方差—协方差分量极大似然估计的通用公式

本文由概括平差函数模型出发，按极大似然做估计原则导出了适用于所有平差函数模型的方差分量估计的通用公式，由Ｋ．Ｋｕｂｉｋ和Ｃ．Ｒ．Ｋｏｃｈ所导出的两个公式都是它的特例。     

Generalyzed variance-covariance propagation law formulae and application to explicit least-squares adjustments

Standard formulae overlook the contribution of a number of terms in the derivation of variance-covariance matrices for parameters in nonlinear least squares adjustment. In a large class of nonlinear mathematical models, these terms can contribute to an important error in the estimation of parameter variances. Improved formulae are derived. A numerical example is given and the use of our improved formula in the case of least-squares adjustment in the explicit case (L=F(X)) is fully documented.     

基于等价条件闭合差的VCE通用解析法

分析指出了现有方差-协方差分量估计(VCE)方法在计算效率与?2统计量统计性质两方面的固有缺陷。利用零空间算子消去概括平差模型中的参数向量,建立了等价条件平差模型。由此定义了等价条件闭合差(ECC),并导出了以ECC表示的?2统计量计算式。进而,基于等价条件闭合差与新构造的可逆方差分量模型提出了方差-协方差分量估计的通用解析法,简称为VCE-ECC法。同时,给出了对应四种基本平差模型的VCE-ECC法简化计算式。实例与仿真结果表明：VCE-ECC法与现有VCE方法的方差-协方差分量估计值在统计意义上无明显差异,并有效地克服了现有VCE方法的固有缺陷。     

A new formula for evaluating the truncation error coefficient

In this paper, a new formula for evaluating the truncation coefficientQ _n is derived from recurrence relations of Legendre polynomials. The present formula has been conveniently processed by an electronic computer, providing the value ofQ _n up to a degreen=49 which are exactly equal to those of Paul (1973).     

Variance-covariance estimation of GPS Networks

Summary It is quite easy to estimate the variance-covariance (VCV) matrix for single session surveys or local networks, but difficult where these local networks are combined together to form a regional network. Our main aim is to develop an appropriate VCV model to combine all the different types of networks, either global, regional or local. By careful estimation and combination of the individual VCVs of the local networks, we can form a unique VCV for local, regional and global networks. Different techniques are used to derive appropriate models for the variancecovariance components of the Global Positioning System (GPS) networks. The VCV models were estimated using homogeneous and heterogeneous data. The variance-covariance components are empirically derived using (a) the covariance of the observations of homogeneous data, (b) a combination of the covariance of the observationsP ^–1 and the covariance of the signal componentsC _ss (for either homogeneous and/or heterogeneous data), (c) only the variances are used to determine the variancecovariance, their covariances being zeros. We compare the solutions of the VCV developed for homogeneous and/or heterogeneous data with other developed VCVs. It was observed that the derived VCV model for the combined homogeneous and/or heterogeneous data of case (b) gives the best estimates in all cases.     

THE CONFORMAL TRANSFORMATION OF A NETWORK OF TRIANGULATION

Abstract

In a recent issue of this Review, an example is given of the conformal transformation of a network of triangulation using Newton's interpolation formula with divided differences. While the application of the method appears to be new, attention should be drawn to the fact that Kruger employed Lagrange's interpolation formula in a discussion and extension of the Schols method in a paper which was published in the Zeitschrift für Vermessungswesen in 1896. A reference to this paper was given at the end of the paper, “Adjustment of the Secondary Triangulation of South Africa”, published in a previous issue of the E.S.R. (iv, 30, 480).     

Probability distribution of eigenspectra and eigendirections of a twodimensional,symmetric rank two random tensor

Let there be given a twodimensional symmetric rank two tensor of random type (examples:strain, stress) which is either directly observed or indirectly estimated from observations by an adjustment procedure. Under the assumption of normalityof tensor components we compute the joint probability density functionas well as the marginal probability density functionsof its eigenspectra (eigenvalues) and eigendirections (orientation parameters). Due to the nonlinearity of the relation between eigenspectra-eigendirections and the random tensor components, via the inverse nonlinear error propagationbiases and aliases of their first and centralized second moments (mean value, variance-covariance) are expressed in terms of Jacobianand Hessianmatrices. The joint probability density function and the first and second moments thus form the fundamental of hypothesis testing and qualify control of eigenspectra (eigenvalues, principal components) and eigendirections (orientation parameters, eigenvectors, principial direction) of a twodimensional, symmetric rank two random tensor.     

