Development of an adjoint-based data assimilation method toward predicting SSE evolution: Two-step optimization of frictional parameters and initial strength on the fault

Development of an adjoint-based data assimilation method toward predicting SSE evolution: Two-step optimization of frictional parameters and initial strength on the fault | 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 Research Article Development of an adjoint-based data assimilation method toward predicting SSE evolution: Two-step optimization of frictional parameters and initial strength on the fault Makiko Ohtani, Nobuki Kame, Masayuki Kano This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5343128/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 10 Jun, 2025 Read the published version in Earth, Planets and Space → Version 1 posted 5 You are reading this latest preprint version Abstract Data assimilation (DA) has tried to incorporate GNSS data into physics-based fault slip models to estimate frictional properties and predict future slip evolution on faults. For unstable slip events such as ordinary fast-slip earthquakes and slow slip events (SSEs), accurately estimating the frictional strength, as well as the frictional parameters, is crucial for reliable slip prediction. However, the frictional strength has not been directly observed, and thus, previous DA studies have often assumed a steady-state strength value for the initial strength to estimate the frictional parameters, which limits the accuracy of long-term slip predictions. In the present study, we propose a new adjoint-based DA method that estimates an appropriate initial frictional strength along with the frictional parameters for assimilating long-term SSEs. The key idea is to impose an additional constraint on DA that assumes the current SSE will recur periodically, though the exact interval is unknown. This approach reflects the observed recurring nature of SSEs. This new method is validated through numerical experiments focusing on long-term Bungo Channel SSEs in southwest Japan. The results demonstrate that our proposed method provides reasonable estimates for both the initial strength and the frictional parameters, enabling accurate predictions of slip evolution and the timing of subsequent SSEs, along with determining the unknown recurrence interval. The method proves effective even with data windows shorter than the recurrence interval, overcoming the limitations of previous DA methods. Data assimilation adjoint method slow slip event Bungo Channel frictional parameters frictional strength twin experiment prediction Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction Slip on the plate boundary has been modeled using the laboratory-derived rate- and state-dependent friction law (RSF; Dieterich, 1979 ; Ruina, 1983 ) to simulate various slip phenomena such as earthquakes, afterslip, and the deep steady slip (e.g. Tse and Rice, 1986 ). Physics-based models have significantly advanced our understanding of fault slip evolution through numerical simulations. However, these simulations often rely on trial-and-error assumptions for frictional parameters to qualitatively reproduce observed fault slip characteristics. In fact, the exact frictional properties of faults remain unknown. Recently, data assimilation (DA) methods have been applied to estimate the frictional properties of RSF faults from crustal deformation data (e.g., Kano et al., 2013 ; Kano et al., 2015 ; van Dinther et al., 2019 ; Hirahara and Nishikori, 2019; Kano et al., 2024 ). DA incorporates observed data into physics-based RSF fault slip models, allowing these models to quantitatively reproduce the observations. Such assimilated models are expected to predict fault slip evolution beyond the data period, similar to weather forecasts. Some previous DA studies have focused on long-term slow slip events (SSEs). SSEs are recurrent, unstable slip events similar to ordinary earthquakes but with a much slower slip velocity. Because of this slow slip characteristic, SSEs are easier to implement DA and can serve as a useful exercise for more challenging ordinary earthquakes. Here we focus on the Bungo Channel SSEs in southwestern Japan as promising targets for DA, following Hirahara and Nishikiori ( 2019 ) (hereafter, HN19) and Kano et al. ( 2024 ) (K24). The Bungo Channel SSEs occur in the deeper part of the seismic region, reaching magnitudes of 6 to 7, with durations of several months to one year and recurrence intervals of 5 to 7 years (Hirose et al., 1999 ; Ozawa et al., 2013 ; Yoshioka et al., 2015 ). While the previous DA studies for the Bungo Channel SSEs have achieved a certain degree of success, they have focused primarily on estimating frictional parameters, leaving the treatment of model variables as a remaining issue. Among the variables of slip velocity V and fault strength Φ on a RSF fault, V can be directly inferred from the crustal deformation data, while Φ lacks relevant observations. Therefore, they set the starting time of DA to a few years before the slip velocity reaches its peak and fixed the initial value of Φ at the time to satisfies the steady-state condition dΦ/d t = 0 for the initial guesses of the parameters. While this approach enabled estimation of frictional parameters, the assumed Φ in such a manner raises issues in predicting slip. As will be shown later, there is no unique combination of the initial Φ and frictional parameters that can explain observation data in a limited time window. The resulting trade-off between their estimations in DA leads to significant differences in predicted slip evolution beyond the assimilation window, highlighting the importance of accurately estimating initial value of Φ for reliable slip prediction. This trade-off and misprediction issue cannot be resolved using existing DA methods that relies only on fitting the data. K24 was successful for short-term slip prediction, achieved by imposing prior constraints on frictional parameters related to slip instability; however, it was not effective for long-term slip prediction. Therefore, to accurately estimate Φ in addition to the frictional parameters and to obtain reliable slip predictions, we propose a new method based on the adjoint method, a variational DA approach. The key to the proposed method is to introduce an additional constraint; “the current SSE should recur periodically in the future though we do not know its recurrence period,” reflecting the observed recurring nature of SSEs. We refer to this new method as "two-step optimization method." In the present study, we conduct numerical experiments, called as “twin experiments”, commonly used to evaluate DA method. A twin experiment employs a physics-based model to construct the true state of the SSE slip model and the observational data is generated from it. The same physics-based model is used as the assimilation model in DA. Through twin experiments, we can evaluate our new DA method under ideal conditions, independent of the model quality. In the following sections, we first describe the settings for twin experiments including the physics-based Bungo Channel SSE model, true state of the employing SSE model, and the synthetic observational data, that is consistent with HN19. Next, we review the adjoint method used in previous DA studies for fault slip and highlight the non-uniqueness in DA estimates arising from a trade-off relationship. We particularly emphasize the importance of accurately estimating the initial value of Φ for slip prediction. We then present our new "two-step optimization method" to address this issue. In twin experiments, our method is shown to effectively constrain both Φ and the frictional parameters, enabling reliable predictions of subsequent SSEs. 2 Settings for twin experiments 2.1 Physics-based model for Bungo Channel SSEs We employ a physics-based model that is identical to that proposed in HN19. We assume a flat rectangular fault with a dip angle of 15°, subducting at a relative plate velocity of V pl = 6.5 cm/yr, and a circular SSE patch on the fault (Fig. 1 ). The fault is discretized into N = 60 × 50 cells (in the X’ 1 and X’ 2 directions, respectively), each with a size of 2 km × 2 km. The slip is assumed to be uniform in each cell, and the boundary integral equation method is used to simulate the spatiotemporal evolution of slip. Following the quasi-dynamic approximation proposed by Rice ( 1993 ), the shear traction τ i ( t ) in the dip direction on the i -th cell ( i = 1, …, N ) at time t is described by $$\:{\tau\:}_{i}\left(t\right)=\sum\:_{j=1}^{N}{K}_{ij}\left({u}_{j}\left(t\right)-{V}_{pl}t\right)-\frac{G}{2{V}_{s}}{V}_{i}\left(t\right)\:\:,$$ 1 where u i and V i are the slip and the slip velocity in the dip direction of the i -th cell, respectively. K ij is the slip response function representing the change in shear traction at the i -th cell due to a unit slip in the dip direction at the j -th cell. The first term on the right-hand side accounts for the shear traction change due to the slip deficit compared to the steady plate subduction. The second term approximates the inertial effect of the elastic continuum, known as radiation damping (Rice, 1993 ). In modeling within an elastic homogeneous half-space medium, K ij is calculated following Okada ( 1992 ) with a Poisson's ratio of 0.25, a rigidity G = 40 GPa, and an S wave velocity V s = 3.0 km/s. We assume that the friction on the slip surface obeys the RSF law derived in the laboratory (Dieterich, 1979 ; Ruina, 1983 ). The shear traction τ i in Eq. ( 1 ) should be quasi-dynamically equal to the friction, as expressed in Eq. ( 2 ) (Nakatani, 2001 ): $$\:{\tau\:}_{i}\left(t\right)={A}_{i}ln\left(\frac{{V}_{i}\left(t\right)}{{V}_{*}}\right)+{{\Phi\:}}_{i}\left(t\right),\:\:{{\Phi\:}}_{i}\left(t\right)={\tau\:}_{*}+{B}_{i}ln\left(\frac{{\theta\:}_{i}\left(t\right)}{{\theta\:}_{*}}\right),$$ 2 where τ * and θ * are the steady-state values of friction and state variable at the reference slip rate V * = V pl . The frictional strength Φ is related to the conventional state variable θ , which is interpreted as the amount of frictional bonds at the real contact area of the slip interface. In the present paper, Φ and θ are used interchangeably, as they are in one-to-one correspondence. For the evolution law of θ , we employ the aging law, which has the following form (Ruina, 1983 ): $$\:\frac{d{\theta\:}_{i}\left(t\right)}{dt}=1-\frac{{V}_{i}\left(t\right){\theta\:}_{i}\left(t\right)}{{L}_{i}}\:\:.$$ 3 The frictional parameters A , B , and L are empirical constants, with L referred to as the characteristic slip distance. These parameters are assumed to be uniform both inside and outside the SSE patch. Specifically, A i = A in , B i = B in , and L i = L in for cells inside the patch, and A i = A out , B i = B out , and L i = L out for the cells outside the patch. We numerically solve the system of equations ( 1 )—(3) to obtain the fault slip evolution using the DOP853 code, which is an explicit Runge-Kutta algorithm with adaptive step size control and dense output (Hairer et al., 1993 ). The dense output technique allows us to obtain outputs at arbitrary times necessary for DA. The model variables at time t are represented in matrix-vector form as x ( t ) = ( V ( t ), θ ( t )), where V ( t ) = ( V 1 ( t ), V 2 ( t ),…, V N ( t )) and θ ( t ) = ( θ 1 ( t ), θ 2 ( t ),…, θ N ( t )). The subscripts indicate the fault cell numbers. In DA, we fix the frictional parameters outside the SSE patch and only estimate the parameters within the patch, C = ( A in , A in – B in , L in ), following HN19. Along with x ( t ), we optimize the model state Z ( t ) ≡ ( x ( t ), C ) T to fit the time series of observational data. The superscript T denotes the transpose of the vector. 2.2 True state of Bungo Channel SSEs model In the following, we set up an true state Z true to be referenced in twin experiments as employed in HN19. Regarding frictional parameters, we assume A and L are uniform on the entire fault, with A in true = A out = 100 kPa and L in true = L out = 40 mm. The parameters B in true and B out are 150 kPa and 50 kPa inside and outside the SSE patch, respectively. In other words, the true value of C is set to C true = ( A in true , A in true – B in true , L in true ) = (100 kPa, – 50 kPa, 40 mm). The simulated fault slip evolution shows recurrent transient increases in slip velocity within the circular patch that can be regarded as SSEs. We reset the time origin t = 0 at the point when the slip velocity turns to increase during the stable SSE cycles that occur after the influence of initial condition has disappeared. Then, we use the first 8000 days as the true state, Z true ( t ), that includes three times of SSE occurrence with the interval of 2620 days. The slip velocity at the point at the center of the SSE patch peaks at T SSE _ 1 true = 727 days, T SSE _ 2 true = 3347 days, and T SSE _ 3 true = 5967 days. The duration, recurrence interval, and maximum slip velocity are similar to those of the observed Bungo Channel SSEs. For details, see section 3.2.1 (LSSE model 1) in HN19. 2.3 Synthetic observational data From the true state Z true ( t ), we generate synthetic time series of the observational data. Following HN19, we select N obs = 93 GEONET stations around the Bungo Channel (Fig. 1 b) as the observation points where the surface displacement rates are available. The displacement rate \(\:{\mathbf{V}}_{i}^{obs}\) ( t ) ( three components: horizontal and vertical) at the i -th station ( i = 1, …, N obs ) is related to the fault slip velocity V j ( t ) at j -th cell ( j = 1, …, N ) by Eq. ( 4 ) with observation matrix H ij , calculated following Okada ( 1992 ), assuming a homogeneous elastic half-space medium; $$\:{\mathbf{V}}_{i}^{obs}\left(t\right)=\sum\:_{j=1}^{N}{\mathbf{H}}_{ij}\left({V}_{j}\left(t\right)-{V}_{pl}\right)\:.