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。     

方差-协方差分量估计的概括平差因子法

针对现有方差-协方差分量估计(variance-covariance component estimation,VCE)理论存在的问题,通过引入间接平差的平差因子概念,定义并研究了基于概括平差模型的概括平差因子、概括闭合差及其方差阵,进而利用二次期望公式提出了基于概括平差因子的VCE新方法。该方法适用于概括平差模型所归纳的4种函数模型形式,并通过概括平差因子揭示了平差函数模型与VCE是否存在解析估计形式的关系。实例计算结果表明,现有迭代型VCE方法改变了LS估计量的统计性质,而VCE新方法解析估计具有LS统计性质,且无需初值。     

等价条件闭合差最小范数分量的GNSS坐标时间序列噪声估计

为了精确高效地求解全球导航卫星系统（global navigation satellite system,GNSS）坐标时间序列噪声分量,结合等价条件平差模型以及最小范数二次无偏估计法,提出等价条件闭合差最小范数分量估计方法。首先,采用等价条件闭合差构造二次型方差估计公式,结合不变性、无偏性、最小范数准则等条件,导出基于等价条件闭合差的方差-协方差分量最小范数估计公式;然后,采用最小二乘方差分量估计（least-squares variance component estimation,LS-VCE）法、最小范数二次无偏估计法（minimum norm quadratic unbiased estimate,MINQUE）验证所提方法的正确性及有效性。通过模拟时间序列和北美GNSS站坐标时间序列的噪声估计结果发现,所提方法与LS-VCE法和MINQUE法的估计效果一致,但计算时间相较于LS-VCE法减少了70%以上。     

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。     

各种平差模型的方差分量估计

本文从附有限制条件的条件平差(the Condition Adjustment with Constraints)的数学模型(简称CAC模型)出发,推导了方差分量估计的特征方程,并给出了方差分量估值的方差—协方差矩阵。由于间接平差、条件平差、附有条件的间接平差、附有参数的条件平差均为CAC模型的特例,它们的特征方程很容易由CAC模型的特征方程简化而来,这就充分显示了现有各种平差模型方差分量估计公式之间的区别与联系,有利于建立在各种平差模型下关于方差分量估计的整体概念。     

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。     

基于等效残差的方差-协方差分量估计

首先概括性地阐述方差-协方差分量估计(VCE)理论的发展历史与研究现状;利用正交分解提取出等效残差,建立VCE的基本方程,在此基础上阐述VCE算法的局部最优特性和可能出现负定协方差阵估计结果等两大问题,并分析可能的解决方案及其复杂性;导出初值给定的Helmert,最小二乘和MINQUE VCE的线性逼近估计公式,证明基于等效残差的估计公式与已有VCE公式等价,各种估计方法在本质上是一致的;最后,用两个算例验证本文的观点.     

方差—协方差—分量估计

本文从间接平差出发推导了含有检验矩阵的方差—协方差一分量估计的特征方程,并详细讨论了估值的统计特性。作为协方差估计的特例,对方差估计的若干公式进行了探讨,指出了它们之间的相互联系及各自的统计特性。最后给出了适合条件平差的方羞—协方差一分量估计公式。     

基于条件方程的方差分量估计公式

著名的Helmert方差分量估计公式是基于间接观测平差模型导出的。基于条件观测平差模型导出了方差分量估计公式并给出了实际应用范例,且对两种模型的方差分量估计公式的等价性进行了理论证明。算例表明,文中的估计公式能正确地估计出各类观测值的方差因子。     

扩展平差模型下的Helmert方差分量估计

本文推导了附有条件的间接平差和附有未知数的条件平差模型(简称为扩展平差模型)下的Helmert方差分量估计公式。并证明了对同一平差问题两种模型下方差分量估计公式的等价性。     

基于等效残差积探测粗差的方差-协方差分量估计

已有方差-协方差分量估计(VCE)的粗差探测是对残差检验,而VCE的本质是利用残差的二阶量来估计它的二阶中心矩,因此更合理的方法应该对残差二阶量检验.由最小二乘残差估值导出了一组服从标准正态分布的等效残差;然后从等效残差的VCE基本方程出发,分别采用χ2统计量和正态积Np统计量检验等效残差的平方及其乘积.结果表明,采用...     

Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data

The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness.     

Least-squares variance component estimation

Least-squares variance component estimation (LS-VCE) is a simple, flexible and attractive method for the estimation of unknown variance and covariance components. LS-VCE is simple because it is based on the well-known principle of LS; it is flexible because it works with a user-defined weight matrix; and it is attractive because it allows one to directly apply the existing body of knowledge of LS theory. In this contribution, we present the LS-VCE method for different scenarios and explore its various properties. The method is described for three classes of weight matrices: a general weight matrix, a weight matrix from the unit weight matrix class; and a weight matrix derived from the class of elliptically contoured distributions. We also compare the LS-VCE method with some of the existing VCE methods. Some of them are shown to be special cases of LS-VCE. We also show how the existing body of knowledge of LS theory can be used to one’s advantage for studying various aspects of VCE, such as the precision and estimability of VCE, the use of a-priori variance component information, and the problem of nonlinear VCE. Finally, we show how the mean and the variance of the fixed effect estimator of the linear model are affected by the results of LS-VCE. Various examples are given to illustrate the theory.     

组合导航抗差自适应卡尔曼滤波

针对组合导航系统的定位精度与稳定性要求不断提高的现状,该文引入一种观测噪声协方差与抗差自适应相结合的Kalman滤波算法。利用新息向量和移动窗口协方差分析法,动态自适应修正观测噪声协方差阵;通过分析基于状态不符值、方差分量的统计量构造的自适应因子所存在的问题,提出一种由预测残差向量构造的自适应因子。仿真结果表明,该方法能够有效抑制观测异常对组合导航定位精度的影响。     

偏差改正的Partial EIV模型方差分量估计

针对Partial EIV模型的方差分量估计中未考虑参数估值偏差所带来的影响,将Partial EIV模型视为非线性函数得到参数估值的偏差及二阶近似协方差表达式,计算得到偏差改正后的参数估值,结合方差分量估计方法,更新由参数估值影响的矩阵变量,给出了基于偏差改正的方差分量估计迭代方法。试验结果表明,参数估值及其协方差主要受参数估值偏差大小的影响,加入偏差改正能够得到更加合理的参数估值及方差分量估值,偏差改正后的方差分量估值可更加合理地评估参数估值的精度信息。     

中国大地测量数据处理60年重要进展第一部分：函数模型和随机模型进展

大地测量函数模型与随机模型是大地测量数据处理必须要涉及的模型。经过60年不懈努力,中国学者根据大地测量应用实际,构建和改进了许多函数模型,如广义测量平差模型、等价观测方程、非线性平差模型等;在方差分量估计和误差检验方面做了大量创新性工作,如基于Bayes估计的方差分量估计、拟准误差检验等,丰富了测量平差理论与方法。这一部分重点介绍建国60年来中国学者在大地测量数据处理函数模型、随机模型和误差检验方面所取得的成就。     

随机模型误差对函数模型选择的影响

测量平差问题中,函数模型误差和随机模型误差通常是同时存在的。本文揭示了两者的关系,着重讨论了两者的相互作用及其影响,特别是随机模型误差对函数模型选择的影响。同时也讨论了用来改善随机模型的“方差分量估计技术”对函数模型误差的吸收作用。最后得出了一些有益的结论。     

Variance component estimation uncertainty for unbalanced data: application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous data set comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity.