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1.
Vahid Mahboub 《Studia Geophysica et Geodaetica》2016,60(2):195-209
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. 相似文献
2.
Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares 总被引：2，自引：0，他引：2
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. 相似文献
3.
以旋转椭球体面上某点为原点建立一个大地坐标单位活动坐标架. 通过平移, 使活动坐标架的原点与以椭球中心为原点的笛卡尔单位标架的原点相重合. 然后再通过两次标架旋转, 使活动坐标架与笛卡尔单位标架完全重合. 本文给出了使两个单位标架相重合的转换关系式, 以及该点位移在两个单位标架中的坐标转换式； 在此基础上, 考虑该点的位移及活动坐标架皆为该点大地坐标的函数, 经复杂推导, 分别给出了该点位移向量的微分在大地坐标系中的分量以及该点分别沿坐标曲线的弧微分表达式, 继而导出了该点的位移梯度矩阵; 最后推导出了椭球坐标系的应变张量与转动张量表达式, 并对转动张量的几何含义进行了较详细的解释, 且采用曲面理论对球面与椭球面的应变张量间的内在关系进行了讨论. 相似文献
4.
Expression of strain and rotation tensor in geodetic coordinates 总被引：1，自引：0，他引：1
In this paper, unit moving trihedron is first constructed for a point on the surface of a revolution ellipsoid. Via translation, the origin of the trihedron coincides with that of Cartesian coordinates established at the center of the ellipsoid, and then through two coordinate rotations, the trihedron completely coincides with the Cartesian coordi-nates. Transformation formulae between the moving trihedron and unit Cartesian coordinate frameworks as well as transformation of point displacement between two unit coordinate frameworks are presented. Based on the above transformation formulae between two different coordinate frameworks, due to the fact that the displacement and moving trihedron of the point are both functions of the geodetic coordinates, components in the corresponding axis for differential of displacement vector and geodetic curves arc differential at the point in geodetic system can be obtained through complicated derivation. Displacement gradient matrix at the point in geodetic system is also given. Finally, expressions of strain and rotation tensor in geodetic coordinates are presented. Geometric meanings of the rotation tensor are explained in detail. The intrinsic relationship between strain tensors of sphere and ellip-soid are also discussed. 相似文献
5.
Variance component estimation in errors-in-variables models and a rigorous total least-squares approach 总被引：1，自引：0，他引：1
Vahid Mahboub 《Studia Geophysica et Geodaetica》2014,58(1):17-40
A method for variance component estimation (VCE) in errors-in-variables (EIV) models is proposed, which leads to a novel rigorous total least-squares (TLS) approach. To achieve a realistic estimation of parameters, knowledge about the stochastic model, in addition to the functional model, is required. For an EIV model, the existing TLS techniques either do not consider the stochastic model at all or assume approximate models such as those with only one variance component. In contrast to such TLS techniques, the proposed method considers an unknown structure for the stochastic model in the adjustment of an EIV model. It simultaneously predicts the stochastic model and estimates the unknown parameters of the functional model. Moreover the method shows how an EIV model can support the Gauss-Helmert model in some cases. To make the VCE theory into EIV model more applicable, two simplified algorithms are also proposed. The proposed methods can be applied to linear regression and datum transformation. We apply these methods to these examples. In particular a 3-D non-linear close to identical similarity transformation is performed. Two simulation studies besides an experimental example give insight into the efficiency of the algorithms. 相似文献
6.
We present formulas for direct closed-form transformation between geodetic coordinates (φ, λ, h) and ellipsoidal coordinates
(β, λ, u) for any oblate ellipsoid of revolution. These will be useful for those dealing with ellipsoidal representations
of the Earth’s gravity field or other oblate ellipsoidal figures. The numerical stability of the transformations for near-polar
and near-equatorial regions is also considered. 相似文献
7.
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. 相似文献
8.
Effect of common point selection on coordinate transformation parameter determination 总被引：2，自引：0，他引：2
The use of satellite positioning techniques commonly requires a transformation from a Conventional Terrestrial coordinate
system to a Geodetic coordinate system, or vice versa. For such a transformation, the main problem is the determination of
transformation parameters between these coordinate systems. The transformation parameters are estimated by a least-squares
process using “common” points, i.e., those points whose coordinates are known in both systems. Therefore, the precision of
so estimated transformation parameters is closely related to certain characteristics of the common points. In this contribution,
we have formulated some theoretical relations between the transformation parameters and the number and the distribution of
common points, and corroborated the theoretical results numerically, using a simulated geodetic network. 相似文献
9.
An algorithmic approach to the total least-squares problem with linear and quadratic constraints 总被引：5，自引：0，他引：5
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. 相似文献
10.