$$ 4 Observation is taken every 5 days during the period t = 0 to 2620 days, resulting in a time series with N step = 524 steps. The length of the observation period matches the recurrence interval of SSE, which is 2620 days. The effect of time width of the data window will be discussed later. At each time step t k ( k = 0, …, N step ), we calculate \(\:{\mathbf{V}}_{i}^{obs}\left({t}_{k}\right)\) at each station using the simulated Z true ( t k ). Observational noises \(\:{\mathbf{r}}_{i}^{obs}\left({t}_{k}\right)\) , which follows a normal distribution with a standard deviation of 3 mm/yr in the horizontal components and 9 mm/yr in the vertical component (as in HN19), is then added to \(\:{\mathbf{V}}_{i}^{obs}\left({t}_{k}\right)\) to obtain the time series data \(\:{\mathbf{d}}_{{t}_{k}}\) for k = 0, …, N step . As an example, we show the synthetic observational data at station P (red circle in Fig. 1 b) in Fig. 1 (c). 3 DA Method 3.1 Ordinary adjoint method applied to RSF fault slip and challenges in slip prediction This section describes the ordinary adjoint method that has been applied to the RSF fault with the crustal deformation data and demonstrates the problem in predicting future slip that arises in such DA. DA optimize Z ( t ) to minimize the cost function J , which is defined as the weighted sum of misfits between the observed data and the numerical predicts from the physics-based model over the time series. We set J as $$\:J=\frac{1}{2}\sum\:_{k=0}^{{N}^{step}}\sum\:_{i=1}^{{N}^{obs}}{\left({\mathbf{V}}_{i}^{obs}\left({t}_{k}\right)-{\mathbf{d}}_{{t}_{k}}\right)}^{T}{\mathbf{R}}_{{t}_{k}}^{-1}\left({\mathbf{V}}_{i}^{obs}\left({t}_{k}\right)-{\mathbf{d}}_{{t}_{k}}\right)\:\:,$$ 5 V i obs ( t k ) is related to Z ( t k ) by Eq. ( 4 ) and \(\:{\mathbf{R}}_{{t}_{k}}\) denotes the observational error covariance matrix at time step k . We assume that there are no correlations between the observational errors across different directions or between those at different stations. The optimization is performed based on the gradient of J with respect to the initial variables and parameters, ∂ J /∂ Z ( t 0 ), where t 0 is the initial time for DA. The misfit information at each time is propagated backward by solving the adjoint equation and then integrated to obtain ∂ J /∂ Z ( t 0 ). We follow the formulation provided by Kano et al. ( 2015 ) to compute ∂ J /∂ Z ( t 0 ); see Appendix A in Kano et al. ( 2015 ) for the details. We use the second-order Runge-Kutta method for backward time integration, with a time step same as that for the observational data, with 5 days interval. For optimization, we use the quasi-Newton method of L-BFGS-B (Zhu et al., 1994 ; Byrd et al., 1995 ) utilizing the above-derived gradient ∂ J /∂ Z ( t 0 ). We begin optimization from initial guess Z ( t 0 ) = Z (0) ( t 0 ) and iteratively update Z ( t 0 ). In Fig. 2 , we show some representative assimilation results to demonstrate the significance of the initial distribution of the frictional strength, Φ ( t 0 ), in predicting slip. Here, we set initial assimilation time as t 0 = 0. We optimize only the frictional parameters C using initial guesses of (0.75 A in true , 0.75( A in true – B in true ), 1.25 L in true ), while fixing the initial variables x ( t 0 ). Among x ( t 0 ), for simplicity, we assume V ( t 0 ) to be the same as that of the true model, V ( t 0 ) = V true ( t 0 ). In Case 1, we assume the correct value also for Φ ( t 0 ); Φ ( t 0 ) = Φ true ( t 0 ). On the other hand, in Case 2, we assume a steady-state value respect to the velocity; Φ ( t 0 ) = Φ ss ( V ( t 0 )), which is attained when d Φ ss ( V ( t 0 ))/d t = 0 under the initially assumed C . The latter assumption has often been employed in previous studies. The values of Φ ( t 0 ) for each case is shown in Fig. 2 (a). In both Cases 1 and 2, the surface displacement rate is successful to fit the observational data (Fig. 2 b dots) and is in good agreement with the true state (red line). However, after the data period, Case 1 succeeds to predict the surface displacement rate, while Case 2 fails. The estimated C is (0.991 A in true , 0.995( A in true – B in true ), 1.013 L in true ) and (0.663 A in true , 1.055( A in true – B in true ), 0.951 L in true ) in Cases 1 and 2, respectively: Case 1 succeeds in estimating C close to its true values, while Case 2 fails. This suggests that resolving Z ( t 0 ) from the observational data within this time window is challenging due to the inherent trade-off characteristics in the RSF model. This is a disadvantage in predicting future slip. 3.2 Our proposed method: Two-step optimization method To resolve the trade-off between the strength and frictional parameters, we introduce an additional constraint that SSEs recur and produce periodic time-series for slip velocity and strength. In other words, we search for a solution that produces identical snapshots of the slip velocity and the strength over the fault at both the initial time t = t 0 and a certain unknown elapsed time t m after the occurrence of an SSE. The timing t m must be determined through the DA optimization process. The constraint may be acceptable, reflecting the observed recurring slip histories of the Bungo Channel SSEs. Figure 3 shows a schematic diagram of the proposed method. This method involves two-step optimizations for updating C and Φ ( t 0 ), which we have named the “two-step optimization method”. As noted in the Introduction, the slip velocity can be directly inferred from crustal deformation data. Therefore, we can expect the initial slip velocity to be accurately estimated without DA and assume V ( t 0 ) = V true ( t 0 ), for simplicity. The DA process starts with initial guesses C = C (0) and Φ ( t 0 ) = Φ (0) ( t 0 ) and iteratively updates them to obtain C ( p ) and Φ ( p ) ( t 0 ) after the p -th iteration. Each iteration involves two optimization: One is to update the frictional parameters C while keeping Φ ( t 0 ), and the other one is to update Φ ( t 0 ) while keeping C fixed. These two steps—updating C and Φ ( t 0 )—are applied alternately until the vales no longer change significantly. In the following, we explain how to update C ( p – 1) and Φ ( p – 1) ( t 0 ) to C ( p ) and Φ ( p ) ( t 0 ). As the result of ( p – 1)-th iteration, we obtain the slip evolution through forward time integration. For simplicity, let us focus on a specific cell at the center of the SSE patch and find the time t m when the initial slip velocity reappears after the occurrence of a SSE; V ( p−1 ) center ( t m ) = V center ( t 0 ) (Fig. 3 b, top). Since we are searching for a periodic solution, it is expected that the strength distribution at t = t m provides a better estimation of the initial strength distribution. Therefore, we update Φ ( t 0 ) to Φ ( p ) ( t 0 ) = Φ New ≡ Φ ( p – 1) ( t m ) (Fig. 3 b, bottom). The slip evolution predicted by the forward time integration with Φ ( p ) and C ( p – 1) no longer explains the observations. Therefore, we optimize C by applying the ordinary adjoint method while keeping the initial condition fixed at Φ ( p ) ( t 0 ). This optimization is an iterative process described in the previous subsection. We apply the constraint of 0.5 ≦ A in / A in true , ( A in – B in )/( A in true – B in true ), and L in / L in true ≦1.5, and update C , resulting in C ( p ) . Then we obtain the updated set of C and Φ ( t 0 ); specifically, C ( p ) and Φ ( p ) ( t 0 ). For the update of Φ ( t 0 ) from p = 0 to p = 1, we set Φ (1) ( t 0 ) = Φ (0) ( t 0 ) to adjust the number of updates. For the initial guess of the parameters C (0) , we set the initial guess Φ (0) ( t 0 ) such that satisfies d Φ ( t 0 )/d t = 0 with the initial slip velocity V ( t 0 ) and the parameters C (0) . In Fig. 3 (a), we denote such Φ ( t 0 ) as Φ ss ( V ( t 0 )). The two-step optimization is applied until the convergence condition (Eq. 6 ) is satisfied. $$\:\left\{\begin{array}{c}\left({\mathbf{C}}^{\left(p\right)}-{\mathbf{C}}^{\left(p-1\right)}\right)<\:ϵ{\mathbf{C}}^{\left(p-1\right)}\\\:\left({\varvec{\Phi\:}}^{New}\left({t}_{0}\right)-{\varvec{\Phi\:}}^{\left(p\right)}\left({t}_{0}\right)\right)<\:ϵ{\varvec{\Phi\:}}^{\left(p\right)}\left({t}_{0}\right)\end{array}\right.\:,$$ 6 with ε = 1.0×10 − 3 . If this condition is satisfied after the p -th iteration, we adopt C ( p ) and Φ ( p ) ( t 0 ) as the final estimates for C and Φ ( t 0 ). This is the way of incorporating the recursive nature of SSEs into the assimilation process. In the next section, we show our new method is successful when applying to the synthetic data for a whole cycle of SSE (Experiments 1 and 2). Additionally, the method is applicable to cases with a data window width of less than one cycle of SSE (Experiment 3). 4 Result and Discussions 4.1 Numerical Experiment 1: A successful example of Estimation We show the result of a twin experiment with the initial guess of C (0) = (0.750 A true , 0.500( A – B ) true , 1.250 L true ). Let us refer to this experiment as Exp. 1. In this experiment, it takes 14 iterations to satisfy the condition for convergence (Eq. 6 ). Figure 4 shows Z ( p ) ( t 0 ), the estimated state after each iteration. As the number of iteration p increases, the cost function J decreases (top panel of Fig. 4 a), the normalized frictional parameters converge to 1 (middle), and the normalized strength at the center of the SSE patch converges to 1 (bottom). The value of J decreases with fluctuations rather than monotonically. This is due to the alternating exploration of strength and parameters, which does not guarantee that the results of the optimization will produce lower J value than those of the previous iteration. However, as a result, J gradually decreases, allowing for a successful estimate of the state to be very close to the true one. The final estimated values of the frictional parameters are C = (0.999 A true , 0.997( A – B ) true , 0.996 L true ). They are estimated within a residual of 0.4% from the true values. In Fig. 4 (b), we also show the strength distribution at each iteration number p = 0, 6, 10, 13, and 14 as the difference from the true distribution (Fig. 2 a, left). The cases for p = 13 and p = 14 look similar, confirming that the strength has converged to a stable distribution. With the estimated values of Z ( p ) ( t 0 ), the physics-based model provides the evolution of the slip. Figure 5 shows the calculated slip evolutions for t = 0—2620 days for the estimated values Z ( p ) ( t 0 ) ( p = 0, …, 14). The top panel shows the surface displacement rate in the X 2 direction at the observation station P. The middle and bottom panels show the slip velocity and the strength at the center of the SSE patch, respectively. In all panels, the timing of the SSE (peak velocity) appears consistent with that of the true state (red lines). As the number of iterations increases, the amplitudes approach the true values. We can also calculate the slip evolution for the time period beyond the data range, t > 2620 days (Fig. 6 ). In this figure, we check the predictability of the future slip evolution. The surface displacement at the observation station P predicted from the final estimate Z (14) ( t 0 ) in Exp. 1 (green dashed line) shows good agreement with the true values (red line), and the two subsequent SSEs after the observation period show good agreement. Note that the predicted slip using the final optimized values is not completely periodic. This may be due to the artificial noise and the non-uniform distribution of the observation stations. In summary, we propose a two-step optimization method and confirm that it works well enough to find the initial values of the variables, frictional parameters, and slip evolution close to the true values. 4.2 Numerical Experiment 2: Dependence on initial guesses In the previous subsection, we have demonstrated how the two-step optimization method works to provide an acceptable prediction of future slip with a specific initial guess. However, this result does not ensure successful data assimilation for arbitrary initial guesses due to the non-linearity of the RSF fault system. Therefore, we next conduct an experiment to explore DA solutions by starting from variable initial guesses within a specified range (Exp. 2). We set the initial guesses for the frictional parameters C (0) as each combination of A in / A in true = (0.75, 1.00, 1.25), ( A in – B in )/( A in true – B in true ) = (0.50, 0.75, 1.00, 1.25, 1.50), and L in / L in true = (0.50, 0.75, 1.00, 1.25, 1.50). Then we perform a total of 75 assimilations for each initial guess applying the two-step optimization method. We find that some cases converge to an optimal value for Z ( t 0 ) (referred to as “converged case”), while others do not (see Table 1 ). Negative cases typically result in a more stable slip compared to the true state during the iterations. Consequently, the condition V center ( t m ) = V center ( t 0 ) is not met in the resulting time series, indicating a failure of this method. Another negative case occurs when the parameters move to the edge of the allowable range, during the iteration. This situation requires an extremely small time step for forward time integration, resulting in a high computational cost that cannot be completed. These cases can be rejected as ones with bad initial guesses. Table 1 ; Converged or not converged cases in Exp. 2; A in = 0.75 ( A in – B in )༼ L in 0.50 0.75 1.00 1.25 1.50 0.50 ○ × ○ ○ ○ 0.75 × ○ ○ ○ ○ 1.00 × ○ ○ ○ ○ 1.25 ○ × ○ ○ ○ 1.50 ○ × × ○ ○ A in = 1.00 ( A in – B in )༼ L in 0.50 0.75 1.00 1.25 1.50 0.50 × ○ ○ × × 0.75 × ○ ○ ○ ○ 1.00 × ○ ○ ○ ○ 1.25 × ○ ○ ○ ○ 1.50 × × × ○ ○ A in = 1.25 ( A in – B in )༼ L in 0.50 0.75 1.00 1.25 1.50 0.50 ○ ○ ○ ○ ○ 0.75 × × ○ ○ ○ 1.00 × ○ ○ ○ ○ 1.25 × × ○ ○ ○ 1.50 × × × ○ ○ The converged and not converged cases, starting from the various initial guesses in Exp. 2. The initial guess A in , A in – B in , and L in are indicated, normalized by the true values. The converged cases are shown with ○, and the others are with ×. Figure 6 compares the results of all converged cases from Exp. 2. The slip evolutions using all final optimized values are shown in Figs. 6 (a) and (b) with gray lines. For the estimation period from t = 0 to 2620 days, the lines overlap and adequately explain both the observational data and the true state. The time T SSE _ 1 , indicating the timing of the peak velocity at the center of SSE patch, ranges between 709 