参数估计的统一模型——广义Gauss-Markov模型

设计矩阵可以是秩亏阵,观测值的协方差阵可以是奇异阵的广义 Gauss-Markov模型(简称广义G-M模型),它是一种形式简单的统一模型。本文从最小二乘估计V~TQ~-V=min 出发,研究广义G-M 模型的参数估计理论和方法。说明了V~TQV~-=min与 Rao及Bjeharmmar等的平差原则一致。并对广义 G-M模型之解及其性质进行了系统讨论。     

Stochastic estimation of tropospheric path delays in global positioning system geodetic measurements

Water vapor radiometric (WVR) and surface meteorological (SM) measurements taken during three Global Positioning System (GPS) geodetic experiments are used to calculate process noise levels for random walk and first-order Gauss-Markov temporal models of tropospheric path delays. Entire wet and combined wet and dry zenith delays at each network site then are estimated simultaneously with the geodetic parameters without prior calibration. The path delays and corresponding baseline estimates are compared to those obtained with calibrated data and stochastic residual delays. In this manner, the marginal utility of a priori tropospheric calibration is assessed given the ability to estimate the path delays directly using only theGPS data. Estimation of total zenith path delays with appropriate random walk or Gauss-Markov models yields baseline repeatabilities of a few parts in 10⁸. This level of geodetic precision, and accuracy as suggested by analyses on collocated baselines estimated independently by very long baseline interferometry, is comparable to or better than that obtained after path delay calibration usingWVR and/orSM measurements. Results suggest thatGPS data alone have sufficient strength to resolve centimeter-level zenith path delay fluctuations over periods of a few minutes.     

An algorithm for the inverse of a multivariate homogeneous polynomial of degreen

Summary Various geodetic problems (the free nonlinear geodetic boundary value problem, the computation of Gauß-Krüger coordinates or UTM coordinates, the problem of nonlinear regression) demand theinversion of an univariate, bivariate, trivariate, in generalmultivariate homogeneous polynomial of degree n. The new algorithm which is oriented towardsSymbolic Computer Manipulation is based upon the algebraic power base computation with respect toKronecker-Zehfu product structure leading to the solution of a system oftriangular matrix equations: Only the first row of the inverse triangular matrix has to be computed. TheSymbolic Computer Manipulation program of the GKS algorithm is available from the authors.     

The free nonlinear boundary value problem of physical geodesy

Summary Within potential theory of Poisson-Laplace equation the boundary value problem of physical geodesy is classified asfree andnonlinear. For solving this typical nonlinear boundary value problem four different types of nonlinear integral equations corresponding to singular density distributions within single and double layer are presented. The characteristic problem of free boundaries, theproblem of free surface integrals, is exactly solved bymetric continuation. Even in thelinear approximation of fundamental relations of physical geodesy the basic integral equations becomenonlinear because of the special features of free surface integrals.     

统一方差分量估计公式

本文提出了一种适用于协方差阵奇异或非奇异、设计阵列满秩或降秩时的方差分量估计方法;其公式推导简单,形式统一,不需解线性方程组,同时可保证迭代计算方差分量的非负要求。作为特例还和Helmert法及MINQUE法作了比较。最后讨论了用真误差进行方差分量估计的计算公式,并给出了一个测距误差分析的实例。     

非线性混合边值问题及其抗差解

本文从函数模型和随机模型两方面入手综合讨论大地测量混合边值问题及其解。函数模型方面，首先对混合边值问题进行了一般描述；继而分别建立了垂直边值和水平边值问题的二阶近似模型；随机模型方面，考虑到重力场的随机模型与随机观测量及其重力场本身的随机性有关，任何假设的理想模型都可能偏离实际模型。故文章中介绍了能抵制随机模型偏差影响的抗差估计原理。如果将重力场看成随机场，则建议采用抗差拟合推估解。     

随机Poisson方程Dirichlet大地边值问题的随机积分解

利用随机微分方程理论,给出了随机Poisson方程Dirichlet大地边值问题的随机积分解,讨论了随机与确定边值问题间的关联。对应视为随机过程的函数,若采用确定性边值问题求解,不确定性影响将被直接带入最终解中;若采用随机积分解,则类似Gauss白噪声的影响将被滤掉,这对进一步提高重力场的求解精度具有重要影响。     

