Adjustment theory: an introduction
Adjustment theory: an introduction
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DOI：
10.59490/tb.95
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发表时间：
2000-12
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影响因子：
--
通讯作者：
P. Teunissen
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文献类型：
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作者：
P. Teunissen
平差理论可以看作是数学大地测量学的一部分，它处理冗余测量的最佳组合以及未知参数的估计。它对大地测量学家来说是必不可少的，其意义相当于力学对土木工程师或机械工程师的意义。从历史上看，第一个结合冗余测量的方法起源于对大地测量学和天文学中三个问题的研究，即确定地球的大小和形状，解释木星和土星运动的长期不平等，以及找到月球运动的数学表示。如今，调整方法用于各种各样的大地测量应用，例如从测量和导航到遥感和全球定位。进行冗余测量的两个主要原因是希望提高计算结果的准确性和能够检查错误的要求。由于测量中固有的不确定性，测量冗余通常导致不一致的方程组。如果没有额外的准则，这样的方程组不是唯一可解的。在这门平差理论的入门课程中，我们将开发并介绍解决不一致方程组的方法。主要原理是最小二乘平差及其统计特性。不相容的方程组可以有许多不同的形式。它们可以以参数形式、隐式形式或这两种形式的组合给出。在每一种情况下，最小二乘法的原理都是一样的。然而，解决方案的算法实现将有所不同。根据手头的应用，人们也可能希望选择在一个单一步骤中或以逐步的方式获得解决方案。这导致需要以分区形式来制定方程组。存在不同的分区，测量分区、参数分区、或测量和参数两者的分区。分区的选择也影响解决方案的算法实现。在这篇介绍性的文章中，强调了调整的方法，尽管给出了各种例子来说明理论。所讨论的方法是解决大地测量中各种平差问题的基础。
Adjustment theory can be regarded as the part of mathematical geodesy that deals with the optimal combination of redundant measurements together with the estimation of unknown parameters. It is essential for a geodesist, its meaning comparable to what mechanics means to a civil engineer or a mechanical engineer. Historically, the first methods of combining redundant measurements originate from the study of three problems in geodesy and astronomy, namely to determine the size and shape of the Earth, to explain the long-term inequality in the motions of Jupiter and Saturn, and to find a mathematical representation of the motions of the Moon. Nowadays, the methods of adjustment are used for a much greater variety of geodetic applications, ranging from, for instance, surveying and navigation to remote sensing and global positioning. The two main reasons for performing redundant measurements are the wish to increase the accuracy of the results computed and the requirement to be able to check for errors. Due to the intrinsic uncertainty in measurements, measurement redundancy generally leads to an inconsistent system of equations. Without additional criteria, such a system of equations is not uniquely solvable. In this introductory course on adjustment theory, methods are developed and presented for solving inconsistent systems of equations. The leading principle is that of least-squares adjustment together with its statistical properties. The inconsistent systems of equations can come in many different guises. They could be given in parametric form, in implicit form, or as a combination of these two forms. In each case the same principle of least-squares applies. The algorithmic realizations of the solution will differ however. Depending on the application at hand, one could also wish to choose between obtaining the solution in one single step or in a step-wise manner. This leads to the need of formulating the system of equations in partitioned form. Different partitions exist, measurement partitioning, parameter partitioning, or a partitioning of both measurements and parameters. The choice of partitioning also affects the algorithmic realization of the solution. In this introductory text the methodology of adjustment is emphasized, although various samples are given to illustrate the theory. The methods discussed form the basis for solving different adjustment problems in geodesy.