and 747 days, with a precision of 20 days relative to the true state ( T SSE _ 1 true = 727 days). The predicted slip for t > 2620 days diverges over time, although the timings of the subsequent SSEs are all in phase with the true model. The predicted timings of the second and third SSEs are T SSE _ 2 = 3321 to 3346 days and T SSE _ 3 = 5926 to 5970 days, with deviations of 26 days and 41 days from the true state ( T SSE _ 2 true and T SSE _ 3 rue ), respectively. Generally, the timings of the peak velocities are accurately estimated and predicted, although the magnitudes tend to diverge from the true values (Figs. 6 a and 6 b). The converged optimized sets of the normalized frictional parameters, A in / A in true , ( A in – B in )/( A in true – B in true ), and L in / L in true are shown in Fig. 6 (c) along with the value of the cost function J . The optimized parameter sets distribute along a line that passes close to the true values of ( A in / A in true , ( A in – B in )/( A in true – B in true ), L in / L in true ) = (1, 1, 1). This alignment is a consequence of the characteristics of the RSF fault system. Figure 7 shows the recurrence interval and peak slip velocity of the stable SSE cycle obtained by performing forward simulations for each set of parameters. We can confirm that the optimized frictional parameters (black dots) produce recurrence intervals and peak slip velocities that match those of the true state. This indicates that multiple parameter sets satisfy the imposed periodicity constraint. The proposed method aims to resolve the trade-off between initial strength and frictional parameters. It effectively picks up solutions where SSEs occur periodically. However, as noted, a trade-off for frictional parameters still exists. Among the converged cases, A in and L in are estimated within 30% of the true values, while A in – B in is more precisely determined, within 12% of the true value (Fig. 6 c). Note that even with a set of the parameters not very close to the true values, the predictions for the subsequent two SSEs are still within 26 days and 41 days from the true model, respectively. 4.3 Numerical Experiment 3: Estimation with a short data window less than a single SSE interval The proposed method can be applied to a data window that is shorter than the interval of periodic SSEs. In this section (Exp. 3), we extract the period of t = 100—1400 days from the synthetic observational data previously used and employ it as the observational data for 1300 days, corresponding to half of the SSE period. We use the data from t = 100 to exclude the specific time t = 0, where dΦ/dt = 0 at the center of the SSE patch. We set the initial time for estimation as t 0 = 100 days and apply the two-step optimization method. A twin experiment is conducted with the initial guess C (0) = (1.00 A in true , 1.25( A in true – B in true ), 1.25 L in true ) and Φ (0) ( t 0 ) that satisfies the condition d Φ ( t 0 )/d t = 0 . The assimilation interval is set to 5 days, and N step = 260. As a result, the parameters are estimated to be C = (1.063 A in true , 0.992( A in true – B in true ), 0.955 L in true ). In Fig. 8 , the assimilated surface displacement rates (blue line) are in good agreement with the observational data from t = 100 to 1400 days (dots). It also agrees with the true state (red line) particularly in the phase of SSEs both in the periods of assimilation and prediction. The first, second, and third SSE reach their peak slip velocity at T SSE_ 1 = 727 days, T SSE_ 2 = 3353 days, and T SSE_ 3 = 5990 days, respectively. They deviate from the true model by only 4 days, 6 days, and 23 days, respectively. The estimation is comparable to the results from Exp. 2, which utilized observational data from the entire cycle of SSE. This experiment demonstrates that a short data window is sufficient to estimate the recurrence interval of ongoing SSEs. We note that the data should include the periods of both velocity increase and decrease during a SSE for accurate estimation. 5 Future Perspectives In this study, we have demonstrated the importance of estimating frictional strength in slip prediction. Since direct observations of frictional strength are lacking, we have constrained the strength by assuming periodic occurrence of SSEs, which significantly limits the application. In future work, we aim to apply our developed method to actual observational data for the Bungo Channel SSEs, as in K24. Although the Bungo Channel SSEs recur at intervals of 5 to 7 years, they are not purely cyclic, meaning our method cannot be applied directly. However, the variability may fundamentally be due to factors such as heterogeneity of fault surface or stress perturbations, which are not accounted for in the simple physical model employed in this study. While we need to consider how to address this problem, as a first step we intend to apply our method to a period including a single SSE, similar to K24. By applying the method to multiple datasets from different periods, each including a single SSE, we aim to investigate the variability in the estimated recurrence periods, frictional properties, and frictional strength distributions. Such comparisons potentially lead to a deeper understanding of the mechanisms behind these variations and may offer insights into elements missing from the employed simple model. Alternatively, Kame et al. ( 2014 ) showed the possibility of monitoring frictional strength using acoustic methods, such as seismic reflection surveys. If such monitoring becomes available in the future, incorporating frictional strength monitoring along with crustal deformation data in DA may be enable daily fault slip prediction, similar to weather forecasting. 6 Conclusions In this study, we developed a new adjoint-based data assimilation (DA) method specifically designed to estimate both the initial frictional strength and the frictional parameters of fault slip models for slow slip events (SSEs). While previous DA studies focusing on SSEs successfully estimated frictional parameters that can explain the data, they often relied on fixed initial values for frictional strength, limiting the accuracy of long-term slip predictions. Our approach addresses this limitation by incorporating an additional constraint assuming the periodic recurrence of SSEs, thus effectively resolving the trade-off between the initial frictional strength and the frictional parameters. The proposed method was tested through twin experiments focusing on the Bungo Channel SSEs in southwestern Japan. The results demonstrated that our adjoint-based approach effectively estimates the initial frictional strength, which is critical for improving the prediction of subsequent SSE occurrences. Additionally, estimations using various initial guesses for the frictional parameters consistently provided results close to the true values, highlighting the robustness of the proposed method. We would like to emphasize that while our method assumes periodicity, it does not require presupposing a specific value for the period itself; the recurrence interval is determined through optimization. Numerical experiments have demonstrated that the proposed method provides reasonable estimates even with data windows shorter than the SSE recurrence interval. Our results suggest that data from a fragment of the cycle still contains information about periodicity. In future studies, we will apply this method to real data. Even when the histories of SSEs are unknown, this method could potentially estimate the recurrence interval and predict the occurrence of the next SSE. Additionally, the approach to ‘periodicity’ used in this study may also be useful for DA focusing on ordinary earthquakes with similar recurrent characteristics to SSEs. Declarations Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. List of abbreviations DA Data Assimilation SSE Slow Slip Event EnKF Ensemble Kalman Filter RSF rate- and state-friction GEONET Global Navigation Satellite System Earth Observation Network System Availability of data and materials Not applicable. No datasets were used in the present study. Competing interests The authors declare no competing interests. Funding The present study was supported by JSPS KAKENHI grants JP20K14574 and JP21K03694. Authors' contributions M.O. designed the research, developed the method, performed the numerical experiments, and drafted the manuscript. All authors discussed the results and approved the final manuscript. Acknowledgements The present study was supported by JSPS KAKENHI grants JP20K14574 and JP21K03694. This research was conducted using the FUJITSU Supercomputer PRIMEHPC FX1000 and FUJITSU Server PRIMERGY GX2570 (Wisteria/BDEC-01) at the Information Technology Center, The University of Tokyo. References Byrd, R. H., Lu, P., Nocedal, J., Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on scientific computing, 16(5), 1190-1208. https://doi.org/10.1137/0916069 . Dieterich, J. H. (1979) Modeling of rock friction: 1. Experimental results and constitutive equations. J. Geophys. Res., Solid Earth, 84(5):2161–2168. https://doi.org/10.1029/JB084 iB05p 02161 Hairer, E, Nørsett, S. P., Wanner, G. (1993) Solving ordinary differential equations I: —nonstiff problems. Springer series in computational mathematics, vol 8, 2nd edn., Springer Berlin, Heidelberg Hirahara, K., Nishikiori, K. (2019) Estimation of frictional properties and slip evolution on a long-term slow slip event fault with the ensemble Kalman filter: numerical experiments. Geophysical Journal International, 219(3), 2074-2096. https://doi.org/10.1093/gji/ggz415 Hirose, H., Hirahara, K., Kimata, F., Fujii, N., Miyazaki, S. I. (1999) A slow thrust slip event following the two 1996 Hyuganada earthquakes beneath the Bungo Channel, southwest Japan. Geophysical Research Letters, 26(21), 3237-3240. https://doi.org/10.1029/1999GL010999 Kame, N., Nagata, K., Nakatani, M., & Kusakabe, T. (2014). Feasibility of acoustic monitoring of strength drop precursory to earthquake occurrence. Earth, Planets and Space, 66, 41. https://doi.org/10.1186/1880-5981-66-41 Kano, M., Miyazaki, S. I., Ito, K., & Hirahara, K. (2013). An adjoint data assimilation method for optimizing frictional parameters on the afterslip area. Earth, Planets and Space, 65, 1575-1580. https://doi.org/10.5047/eps.2013.08.002 Kano, M., Miyazaki, S. I., Ishikawa, Y., Hiyoshi, Y., Ito, K., Hirahara, K. (2015) Real data assimilation for optimization of frictional parameters and prediction of afterslip in the 2003 Tokachi-oki earthquake inferred from slip velocity by an adjoint method. Geophysical Journal International, 203(1), 646-663. https://doi.org/10.1093/gji/ggv289 Kano, M., Tanaka, Y., Sato, D., Iinuma, T., Hori, T. (2024). Data assimilation for fault slip monitoring and short-term prediction of spatio-temporal evolution of slow slip events: application to the 2010 long-term slow slip event in the Bungo Channel, Japan. Earth, Planets and Space, 76(1), 1-12. https://doi.org/10.1186/s40623-024-02004-9 Nakatani, M. (2001) Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology. Journal of Geophysical Research: Solid Earth, 106(B7), 13347-13380. https://doi.org/10.1029/2000JB900453 Okada, Y. (1992) Internal deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Am., 82(2):1018–1040 Ozawa, S., Yarai, H., Imakiire, T., Tobita, M. (2013) Spatial and temporal evolution of the long-term slow slip in the Bungo Channel, Japan. Earth, Planets and Space, 65(2), 67-73. https://doi.org/10.5047/eps.2012.06.009 Rice, J. R. (1993) Spatio‐temporal complexity of slip on a fault. Journal of Geophysical Research: Solid Earth, 98(B6), 9885-9907. https://doi.org/10.1029/93JB00191 Ruina, A. (1983) Slip instability and state variable friction laws. Journal of Geophysical Research: Solid Earth, 88(B12), 10359-10370. https://doi.org/10.1029/JB088iB12p10359 . Tse, S. T., Rice, J. R. (1986) Crustal earthquake instability in relation to the depth variation of frictional slip properties. Journal of Geophysical Research: Solid Earth, 91(B9), 9452-9472. https://doi.org/10.1029/JB091iB09p09452 van Dinther, Y., Künsch, H. R., Fichtner, A. (2019) Ensemble data assimilation for earthquake sequences: probabilistic estimation and forecasting of fault stresses. Geophysical Journal International, 217(3), 1453-1478. https://doi.org/10.1093/gji/ggz063 Yoshioka, S., Matsuoka, Y., Ide, S. (2015) Spatiotemporal slip distributions of three long-term slow slip events beneath the Bungo Channel, southwest Japan, inferred from inversion analyses of GPS data. Geophysical Journal International, 201(3), 1437-1455. https://doi.org/10.1093/gji/ggv022 Zhu, C., Byrd, R.H., Lu, P., Nocedal, J. (1994) L-BFGS-B: a limited memory FORTRAN code for solving bound constrained optimization problems. Tech. Report, NAM-11, EECS Department, Northwestern University. Supplementary Files Graphicalabstract.png Cite Share Download PDF Status: Published Journal Publication published 10 Jun, 2025 Read the published version in Earth, Planets and Space → Version 1 posted Editorial decision: Major Revision 28 Jan, 2025 Reviewers agreed at journal 06 Nov, 2024 Reviewers invited by journal 06 Nov, 2024 Editor assigned by journal 29 Oct, 2024 First submitted to journal 27 Oct, 2024 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. 