Data-snooping procedure applied to errors-in-variables models

The theory of Baarda’s data snooping — normal and F tests respectively based on the known and unknown posteriori variance — is applied to detect blunders in errors-invariables (EIV) models, in which gross errors are in the vector of observations and/or in the coefficient matrix. This work is a follow-up to an earlier work in which we presented the formulation of the weighted total least squares (WTLS) based on the standard least squares theory. This method allows one to directly apply the existing body of knowledge of the least squares theory to the errors-in-variables models. Among those applications, data snooping methods in an EIV model are of particular interest, which is the subject of discussion in the present contribution. This paper generalizes the Baarda’s data snooping procedure of the standard least squares theory to an EIV model. Two empirical examples, a linear regression model and a 2-D affine transformation, using simulated and real data are presented to show the efficacy of the presented formulation. It is highlighted that the method presented is capable of detecting outlying equations (rather than outlying observations) in a straightforward manner. Further, the WTLS method can be used to handle different TLS problems. For example, the WTLS problem for the conditions and mixed models, the WTLS problem subject to constraints and variance component estimation for an EIV model can easily be established. These issues are in progress for future publications.     

A robust weighted total least squares algorithm and its geodetic applications

Total least squares (TLS) can solve the issue of parameter estimation in the errors-invariables (EIV) model, however, the estimated parameters are affected or even severely distorted when the observation vector and coefficient matrix are contaminated by gross errors. Currently, the use of existing robust TLS (RTLS) methods for the EIV model is unreasonable. Original residuals are directly used in most studies to construct the weight factor function, thus the robustness for the structure space is not considered. In this study, a robust weighted total least squares (RWTLS) algorithm for the partial EIV model is proposed based on Newton-Gauss method and the equivalent weight principle of general robust estimation. The algorithm utilizes the standardized residuals to construct the weight factor function and employs the median method to obtain a robust estimator of the variance component. Therefore, the algorithm possesses good robustness in both the observation and structure spaces. To obtain standardized residuals, we use the linearly approximate cofactor propagation law for deriving the expression of the cofactor matrix of WTLS residuals. The iterative procedure and precision assessment approach for RWTLS are presented. Finally, the robustness of RWTLS method is verified by two experiments involving line fitting and plane coordinate transformation. The results show that RWTLS algorithm possesses better robustness than the general robust estimation and the robust total least squares algorithm directly constructed with original residuals.     

Bayesian inference for the Errors-In-Variables model

We discuss the Bayesian inference based on the Errors-In-Variables (EIV) model. The proposed estimators are developed not only for the unknown parameters but also for the variance factor with or without prior information. The proposed Total Least-Squares (TLS) estimators of the unknown parameter are deemed as the quasi Least-Squares (LS) and quasi maximum a posterior (MAP) solution. In addition, the variance factor of the EIV model is proven to be always smaller than the variance factor of the traditional linear model. A numerical example demonstrates the performance of the proposed solutions.     

火山Mogi模型形变反演的虚拟观测法及方差分量估计

针对Mogi模型垂直位移与水平位移联合反演中的病态问题,改进火山形变总体最小二乘(Total Least Squares,TLS)联合反演的虚拟观测法,并使用方差分量估计(Variance Components Estimation,VCE)方法确定病态问题的正则化参数.将附有先验信息的参数作为观测方程,与垂直位移和水平位移的观测方程联合解算,推导了三类观测方程联合反演的求解公式及基于总体最小二乘方差分量估计确定正则化参数的表达式,给出了算法的迭代流程.通过算例实验,研究了总体最小二乘联合反演的虚拟观测法在火山Mogi模型形变反演中的应用;算例结果表明,三类数据的联合平差及方差分量估计方法可以确定权比因子并得到修正后的压力源参数,具有一定的实际参考价值.     

Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares

The errors-in-variables (EIV) model is a nonlinear model, the parameters of which can be solved by singular value decomposition (SVD) method or the general iterative algorithm. The existing formulae for covariance matrix of total least squares (TLS) parameter estimates don’t fully consider the randomness of quantities in iterative algorithm and the biases of parameter estimates and residuals. In order to reflect more reasonable precision information for TLS adjustment, the derivative-free unscented transformation with scaled symmetric sampling strategy, i.e. scaled unscented transformation (SUT), is introduced and implemented. In this contribution, we firstly discuss the existing various solutions of TLS adjustment and covariance matrices of TLS parameter estimates and derive the general first-order approximate cofactor matrices of random quantities in TLS adjustment. Secondly, based on the combination of TLS iterative algorithm and calculation process of SUT, we design the two SUT algorithms to calculate the biases and the second-order approximate covariance matrices. Finally, the straight line fitting model and plane coordinate transformation model are used to demonstrate that applying SUT for precision estimation of TLS adjustment is feasible and effective.     

An algorithmic approach to the total least-squares problem with linear and quadratic constraints

Proper incorporation of linear and quadratic constraints is critical in estimating parameters from a system of equations. These constraints may be used to avoid a trivial solution, to mitigate biases, to guarantee the stability of the estimation, to impose a certain “natural” structure on the system involved, and to incorporate prior knowledge about the system. The Total Least-Squares (TLS) approach as applied to the Errors-In-Variables (EIV) model is the proper method to treat problems where all the data are affected by random errors. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear constraints and TLS problems with a quadratic constraint. In this contribution, a new algorithm is presented to solve TLS problems with both linear and quadratic constraints. The new algorithm is developed using the Euler-Lagrange theorem while following an optimization process that minimizes a target function. Two numerical examples are employed to demonstrate the use of the new approach in a geodetic setting.     

A weighted least-squares solution to a 3-D symmetrical similarity transformation without linearization

A weighted least-squares (WLS) solution to a 3-D non-linear symmetrical similarity transformation within a Gauss-Helmert (GH) model, and/or an errors-in-variables (EIV) model is developed, which does not require linearization. The geodetic weight matrix is the inverse of the observation dispersion matrix (second-order moment). We suppose that the dispersion matrices are non-singular. This is in contrast to Procrustes algorithm within a Gauss-Markov (GM) model, or even its generalized algorithms within the GH and/or EIV models, which cannot accept geodetic weights. It is shown that the errors-invariables in the source system do not affect the estimation of the rotation matrix with arbitrary rotational angles and also the geodetic weights do not participate in the estimation of the rotation matrix. This results in a fundamental correction to the previous algorithm used for this problem since in that algorithm, the rotation matrix is calculated after the multiplication by row-wise weights. An empirical example and a simulation study give insight into the efficiency of the proposed procedure.     

Variance-covariance component estimation for structured errors-in-variables models with cross-covariances

Studia Geophysica et Geodaetica - In this contribution, an iterative algorithm for variance-covariance component estimation based on the structured errors-in-variables (EIV) model is proposed. We...     

Effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data

This paper investigates the effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data. The reservoir parameters are related to electrical resistivity by the Poupon model and to elastic moduli and density by the Xu-White model. To handle uncertainty in the rock-physics models, we consider their outputs to be random functions with modes or means given by the predictions of those rock-physics models and we consider the parameters of the rock-physics models to be random variables defined by specified probability distributions. Using a Bayesian framework and Markov Chain Monte Carlo sampling methods, we are able to obtain estimates of reservoir parameters and information on the uncertainty in the estimation. The developed method is applied to a synthetic case study based on a layered reservoir model and the results show that uncertainty in both rock-physics models and in their parameters may have significant effects on reservoir parameter estimation. When the biases in rock-physics models and in their associated parameters are unknown, conventional joint inversion approaches, which consider rock-physics models as deterministic functions and the model parameters as fixed values, may produce misleading results. The developed stochastic method in this study provides an integrated approach for quantifying how uncertainty and biases in rock-physics models and in their associated parameters affect the estimates of reservoir parameters and therefore is a more robust method for reservoir parameter estimation.     

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

A fast method with tunable accuracy is proposed to estimate errors and uncertainties in pore-scale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another “equivalent” sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an open-source code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics.     