Deterministic and stochastic parameters in inertial survey adjustment

In the adjustment of inertial position surveys the additional parameters describing the systematic errors of individual traverses can be considered as deterministic or stochastic. The paper deals with various aspects of the deterministic or stochastic approach by way of a standard functional model. If purely deterministic parameters are set up, the solvability of the least squares problem depends on redundant observations like coordinate discrepancies of forward and backward runs or coordinate differences at cross-over points of traverse networks. Inequalities are presented to handle the configuration problem for any net and for several ways of introducing parameter sets. Also condition equations being geometrically explainable are developed solving the datum problem in free adjustment applications. Based on the Ebersberger Forst campaigns with a large amount of Ferranti, Honeywell and Litton data, numerical investigations into the stochastic properties of the additional parameters and the observations follow. It turns out that additional parameters for Honeywell and Litton data can be considered as stochastic parameters while for Ferranti data significant azimuth and time dependent effects can be found. The investigations of true errors show that in case of the deterministic adjustment approach a diagonal covariance matrix can be introduced and in case of stochastic additional parameters a first order Gauss-Markov process serves as a good approximation for the stochastic behaviour of the observations.     

四种改进积分法的低空扰动引力计算

针对Stokes积分方法计算扰动引力中计算点从空中趋近地面时存在积分奇异和不连续的问题,该文提出了去中央奇异点法、奇异点积分值修正法、中央格网加密算法和改进积分式法4种改进Stokes积分的计算公式,并进行了实验计算。计算结果表明:近地空间范围内,4种改进算法都能在一定程度上改进原始积分的奇异性问题;相同条件下,奇异点积分值修正法和改进积分式法计算精度最高,适宜于低空计算;改进积分式法通过理论推导,得到了从球外部到球面统一、连续且无奇异的改进Stokes积分公式,理论严谨。     

Construction of Green's function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition

Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e <sup>2</sup> <sub>0</sub>)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity. A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme recommended so far to solve this boundary-value problem is thus irrelevant. Received: 5 June 1997 / Accepted: 20 February 1998     

Finite element analysis of quasi-static earthquake displacement fields observed by GPS

In order to investigate the ability of numerical techniques in computing seismic displacement fields, a forward problem of dislocation theory is first solved by both analytical and numerical techniques. Convergency of the numerical solution and a comparison with the results of a well-known analytical solution are presented for a particular problem. The results are then used when the numerical technique is applied to the same problem without any information on the dislocation source. The latter results may be applied in an inverse dislocation problem. Numerical techniques are in particular expected to be flexible enough to (1) include internal discontinuities, (2) apply geodetic observations as boundary conditions and (3) model lateral, in addition to radial, material heterogeneities. A model with lateral variation of elastic parameters is considered in the last problem. Acknowledgments.The authors are most grateful to Y. Okada, National Research Institute for Earth Science and Disaster Prevention, Japan, for providing the Fortran code of his analytical solution.     

利用基本矩阵重构地面影像位姿信息

在地面采集的盲区影像往往缺少位姿信息,进而与空中影像联合处理时造成分层错乱。因此为提高倾斜模型精细度,本文通过约束对极奇异矩阵的方法重构补拍影像的位置姿态。首先进行空地影像特征匹配,得出两者的基本矩阵,规范化空中影像投影矩阵,地面影像投影矩阵则由基本矩阵分解成的反对称矩阵和非奇异矩阵组成;然后利用空中影像POS信息消除该投影矩阵对的多义性,从而得出地面影像位置姿态的准确值。试验结果表明,该方法能够在无像控条件下自动计算出地面盲区补拍影像的位姿信息且与空中影像配准良好,在保证模型精细的前提下提高了作业效率。     

Improved second order design and the datum choice

It is shown that also in a rank deficient Gauss-Markov model higher weights of the observations automatically improve the precision of the estimated parameters as long as they are computed in thesame datum. However, the amount of improvement in terms of the trace of the dispersion matrix isminimum for the so-called “free datum” which corresponds to the pseudo-inverse normal equations matrix. This behaviour together with its consequences is discussed by an example with special emphasis on geodetic networks for deformation analysis.     