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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-5343128","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":374733208,"identity":"3fd1410e-900a-4fb9-8f21-3c23b7eebdb8","order_by":0,"name":"Makiko Ohtani","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIie2QMUvEMBTHXwjUJZI1oX6IJwcVMdxnUQKdOzkd8lw6Hbr2wM1PcN+gpXBu17XQQYrrbS6CJ9hw4J1g7hwF8xseCY8f/38CEAj8RZQbaBCAEaOdBfcrboXpl6J+qUCNbn5TvMj4vnrNsmaEzSOxWW5uJAiEtwkcnXkU/VDzuMAuwbYnNs9TpUkgmy6An9PPCrYWYoGdwbYi1ue1wucVwjEBx9Kr8HeByx2lHFI+9ivRkFIm2Ny6YhuF70vRhU0uBNqRbhlVxTLVM4qy+mShvG+R6uqlE+vx6V3z1PfTayMl8Hm/mhjr+7Et6hJKFm3OQyVl8aAiXfX19j4+rAQCgcA/4RMX9FatoziD1wAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0001-5321-502X","institution":"Kyoto University Graduate School of Science Faculty of Science: Kyoto Daigaku Rigaku Kenkyuka Rigakubu","correspondingAuthor":true,"prefix":"","firstName":"Makiko","middleName":"","lastName":"Ohtani","suffix":""},{"id":374733209,"identity":"74351338-dd51-4484-be22-c8b96dafceaa","order_by":1,"name":"Nobuki Kame","email":"","orcid":"","institution":"University of Tokyo: Tokyo Daigaku","correspondingAuthor":false,"prefix":"","firstName":"Nobuki","middleName":"","lastName":"Kame","suffix":""},{"id":374733210,"identity":"b9093848-6ad8-40c1-be21-837464740c17","order_by":2,"name":"Masayuki Kano","email":"","orcid":"","institution":"Tohoku University Graduate School of Science Faculty of Science: Tohoku Daigaku Daigakuin Rigaku Kenkyuka Rigakubu","correspondingAuthor":false,"prefix":"","firstName":"Masayuki","middleName":"","lastName":"Kano","suffix":""}],"badges":[],"createdAt":"2024-10-28 00:43:48","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5343128/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5343128/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1186/s40623-025-02222-9","type":"published","date":"2025-06-10T15:57:20+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":69098419,"identity":"1cca1a48-9478-4105-99ec-94211a8fdf84","added_by":"auto","created_at":"2024-11-15 15:13:31","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":355132,"visible":true,"origin":"","legend":"\u003cp\u003eGeometry of the fault and the observation stations;\u003c/p\u003e\n\u003cp\u003e(a) Geometry of the Bungo SSE fault model. The fault is 120 km in the strike direction and 100 km in the dip direction (blue), and a circular SSE patch with a radius of 35 km (yellow) is set on the fault. We set \u003cem\u003eX\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003cem\u003eX\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e plane on the ground surface and \u003cem\u003eX\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e axis perpendicular to the ground surface upward positive. We also set \u003cem\u003eX’\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003cem\u003eX’\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e plane on the slip interface with the origin O at the center of the fault, which locates 25 km in depth. (b) Map showing the rectangular fault (blue) with the SSE patch (yellow) projected onto the ground surface. Observation stations of GEONET used in the present paper are also indicated with light-blue and red circles. The synthetic observation data at station P (indicated by the red circle) is shown in panel (c) as dots, along with the true state (red line) employed in the numerical experiments in this study.\u003c/p\u003e","description":"","filename":"Fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/e912c21a9856375e4834ffa9.png"},{"id":69097758,"identity":"7bf31e01-5d98-4e7f-9adf-15ecb95408b8","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":109255,"visible":true,"origin":"","legend":"\u003cp\u003eTrade-off relation of the initial strength distribution and the frictional parameters;\u003c/p\u003e\n\u003cp\u003e(a) Initial strength distributions [MPa] on the fault. The left panel shows the true state, which is also used in Case 1. The right panel is the one used in Case 2. (b) The surface displacement rate [10\u003csup\u003e-8\u003c/sup\u003e m/s] in \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e direction at the observation station P (Figure 1b). The true state is shown with the red line, while the synthetic observational data generated from the true state is shown with dots, which almost overwrap with the red line. Additionally, those estimated and predicted in Cases 1 and 2 are shown with blue and light blue lines, respectively.\u003c/p\u003e","description":"","filename":"Fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/31a7aad67f4dbe771d0c2983.png"},{"id":69097760,"identity":"d35726d0-0495-4803-b41b-8be851e2b534","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":386020,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic view of the two-step optimization method;\u003c/p\u003e\n\u003cp\u003e(a) Flow of the two-step optimization method, estimating the C and Φ(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). (b) illustrates how to renew the initial strength distribution Φ(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) in each iteration. Each line indicates the time evolution of the velocity (top) and the strength (bottom). After the \u003cem\u003ep\u003c/em\u003e-th iteration, the time evolutions V\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e) and Φ\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e) are calculated as the functions of C\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e, V\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), and Φ\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e).\u003c/p\u003e","description":"","filename":"Fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/c5221f1d75917e546f654e25.png"},{"id":69098421,"identity":"6fadd661-75f3-4276-8e4d-fa067d27691c","added_by":"auto","created_at":"2024-11-15 15:13:31","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":188858,"visible":true,"origin":"","legend":"\u003cp\u003eUpdate of the model state in the estimation of Exp. 1;\u003c/p\u003e\n\u003cp\u003e(a) The estimated values after the \u003cem\u003ep\u003c/em\u003e-th iteration (\u003cem\u003ep \u003c/em\u003e= 0, …, 14) of the cost function (top), frictional parameters (middle), and the initial frictional strength at the center of the SSE patch (bottom) in Exp. 1. The dashed lines indicate the values of the true state. (b) The difference of the estimated initial strength distributions [kPa] on the fault plane after the \u003cem\u003ep\u003c/em\u003e-th iteration and the true values, \u003cstrong\u003eΦ\u003c/strong\u003e\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) – \u003cstrong\u003eΦ\u003c/strong\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). The values for \u003cem\u003ep\u003c/em\u003e = 0, 6, 10, 13, and 14 are shown. \u003cem\u003ep\u003c/em\u003e = 0 refers to the initial guess.\u003c/p\u003e","description":"","filename":"Fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/f03a3d28782a3fa5f732fc46.png"},{"id":69098577,"identity":"173516bf-8726-46d7-a656-82c740801f5f","added_by":"auto","created_at":"2024-11-15 15:21:31","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":177084,"visible":true,"origin":"","legend":"\u003cp\u003eThe estimated time evolution of the surface displacement rate and the model variables (Exp. 1);\u003c/p\u003e\n\u003cp\u003eThe time evolution of the surface displacement rate [10\u003csup\u003e-9\u003c/sup\u003e m/s] in \u003cem\u003eX\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e direction at the station P (top), slip velocity [10\u003csup\u003e-8\u003c/sup\u003e m/s] at the center of the SSE patch (middle), and the frictional strength [MPa] at the center of the SSE patch (bottom). Each line represents the results calculated using the initial model state Z\u003csup\u003e(\u003c/sup\u003e\u003csup\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) estimated after the \u003cem\u003ep\u003c/em\u003e-th iteration. In each panel, the time evolution of the true state is shown with a red line. The top panel also displays the synthetic observational data as dots, while the middle panel includes a dashed line indicating \u003cem\u003eV\u003c/em\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003epl\u003c/em\u003e\u003c/sub\u003e.\u003c/p\u003e","description":"","filename":"Fig5.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/b4f813f319d077c8aa96d69f.png"},{"id":69097764,"identity":"41f68e48-5222-4975-95a6-780002e013c3","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":410941,"visible":true,"origin":"","legend":"\u003cp\u003eResultant slip evolution and the final estimated values of C obtained in Exp. 2;\u003c/p\u003e\n\u003cp\u003e(a) The surface displacement rate [10\u003csup\u003e-9\u003c/sup\u003e m/s] in the \u003cem\u003eX\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e direction at station P. All the estimated displacement rates obtained from the converged cases in Exp. 2 are shown with gray lines. The red and green dashed lines represent the true case and the result from Exp. 1, respectively. (b) Magnified view of the time around each slip peak (SSE) from (a). (c) All the final estimated values of \u003cstrong\u003eC\u003c/strong\u003e obtained in Exp. 2. Each dot represents the final estimated values, normalized by the true values. The color of each dot indicates the value of the cost function, calculated using the estimated model state.\u003c/p\u003e","description":"","filename":"Fig6.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/c25a62f3bfc1f34b42304977.png"},{"id":69097762,"identity":"a1398b60-593c-417a-9724-9bc3401c7bfb","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":234864,"visible":true,"origin":"","legend":"\u003cp\u003eCharacteristic of the stable cycle;\u003c/p\u003e\n\u003cp\u003e(a) The recurrence interval [days] and (b) the peak slip velocity [10\u003csup\u003e-8\u003c/sup\u003e m/s] at the center of the SSE patch, obtained through forward time integration. In each panel, the color bars are white at the true values: (a) 2620 days and (b) 1.48 × 10\u003csup\u003e-8 \u003c/sup\u003em/s. The optimized \u003cstrong\u003eC\u003c/strong\u003e values in Exp. 2 are also plotted with black dots at the corresponding estimated value.\u003c/p\u003e","description":"","filename":"Fig7.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/22133e5555c950506028af75.png"},{"id":69097759,"identity":"c8aed68e-e4fb-4c21-acb4-c6298de5cfaf","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":118272,"visible":true,"origin":"","legend":"\u003cp\u003eResultant slip evolution obtained in Exp. 3;\u003c/p\u003e\n\u003cp\u003eSame as Figure 6(a) but for the Exp. 3.\u003c/p\u003e","description":"","filename":"Fig8.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/14cc50d52757a74f0737a449.png"},{"id":84726879,"identity":"b629a9d1-ba7d-4ea0-a739-7c86a67e1367","added_by":"auto","created_at":"2025-06-16 16:08:57","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2953019,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/6859f6c8-15bd-48a2-8ce4-c7a1a3d0cf41.pdf"},{"id":69097766,"identity":"bc7dd932-de8b-4996-bff2-5666f75b06e8","added_by":"auto","created_at":"2024-11-15 15:05:31","extension":"png","order_by":12,"title":"","display":"","copyAsset":false,"role":"supplement","size":299572,"visible":true,"origin":"","legend":"","description":"","filename":"Graphicalabstract.png","url":"https://assets-eu.researchsquare.com/files/rs-5343128/v1/525a177958d1d34173acaca4.png"}],"financialInterests":"","formattedTitle":"Development of an adjoint-based data assimilation method toward predicting SSE evolution: Two-step optimization of frictional parameters and initial strength on the fault","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eSlip on the plate boundary has been modeled using the laboratory-derived rate- and state-dependent friction law (RSF; Dieterich, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1979\u003c/span\u003e; Ruina, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1983\u003c/span\u003e) to simulate various slip phenomena such as earthquakes, afterslip, and the deep steady slip (e.g. Tse and Rice, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). Physics-based models have significantly advanced our understanding of fault slip evolution through numerical simulations. However, these simulations often rely on trial-and-error assumptions for frictional parameters to qualitatively reproduce observed fault slip characteristics. In fact, the exact frictional properties of faults remain unknown.\u003c/p\u003e \u003cp\u003eRecently, data assimilation (DA) methods have been applied to estimate the frictional properties of RSF faults from crustal deformation data (e.g., Kano et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Kano et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; van Dinther et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Hirahara and Nishikori, 2019; Kano et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). DA incorporates observed data into physics-based RSF fault slip models, allowing these models to quantitatively reproduce the observations. Such assimilated models are expected to predict fault slip evolution beyond the data period, similar to weather forecasts.\u003c/p\u003e \u003cp\u003eSome previous DA studies have focused on long-term slow slip events (SSEs). SSEs are recurrent, unstable slip events similar to ordinary earthquakes but with a much slower slip velocity. Because of this slow slip characteristic, SSEs are easier to implement DA and can serve as a useful exercise for more challenging ordinary earthquakes. Here we focus on the Bungo Channel SSEs in southwestern Japan as promising targets for DA, following Hirahara and Nishikiori (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) (hereafter, HN19) and Kano et al. (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) (K24). The Bungo Channel SSEs occur in the deeper part of the seismic region, reaching magnitudes of 6 to 7, with durations of several months to one year and recurrence intervals of 5 to 7 years (Hirose et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Ozawa et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Yoshioka et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2015\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eWhile the previous DA studies for the Bungo Channel SSEs have achieved a certain degree of success, they have focused primarily on estimating frictional parameters, leaving the treatment of model variables as a remaining issue. Among the variables of slip velocity \u003cem\u003eV\u003c/em\u003e and fault strength Φ on a RSF fault, \u003cem\u003eV\u003c/em\u003e can be directly inferred from the crustal deformation data, while Φ lacks relevant observations. Therefore, they set the starting time of DA to a few years before the slip velocity reaches its peak and fixed the initial value of Φ at the time to satisfies the steady-state condition dΦ/d\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 for the initial guesses of the parameters. While this approach enabled estimation of frictional parameters, the assumed Φ in such a manner raises issues in predicting slip. As will be shown later, there is no unique combination of the initial Φ and frictional parameters that can explain observation data in a limited time window. The resulting trade-off between their estimations in DA leads to significant differences in predicted slip evolution beyond the assimilation window, highlighting the importance of accurately estimating initial value of Φ for reliable slip prediction.