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[SCG56-P03] Estimation of a Strain-rate Field from Geodetic Data Using a Sparse Modeling
Keywords:Strain-rate field, Sparse modeling, Arima-Takatsuki Fault Zone
We observe active seismicity and crustal deformation in Japanese Islands. Since earthquake occurrences are closely related to strain concentration, it is important to accurately estimate a strain-rate field. Many studies have been conducted to estimate spatially continuous deformation fields from spatially discrete geodetic data such as GNSS. For example, Okazaki et al. (2021) estimated the strain-rate field in Japan by taking the smoothness of the velocity field as a priori information (L2 regularization). In this case, however, it is difficult to estimate localized strain rates nearby fault zones due to the smoothness constraint. On the other hand, Nishimura et al. (2018) divided western Japan into 14 blocks and found localized strain rates along active fault zones selected as block boundaries. However, there is arbitrariness in the selection of active faults used as the block boundary. In other words, previous studies have difficulties in properly estimating localized strain rates near fault zones. To tackle this problem, we introduce a sparsity constraint that corresponds with L1-norm minimization. In this study, for simplicity, we make the formulation for 1D space.
We express a velocity field by the superposition of cubic B-spline functions. The objective function we define is composed of 3 terms: residuals between observed and estimated velocities, L1-norm and L2-norm of second derivatives of velocity fields. The L1-norm and L2-norm correspond to sparsity and smoothness constraints, respectively. This objective function is called ‘Elastic net’, which is a type of sparse modeling. The weights of these terms are specified by hyperparameters. To determine the optimum values of the hyperparameters, we use the leave-one-out cross-validation method. After determining the optimum values of hyperparameters, we obtain the optimum values of the expansion coefficients of the cubic B-spline functions by minimizing the objective function.
To investigate the validity and limitation of the proposed method, we performed a synthetic test. Deformation fields were given by steady slip on a buried strike-slip fault. We took a profile perpendicular to the fault strike, and put observation points with an approximately constant interval. We generated synthetic velocity data by adding a Gaussian error to the true velocity at each observation point. On this setting, we investigated how well the proposed method reproduces the true strain-rate profile by changing the locking depth and the slip rate of the fault and the magnitude of the error. As a result, we found that (1) the proposed method outperformed the L2 regularization, when the error (variance) was less than 0.2 mm2/yr2; (2) the outperformance is clearer for larger intervals of observation points.
Next, we applied the proposed method to GNSS data across the Arima-Takatsuki Fault Zone. The data used are the daily coordinates of the GSI F5 solution [Altamimi et al., 2016] for 10 years since January 1, 2001. The observation error for most stations was about or less than 0.2 mm2/yr2. The proposed method estimated the peak strain rate near the fault by 10% larger than the L2 regularization. This indicates that a more localized strain rate near the fault was obtained by the proposed method. Fitting the analytical solution to the estimated strain-rate profile, the locking depth was obtained to be 24 km, which was shallower than 29 km obtained by the L2 regularization.
We express a velocity field by the superposition of cubic B-spline functions. The objective function we define is composed of 3 terms: residuals between observed and estimated velocities, L1-norm and L2-norm of second derivatives of velocity fields. The L1-norm and L2-norm correspond to sparsity and smoothness constraints, respectively. This objective function is called ‘Elastic net’, which is a type of sparse modeling. The weights of these terms are specified by hyperparameters. To determine the optimum values of the hyperparameters, we use the leave-one-out cross-validation method. After determining the optimum values of hyperparameters, we obtain the optimum values of the expansion coefficients of the cubic B-spline functions by minimizing the objective function.
To investigate the validity and limitation of the proposed method, we performed a synthetic test. Deformation fields were given by steady slip on a buried strike-slip fault. We took a profile perpendicular to the fault strike, and put observation points with an approximately constant interval. We generated synthetic velocity data by adding a Gaussian error to the true velocity at each observation point. On this setting, we investigated how well the proposed method reproduces the true strain-rate profile by changing the locking depth and the slip rate of the fault and the magnitude of the error. As a result, we found that (1) the proposed method outperformed the L2 regularization, when the error (variance) was less than 0.2 mm2/yr2; (2) the outperformance is clearer for larger intervals of observation points.
Next, we applied the proposed method to GNSS data across the Arima-Takatsuki Fault Zone. The data used are the daily coordinates of the GSI F5 solution [Altamimi et al., 2016] for 10 years since January 1, 2001. The observation error for most stations was about or less than 0.2 mm2/yr2. The proposed method estimated the peak strain rate near the fault by 10% larger than the L2 regularization. This indicates that a more localized strain rate near the fault was obtained by the proposed method. Fitting the analytical solution to the estimated strain-rate profile, the locking depth was obtained to be 24 km, which was shallower than 29 km obtained by the L2 regularization.