Data gaps in finite-dimensional boundary value problems for satellite gradiometry

 In the framework of a boundary value problem (BVP), when areas on the boundary are void of data the solution of the problem becomes undetermined and clearly more difficult. Physically, this could be the situation in which a gradiometer on a satellite on a perfectly circular orbit covers a sphere with measured second radial derivatives: if the satellite orbit is not polar, there are caps at satellite altitude which are not covered by data. A solution is presented based on an iterative algorithm, under the hypothesis of using a finite-dimensional model as is usually done in the time-wise approach. The convergence of the iterative solution is proved and a numerical example is shown to confirm the theoretical result. Received: 14 August 2000 / Accepted: 12 April 2001     

M-estimator for the 3D symmetric Helmert coordinate transformation

The M-estimator for the 3D symmetric Helmert coordinate transformation problem is developed. Small-angle rotation assumption is abandoned. The direction cosine matrix or the quaternion is used to represent the rotation. The \(3 \times 1\) multiplicative error vector is defined to represent the rotation estimation error. An analytical solution can be employed to provide the initial approximate for iteration, if the outliers are not large. The iteration is carried out using the iterative reweighted least-squares scheme. In each iteration after the first one, the measurement equation is linearized using the available parameter estimates, the reweighting matrix is constructed using the residuals obtained in the previous iteration, and then the parameter estimates with their variance-covariance matrix are calculated. The influence functions of a single pseudo-measurement on the least-squares estimator and on the M-estimator are derived to theoretically show the robustness. In the solution process, the parameter is rescaled in order to improve the numerical stability. Monte Carlo experiments are conducted to check the developed method. Different cases to investigate whether the assumed stochastic model is correct are considered. The results with the simulated data slightly deviating from the true model are used to show the developed method’s statistical efficacy at the assumed stochastic model, its robustness against the deviations from the assumed stochastic model, and the validity of the estimated variance-covariance matrix no matter whether the assumed stochastic model is correct or not.     

OUTLIER DETECTION AND RELIABILITY OF ADJUSTMENT MODELS WITH SINGULAR COVARIANCE MATRICES

Up to now,outlier detection and reliability theory are generallybased on the regular Gauss-Markov models,in which the covariance matrix of ob-servations is positively definite.For the adjustment models with singular covari-ance matrix,the statistics for outlier detection are derived by the authors.Thecorresponding reliability theory is developed.And the application of the theory isdemonstrated with a practical example.     

一种新的病态问题奇异值修正方法

针对测量平差中的病态最小二乘问题,提出了统一的奇异值修正公式,以此为基础提出一种新的奇异值修正法.所提方法克服了现有方法需要确定奇异值截断阈值或者修正阈值的缺陷,基本没有增加额外的计算量,计算简单快捷精度高. 另外,所提方法普适性强,对方程组系数矩阵的维数和是否满秩没有特殊的要求,可以适用于多种类型平差方程组的求解. 以两个病态方程为例对所提方法进行了数值验证,并将计算结果与最小二乘解和奇异值截断解进行了比较,结果表明,所提方法可以获得精度更高的计算结果.      

具有先验信息的区域增强系统卫星轨道确定方法

针对中国实时精密定位服务系统的建设,提出了一种具有全球先验信息的导航卫星轨道确定方法,其关键点在于先验轨道信息权函数的确定。首先在预报轨道误差特性分析的基础上,给出了采用一阶高斯-马尔科夫模型进行先验信息权函数的建立方法;然后对先验信息和区域观测数据赋予相应权,进行联合定轨。     

Through-Wall Imaging via a Linear Inverse Scattering Algorithm

A through-wall imaging problem for a 2-D scalar geometry is addressed. It is cast as an inverse scattering problem and tackled under the linear model of the electromagnetic scattering that is provided by the Born approximation. A truncated singular value decomposition inversion scheme is exploited, and the performances that are achievable by such an inversion scheme are assessed by exploiting synthetic data. The cases of weakly and strongly scattering objects are both considered. Finally, an example of reconstruction that is obtained by exploiting experimental data is presented.     

On symmetrical three-dimensional datum conversion

A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.