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。     

一、P20 一般性问题

CH20081939基于方差分量估计的拟合推估及其在GIS误差纠正的应用=Variance Component Estimation Based Collocation and Its Application in GIS Error Fitting/杨元喜,张菊清,张亮(西安测绘研究所)//测绘学报.-2008,37(2).-152～157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。图2表1参22     

无缝线性回归与预测模型

建立回归模型常采用最小二乘方法并忽略自变量观测误差。尽管同时顾及自变量和因变量观测误差的总体最小二乘方法近年来得到了广泛研究,但在模型预测时,依然忽略了待预测自变量的观测误差。对此,本文提出了一种严格考虑所有变量观测误差的无缝线性回归和预测模型,该模型将回归模型的建立和因变量预测联合处理,在建立回归模型过程中对待预测自变量的观测误差进行估计并修正,从而提高了模型预测效果。理论证明,现有的几种线性回归模型都是无缝线性回归和预测模型的特例。试验结果表明,无缝线性回归和预测模型的预测效果优于现有的几种模型,尤其在变量观测误差相关性较大时,无缝模型对预测效果的改善更为显著。     

灰色模型参数估计改进及其在变形预报中应用

经典的灰色模型在建模过程中,参数估计是依据最小二乘准则得到的。当观测数据中存在着较大随机波动时,导致系数矩阵中存在较大误差,这种估计方法可能影响参数的可靠性,进而影响该模型应用。针对这一问题,本文提出运用整体最小二乘法改进估计参数,该方法在解算法方程时,顾及了系数阵和观测向量同时存在的误差,因而可以改善参数的估计精度,提高模型预报精度。最后,结合矿山测量工程实际,证实了这种参数估计方法能提高老采空区残余沉降的预测精度。     

一种Kalman滤波系统误差及其协方差矩阵的半参数估计方法

提出用半参数估计理论来解决系统误差对Kalman滤波解的影响问题。即用半参数模型中的非参数分量表达观测模型和动力学模型中未知的系统误差,在移动的窗口内,基于观测残差和状态向量预测残差拟合模型系统误差,进而修正相应的观测向量和状态预测向量的协方差矩阵,以消除系统误差对滤波的影响。同时这种方法还有明显的优点,就是在滤波过程中不需要对系统误差做任何假设。文中推导了基于正则核估计来解算导航系统半参数模型的相应公式,并根据一个模拟的算例,证明了算法的有效性。     

利用基于抗差垂直向方差分量估计的GPS-InSAR数据融合方法反演三维形变场

GPS-InSAR数据融合解算三维形变场模型易受观测值粗差影响,且基于方差分量估计的定权方法不具备抵御粗差能力,计算效率低下。鉴于此,本文提出了一种基于抗差垂直向方差分量估计的GPS-InSAR融合解算模型,利用方差分量估计方法及抗差估计理论,通过对观测值最优化分类并进行选权迭代,精确分配权重,进而有效计算三维形变场。试验结果表明,该方法能有效抵御观测值粗差不利影响,提高三维形变场反演精度,提升逐点式计算的三维形变场效率。     

随机模型对解算西安流动SLR站坐标的影响

合理的参数估计及精度评定不仅需要可靠的函数模型，而且需要正确的随机模型。从权函数和粗差编辑两方面，研究了不同随机模型对西安流动卫星激光测距（satellite laser ranging，SLR）站坐标解算的影响，采用全球Lageos-1卫星观测数据计算了西安流动SLR站坐标。计算结果表明:①西安流动SLR站的观测精度和坐标解算精度均达到厘米级。②随机模型直接影响SLR站坐标的解算结果及可靠性；对于相同的计算弧段，抗差方差分量估计得到的站坐标精度最高、结果最稳定，残差加权均方差最小，观测资料利用率也最高；对于相同的计算方案，采用的SLR数据越多，坐标估计精度越高。     

航空矢量重力测量中光纤陀螺随机漂移误差实时补偿方法

光纤陀螺随机漂移误差是影响航空矢量重力测量系统姿态解算精度的关键因素。建立模型并在输出中对其补偿是抑制该项误差的有效方法。针对传统ARMA模型只能对平稳随机漂移误差建模,且模型无法满足实时滤波需求的问题,本文引入适用于非平稳随机漂移误差的ARIMA模型,同时给出详细的建模过程,并提出采用实时平均算法消除原始采样序列中常值分量的思路,实现了随机漂移误差的实时Kalman滤波估计。基于本文所提出的模型和实时滤波算法,对光纤陀螺实测数据进行分析,结果表明处理后信号中随机漂移误差的方差减小了46.5%。Allan方差分析结果表明,滤波后角度随机游走系数和角速率随机游走系数分别降低了约50%和40%。本文的结果说明ARIMA模型能够准确描述陀螺的非平稳随机漂移误差。基于实时平均算法的Kalman滤波可实现随机漂移误差的在线估计,有望提高航空矢量重力测量系统的姿态解算精度。     

