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。     

具有奇异协方差阵观测值的平差方法及应用

本文论述了具有奇异协方差阵观测值的平差原则，归结为V~TQ~-V=min总结成直接和间接两种解法。推导了间接平差和条件平差全部公式。举例说明了具有奇异协方差阵观测值的三角网平差和相关分组平差的实施。最后对Bierhammar提出的解法作了讨论。     

加权混合估计中权值的确定方法

综合了大地测量中各种异方差多源观测模型和联合平差方法,说明了混合估计方法可以用于测量数据融合,平衡附加信息和样本信息对参数估计的影响。通过求取权值使参数估计的协方差阵的迹最小的方法,给出了一个权的最优选择方法。本文扩展了已有的加权混合估计方法,使得新方法中的权不受验前单位权方差的限制,能有效应用于大地测量数据处理。     

基于等价方差-协方差的粗差探测敏感度分析及实现

选权迭代法在面对独立观测量中的粗差时能够表现出良好的探测效果,但由于其只是一种基于独立观测值的稳健估计法,没有考虑到观测值之间的相关性[1]。而现有的等价权函数虽然都满足稳健估计的要求,但由于所构造的等价权阵不对称,使得最后平差结果严重偏离实际情况。本文介绍在传统稳健估计的基础上,在定权时充分考虑相关观测值之间相关性的不变性,构造对称的方差—协方差阵不断扩大,并通过VB进行编程及实例分析,发现该方法对粗差的敏感度非常强,探测精度很高。     

秩亏水准网平差公式的一维表达式

秩亏水准网按附中条件法平差的法议程系数和参数先验权阵具有对称特性,利用此特性和水准网附加矩阵的特殊形式,以及文献[2]中给出的线性议程组未知数及其函数,系数阵逆阵计算的一维公式,可导出秩亏水准网按中条件法平差的一维平差计算公式,使秩亏水准网平差计算和程序设计简单易行。     

GPS重复基线粗差探测方法比较及实例分析

阐述了在传统稳健的基础上,针对GPS重复基线含有粗差时,比较各大选权迭代方法与等价方差-协方差稳健估计方法分别进行粗差探测,并通过VB进行编程及实例分析,论证等价方差-协方差稳健估计方法通过充分合理定权,不断构造严格对称的方差-协方差阵,并对其进行不断扩大,从而能达到对重复基线高效精准的探测.     

最小二乘原理的进一步推广

本文根据参数估计的原理，讨论在信号X和噪声Δ的协方差Cov（X，Δ）≠0的情况下的滤波问题。导出了更一般的“广义最小二乘原理”，并根据所导出的广义最小二乘原理，得到了滤波问题的最小二乘平差算法及最小二乘平差问题的滤波算法。     

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。     

卫星重力径向梯度数据的最小二乘配置调和分析

本文深入研究了利用卫星重力梯度径向分量确定地球引力场位系数的最小二乘配置(LSC)调和分析方法。首先论述了最小二乘配置法的原理,推导了扰动引力梯度观测量与球谐系数之间的协方差和自协方差矩阵,在扰动引力梯度观测数据为等经差规则网格数据的情况下,引力位与扰动引力梯度之间的协方差矩阵具有分块Toeplitz循环阵的结构,有效的利用FFT变换技术将其降阶;研究利用截断奇异值分解法（TSVD）解决协方差阵的病态性问题;最后得到了引力梯度径向分量的最小二乘配置调和分析的完整计算公式。模拟试算结果表明,基于TSVD的最小二乘配置调和分析方法,能够以较高的精度还原全球重力场,验证了本文算法的有效性和实用性。     

近代平差理论的扩充

本文在推证了用广义逆表达的随机参数最优线性无偏估计公式基础上,导出了滤波、推估和配置等广义估计公式,其特点是观测值的协方差阵不要求满秩。最后,作者提出了参数为随机的自由网平差新方法。     

基于方差分量估计的拟合推估及其在GIS误差纠正的应用

拟合推估解算必须首先求得信号向量的方差协方差矩阵,该协方差矩阵一般通过选定的协方差函数,并通过已测点数据进行拟合得到。显然观测噪声的先验方差协方差阵与拟合得到的随机信号的方差协方差矩阵必须相互协调,即观测噪声向量和信号向量的权矩阵所对应的方差因子应该一致,否则将对固定效应和随机效应参数的估计带来系统性的影响。应用方差分量估计来协调拟合推估模型中观测噪声和信号向量的随机模型,并分别从极大似然估计、MINQUE估计、赫尔默特方差分量估计三方面构建了拟合推估模型的方差分量解,最后利用新提出的理论与方法,对一幅实际的扫描地形图进行误差纠正,结果表明基于方差分量估计的拟合推估法能够提高扫描地形图的精度。     