Bayesian inference for the Errors-In-Variables model 总被引：1，自引：0，他引：1
Xing Fang Bofeng Li Hamza Alkhatib Wenxian Zeng Yibin Yao 《Studia Geophysica et Geodaetica》2017,61(1):35-52
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. 相似文献
11.
12.
Complex aquifer systems are often modeled with quasi-three-dimensional models, which consider two-dimensional horizontal flow in the aquifers and one-dimensional vertical flow through aquitards. When the aquifer system consists of a phreatic aquifer and one or more semiconfined aquifers connected by aquitards, the discrete model consists of a nonlinear system of algebraic equations, because the transmissivity of the phreatic aquifer depends on the phreatic head. If the water extraction is very high, the phreatic aquifer can be depleted and the equations of the model must be modified accordingly. There are not simple and general criteria to state if the phreatic aquifer is depleted before solving the system of equations. Therefore, the iterative procedures (e.g., relaxation methods), used to find the solution to the forward problem, must handle these particular conditions and can suffer several problems of convergence. These problems can be caused by the choice of the initial head values or of the relaxation coefficient of the iterative algorithms; however, they can also be caused by the nonexistence or nonuniqueness of the solution to the system of nonlinear equations. The study of existence and uniqueness of the general problem is very difficult and, therefore, we consider a simplified problem, for which the discrete model can be handled analytically. The results of the numerical experiments show that the solution to the forward problem can be nonunique. Only for some cases it is possible to invoke physical arguments to eliminate tentative solutions. 相似文献
13.
Gin-Show Liou 《地震工程与结构动力学》1991,20(12):1115-1125
Presented is a systematic procedure for generating impedance (or compliance) matrices for foundations with arbitrary shapes, resting on an elastic half-space medium. A technique to decompose prescribed harmonic tractions on the half-space medium is employed to solve analytically the differential wave equations in cylindrical coordinates. However, the interaction stresses due to the vibration of a foundation with arbitrary shape are described in rectangular coordinates, and assumed to be piecewise constant in the region of the arbitrary shape. A coordinate transformation matrix is introduced for the piecewise constant tractions in order to use the solution of the differential wave equations in cylindrical coordinates. Finite element modelling is assumed in rectangular coordinates for the foundation itself. The impedance matrix is then obtained for the finite element model, using a variational principle and the reciprocal theorem. A simple example of a rigid square plate resting on a half-space medium and subjected to vertical excitation is used to demonstrate the efficiency and effectiveness of the procedure. Some numerical aspects are investigated and some possible extensions of the procedure are also discussed. 相似文献
14.
Giuseppe Muscolino 《地震工程与结构动力学》1990,19(2):205-216
A method to evaluate the dynamic response of composite multiply connected primary-secondary systems is presented. For this purpose the nodal equations of motion of the composite system are given in ‘total’ and ‘relative’ coordinates. In the framework of the component-mode synthesis method a coordinate transformation, here defined as an admissible one, is used to reduce the nodal equations of motion. This coordinate transformation is theoretically and numerically compared with the coordinate transformation usually used in the literature, which generally gives larger errors with respect to the former when a reduced number of nodes is considered. 相似文献
15.
A modal superposition pseudo-force method for the dynamic analysis of structural systems with non-proportional damping is presented. The method combines the advantages of the classical modal superposition method and the pseudo-force method. When the system damping is non-proportional, the dynamic equilibrium equations in generalized coordinates are coupled through the damping terms. In the present method, these coupled equations are solved by an iterative process in which the coupling terms are treated as pseudo-forces. A proof of the convergence of the iterative process is given. Numerical examples show the good convergence characteristics of the process and the good accuracy of the obtained results. 相似文献
16.
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. 相似文献
17.
Mikael Mortensen Hans Petter Langtangen Garth N. Wells 《Advances in water resources》2011,34(9):1082-1101
Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model. 相似文献
18.
Data-snooping procedure applied to errors-in-variables models 总被引：1，自引：0，他引：1
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. 相似文献
19.
A three-dimensional non-hydrostatic numerical model for simulation of the free-surface stratified flows is presented. The model is a non-hydrostatic extension of free-surface primitive equation model with a general vertical coordinate and horizontal orthogonal curvilinear coordinates. The model equations are integrated with mode-splitting technique and decomposition of pressure and velocity fields on hydrostatic and non-hydrostatic components. The model was tested against laboratory experiments on the steep wave transformation over the longshore bar, solitary wave impact on the vertical wall, the collapse of the mixed region in the thin pycnocline, mixing in the lock-exchange flows and water exchange through the sea strait. The agreement is generally fair.Responsible Editor: Hans Burchard 相似文献