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.     

A structured and constrained Total Least-Squares solution with cross-covariances

A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously.     

An algorithm for the minimum-length least-squares solution of a set of observation equations

Summary A new computer-oriented algorithm GSO is presented for solving overdetermined systems of linear observation equations according to the principle of the least-squares method. The matrix of the system of observation equations may be of deficient rank. In this case the algorithm leads to the vector of unknowns with a minimum Euclidean norm. Alternatively, it is possible to minimize the norm of a subvector formed by a selected group of unknowns. The weight coefficient matrix, corresponding to the vector (subvector) of unknows, has the least possible trace. The algorithm GSO is based on the Gram-Schmidt Orthogonalization of suitably defined augmented matrices. The establishing and solving of normal equations is not necessary. Apart from the unknowns and residuals, GSO also determines the factorized weight coefficient matrices of the adjusted values.Presented at the I.A.G. International Symposium on Optimization of Design and Computation of Control Networks, Sopron, Hungary 1977.     

CT系统的能谱估计及射束硬化校正算法

本文给出对CT系统X射线能谱的估计算法和单成分被测物体CT成像的射束硬化校正算法，并给出和分析了数值模拟结果。     

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.     

FHD分量质子磁力仪标定技术

FHD分量质子磁力仪利用补偿测量方法（Nelson方法）测量水平分量,利用偏置测量方法（Sensor方法）测量磁偏角的相对变化,由于装置存在误差和长期漂移,会对观测数据的长期稳定性产生影响,为了消除或校正这些影响,对观测误差进行分析,提出对相关影响因素进行校正的方法和步骤,并提出观测系统漂移量的判断和估算方法,以及在标定过程中应注意的问题,对FHD质子磁力仪的标定效果和质量起到较好的监控作用.     

Reverse-time migration and amplitude correction in the angle-domain based on Poynting vector

We propose a method based on the Poynting vector that combines angle-domain imaging and image amplitude correction to overcome the shortcomings of reverse-time migration that cannot handle different angles during wave propagation. First, the local image matrix (LIM) and local illumination matrix are constructed, and the wavefield propagation directions are decomposed. The angle-domain imaging conditions are established in the local imaging matrix to remove low-wavenumber artifacts. Next, the angle-domain common image gathers are extracted and the dip angle is calculated, and the amplitude-corrected factors in the dip angle domain are calculated. The partial images are corrected by factors corresponding to the different angles and then are superimposed to perform the amplitude correction of the final image. Angle-domain imaging based on the Poynting vector improves the computation efficiency compared with local plane-wave decomposition. Finally, numerical simulations based on the SEG/EAGE velocity model are used to validate the proposed method.     

Simultaneous identification of parameter,initial condition,and boundary condition in groundwater modelling

In this paper, we perform an inverse method to simultaneously estimate aquifer parameters, initial condition, and boundary conditions in groundwater modelling. The parameter estimation is extended to a complete inverse problem that makes the calibrated groundwater flow model more realistic. The adjoint state method, the gradient search method, and the least square error algorithm are combined to build the optimization procedure. Horizontal two‐dimensional groundwater flow in a confined aquifer is exemplified to demonstrate the correlation between unknowns, the contribution of observation, as well as the suitability of applying the inverse method. The correlation analysis shows the connection between storage coefficient and initial condition. Besides, transmissivity and boundary conditions are also highly correlated. More observations at different location and time are necessary to provide sufficient information. A time series of unsteady head is requested for estimation of storage coefficient and initial condition. Observation near boundary is very effective for boundary condition estimation. The observation at pumping well mostly contributes to the estimation of transmissivity. According to all observations, it is possible to identify parameters, initial condition, and boundary condition simultaneously. Furthermore, the results not only illustrate the traditional assumption of known boundary condition but also initial condition, which may cause an incorrect estimation. Copyright © 2009 John Wiley & Sons, Ltd.     

广义反射-透射系数算法的无量纲化

水平层状介质中瑞利波频散曲线的计算一直是受关注的问题.陈晓非提出的广义反射-透射系数方法虽然具备很好的精确性和稳定性,但是算法中起决定作用的矩阵E存在有量纲元素,并且不同元素之间的数量级差异很大.本文作者通过把广义反射-透射系数算法无量纲化,得到了简洁的瑞利波频散函数的公式体系.分别计算了改进后算法和原算法中矩阵E的最大模元素与最小模元素的比值ε,发现前者的矩阵E中最大模元素与最小模元素最多只相差1个数量级,而后者的矩阵E中最大模元素与最小模元素之间最大达11个数量级的差异,尽管后者可以通过选取变量的单位减小ε的值,但新算法精度更高,从而完善了利用广义反射-透射系数方法求解瑞利波频散曲线问题的理论和算法.     