附有约束最小二乘配置的信号与观测误差的方差分量估计

介绍方差-协方差分量估计理论的研究和发展情况,讨论最小二乘配置模型信号与观测误差的方差分量估计问题。在实际应用中,考虑到未知参数间存在几何或物理约束,针对附有约束条件最小二乘配置的方差分量估计的问题,基于Helmert方差分量估计原理,导出相应的计算公式。模拟算例结果表明,利用约束条件能够改善方差分量的估计精度,验证方法的有效性。     

天文大地网与GPS2000网联合平差数据处理方法

介绍了天文大地网与GPS2000网联合平差数据处理中的主要数学模型、大规模稀疏矩阵的有效解算方法，用Helmert方差分量估计算法进行地面网的方差分量估计，并用该方法对我国天文大地网的观测数据及空间网数据进行联合平差，得出全网平均点位的点位中误差为o．11m，点位精度95．5％优于0．3m。     

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。     

A New Method for Identifying the Model Error of Adjustment System

IntroductionAdjust ment process deals with the problemsfor esti mating parameters and assessing precisionbased on observations with errors . The researchon adjust ment system involves building mathmodel , determining the opti mization rule andstudying adjust ment arithmetic . Math model ofadjust ment system(also called adjust ment mod-el)is composed of functional and stochastic mod-els . The former describes the relationship be-tween observations and the expectation of pa-rameters . The latte…     

A new method for indentifying the model error of adjustment system

Some theory problems affecting parameter estimation are discussed in this paper. Influence and transformation between errors of stochastic and functional models is pointed out as well. For choosing the best adjustment model, a formula, which is different from the literatures existing methods, for estimating and identifying the model error, is proposed. On the basis of the proposed formula, an effective approach of selecting the best model of adjustment system is given.     

Solution of the weighted symmetric similarity transformations based on quaternions

A new method through Gauss–Helmert model of adjustment is presented for the solution of the similarity transformations, either 3D or 2D, in the frame of errors-in-variables (EIV) model. EIV model assumes that all the variables in the mathematical model are contaminated by random errors. Total least squares estimation technique may be used to solve the EIV model. Accounting for the heteroscedastic uncertainty both in the target and the source coordinates, that is the more common and general case in practice, leads to a more realistic estimation of the transformation parameters. The presented algorithm can handle the heteroscedastic transformation problems, i.e., positions of the both target and the source points may have full covariance matrices. Therefore, there is no limitation such as the isotropic or the homogenous accuracy for the reference point coordinates. The developed algorithm takes the advantage of the quaternion definition which uniquely represents a 3D rotation matrix. The transformation parameters: scale, translations, and the quaternion (so that the rotation matrix) along with their covariances, are iteratively estimated with rapid convergence. Moreover, prior least squares (LS) estimation of the unknown transformation parameters is not required to start the iterations. We also show that the developed method can also be used to estimate the 2D similarity transformation parameters by simply treating the problem as a 3D transformation problem with zero (0) values assigned for the z-components of both target and source points. The efficiency of the new algorithm is presented with the numerical examples and comparisons with the results of the previous studies which use the same data set. Simulation experiments for the evaluation and comparison of the proposed and the conventional weighted LS (WLS) method is also presented.     

A new method for indentifying the model error of adjustment system

Some theory problems affecting parameter estimation are discussed in this paper. Influence and transformation between errors of stochastic and functional models is pointed out as well. For choosing the best adjustment model, a formula, which is different from the literatures existing methods, for estimating and identifying the model error, is proposed. On the basis of the proposed formula, an effective approach of selecting the best model of adjustment system is given. Project supported by the Open Research Fund Program of the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (No. 905276031-04-10).     

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.     

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.     

加权整体最小二乘EIV模型与算法——与"加权整体最小二乘EIO模型与算法"一文的讨论

加权整体最小二乘方法是一种能同时顾及EIV（errors-in-variables）模型中系数矩阵和观测向量误差的参数估计方法。根据不同的应用场景，EIV模型则表现出不同的结构特征。"加权整体最小二乘EIO模型与算法"一文采用EIO模型处理EIV模型中的结构化问题^*。为了将其与现有方法进行对比，本文罗列出4种处理EIV模型结构特征的方法，并归纳了8种参数估计公式。同时从精度评定的角度讨论了整体最小二乘解的一阶及更高阶精度近似评定方法。需要强调的是，针对EIV模型及其参数估计理论可以从函数模型、随机模型和参数估计方法3个方面展开研究，但各方法殊途同归。     

基于半参数模型的滑坡段数据估计与预测

在滑坡段数据的参数平差模型的基础上,通过在模型中增加表示模型误差或系统误差的非参数分量,将半参数模型的估计理论应用到滑坡段参数估计与预测中,分离出数据中的系统误差,给出了滑坡稳定系数的预测公式;并以云阳-奉节长江顺层岸滑坡段实测数据为例,将半参数估计方法与最小二乘方法进行了比较分析,验证了半参数估计方法应用到估计和预测滑坡段数据的可行性和有效性。