Increasing numerical efficiency of iterative solution for total least-squares in datum transformations

Cartesian coordinate transformation between two erroneous coordinate systems is considered within the Errors-In-Variables (EIV) model. The adjustment of this model is usually called the total Least-Squares (LS). There are many iterative algorithms given in geodetic literature for this adjustment. They give equivalent results for the same example and for the same user-defined convergence error tolerance. However, their convergence speed and stability are affected adversely if the coefficient matrix of the normal equations in the iterative solution is ill-conditioned. The well-known numerical techniques, such as regularization, shifting-scaling of the variables in the model, etc., for fixing this problem are not applied easily to the complicated equations of these algorithms. The EIV model for coordinate transformations can be considered as the nonlinear Gauss-Helmert (GH) model. The (weighted) standard LS adjustment of the iteratively linearized GH model yields the (weighted) total LS solution. It is uncomplicated to use the above-mentioned numerical techniques in this LS adjustment procedure. In this contribution, it is shown how properly diminished coordinate systems can be used in the iterative solution of this adjustment. Although its equations are mainly studied herein for 3D similarity transformation with differential rotations, they can be derived for other kinds of coordinate transformations as shown in the study. The convergence properties of the algorithms established based on the LS adjustment of the GH model are studied considering numerical examples. These examples show that using the diminished coordinates for both systems increases the numerical efficiency of the iterative solution for total LS in geodetic datum transformation: the corresponding algorithm working with the diminished coordinates converges much faster with an error of at least 10^-5 times smaller than the one working with the original coordinates.     

Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models

This paper concerns efficient uncertainty quantification techniques in inverse problems for Richards’ equation which use coarse-scale simulation models. We consider the problem of determining saturated hydraulic conductivity fields conditioned to some integrated response. We use a stochastic parameterization of the saturated hydraulic conductivity and sample using Markov chain Monte Carlo methods (MCMC). The main advantage of the method presented in this paper is the use of multiscale methods within an MCMC method based on Langevin diffusion. Additionally, we discuss techniques to combine multiscale methods with stochastic solution techniques, specifically sparse grid collocation methods. We show that the proposed algorithms dramatically reduce the computational cost associated with traditional Langevin MCMC methods while providing similar sampling performance.     

Soil acidity adaptive control problem

Problem of soil acidity regularization is modeled as stochastic adaptive control problem with a linear difference equation of the dynamics of a field pH level. Stochastic component in the equation represents an individual time variability of soil acidity of an elementary section. We use Bayesian approach to determine a posteriori probability density function of the unknown parameters of the stochastic transition process. The Kullback–Leibler information divergence is used as a measure of difference between true distribution and its estimation. Algorithm for the construction of an adaptive stabilizing control in such a linear control system is proposed in the paper. Numerical realization of the algorithm is represented for a problem of a field soil acidity control.     

Reliability‐based control algorithms for nonlinear hysteretic systems based on enhanced stochastic averaging of energy envelope

Current reliability‐based control techniques have been successfully applied to linear systems; however, incorporation of stochastic nonlinear behavior of systems in such control designs remains a challenge. This paper presents two reliability‐based control algorithms that minimize failure probabilities of nonlinear hysteretic systems subjected to stochastic excitations. The proposed methods include constrained reliability‐based control (CRC) and unconstrained reliability‐based control (URC) algorithms. Accurate probabilistic estimates of nonlinear system responses to stochastic excitations are derived analytically using enhanced stochastic averaging of energy envelope proposed previously by the authors. Convolving these demand estimates with capacity models yields the reliability of nonlinear systems in the control design process. The CRC design employs the first‐level and second‐level optimizations sequentially where the first‐level optimization solves the Hamilton–Jacobi–Bellman equation and the second‐level optimization searches for optimal objective function parameters to minimize the probability of failure. In the URC design, a single optimization minimizes the probability of failure by directly searching for the optimal control gain. Application of the proposed control algorithms to a building on nonlinear foundation has shown noticeable improvements in system performance under various stochastic excitations. The URC design appears to be the most optimal method as it reduced the probability of slight damage to 8.7%, compared with 11.6% and 19.2% for the case of CRC and a stochastic linear quadratic regulator, respectively. Under the Kobe ground motion, the normalized peak drift displacement with respect to stochastic linear quadratic regulator is reduced to 0.78 and 0.81 for the URC and CRC cases, respectively, at comparable control force levels. Copyright © 2017 John Wiley & Sons, Ltd.     