\u003c/p\u003e \u003cp\u003eThis trade-off and misprediction issue cannot be resolved using existing DA methods that relies only on fitting the data. K24 was successful for short-term slip prediction, achieved by imposing prior constraints on frictional parameters related to slip instability; however, it was not effective for long-term slip prediction. Therefore, to accurately estimate Φ in addition to the frictional parameters and to obtain reliable slip predictions, we propose a new method based on the adjoint method, a variational DA approach. The key to the proposed method is to introduce an additional constraint; \u0026ldquo;the current SSE should recur periodically in the future though we do not know its recurrence period,\u0026rdquo; reflecting the observed recurring nature of SSEs. We refer to this new method as \"two-step optimization method.\" In the present study, we conduct numerical experiments, called as \u0026ldquo;twin experiments\u0026rdquo;, commonly used to evaluate DA method. A twin experiment employs a physics-based model to construct the \u003cem\u003etrue\u003c/em\u003e state of the SSE slip model and the observational data is generated from it. The same physics-based model is used as the assimilation model in DA. Through twin experiments, we can evaluate our new DA method under ideal conditions, independent of the model quality.\u003c/p\u003e \u003cp\u003eIn the following sections, we first describe the settings for twin experiments including the physics-based Bungo Channel SSE model, \u003cem\u003etrue\u003c/em\u003e state of the employing SSE model, and the synthetic observational data, that is consistent with HN19. Next, we review the adjoint method used in previous DA studies for fault slip and highlight the non-uniqueness in DA estimates arising from a trade-off relationship. We particularly emphasize the importance of accurately estimating the initial value of Φ for slip prediction. We then present our new \"two-step optimization method\" to address this issue. In twin experiments, our method is shown to effectively constrain both Φ and the frictional parameters, enabling reliable predictions of subsequent SSEs.\u003c/p\u003e"},{"header":"2 Settings for twin experiments","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Physics-based model for Bungo Channel SSEs\u003c/h2\u003e \u003cp\u003eWe employ a physics-based model that is identical to that proposed in HN19. We assume a flat rectangular fault with a dip angle of 15\u0026deg;, subducting at a relative plate velocity of \u003cem\u003eV\u003c/em\u003e\u003csub\u003epl\u003c/sub\u003e = 6.5 cm/yr, and a circular SSE patch on the fault (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The fault is discretized into \u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;60 \u0026times; 50 cells (in the \u003cem\u003eX\u0026rsquo;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eX\u0026rsquo;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e directions, respectively), each with a size of 2 km \u0026times; 2 km. The slip is assumed to be uniform in each cell, and the boundary integral equation method is used to simulate the spatiotemporal evolution of slip. Following the quasi-dynamic approximation proposed by Rice (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1993\u003c/span\u003e), the shear traction \u003cem\u003eτ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) in the dip direction on the \u003cem\u003ei\u003c/em\u003e-th cell (\u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, \u0026hellip;, \u003cem\u003eN\u003c/em\u003e) at time \u003cem\u003et\u003c/em\u003e is described by\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{\\tau\\:}_{i}\\left(t\\right)=\\sum\\:_{j=1}^{N}{K}_{ij}\\left({u}_{j}\\left(t\\right)-{V}_{pl}t\\right)-\\frac{G}{2{V}_{s}}{V}_{i}\\left(t\\right)\\:\\:,$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e are the slip and the slip velocity in the dip direction of the \u003cem\u003ei\u003c/em\u003e-th cell, respectively. \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eij\u003c/em\u003e\u003c/sub\u003e is the slip response function representing the change in shear traction at the \u003cem\u003ei\u003c/em\u003e-th cell due to a unit slip in the dip direction at the \u003cem\u003ej\u003c/em\u003e-th cell. The first term on the right-hand side accounts for the shear traction change due to the slip deficit compared to the steady plate subduction. The second term approximates the inertial effect of the elastic continuum, known as radiation damping (Rice, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1993\u003c/span\u003e). In modeling within an elastic homogeneous half-space medium, \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eij\u003c/em\u003e\u003c/sub\u003e is calculated following Okada (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1992\u003c/span\u003e) with a Poisson's ratio of 0.25, a rigidity \u003cem\u003eG\u003c/em\u003e\u0026thinsp;=\u0026thinsp;40 GPa, and an S wave velocity \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e = 3.0 km/s.\u003c/p\u003e \u003cp\u003eWe assume that the friction on the slip surface obeys the RSF law derived in the laboratory (Dieterich, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1979\u003c/span\u003e; Ruina, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1983\u003c/span\u003e). The shear traction \u003cem\u003eτ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) should be quasi-dynamically equal to the friction, as expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) (Nakatani, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2001\u003c/span\u003e):\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\tau\\:}_{i}\\left(t\\right)={A}_{i}ln\\left(\\frac{{V}_{i}\\left(t\\right)}{{V}_{*}}\\right)+{{\\Phi\\:}}_{i}\\left(t\\right),\\:\\:{{\\Phi\\:}}_{i}\\left(t\\right)={\\tau\\:}_{*}+{B}_{i}ln\\left(\\frac{{\\theta\\:}_{i}\\left(t\\right)}{{\\theta\\:}_{*}}\\right),$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eτ\u003c/em\u003e\u003csub\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eθ\u003c/em\u003e\u003csub\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sub\u003e are the steady-state values of friction and state variable at the reference slip rate \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003epl\u003c/em\u003e\u003c/sub\u003e. The frictional strength Φ is related to the conventional state variable \u003cem\u003eθ\u003c/em\u003e, which is interpreted as the amount of frictional bonds at the real contact area of the slip interface. In the present paper, Φ and \u003cem\u003eθ\u003c/em\u003e are used interchangeably, as they are in one-to-one correspondence. For the evolution law of \u003cem\u003eθ\u003c/em\u003e, we employ the aging law, which has the following form (Ruina, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1983\u003c/span\u003e):\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\frac{d{\\theta\\:}_{i}\\left(t\\right)}{dt}=1-\\frac{{V}_{i}\\left(t\\right){\\theta\\:}_{i}\\left(t\\right)}{{L}_{i}}\\:\\:.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe frictional parameters \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, and \u003cem\u003eL\u003c/em\u003e are empirical constants, with \u003cem\u003eL\u003c/em\u003e referred to as the characteristic slip distance. These parameters are assumed to be uniform both inside and outside the SSE patch. Specifically, \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003ein\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003ein\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ein\u003c/em\u003e\u003c/sub\u003e for cells inside the patch, and \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003eout\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003eout\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003eout\u003c/em\u003e\u003c/sub\u003e for the cells outside the patch.\u003c/p\u003e \u003cp\u003eWe numerically solve the system of equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)\u0026mdash;(3) to obtain the fault slip evolution using the DOP853 code, which is an explicit Runge-Kutta algorithm with adaptive step size control and dense output (Hairer et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1993\u003c/span\u003e). The dense output technique allows us to obtain outputs at arbitrary times necessary for DA. The model variables at time \u003cem\u003et\u003c/em\u003e are represented in matrix-vector form as \u003cb\u003ex\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e) = (\u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e), \u003cb\u003eθ\u003c/b\u003e (\u003cem\u003et\u003c/em\u003e)), where \u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e) = (\u003cem\u003eV\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e), \u003cem\u003eV\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e),\u0026hellip;, \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e)) and \u003cb\u003eθ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e) = (\u003cem\u003eθ\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e), \u003cem\u003eθ\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e),\u0026hellip;, \u003cem\u003eθ\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e)). The subscripts indicate the fault cell numbers.\u003c/p\u003e \u003cp\u003eIn DA, we fix the frictional parameters outside the SSE patch and only estimate the parameters within the patch, \u003cb\u003eC\u003c/b\u003e = (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e, \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e, \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e), following HN19. Along with \u003cb\u003ex\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e), we optimize the model state \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e) \u0026equiv; (\u003cb\u003ex\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e), \u003cb\u003eC\u003c/b\u003e)\u003csup\u003e\u003cem\u003eT\u003c/em\u003e\u003c/sup\u003e to fit the time series of observational data. The superscript \u003cem\u003eT\u003c/em\u003e denotes the transpose of the vector.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 True state of Bungo Channel SSEs model\u003c/h2\u003e \u003cp\u003eIn the following, we set up an true state \u003cb\u003eZ\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e to be referenced in twin experiments as employed in HN19. Regarding frictional parameters, we assume \u003cem\u003eA\u003c/em\u003e and \u003cem\u003eL\u003c/em\u003e are uniform on the entire fault, with \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = \u003cem\u003eA\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e = 100 kPa and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = \u003cem\u003eL\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e = 40 mm. The parameters \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e and \u003cem\u003eB\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e are 150 kPa and 50 kPa inside and outside the SSE patch, respectively. In other words, the true value of \u003cb\u003eC\u003c/b\u003e is set to \u003cb\u003eC\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e = (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e) = (100 kPa, \u0026ndash; 50 kPa, 40 mm).\u003c/p\u003e \u003cp\u003eThe simulated fault slip evolution shows recurrent transient increases in slip velocity within the circular patch that can be regarded as SSEs. We reset the time origin \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 at the point when the slip velocity turns to increase during the stable SSE cycles that occur after the influence of initial condition has disappeared. Then, we use the first 8000 days as the true state, \u003cb\u003eZ\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e), that includes three times of SSE occurrence with the interval of 2620 days. The slip velocity at the point at the center of the SSE patch peaks at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = 727 days, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = 3347 days, and \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = 5967 days. The duration, recurrence interval, and maximum slip velocity are similar to those of the observed Bungo Channel SSEs. For details, see section 3.2.1 (LSSE model 1) in HN19.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Synthetic observational data\u003c/h2\u003e \u003cp\u003eFrom the true state \u003cb\u003eZ\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e), we generate synthetic time series of the observational data. Following HN19, we select \u003cem\u003eN\u003c/em\u003e\u003csup\u003eobs\u003c/sup\u003e = 93 GEONET stations around the Bungo Channel (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) as the observation points where the surface displacement rates are available. The displacement rate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{V}}_{i}^{obs}\\)\u003c/span\u003e\u003c/span\u003e(\u003cem\u003et\u003c/em\u003e) \u003cb\u003e(\u003c/b\u003ethree components: horizontal and vertical) at the \u003cem\u003ei\u003c/em\u003e-th station (\u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, \u0026hellip;, \u003cem\u003eN\u003c/em\u003e\u003csup\u003eobs\u003c/sup\u003e) is related to the fault slip velocity V\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) at \u003cem\u003ej\u003c/em\u003e-th cell (\u003cem\u003ej\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, \u0026hellip;, \u003cem\u003eN\u003c/em\u003e) by Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) with observation matrix \u003cb\u003eH\u003c/b\u003e\u003csub\u003e\u003cem\u003eij\u003c/em\u003e\u003c/sub\u003e, calculated following Okada (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1992\u003c/span\u003e), assuming a homogeneous elastic half-space medium;\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{\\mathbf{V}}_{i}^{obs}\\left(t\\right)=\\sum\\:_{j=1}^{N}{\\mathbf{H}}_{ij}\\left({V}_{j}\\left(t\\right)-{V}_{pl}\\right)\\:.