Compactly supported radial covariance functions

The Least-squares collocation (LSC) method is commonly used in geodesy, but generally associated with globally supported covariance functions, i.e. with dense covariance matrices. We consider locally supported radial covariance functions, which yield sparse covariance matrices. Having many zero entries in the covariance matrice can both greatly reduce computer storage requirements and the number of floating point operations needed in computation. This paper reviews some of the most well-known compactly supported radial covariance functions (CSRCFs) that can be easily substituted to the usually used covariance functions. Numerical experiments reveals that these finite covariance functions can give good approximations of the Gaussian, second- and third-order Markov models. Then, interpolation of KMS02 free-air gravity anomalies in Azores Islands shows that dense covariance matrices associated with Gaussian model can be replaced by sparse matrices from CSRCFs resulting in memory savings of one-fortieth and with 90% of the solution error less than 0.5 mGal. This article is dedicated to Cerbère.     

顾及框架点坐标误差的三维基准转换严密模型

框架点坐标是由观测数据通过平差得到的,不可避免地受到观测误差的影响。针对原框架和目标框架坐标均存在误差、非公共点与公共点间存在相关性,以及转换系数矩阵中仅部分元素存在误差的实际情况,提出了同时考虑框架内误差以及转换点间相关性的基准转换严密模型,该模型将公共点和非公共点联合处理,同时计算坐标转换参数和所有点的坐标转换值,推导出了新的严格坐标转换公式,该公式为传统坐标转换公式基础上增加一改正量的形式;进一步,推导了原框架和目标框架坐标的方差不一致情况下的坐标转换模型的自适应解法;最后,利用"陆态网络工程"2000个区域站的实测坐标进行坐标转换验证,结果表明,这种严密模型较传统坐标转换模型具有更高的坐标转换精度。     

Gibbs sampler for computing and propagating large covariance matrices

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model. AcknowledgementsThe authors are indebted to Pieter Visser and Pavel Ditmar for providing simulation output that was used in the GOCE error generation experiments. Furthermore, the NASA/NIMA/OSU team is acknowledged for providing public ftp access to the EGM96 error covariance matrix. The two anonymous reviewers are thanked for their valuable comments.     

变二次估计为线性估计的拉直变换法

利用投影变换和拉直变换，得到了方差和协方差估计的一种新模型──线性模型。该模型将方差和协方差分量的二次估计问题变为我们所熟知的线性估计问题。本文介绍了该方法的原理，证明了估计量θ的统计性质。     

Least squares adjustment and collocation

Summary For the estimation of parameters in linear models best linear unbiased estimates are derived in case the parameters are random variables. If their expected values are unknown, the well known formulas of least squares adjustment are obtained. If the expected values of the parameters are known, least squares collocation, prediction and filtering are derived. Hence in case of the determination of parameters, a least squares adjustment must precede a collocation because otherwise the collocation gives biased estimates. Since the collocation can be shown to be equivalent to a special case of the least squares adjustment, the variance of unit weight can be estimated for the collocation also. This estimate gives the scale factor for the covariance matrices being used in the collocation. In addition, the methods of testing hypotheses and establishing confidence intervals for the parameters of the least squares adjustment may be applied to the collocation.     

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

In this contribution, using the example of the Mátern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schön model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Mátern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Mátern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Mátern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).     

病态加权总体最小二乘的广义岭估计解法

针对加权情形下的变量误差(EIV)模型,采用广义岭估计法处理总体最小二乘平差的病态性问题. 结合最优化准则和协方差传播率推导了未知参数的改正数求解公式;根据参数估计值的均方误差最小化原理,通过求偏导数列出广义岭估计中岭参数的迭代解式,并讨论了广义岭参数的含义和作用,给出了确定岭参数的L-曲线法. 通过算例比较分析了加权最小二乘估计、总体最小二乘估计、加权最小二乘岭估计、总体最小二乘岭估计、加权最小二乘的广义岭估计和总体最小二乘广义岭估计,叙述了加权总体最小二乘的广义岭估计的优缺点.