An improved constraint method in optimal estimation of CO_2 from GOSAT SWIR observations

We propose an algorithm that combines a pre-processing step applied to the a priori state vector prior to retrievals,with the modified damped Newton method(MDNM),to improve convergence.The initial constraint vector pre-processing step updates the initial state vector prior to the retrievals if the algorithm detects that the initial state vector is far from the true state vector in extreme cases where there are CO_2 emissions.The MDNM uses the Levenberg-Marquardt parameter y,which ensures a positive Hessian matrix,and a scale factor a,which adjusts the step size to optimize the stability of the convergence.While the algorithm iteratively searches for an optimized solution using observed spectral radiances,MDNM adjusts parameters y and a to achieve stable convergence.We present simulated retrieval samples to evaluate the performance of our algorithm and comparing it to existing methods.The standard deviation of our retrievals adding random noise was less than 3.8 ppmv.After pre-processing the initial estimate when it was far from the true value,the CO_2 retrieval errors in the boundary layers were within 1.2 ppmv.We tested the MDNM algorithm's performance using GOSAT L1b data with cloud screening.Our preliminary validations comparing the results to TCCON FTS measurements showed that the average bias was less than 1.8 ppm and the correlation coefficient was approximately 0.88,which was larger than for the GOSAT L2 product.     

A general weighted total Kalman filter algorithm with numerical evaluation

An applicable algorithm for Total Kalman Filter (TKF) approach is proposed. Meanwhile, we extend it to the case in which we can consider arbitrary weight matrixes for the observation vector, the random design matrix and possible correlation between them. Also the updated dispersion matrix of the predicted unknown is given. This approach makes use of condition equations and straightforward variance propagation rules. It is applicable to data fusion within a dynamic errors-in-variables (DEIV) model, which usually appears in the determination of the position and attitude of mobile sensors. Then, we apply for the first time the TKF algorithm and its extended version named WTKF to a DEIV model and compare the results. The results show the efficiency of the proposed WTKF algorithm. In particular in the case of large weights, WTKF shows approximately 25% improvement in contrast to TKF approach.     

基于BTTB矩阵的快速高精度三维磁场正演

本文改进了一种快速、高精度空间域三维正演算法,用来计算地下场源在水平观测面产生的磁异常ΔT场及其梯度场,以解决传统空间域正演计算效率低的问题.算法采用长方体对场源区域进行剖分,观测点与场源剖分单元体中心点在水平面上的投影重合.改进的算法具有以下三个特点:(1)采用无解析奇点的解析解公式计算磁异常,保证计算精度.(2)通...     

An Estimation Method of Pore Structure and Mineral Moduli Based on Kuster-Toksöz (KT) Model and Biot’s Coefficient

Pore structure and mineral matrix elastic moduli are indispensable in rock physics models. We propose an estimation method of pore structure and mineral moduli based on Kuster-Toksöz model and Biot’s coefficient. In this technique, pore aspect ratios of five different scales from 100 to 10^?4 are considered, Biot’s coefficient is used to determine bounds of mineral moduli, and an estimation procedure combined with simulated annealing (SA) algorithm to handle real logs or laboratory measurements is developed. The proposed method is applied to parameter estimations on 28 sandstone samples, the properties of which have been measured in lab. The water saturated data are used for estimating pore structure and mineral moduli, and the oil saturated data are used for testing these estimated parameters through fluid substitution in Kuster-Toksöz model. We then compare fluid substitution results with lab measurements and find that relative errors of P-wave and S-wave velocities are all less than 5%, which indicates that the estimation results are accurate.     

An improved constraint method in optimal estimation of CO<Subscript>2</Subscript> from GOSAT SWIR observations

We propose an algorithm that combines a pre-processing step applied to the a priori state vector prior to retrievals, with the modified damped Newton method (MDNM), to improve convergence. The initial constraint vector pre-processing step updates the initial state vector prior to the retrievals if the algorithm detects that the initial state vector is far from the true state vector in extreme cases where there are CO₂ emissions. The MDNM uses the Levenberg-Marquardt parameter γ, which ensures a positive Hessian matrix, and a scale factor α, which adjusts the step size to optimize the stability of the convergence. While the algorithm iteratively searches for an optimized solution using observed spectral radiances, MDNM adjusts parameters γ and α to achieve stable convergence. We present simulated retrieval samples to evaluate the performance of our algorithm and comparing it to existing methods. The standard deviation of our retrievals adding random noise was less than 3.8 ppmv. After pre-processing the initial estimate when it was far from the true value, the CO₂ retrieval errors in the boundary layers were within 1.2 ppmv. We tested the MDNM algorithm’s performance using GOSAT L1b data with cloud screening. Our preliminary validations comparing the results to TCCON FTS measurements showed that the average bias was less than 1.8 ppm and the correlation coefficient was approximately 0.88, which was larger than for the GOSAT L2 product.     

