Present day crustal displacement rates can be accurately observed at stations of global navigation satellite system (GNSS), and crustal deformation has been investigated by estimating strain-rate fields from discrete GNSS data. For this purpose, a modified least-square inversion method was proposed by Shen et al. (J Geophys Res 101:27957–27980, 1996). This method offers a simple formulation for simultaneously estimating smooth velocity and strain-rate fields from GNSS data, and it has contributed to clarify crustal deformation fields in many regions all over the world. However, we notice three theoretical points to be examined when we apply the method: mathematical inconsistency between estimated velocity and strain-rate fields, difficulty in objectively determining the optimal value of a hyperparameter that controls smoothness, and inappropriate estimation of uncertainty. In this study, we propose a method of basis function expansion with Akaike’s Bayesian information criterion (ABIC), which overcomes the above difficulties. Application of the two methods to GNSS data in Japan reveals that the inconsistency in the method of Shen et al. is generally insignificant, but could be clear in regions with sparser observation stations such as in islet areas. The method of basis function expansion with ABIC shows a significantly better performance than the method of Shen et al. in terms of the trade-off curve between the residual of fitting and the roughness of velocity field. The estimated strain-rate field with the basis function expansion clearly exhibits a low strain-rate zone in the forearc from the southern Tohoku district to central Japan. We also find that the Ou Backbone Range has several contractive spots around active volcanoes and that these locations well correspond to the subsidence areas detected by InSAR after the 2011 Tohoku-oki earthquake. Thus, the method of basis function expansion with ABIC would serve as an effective tool for estimating strain-rate fields from GNSS data.

The movement and deformation of the Earth’s crust and upper mantle provide critical insights into the evolution of earthquake processes and future earthquake potentials. Crustal deformation can be modeled by dislocation models that represent earthquake faults in the crust as defects in a continuum medium. In this study, we propose a physics-informed deep learning approach to model crustal deformation due to earthquakes. Neural networks can represent continuous displacement fields in arbitrary geometrical structures and mechanical properties of rocks by incorporating governing equations and boundary conditions into a loss function. The polar coordinate system is introduced to accurately model the displacement discontinuity on a fault as a boundary condition. We illustrate the validity and usefulness of this approach through example problems with strike-slip faults. This approach has a potential advantage over conventional approaches in that it could be straightforwardly extended to high dimensional, anelastic, nonlinear, and inverse problems.

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Data-driven machine-learning approaches are being increasingly applied to construct empirical ground-motion models (GMMs). It is a standard practice to divide observational records into learning and test datasets to correctly evaluate the predictive performance of a developed model. However, in this study, we show that division based on records or earthquakes is inappropriate for evaluating the generalization performance on recorded sites when GMMs include site-condition proxies as input variables. Complex models exhibit small residuals at sites used in the training process, but exhibit large residuals at new sites owing to overfitting to the trained sites. As a simple solution, we propose a neural network model that has monotonic dependence on some of the input variables. The model successfully obtains the generalization performance on recorded sites, although it lacks ability to represent oversaturation with input variables suggested in extreme ground-motion ranges. Therefore, alternative methods should be investigated to develop robust data-driven models under general conditions. Dividing the sites into learning and test data would play a fundamental role in developing such robust models.

The effect of organic stabilizers on the structure of supported Pt-Cu nanoparticles was investigated for liquid-phase reduction methods. The crystallite size and copper content of Pt-Cu alloy nanoparticles decreased as the number of carboxylate groups in an employed stabilizer molecule increased (acetate [ succinate [ citrate). The steric and inductive effects of carboxylate groups suppress the reduction and crystal growth of platinum to increase the chance for copper to make alloy with platinum, but at the same time they destabilize metallic copper resulting in a segregation of copper as oxides. In a radiolytic synthesis method, the size and Cu content in Pt-Cu nanoparticles were similarly reduced in the presence of citrate, but the copresent reduction enhancer also had a strong influence. The final particle structure was controlled by the kinetics of reduction of metal precursors and the thermodynamics of metal-stabilizer complexes.