A Bayesian inference framework for fault slip distributions based on ensemble modeling of the uncertainty of underground structure - With a focus on uncertain fault dip

Summary The model prediction errors that originate from the uncertainty of underground structure is often a major contributor of the errors between the data and the model predictions in fault slip estimation using geodetic or seismic waveform data. However, most studies on slip inversions either neglect the model prediction errors or do not distinguish them from observation errors. Several methods that explicitly incorporated the model prediction errors in slip estimation, which has been proposed in the past decade, commonly assumed a Gaussian distribution for the stochastic property of the model prediction errors to simplify the formulation. Moreover, the information on both the slip distribution and the underground structure is expected to be successfully extracted from the data by incorporating the stochastic property of the model prediction errors. In this study, we propose a novel flexible Bayesian inference framework for estimating fault slips that can accurately incorporate non-Gaussian model prediction errors. This method considers the uncertainty of the underground structure, including fault geometry, based on the ensemble modeling of the uncertainty of Green’s functions. Furthermore, the framework allows the estimation of the posterior probability density function (PDF) of the parameters of the underground structure, by calculating the likelihood of each sample in the ensemble. We performed numerical experiments for estimating the slip deficit rate (SDR) distribution on a 2D thrust fault using synthetic data of surface displacement rates, which is the simplest problem setting that can essentially demonstrates the fundamental idea and validate the advantage of the proposed method. In the experiments, the dip angle of the fault plane was the parameter used to characterize the underground structure. The proposed method succeeded in estimating a posterior PDF of SDR that is consistent with the true one, despite the uncertain and inaccurate information of the dip angle. In addition, the method could estimate a posterior PDF of the dip angle that has a strong peak near the true angle. In contrast, the estimation results obtained using a conventional approach, which introduces regularization based on smoothing constraints and does not explicitly distinguish the prediction and observation errors, included a significant amount of bias, which was not noticed in the results obtained using the proposed method. The estimation results obtained using different settings of the parameters suggested that inaccurate prior information of the underground structure with a small variance possibly results in significant bias in the estimated PDFs, particularly the posterior PDFs for SDR, those for the underground structure, and the posterior predicted PDF of the displacement rates. The distribution shapes of the model prediction errors for the representative model parameters in certain observation points are significantly asymmetric with large absolute values of the sample skewness, suggesting that they would not be well-modeled by Gaussian approximations.

Computational Design of Microstructures With Stochastic Property Closures

The present work addresses a stochastic computational solution to define the property closures of polycrystalline materials under uncertainty. The uncertainty in material systems arises from the natural stochasticity of the microstructures as a result of the fluctuations in deformation processes. The microstructural uncertainty impacts the performance of engineering components by causing unanticipated anisotropy in properties. We utilize an analytical uncertainty quantification algorithm to describe the microstructural stochasticity and model its propagation on the volume-averaged material properties. The stochastic solution will be integrated into linear programming to generate the property closure that shows all possible values of the volume-averaged material properties under the uncertainty. We demonstrate example applications for stiffness parameters of α-Titanium, and multi-physics parameters (stiffness, yield strength, magnetostrictive strain) of Galfenol. Significant differences observed between stochastic and deterministic closures imply the importance of considering the microstructural uncertainty when modeling and designing materials.

Multistep-Ahead Prediction of Urban Traffic Flow Using GaTS Model

The mathematical models for traffic flow have been widely investigated for a lot of application, like planning transportation and easing traffic pressure by using statistics and machine learning methods. However, there remains a lot of challenging problems for various reasons. In this research, we mainly focused on three issues: (a) the data of traffic flow are nonnegative, and hereby, finding a proper probability distribution is essential; (b) the complex stochastic property of the traffic flow leads to the nonstationary variance, i.e., heteroscedasticity; and (c) the multistep-ahead prediction of the traffic flow is often of poor performance. To this end, we developed a Gamma distribution-based time series (GaTS) model. First, we transformed the original traffic flow observations into nonnegative real-valued data by using the Box-Cox transformation. Then, by specifying the generalized linear model with the Gamma distribution, the mean and variance of the distribution are regressed by the past data and homochronous terms, respectively. A Bayesian information criterion is used to select the proper Box-Cox transformation coefficients and the optimal model structures. Finally, the proposed model is applied to the urban traffic flow data achieved from Dalian city in China. The results show that the proposed GaTS has an excellent prediction performance and can represent the nonstationary stochastic property well.

Computational Design of Microstructures With Stochastic Property Closures

The present work addresses a stochastic computational solution to define the property closures of polycrystalline materials under uncertainty. The uncertainty in material systems arises from the natural stochasticity of the microstructures and the variations in deformation processes, and impacts the performance of engineering components by causing unanticipated anisotropy in properties. We utilize an analytical uncertainty quantification algorithm to describe the microstructural stochasticity and model its propagation to the volume-averaged material properties. The stochastic solution will be integrated into linear programming to generate the property closure that shows all possible values of the volume-averaged material properties under the uncertainty. We demonstrate example applications for stiffness parameters of a-Titanium, and multi-physics parameters (stiffness, yield strength, magnetostrictive strain) of Galfenol. Significant differences observed between stochastic and deterministic closures imply the importance of considering the microstructural uncertainty when modeling and designing materials.

Numerical Simulation for DC PD in Void

A numerical simulation for DC PD in void is put forward based on the PD physical process. The finite difference method is used to calculate the electric field distribution, and both of the stochastic property and the accumulation of the charge after PD on the void surfaces are considering in the model. The time of PD occurring, the amount of discharge and the voltage across the void are calculated. Meanwhile, the relationship between the DC voltage and the PD time interval or repetition rate is also simulation, the results show that with the increase of the DC voltage, the PD interval corresponding decreases exponentially and the repetition rate increases exponentially.

A Priching Model for Inflation-indexed Bonds

This paper derives the theoretical price of nominal bonds and inflation-indexed bonds through extracting the factors, which are assumed that their stochastic property follows the standard O-U process, in the term structure of nominal interest rates and yields of inflation-indexed bonds by the Principal Component Analysis (PCA). In particular, through reflecting the complex structure of inflation-indexed bonds by accurately applying theoretical price, it brought differentiation from other literatures, and applied this pricing model to Japanese Government Inflation-indexed Bond (JGB) data. The empirical results of above model show that explanation of time series and cross section of Janpan's real and nominal interest rates were outstanding and was found that Fisher hypothesis was rejected in further