三维叠后差分偏移的因子分解法

提出一种三维叠后偏移的一步差分方法，称为因子分解法．差分格式是二阶精度的隐式格式，求解方法与通常一步差分偏移不同，不采用分步法交替求解。ｘ－ｙ方向的二维问题，而是采用因子分解法，将求解的差分方程分解为向前因子和向后因子，从而在一次扫描中同时完成ｘ－ｙ方向的正递归和反递归．为了抑制边界反射，采用了吸收边界条件，给出了理论合成记录和实际记录的偏移结果，数值试验表明该方法具有较好的精度和较高的计算效率．     

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.     

Total least-squares adjustment of condition equations

The usual least-squares adjustment within an Errors-in-Variables (EIV) model is often described as Total Least-Squares Solution (TLSS), just as the usual least-squares adjustment within a Random Effects Model (REM) has become popular under the name of Least-Squares Collocation (without trend). In comparison to the standard Gauss-Markov Model (GMM), the EIV-Model is less informative whereas the REM is more informative. It is known under which conditions exactly the GMM or the REM can be equivalently replaced by a model of condition equations or, more generally, by a Gauss-Helmert Model. Similar equivalency conditions are, however, still unknown for the EIV-Model once it is transformed into such a model of condition equations. In a first step, it is shown in this contribution how the respective residual vector and residual matrix look like if the TLSS is applied to condition equations with a random coefficient matrix to describe the transformation of the random error vector. The results are demonstrated using a numeric example which shows that this approach may be valuable in its own right.     

Simulation of intrinsic random fields of order <Emphasis Type="Italic">k</Emphasis> with a continuous spectral algorithm

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance.     

A least-squares penalty method algorithm for inverse problems of steady-state aquifer models

Based on the generalized Gauss–Newton method, a new algorithm to minimize the objective function of the penalty method in (Bentley LR. Adv Wat Res 1993;14:137–48) for inverse problems of steady-state aquifer models is proposed. Through detailed analysis of the “built-in” but irregular weighting effects of the coefficient matrix on the residuals on the discrete governing equations, a so-called scaling matrix is introduced to improve the great irregular weighting effects of these residuals adaptively in every Gauss–Newton iteration. Numerical results demonstrate that if the scaling matrix equals the identity matrix (i.e., the irregular weighting effects of the coefficient matrix are not balanced), our algorithm does not perform well, e.g., the computation cost is higher than that of the traditional method, and what is worse is the calculations fail to converge for some initial values of the unknown parameters. This poor situation takes a favourable turn dramatically if the scaling matrix is slightly improved and a simple preconditioning technique is adopted: For naturally chosen simple diagonal forms of the scaling matrix and the preconditioner, the method performs well and gives accurate results with low computational cost just like the traditional methods, and improvements are obtained on: (1) widening the range of the initial values of the unknown parameters within which the minimizing iterations can converge, (2) reducing the computational cost in every Gauss–Newton iteration, (3) improving the irregular weighting effects of the coefficient matrix of the discrete governing equations. Consequently, the example inverse problem in Bentley (loc. cit.) is solved with the same accuracy, less computational effort and without the regularization term containing prior information on the unknown parameters. Moreover, numerical example shows that this method can solve the inverse problem of the quasilinear Boussinesq equation almost as fast as the linear one.In every Gauss–Newton iteration of our algorithm, one needs to solve a linear least-squares system about the corrections of both the parameters and the groundwater heads on all the discrete nodes only once. In comparison, every Gauss–Newton iteration of the traditional method has to solve the discrete governing equations as many times as one plus the number of unknown parameters or head observation wells (Yeh WW-G. Wat Resour Res 1986;22:95–108).All these facts demonstrate the potential of the algorithm to solve inverse problems of more complicated non-linear aquifer models naturally and quickly on the basis of finding suitable forms of the scaling matrix and the preconditioner.