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eObservation is taken every 5 days during the period \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 to 2620 days, resulting in a time series with \u003cem\u003eN\u003c/em\u003e\u003csup\u003estep\u003c/sup\u003e = 524 steps. The length of the observation period matches the recurrence interval of SSE, which is 2620 days. The effect of time width of the data window will be discussed later. At each time step \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, \u0026hellip;, \u003cem\u003eN\u003c/em\u003e\u003csup\u003estep\u003c/sup\u003e), we calculate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{V}}_{i}^{obs}\\left({t}_{k}\\right)\\)\u003c/span\u003e\u003c/span\u003e at each station using the simulated \u003cb\u003eZ\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e). Observational noises \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{r}}_{i}^{obs}\\left({t}_{k}\\right)\\)\u003c/span\u003e\u003c/span\u003e, which follows a normal distribution with a standard deviation of 3 mm/yr in the horizontal components and 9 mm/yr in the vertical component (as in HN19), is then added to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{V}}_{i}^{obs}\\left({t}_{k}\\right)\\)\u003c/span\u003e\u003c/span\u003e to obtain the time series data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{d}}_{{t}_{k}}\\)\u003c/span\u003e\u003c/span\u003e for \u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, \u0026hellip;, \u003cem\u003eN\u003c/em\u003e\u003csup\u003estep\u003c/sup\u003e. As an example, we show the synthetic observational data at station P (red circle in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c).\u003c/p\u003e \u003c/div\u003e"},{"header":"3 DA Method","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Ordinary adjoint method applied to RSF fault slip and challenges in slip prediction\u003c/h2\u003e \u003cp\u003eThis section describes the ordinary adjoint method that has been applied to the RSF fault with the crustal deformation data and demonstrates the problem in predicting future slip that arises in such DA. DA optimize \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e) to minimize the cost function \u003cem\u003eJ\u003c/em\u003e, which is defined as the weighted sum of misfits between the observed data and the numerical predicts from the physics-based model over the time series. We set \u003cem\u003eJ\u003c/em\u003e as\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:J=\\frac{1}{2}\\sum\\:_{k=0}^{{N}^{step}}\\sum\\:_{i=1}^{{N}^{obs}}{\\left({\\mathbf{V}}_{i}^{obs}\\left({t}_{k}\\right)-{\\mathbf{d}}_{{t}_{k}}\\right)}^{T}{\\mathbf{R}}_{{t}_{k}}^{-1}\\left({\\mathbf{V}}_{i}^{obs}\\left({t}_{k}\\right)-{\\mathbf{d}}_{{t}_{k}}\\right)\\:\\:,$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eV\u003c/b\u003e \u003csub\u003e \u003cem\u003ei\u003c/em\u003e \u003c/sub\u003e \u003csup\u003eobs\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e) is related to \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e) by Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{R}}_{{t}_{k}}\\)\u003c/span\u003e\u003c/span\u003e denotes the observational error covariance matrix at time step \u003cem\u003ek\u003c/em\u003e. We assume that there are no correlations between the observational errors across different directions or between those at different stations.\u003c/p\u003e \u003cp\u003eThe optimization is performed based on the gradient of \u003cem\u003eJ\u003c/em\u003e with respect to the initial variables and parameters, \u0026part;\u003cem\u003eJ\u003c/em\u003e/\u0026part;\u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), where \u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the initial time for DA. The misfit information at each time is propagated backward by solving the adjoint equation and then integrated to obtain \u0026part;\u003cem\u003eJ\u003c/em\u003e/\u0026part;\u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). We follow the formulation provided by Kano et al. (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) to compute \u0026part;\u003cem\u003eJ\u003c/em\u003e/\u0026part;\u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e); see Appendix A in Kano et al. (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) for the details. We use the second-order Runge-Kutta method for backward time integration, with a time step same as that for the observational data, with 5 days interval. For optimization, we use the quasi-Newton method of L-BFGS-B (Zhu et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Byrd et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1995\u003c/span\u003e) utilizing the above-derived gradient \u0026part;\u003cem\u003eJ\u003c/em\u003e/\u0026part;\u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). We begin optimization from initial guess \u003cb\u003eZ\u003c/b\u003e (\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;\u003cb\u003eZ\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) and iteratively update \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e).\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, we show some representative assimilation results to demonstrate the significance of the initial distribution of the frictional strength, \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), in predicting slip. Here, we set initial assimilation time as \u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0. We optimize only the frictional parameters \u003cb\u003eC\u003c/b\u003e using initial guesses of (0.75\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, 0.75(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u003cem\u003e\u0026ndash; B\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), 1.25\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), while fixing the initial variables \u003cb\u003ex\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). Among \u003cb\u003ex\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), for simplicity, we assume \u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) to be the same as that of the true model, \u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;\u003cb\u003eV\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). In Case 1, we assume the correct value also for \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e); \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) = \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). On the other hand, in Case 2, we assume a steady-state value respect to the velocity; \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) = \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003ess\u003c/sup\u003e(\u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)), which is attained when d\u003cb\u003eΦ\u003c/b\u003e\u003csup\u003ess\u003c/sup\u003e(\u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e))/d\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 under the initially assumed \u003cb\u003eC\u003c/b\u003e. The latter assumption has often been employed in previous studies. The values of \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) for each case is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn both Cases 1 and 2, the surface displacement rate is successful to fit the observational data (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb dots) and is in good agreement with the true state (red line). However, after the data period, Case 1 succeeds to predict the surface displacement rate, while Case 2 fails. The estimated \u003cb\u003eC\u003c/b\u003e is (0.991\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, 0.995(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u003cem\u003e\u0026ndash; B\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), 1.013\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e) and (0.663\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, 1.055(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u003cem\u003e\u0026ndash; B\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), 0.951\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e) in Cases 1 and 2, respectively: Case 1 succeeds in estimating \u003cb\u003eC\u003c/b\u003e close to its true values, while Case 2 fails. This suggests that resolving \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) from the observational data within this time window is challenging due to the inherent \u003cem\u003etrade-off\u003c/em\u003e characteristics in the RSF model. This is a disadvantage in predicting future slip.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Our proposed method: Two-step optimization method\u003c/h2\u003e \u003cp\u003eTo resolve the trade-off between the strength and frictional parameters, we introduce an additional constraint that SSEs recur and produce periodic time-series for slip velocity and strength. In other words, we search for a solution that produces identical snapshots of the slip velocity and the strength over the fault at both the initial time \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and a certain unknown elapsed time \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e after the occurrence of an SSE. The timing \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e must be determined through the DA optimization process. The constraint may be acceptable, reflecting the observed recurring slip histories of the Bungo Channel SSEs.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows a schematic diagram of the proposed method. This method involves two-step optimizations for updating \u003cb\u003eC\u003c/b\u003e and \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), which we have named the \u0026ldquo;two-step optimization method\u0026rdquo;. As noted in the Introduction, the slip velocity can be directly inferred from crustal deformation data. Therefore, we can expect the initial slip velocity to be accurately estimated without DA and assume \u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;\u003cb\u003eV\u003c/b\u003e\u003csup\u003etrue\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), for simplicity. The DA process starts with initial guesses \u003cb\u003eC\u003c/b\u003e\u0026thinsp;=\u0026thinsp;\u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) = \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) and iteratively updates them to obtain \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) after the \u003cem\u003ep\u003c/em\u003e-th iteration. Each iteration involves two optimization: One is to update the frictional parameters \u003cb\u003eC\u003c/b\u003e while keeping \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), and the other one is to update \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) while keeping \u003cb\u003eC\u003c/b\u003e fixed. These two steps\u0026mdash;updating \u003cb\u003eC\u003c/b\u003e and \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)\u0026mdash;are applied alternately until the vales no longer change significantly. In the following, we explain how to update \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e \u0026ndash; 1)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e \u0026ndash; 1)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) to \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs the result of (\u003cem\u003ep\u003c/em\u003e \u0026ndash; 1)-th iteration, we obtain the slip evolution through forward time integration. For simplicity, let us focus on a specific cell at the center of the SSE patch and find the time \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e when the initial slip velocity reappears after the occurrence of a SSE; \u003cem\u003eV\u003c/em\u003e\u003csup\u003e(\u003cem\u003ep\u0026minus;1\u003c/em\u003e)\u003c/sup\u003e\u003csub\u003ecenter\u003c/sub\u003e (\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;\u003cem\u003eV\u003c/em\u003e\u003csub\u003ecenter\u003c/sub\u003e (\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb, top). Since we are searching for a periodic solution, it is expected that the strength distribution at \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003et\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e provides a better estimation of the initial strength distribution. Therefore, we update \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) to \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) = \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e\u003cem\u003eNew\u003c/em\u003e\u003c/sup\u003e\u0026thinsp;\u0026equiv;\u0026thinsp;\u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e \u0026ndash; 1)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e) (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb, bottom).\u003c/p\u003e \u003cp\u003eThe slip evolution predicted by the forward time integration with \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e and \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e \u0026ndash; 1)\u003c/sup\u003e no longer explains the observations. Therefore, we optimize \u003cb\u003eC\u003c/b\u003e by applying the ordinary adjoint method while keeping the initial condition fixed at \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). This optimization is an iterative process described in the previous subsection. We apply the constraint of 0.5\u0026thinsp;≦\u0026thinsp;\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)/(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e ≦1.5, and update \u003cb\u003eC\u003c/b\u003e, resulting in \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e. Then we obtain the updated set of \u003cb\u003eC\u003c/b\u003e and \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e); specifically, \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e). For the update of \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) from \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 to \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, we set \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(1)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) = \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) to adjust the number of updates.