随机模型不正确对方差一致性检验统计量的影响

当平差模型不正确时必然会影响平差结果及其统计性质。文中着重分析了在函数模型正确的前提下，采用不正确的随机模型对方差一致性检验统计量的影响，并给出了具体的影响公式。所得结果对进一步充实测量平差教学内容及实用中检验模型的正确性等方面具有一定的意义。     

Least squares collocation and statistical testing

The authors address the issue of statistical testing in least squares collocation (LSC) in two stages. The first stage concerns the extension and focusing of theLSC equations to the task of statistical testing. The second stage deals with statistical testing titself and is introduced in the second portion of the paper. The paper commences with an overview of the development ofLSC and its relationship to least squares adjustment (LSA). Expressions for the various random variables and their corresponding covariance matrices are derived and in some instances are gleaned from the literature for the following quantities: (i) corrections to the unknown parameters with a priori covariance information; (ii) estimated signal at both the observation and computation points; and (iii) the noise at the observation points. Some of the needed covariance matrices are either obscurely hidden in the literature or not available at all, but, nevertheless are given in the paper. Also given are expressions for the estimated variance factor which forms the basis of various statistical tests. The paper closes with an overview and enumeration of possible statistical tests for detection of outliers in the observations.     

方差分量的极大验后估计

由概括函数模型出发,导出了适应于所有平差模型的方差分量的极大验后估计,这种估计实际上是先验方差与样本方差的凸组合。文中还给出了一个简单的数例,说明这种方法的可行性。     

地图数字化中基于点位坐标的统一平差模型

探讨了地图数字化过程中采用平差方法来建立坐标计算的统一模型。将测得的角度、边长等数据化为方向、距离观测值列立误差方程，通过调用统一的平差迭代计算程序来计算未知点坐标及精度。详细给出了基于平差的11种常用坐标计算方法，并通过实例进行了说明。     

等价条件平差模型的方差-协方差分量最小二乘估计方法

提出等价条件闭合差的方差-协方差分量最小二乘估计方法,简称LSV-ECM法。首先,利用等价条件平差模型建立了基于等价条件闭合差二次型的方差-协方差分量估计方程,由矩阵半拉直算子将其变换为线性Gauss-Markov形式,进而通过最小二乘准则导出了具有模型通用性、形式简洁性且满足无偏性和最优性的方差-协方差分量估计公式。其次,证明了LSV-ECM方法与残差型VCE方法的等价性,并在此基础上通过计算复杂度定量分析了所提方法的计算高效性。最后,通过边角网平差和中国区域GNSS站坐标时序建模及其结果分析,验证了所提新方法的正确性和计算高效性。     

基于等价方差-协方差抗差估计在GPS网平差中的应用

在GPS网平差中,其观测值为三维基线向量,基线向量3个分量间存在较强的相关性,在平差时必须考虑。根据等价方差-协方差理论,本文以GPS网平差为例,对比IGG-2方案,说明了等价方差-协方差理论在相关观测抗差估计中的优越性。     

不等式约束加权整体最小二乘的凝聚函数法

误差向量的方差-协方差阵是一般对称正定矩阵下的附不等式约束加权整体最小二乘平差模型,研究了其参数估计和精度评定问题。首先,将残差平方和极小化函数在整体最小二乘准则下转化为只包含模型参数的目标函数,同时将所有的不等式约束表示成一个等价的凝聚约束函数,并运用乘子罚函数策略将不等式约束加权整体最小二乘平差问题转化为相应的无约束最优化问题,并用BFGS方法求解。然后,将误差方程和约束函数线性展开,推导了最优解和观测量间的近似线性函数关系,运用方差-协方差传播律得到了最优解的近似方差。最后,用数值实例验证了方法的有效性和可行性。     

LONG LINES ON THE EARTH

Abstract

The Arc of the Geodesic.—In the first part of this paper a method was given for computing the azimuth of a geodesic. The method gives the convergence of the geodesic correctly up to the second power of e the eccentricity. The formula (9), however, also depends on the assumption that σ, the arc-length of the geodesic, can be obtained with sufficient accuracy from the Supplemental Dalby Theorem, that is to say, by a purely spherical computation. It is, therefore, needful to show that this supposition is justifiable; a means must in fact be indicated for verifying the assumption.     

THEORETIC RESEARCH ON ROBUSTIFIED LEAST SQUARES ESTIMATOR BASED ON EQUIVALENT VARIANCE-COVARIANCE

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New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

方差-协方差分量估计公式等价性的进一步证明

通过附有条件的间接平差模型进一步证明各类方差-协方差分量估计公式之间的等价性。