is allowed to vary from 0 to 2 m, and the rake angle is from 60\u0026deg; to 120\u0026deg;. The slip duration is allowed to vary from 1.6 to 16.0 at a 1.6-s interval, and the rupture velocity of each subfault can vary from 0.1 to 3.0 km/s.\u003c/p\u003e\u003cp\u003eWe also adopt some model constraints to stabilize the inversions, including slip smoothing (minimizing the difference between the slip in adjacent subfaults), temporal smoothing (compressing the irresolvable roughness in rupture fronts), and seismic moment fitness (minimizing the difference between the total moment obtained and a priori one). We set the weight between the constraint and waveform fitness to 0.1 (e.g., Ji, et al.\u003csup\u003e\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e\u003c/sup\u003e; Shao, et al.\u003csup\u003e\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e\u003c/sup\u003e; Liu and Yao\u003csup\u003e\u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e\u003c/sup\u003e).\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eBoundary integral equation method\u003c/span\u003e\u003c/p\u003e\u003cp\u003eWe adopt the boundary integral equation method (BIEM) to numerically simulate fault slip evolution, including nucleation, propagation and termination.\u003c/p\u003e\u003cp\u003eThe extended BIEM of Zhang \u0026amp; Chen\u003csup\u003e\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e,\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/sup\u003e comes from the representation theorem,\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{u}_{n}\\left(x,t\\right)={\\int\\:}_{-\\infty\\:}^{\\infty\\:}d\\tau\\:{\\iint\\:}_{\\varSigma\\:}\\left[{u}_{i}\\left(\\xi\\:,\\tau\\:\\right)\\right]{c}_{ujpq}{\\nu\\:}_{j}\\partial\\:{G}_{np}\\left(x,t-\\tau\\:;\\xi\\:,0\\right)/\\partial\\:{\\xi\\:}_{q}d\\varSigma\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e.\u003c/p\u003e\u003cp\u003eSubstituting Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) into the isotropic and homogeneous constitutive relationship \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\tau\\:}_{ij}=\\lambda\\:{\\delta\\:}_{ij}{\\partial\\:}_{m}{u}_{m}+\\mu\\:\\left({\\partial\\:}_{j}{u}_{i}+{\\partial\\:}_{i}{u}_{j}\\right)\\)\u003c/span\u003e\u003c/span\u003e and through regularization and discretization, we obtain the following discrete boundary integral equation for updating the shear stress field on a planar fault:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{\\varvec{\\tau\\:}}^{\\varvec{i}\\varvec{j}\\varvec{k}}={\\varvec{\\tau\\:}}_{0}+{\\sum\\:}_{\\varvec{l},\\varvec{m},\\varvec{n}}{\\varvec{C}}^{\\varvec{i}\\varvec{j}\\varvec{k},\\varvec{l}\\varvec{m}\\varvec{n}}{\\varvec{V}}^{\\varvec{l}\\varvec{m}\\varvec{n}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\tau\\:}^{ijk}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\tau\\:}_{0}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}^{lmn}\\)\u003c/span\u003e\u003c/span\u003e are the stress, initial stress and slip velocity, respectively. The superscripts \u0026lsquo;\u003cem\u003eijk\u003c/em\u003e\u0026rsquo; denote the spatial grids of the fault plane and the temporal steps of the field points, while \u0026lsquo;\u003cem\u003elmn\u003c/em\u003e\u0026rsquo; denote those of the source points. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}^{ijk,lmn}\\)\u003c/span\u003e\u003c/span\u003e is the kernel that links slip velocity to stress. The detailed expression of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}^{ijk,lmn}\\)\u003c/span\u003e\u003c/span\u003e is presented by Zhang \u0026amp; Chen\u003csup\u003e\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/sup\u003e. For our full-space model, we rewrite Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) as,\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{\\varvec{\\tau\\:}}^{\\varvec{i}\\varvec{j}\\varvec{k}}={\\varvec{T}}_{\\varvec{e}}+{\\varvec{\\sigma\\:}}_{\\varvec{r}}+{\\sum\\:}_{\\varvec{l},\\varvec{m},\\varvec{n}}{\\varvec{C}}^{\\varvec{i}\\varvec{j}\\varvec{k},\\varvec{l}\\varvec{m}\\varvec{n}}{\\varvec{V}}^{\\varvec{l}\\varvec{m}\\varvec{n}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{e}={\\tau\\:}_{0}-{\\sigma\\:}_{r}\\)\u003c/span\u003e\u003c/span\u003e. In our simulations, we apply the slip-weakening friction law\u003csup\u003e\u003cspan additionalcitationids=\"CR52\" citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/sup\u003e. The slip-weakening friction law is as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\varvec{T}\\left(\\varvec{D}\\right)=\\left\\{\\begin{array}{c}\\left(1-\\frac{\\varvec{D}}{{\\varvec{D}}_{\\mathbf{c}}}\\right)({\\varvec{\\sigma\\:}}_{\\mathbf{u}}-{\\varvec{\\sigma\\:}}_{\\mathbf{r}})+{\\varvec{\\sigma\\:}}_{\\mathbf{r}},\\:\\:D\u0026lt;{\\varvec{D}}_{\\mathbf{c}}\\:\\\\\\:{\\varvec{\\sigma\\:}}_{\\mathbf{r}},\\:\\:D\\ge\\:{\\varvec{D}}_{\\mathbf{c}}\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e is the peak shear strength, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{\\text{r}}\\)\u003c/span\u003e\u003c/span\u003e is the residual stress, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:D\\)\u003c/span\u003e\u003c/span\u003e is slip and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e is the critical slip distance. Once the rupture is triggered, the patch strength drops from peak strength \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e to a small residual value, denoted \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{\\text{r}}\\)\u003c/span\u003e\u003c/span\u003e. We denote the breakdown stress drop \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({\\sigma\\:}_{\\text{u}}-{\\sigma\\:}_{\\text{r}})\\)\u003c/span\u003e\u003c/span\u003e as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e and the dynamic stress drop \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({\\sigma\\:}_{0}-{\\sigma\\:}_{\\text{r}})\\)\u003c/span\u003e\u003c/span\u003e