\u003c/p\u003e \u003cp\u003eFor the initial guess of the parameters \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e, we set the initial guess \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) such that satisfies d\u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)/d\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cb\u003e0\u003c/b\u003e with the initial slip velocity \u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) and the parameters \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a), we denote such \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) as \u003cb\u003eΦ\u003c/b\u003e\u003csub\u003ess\u003c/sub\u003e(\u003cb\u003eV\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)). The two-step optimization is applied until the convergence condition (Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) is satisfied.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:\\left\\{\\begin{array}{c}\\left({\\mathbf{C}}^{\\left(p\\right)}-{\\mathbf{C}}^{\\left(p-1\\right)}\\right)\u0026lt;\\:ϵ{\\mathbf{C}}^{\\left(p-1\\right)}\\\\\\:\\left({\\varvec{\\Phi\\:}}^{New}\\left({t}_{0}\\right)-{\\varvec{\\Phi\\:}}^{\\left(p\\right)}\\left({t}_{0}\\right)\\right)\u0026lt;\\:ϵ{\\varvec{\\Phi\\:}}^{\\left(p\\right)}\\left({t}_{0}\\right)\\end{array}\\right.\\:,$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewith \u003cem\u003eε\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e. If this condition is satisfied after the \u003cem\u003ep\u003c/em\u003e-th iteration, we adopt \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) as the final estimates for \u003cb\u003eC\u003c/b\u003e and \u003cb\u003eΦ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e).\u003c/p\u003e \u003cp\u003eThis is the way of incorporating the recursive nature of SSEs into the assimilation process. In the next section, we show our new method is successful when applying to the synthetic data for a whole cycle of SSE (Experiments 1 and 2). Additionally, the method is applicable to cases with a data window width of less than one cycle of SSE (Experiment 3).\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Result and Discussions","content":"\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Numerical Experiment 1: A successful example of Estimation\u003c/h2\u003e \u003cp\u003eWe show the result of a twin experiment with the initial guess of \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e = (0.750\u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e, 0.500(\u003cem\u003eA\u003c/em\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e)\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e, 1.250\u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e ). Let us refer to this experiment as Exp. 1. In this experiment, it takes 14 iterations to satisfy the condition for convergence (Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows \u003cb\u003eZ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), the estimated state after each iteration. As the number of iteration \u003cem\u003ep\u003c/em\u003e increases, the cost function \u003cem\u003eJ\u003c/em\u003e decreases (top panel of Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea), the normalized frictional parameters converge to 1 (middle), and the normalized strength at the center of the SSE patch converges to 1 (bottom). The value of \u003cem\u003eJ\u003c/em\u003e decreases with fluctuations rather than monotonically. This is due to the alternating exploration of strength and parameters, which does not guarantee that the results of the optimization will produce lower \u003cem\u003eJ\u003c/em\u003e value than those of the previous iteration. However, as a result, \u003cem\u003eJ\u003c/em\u003e gradually decreases, allowing for a successful estimate of the state to be very close to the true one. The final estimated values of the frictional parameters are \u003cb\u003eC\u003c/b\u003e = (0.999\u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e, 0.997(\u003cem\u003eA\u003c/em\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e)\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e, 0.996\u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003etrue\u003c/em\u003e\u003c/sub\u003e). They are estimated within a residual of 0.4% from the true values. In Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(b), we also show the strength distribution at each iteration number \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, 6, 10, 13, and 14 as the difference from the true distribution (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea, left). The cases for \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;13 and \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;14 look similar, confirming that the strength has converged to a stable distribution.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWith the estimated values of \u003cb\u003eZ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), the physics-based model provides the evolution of the slip. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the calculated slip evolutions for \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0\u0026mdash;2620 days for the estimated values \u003cb\u003eZ\u003c/b\u003e\u003csup\u003e(\u003cem\u003ep\u003c/em\u003e)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) (\u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, \u0026hellip;, 14). The top panel shows the surface displacement rate in the \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e direction at the observation station P. The middle and bottom panels show the slip velocity and the strength at the center of the SSE patch, respectively. In all panels, the timing of the SSE (peak velocity) appears consistent with that of the true state (red lines). As the number of iterations increases, the amplitudes approach the true values.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe can also calculate the slip evolution for the time period beyond the data range, \u003cem\u003et\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;2620 days (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). In this figure, we check the predictability of the future slip evolution. The surface displacement at the observation station P predicted from the final estimate \u003cb\u003eZ\u003c/b\u003e\u003csup\u003e(14)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) in Exp. 1 (green dashed line) shows good agreement with the true values (red line), and the two subsequent SSEs after the observation period show good agreement. Note that the predicted slip using the final optimized values is not completely periodic. This may be due to the artificial noise and the non-uniform distribution of the observation stations. In summary, we propose a two-step optimization method and confirm that it works well enough to find the initial values of the variables, frictional parameters, and slip evolution close to the true values.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Numerical Experiment 2: Dependence on initial guesses\u003c/h2\u003e \u003cp\u003eIn the previous subsection, we have demonstrated how the two-step optimization method works to provide an acceptable prediction of future slip with a specific initial guess. However, this result does not ensure successful data assimilation for arbitrary initial guesses due to the non-linearity of the RSF fault system. Therefore, we next conduct an experiment to explore DA solutions by starting from variable initial guesses within a specified range (Exp. 2).\u003c/p\u003e \u003cp\u003eWe set the initial guesses for the frictional parameters \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e as each combination of \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = (0.75, 1.00, 1.25), (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)/(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e) = (0.50, 0.75, 1.00, 1.25, 1.50), and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = (0.50, 0.75, 1.00, 1.25, 1.50). Then we perform a total of 75 assimilations for each initial guess applying the two-step optimization method. We find that some cases converge to an optimal value for \u003cb\u003eZ\u003c/b\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) (referred to as \u0026ldquo;converged case\u0026rdquo;), while others do not (see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Negative cases typically result in a more stable slip compared to the true state during the iterations. Consequently, the condition \u003cem\u003eV\u003c/em\u003e\u003csub\u003ecenter\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;\u003cem\u003eV\u003c/em\u003e\u003csub\u003ecenter\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) is not met in the resulting time series, indicating a failure of this method. Another negative case occurs when the parameters move to the edge of the allowable range, during the iteration. This situation requires an extremely small time step for forward time integration, resulting in a high computational cost that cannot be completed. These cases can be rejected as ones with bad initial guesses.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e; Converged or not converged cases in Exp. 2;\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003e\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e = 0.75\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)༼\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003e\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e = 1.00\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)༼\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003e\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e = 1.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)༼\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026times;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e○\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"7\"\u003eThe converged and not converged cases, starting from the various initial guesses in Exp. 2. The initial guess \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e, \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e, and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e are indicated, normalized by the true values. The converged cases are shown with ○, and the others are with \u0026times;.\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e compares the results of all converged cases from Exp. 2. The slip evolutions using all final optimized values are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e (a) and (b) with gray lines. For the estimation period from \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 to 2620 days, the lines overlap and adequately explain both the observational data and the true state. The time \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, indicating the timing of the peak velocity at the center of SSE patch, ranges between 709 and 747 days, with a precision of 20 days relative to the true state (\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e = 727 days). The predicted slip for \u003cem\u003et\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;2620 days diverges over time, although the timings of the subsequent SSEs are all in phase with the true model. The predicted timings of the second and third SSEs are \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e = 3321 to 3346 days and \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e = 5926 to 5970 days, with deviations of 26 days and 41 days from the true state (\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e and \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e_\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e\u003csup\u003erue\u003c/sup\u003e), respectively. Generally, the timings of the peak velocities are accurately estimated and predicted, although the magnitudes tend to diverge from the true values (Figs.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eThe converged optimized sets of the normalized frictional parameters, \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)/(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(c) along with the value of the cost function \u003cem\u003eJ\u003c/em\u003e. The optimized parameter sets distribute along a line that passes close to the true values of (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, (\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e)/(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e/\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e) = (1, 1, 1). This alignment is a consequence of the characteristics of the RSF fault system. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the recurrence interval and peak slip velocity of the stable SSE cycle obtained by performing forward simulations for each set of parameters. We can confirm that the optimized frictional parameters (black dots) produce recurrence intervals and peak slip velocities that match those of the true state. This indicates that multiple parameter sets satisfy the imposed periodicity constraint.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe proposed method aims to resolve the trade-off between initial strength and frictional parameters. It effectively picks up solutions where SSEs occur periodically. However, as noted, a trade-off for frictional parameters still exists. Among the converged cases, \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e are estimated within 30% of the true values, while \u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e is more precisely determined, within 12% of the true value (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ec). Note that even with a set of the parameters not very close to the true values, the predictions for the subsequent two SSEs are still within 26 days and 41 days from the true model, respectively.