基于统计岩石物理模型的各向异性页岩储层参数反演

提出了各向异性页岩储层统计岩石物理反演方法.通过统计岩石物理模型建立储层物性参数与弹性参数的定量关系,使用测井数据及井中岩石物理反演结果作为先验信息,将地震阻抗数据定量解释为储层物性参数、各向异性参数的空间分布.反演过程在贝叶斯框架下求得储层参数的后验概率密度函数,并从中得到参数的最优估计值及其不确定性的定量描述.在此过程中综合考虑了岩石物理模型对复杂地下介质的描述偏差和地震数据中噪声对反演不确定性的影响.在求取最大后验概率过程中使用模拟退火优化粒子群算法以提高收敛速度和计算准确性.将统计岩石物理技术应用于龙马溪组页岩气储层,得到储层泥质含量、压实指数、孔隙度、裂缝密度等物性,以及各向异性参数的空间分布及相应的不确定性估计,为页岩气储层的定量描述提供依据.     

Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method

This paper presents a novel nonlinear finite element (FE) model updating framework, in which advanced nonlinear structural FE modeling and analysis techniques are used jointly with the extended Kalman filter (EKF) to estimate time‐invariant parameters associated to the nonlinear material constitutive models used in the FE model of the structural system of interest. The EKF as a parameter estimation tool requires the computation of structural FE response sensitivities (total partial derivatives) with respect to the material parameters to be estimated. Employing the direct differentiation method, which is a well‐established procedure for FE response sensitivity analysis, facilitates the application of the EKF in the parameter estimation problem. To verify the proposed nonlinear FE model updating framework, two proof‐of‐concept examples are presented. For each example, the FE‐simulated response of a realistic prototype structure to a set of earthquake ground motions of varying intensity is polluted with artificial measurement noise and used as structural response measurement to estimate the assumed unknown material parameters using the proposed nonlinear FE model updating framework. The first example consists of a cantilever steel bridge column with three unknown material parameters, while a three‐story three‐bay moment resisting steel frame with six unknown material parameters is used as second example. Both examples demonstrate the excellent performance of the proposed parameter estimation framework even in the presence of high measurement noise. Copyright © 2015 John Wiley & Sons, Ltd.     

The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology

This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.     

Facies-constrained prestack seismic probabilistic inversion driven by rock physics

Seismic Rock physics plays a bridge role between the rock moduli and physical properties of the hydrocarbon reservoirs. Prestack seismic inversion is an important method for the quantitative characterization of elasticity, physical properties, lithology and fluid properties of subsurface reservoirs. In this paper, a high order approximation of rock physics model for clastic rocks is established and one seismic AVO reflection equation characterized by the high order approximation(Jacobian and Hessian matrix) of rock moduli is derived. Besides, the contribution of porosity, shale content and fluid saturation to AVO reflectivity is analyzed. The feasibility of the proposed AVO equation is discussed in the direct estimation of rock physical properties. On the basis of this, one probabilistic AVO inversion based on differential evolution-Markov chain Monte Carlo stochastic model is proposed on the premise that the model parameters obey Gaussian mixture probability prior model. The stochastic model has both the global optimization characteristics of the differential evolution algorithm and the uncertainty analysis ability of Markov chain Monte Carlo model. Through the cross parallel of multiple Markov chains, multiple stochastic solutions of the model parameters can be obtained simultaneously, and the posterior probability density distribution of the model parameters can be simulated effectively. The posterior mean is treated as the optimal solution of the model to be inverted.Besides, the variance and confidence interval are utilized to evaluate the uncertainties of the estimated results, so as to realize the simultaneous estimation of reservoir elasticity, physical properties, discrete lithofacies and dry rock skeleton. The validity of the proposed approach is verified by theoretical tests and one real application case in eastern China.