Gauss-Markov model with random parameters to adjust results of surveys of geodetic control networks

IF 0.3 Q4 REMOTE SENSING Reports on Geodesy and Geoinformatics Pub Date : 2021-06-01 DOI:10.2478/rgg-2021-0001
M. Banaś, J. Czaja, J. Dąbrowski
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Abstract

Abstract Alignment of an engineering object project in the field is always conducted at the points of the geodetic control network, the coordinates of which are determined on the basis of the results of its elements survey and with connection to the national spatial reference system. The points of the national spatial reference system determined on the basis of previous surveys have specified coordinates with adequate accuracy, which is included in their covariance matrix. The coordinates of the geodetic control network points are determined more accurately than the points of the national spatial reference system and this means that the results of surveys of the geodetic control network have to be adequately incorporated into the coordinates of the reference points. In order to perform this incorporation, it may be assumed that the coordinates of the reference points are random, that is, they have a covariance matrix, which should be used in the process of adjusting the results of the geodetic control network observation. This research paper presents the principles for the estimation of the Gauss-Markov model parameters applied in case of those geodetic control networks in which the coordinates of the reference points have random character. On the basis of the observation equations δ + AX = L for the geodetic control network and using the weighting matrix P and the matrix of conditional covariances (P−1 + ACXAT) for the observation vector L, the parameter vector X is estimated in the form of the derived formula X^=(CX−1+ATPA)−1ATP⋅L {\bf{\hat X}} = {\left( {{\bf{C}}_X^{ - 1} + {{\bf{A}}^T}{\bf{PA}}} \right)^{ - 1}}{{\bf{A}}^T}{\bf{P}} \cdot {\bf{L}} . The verification of these estimation principles has been illustrated by the example of a fragment of a levelling geodetic control network consisting of three geodetic control points and two reference points of the national spatial reference system. The novel feature of the proposed solution is the application of covariance matrices of the reference point coordinates to adjust the results of the survey of geodetic control networks and to determine limit standard deviations for the estimated coordinates of geodetic control network points.
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带随机参数的高斯-马尔可夫模型对大地测控网测量结果的调整
摘要野外工程对象项目的对中总是在大地测量控制网的点上进行,这些点的坐标是根据其要素测量结果并与国家空间参考系统相连接确定的。根据以前的调查确定的国家空间参考系统的点具有足够精确的指定坐标,这包括在它们的协方差矩阵中。大地测量控制网点的坐标比国家空间参考系统点的坐标确定得更精确,这意味着大地测量控制网的测量结果必须充分纳入参考点的坐标。为了进行这种合并,可以假设参考点的坐标是随机的,即它们有一个协方差矩阵,在调整大地测控网观测结果的过程中使用协方差矩阵。本文研究了参考点坐标具有随机性的大地测量控制网中高斯-马尔可夫模型参数估计的原理。根据大地测量控制网的观测方程δ + AX = L,利用观测向量L的加权矩阵P和条件协方差矩阵(P−1 + ACXAT),对参数向量X进行估计,推导公式为X^=(CX−1+ATPA)−1ATP·L {\bf{\hat X}}={\left ({{\bf{C}}_X^ - 1{+} ^{{\bf{A}}T }{\bf{PA}}}\right)^ {- 1}}{{\bf{A}}^T }{\bf{P}}\cdot。以一个由三个大地测量控制点和两个国家空间参考系统参考点组成的水准测量控制网片段为例,说明了这些估计原理的验证。该方法的新颖之处在于利用参考点坐标的协方差矩阵对大地测控网测量结果进行调整,并确定大地测控网测点估计坐标的极限标准差。{\bf{L}}
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来源期刊
自引率
28.60%
发文量
5
审稿时长
12 weeks
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