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Numerical Experiment 3: Estimation with a short data window less than a single SSE interval\u003c/h2\u003e \u003cp\u003eThe proposed method can be applied to a data window that is shorter than the interval of periodic SSEs. In this section (Exp. 3), we extract the period of \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;100\u0026mdash;1400 days from the synthetic observational data previously used and employ it as the observational data for 1300 days, corresponding to half of the SSE period. We use the data from \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;100 to exclude the specific time \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, where dΦ/dt\u0026thinsp;=\u0026thinsp;0 at the center of the SSE patch. We set the initial time for estimation as \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;100 days and apply the two-step optimization method. A twin experiment is conducted with the initial guess \u003cb\u003eC\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e = (1.00\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, 1.25(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), 1.25\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e ) and \u003cb\u003eΦ\u003c/b\u003e\u003csup\u003e(0)\u003c/sup\u003e(\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e) that satisfies the condition d\u003cb\u003eΦ\u003c/b\u003e (\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)/d\u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cb\u003e0\u003c/b\u003e. The assimilation interval is set to 5 days, and \u003cem\u003eN\u003c/em\u003e\u003csup\u003e\u003cem\u003estep\u003c/em\u003e\u003c/sup\u003e = 260.\u003c/p\u003e \u003cp\u003eAs a result, the parameters are estimated to be \u003cb\u003eC\u003c/b\u003e = (1.063\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e, 0.992(\u003cem\u003eA\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e \u0026ndash; \u003cem\u003eB\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e), 0.955\u003cem\u003eL\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e\u003csup\u003etrue\u003c/sup\u003e). In Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the assimilated surface displacement rates (blue line) are in good agreement with the observational data from \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;100 to 1400 days (dots). It also agrees with the true state (red line) particularly in the phase of SSEs both in the periods of assimilation and prediction. The first, second, and third SSE reach their peak slip velocity at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE_\u003c/em\u003e1\u003c/sub\u003e = 727 days, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE_\u003c/em\u003e2\u003c/sub\u003e = 3353 days, and \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eSSE_\u003c/em\u003e3\u003c/sub\u003e = 5990 days, respectively. They deviate from the true model by only 4 days, 6 days, and 23 days, respectively. The estimation is comparable to the results from Exp. 2, which utilized observational data from the entire cycle of SSE. This experiment demonstrates that a short data window is sufficient to estimate the recurrence interval of ongoing SSEs. We note that the data should include the periods of both velocity increase and decrease during a SSE for accurate estimation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5 Future Perspectives","content":"\u003cp\u003eIn this study, we have demonstrated the importance of estimating frictional strength in slip prediction. Since direct observations of frictional strength are lacking, we have constrained the strength by assuming periodic occurrence of SSEs, which significantly limits the application. In future work, we aim to apply our developed method to actual observational data for the Bungo Channel SSEs, as in K24. Although the Bungo Channel SSEs recur at intervals of 5 to 7 years, they are not purely cyclic, meaning our method cannot be applied directly. However, the variability may fundamentally be due to factors such as heterogeneity of fault surface or stress perturbations, which are not accounted for in the simple physical model employed in this study. While we need to consider how to address this problem, as a first step we intend to apply our method to a period including a single SSE, similar to K24. By applying the method to multiple datasets from different periods, each including a single SSE, we aim to investigate the variability in the estimated recurrence periods, frictional properties, and frictional strength distributions. Such comparisons potentially lead to a deeper understanding of the mechanisms behind these variations and may offer insights into elements missing from the employed simple model.\u003c/p\u003e \u003cp\u003eAlternatively, Kame et al. (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) showed the possibility of monitoring frictional strength using acoustic methods, such as seismic reflection surveys. If such monitoring becomes available in the future, incorporating frictional strength monitoring along with crustal deformation data in DA may be enable daily fault slip prediction, similar to weather forecasting.\u003c/p\u003e"},{"header":"6 Conclusions","content":"\u003cp\u003eIn this study, we developed a new adjoint-based data assimilation (DA) method specifically designed to estimate both the initial frictional strength and the frictional parameters of fault slip models for slow slip events (SSEs). While previous DA studies focusing on SSEs successfully estimated frictional parameters that can explain the data, they often relied on fixed initial values for frictional strength, limiting the accuracy of long-term slip predictions. Our approach addresses this limitation by incorporating an additional constraint assuming the periodic recurrence of SSEs, thus effectively resolving the trade-off between the initial frictional strength and the frictional parameters.\u003c/p\u003e \u003cp\u003eThe proposed method was tested through twin experiments focusing on the Bungo Channel SSEs in southwestern Japan. The results demonstrated that our adjoint-based approach effectively estimates the initial frictional strength, which is critical for improving the prediction of subsequent SSE occurrences. Additionally, estimations using various initial guesses for the frictional parameters consistently provided results close to the true values, highlighting the robustness of the proposed method.\u003c/p\u003e \u003cp\u003eWe would like to emphasize that while our method assumes periodicity, it does not require presupposing a specific value for the period itself; the recurrence interval is determined through optimization. Numerical experiments have demonstrated that the proposed method provides reasonable estimates even with data windows shorter than the SSE recurrence interval. Our results suggest that data from a fragment of the cycle still contains information about periodicity. In future studies, we will apply this method to real data. Even when the histories of SSEs are unknown, this method could potentially estimate the recurrence interval and predict the occurrence of the next SSE. Additionally, the approach to \u0026lsquo;periodicity\u0026rsquo; used in this study may also be useful for DA focusing on ordinary earthquakes with similar recurrent characteristics to SSEs.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eList of abbreviations\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDA\u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Data Assimilation\u003c/p\u003e\n\u003cp\u003eSSE \u0026nbsp; \u0026nbsp;\u0026nbsp;\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Slow Slip Event\u003c/p\u003e\n\u003cp\u003eEnKF\u0026nbsp;\u0026nbsp;\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Ensemble Kalman Filter\u003c/p\u003e\n\u003cp\u003eRSF\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;rate- and state-friction\u003c/p\u003e\n\u003cp\u003eGEONET\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Global Navigation Satellite System Earth Observation Network System\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable. No datasets were used in the present study.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe present study was supported by JSPS KAKENHI grants JP20K14574 and JP21K03694.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors\u0026apos; contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eM.O. designed the research, developed the method, performed the numerical experiments, and drafted the manuscript. All authors discussed the results and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe present study was supported by JSPS KAKENHI grants JP20K14574 and JP21K03694. This research was conducted using the FUJITSU Supercomputer PRIMEHPC FX1000 and FUJITSU Server PRIMERGY GX2570 (Wisteria/BDEC-01) at the Information Technology Center, The University of Tokyo.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003cp\u003eByrd, R. H., Lu, P., Nocedal, J., Zhu, C. (1995). A limited memory algorithm for bound constrained optimization.\u0026nbsp;SIAM Journal on scientific computing,\u0026nbsp;16(5), 1190-1208.\u0026nbsp;\u003cbr\u003e\u0026nbsp;\u003ca href=\"https://doi.org/10.1137/0916069\"\u003ehttps://doi.org/10.1137/0916069\u003c/a\u003e.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDieterich, J. H. (1979) Modeling of rock friction: 1. Experimental results and constitutive equations. J. Geophys. 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Report, NAM-11, EECS Department, Northwestern University.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"earth-planets-and-space","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"epsp","sideBox":"Learn more about [Earth, Planets and Space](http://earth-planets-space.springeropen.com)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/epsp/default.aspx","title":"Earth, Planets and Space","twitterHandle":"@SpringerOpen","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Data assimilation, adjoint method, slow slip event, Bungo Channel, frictional parameters, frictional strength, twin experiment, prediction","lastPublishedDoi":"10.21203/rs.3.rs-5343128/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5343128/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Data assimilation (DA) has tried to incorporate GNSS data into physics-based fault slip models to estimate frictional properties and predict future slip evolution on faults. For unstable slip events such as ordinary fast-slip earthquakes and slow slip events (SSEs), accurately estimating the frictional strength, as well as the frictional parameters, is crucial for reliable slip prediction. However, the frictional strength has not been directly observed, and thus, previous DA studies have often assumed a steady-state strength value for the initial strength to estimate the frictional parameters, which limits the accuracy of long-term slip predictions. In the present study, we propose a new adjoint-based DA method that estimates an appropriate initial frictional strength along with the frictional parameters for assimilating long-term SSEs. The key idea is to impose an additional constraint on DA that assumes the current SSE will recur periodically, though the exact interval is unknown. This approach reflects the observed recurring nature of SSEs. This new method is validated through numerical experiments focusing on long-term Bungo Channel SSEs in southwest Japan. The results demonstrate that our proposed method provides reasonable estimates for both the initial strength and the frictional parameters, enabling accurate predictions of slip evolution and the timing of subsequent SSEs, along with determining the unknown recurrence interval. The method proves effective even with data windows shorter than the recurrence interval, overcoming the limitations of previous DA methods.","manuscriptTitle":"Development of an adjoint-based data assimilation method toward predicting SSE evolution: Two-step optimization of frictional parameters and initial strength on the fault","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-15 15:05:26","doi":"10.21203/rs.3.rs-5343128/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revision","date":"2025-01-28T19:58:43+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2024-11-06T13:13:37+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-11-06T08:50:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-10-30T02:48:32+00:00","index":"","fulltext":""},{"type":"submitted","content":"Earth, Planets and Space","date":"2024-10-27T20:42:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"earth-planets-and-space","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"epsp","sideBox":"Learn more about [Earth, Planets and Space](http://earth-planets-space.springeropen.com)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/epsp/default.aspx","title":"Earth, Planets and Space","twitterHandle":"@SpringerOpen","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1dd09ea8-ce6d-4474-a5f1-9b6695872195","owner":[],"postedDate":"November 15th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-06-16T16:07:28+00:00","versionOfRecord":{"articleIdentity":"rs-5343128","link":"https://doi.org/10.1186/s40623-025-02222-9","journal":{"identity":"earth-planets-and-space","isVorOnly":false,"title":"Earth, Planets and Space"},"publishedOn":"2025-06-10 15:57:20","publishedOnDateReadable":"June 10th, 2025"},"versionCreatedAt":"2024-11-15 15:05:26","video":"","vorDoi":"10.1186/s40623-025-02222-9","vorDoiUrl":"https://doi.org/10.1186/s40623-025-02222-9","workflowStages":[]},"version":"v1","identity":"rs-5343128","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5343128","identity":"rs-5343128","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}