as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{e}}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eUsing Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and the slip-weakening frictional law (7), the slip rate and stress at each simulation point at each time step can be obtained. In the simulations of SSAR and sub-Rayleigh ruptures, we maintain the wave velocities and nucleation zone radius of 5 km for both rupture types. We use a P-wave velocity of 6 km/s, an S-wave velocity of 3.464 km/s, and a density \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\rho\\:\\)\u003c/span\u003e\u003c/span\u003e of 2670 kg/m\u0026sup3;. By varying the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{e}}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{u}}\\)\u003c/span\u003e\u003c/span\u003e parameters, we simulate a SSAR and a sub-Rayleigh rupture. Our previous work\u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e outlines the parameter domains corresponding to each rupture type. Specific parameter settings are detailed in Supplementary Table\u0026nbsp;1. The rupture is initiated by increasing the initial stress in the nucleation zone just above the yield strength, with its evolution governed by the slip-weakening law.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eCompeting interests\u003c/h2\u003e\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAuthor contributions\u003c/h2\u003e\u003cp\u003eXueting Wei, Xiaofei Chen and Wei Liu proposed the idea, designed the experiments, analysed the data, and wrote the paper.\u003c/p\u003e\u003ch2\u003eAcknowledgments\u003c/h2\u003e\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (Grants 42304064, 42304061, 92155307 and 41874054), the National Key Research and Development Program of China (No. 2021YFC3000702), Guangdong Provincial Key Laboratory of Geophysical High-resolution Imaging Technology (2022B1212010002), Shenzhen Science and Technology Program (KQTD20170810111725321) and Center for Computational Science and Engineering at Southern University of Science and Technology.\u003c/p\u003e\u003ch2\u003eData availability\u003c/h2\u003e\u003cp\u003eThe F-net waveforms used in this work are available via the website of the National Research Institute for Earth Science and Disaster Resilience (NIED), Japan (at the following URL: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://www.hinet.bosai.go.jp/\u003c/span\u003e\u003cspan address=\"http://www.hinet.bosai.go.jp/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e). The VLFs catalogue are attributed to Takemura, et al.\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e. The regular earthquake catalog is from the U.S. Geological Survey (USGS) (at the following URL: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://earthquake.usgs.gov/earthquakes/search\u003c/span\u003e\u003cspan address=\"https://earthquake.usgs.gov/earthquakes/search\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e), and the corresponding focal mechanism solutions are provided by the NIED (at the following URL: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://www.hinet.bosai.go.jp/\u003c/span\u003e\u003cspan address=\"http://www.hinet.bosai.go.jp/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e\u003ch2\u003eCode availability\u003c/h2\u003e\u003cp\u003eThe code for simulation of earthquake ruptures in this work is available online (at the following URL: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.5281/zenodo.5835606\u003c/span\u003e\u003cspan address=\"10.5281/zenodo.5835606\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAndo, M. 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Am.\u003c/em\u003e 108, 1947\u0026ndash;1961 (2018). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1785/0120170309\u003c/span\u003e\u003cspan address=\"10.1785/0120170309\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7074847/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7074847/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSlow earthquakes represent a distinct class of fault slip phenomena characterized by prolonged source duration and reduced seismic radiation compared to regular earthquakes. While essential for understanding seismic cycles and megathrust earthquakes, the mechanisms governing their rupture processes remain elusive. Here we present the finite-fault inversions of very low-frequency earthquakes (VLFs) in the Nankai Trough, a subtype of slow earthquakes producing detectable seismic waves while maintaining resolvable rupture dimensions. Comparison with two nearby regular earthquakes reveals that VLFs exhibit a distinct two-stage process: (1) rapidly accelerating slip confined to hypocentral regions, and (2) progressive deceleration and spontaneous termination within the initial slip zone, without lateral propagation. Source dynamic simulations reconcile these observations through a slip-weakening friction model, suggesting that a low slip-weakening rate facilitates slow self-arresting ruptures consistent with VLFs. Our findings provide observational insights into VLFs rupture dynamics, suggesting that slow earthquakes may experience limited propagation and self-arresting evolutions.\u003c/p\u003e","manuscriptTitle":"Characterizing spatially restricted rupture in Nankai shallow very low-frequency earthquakes","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-18 11:51:13","doi":"10.21203/rs.3.rs-7074847/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"2172b61a-5b75-45b0-9244-e859e96b8602","owner":[],"postedDate":"July 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":51530563,"name":"Earth and environmental sciences/Solid Earth sciences/Seismology"},{"id":51530564,"name":"Earth and environmental sciences/Solid Earth sciences/Geophysics"}],"tags":[],"updatedAt":"2025-11-06T09:45:38+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-18 11:51:13","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7074847","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7074847","identity":"rs-7074847","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}