Chapter 2Least-Squares Adjustments

Least-squares adjustment is useful for estimating parameters and carrying out objective quality control of measurements by processing observations according to a mathematical model and well-defined rules. The objectivity of least-squares quality control is especially useful in surveying when depositing or exchanging observations or verifying the internal accuracy of a survey. Least-squares solutions require redundant observations, i.e., more observations are required than are necessary to determine a set of unknowns exactly. This chapter contains compact but complete derivations of least-squares algorithms. For additional in-depth study of adjustments we recommend Grafarend (2006).

First, the statistical nature of measurements is analyzed, followed by a discussion of stochastic and mathematical models. The mixed adjustment model is derived in detail, and the observation equation and the condition equation models are deduced from the mixed model through appropriate specification. The cases of observed and weighted parameters are presented as well. A special section is devoted to minimal and inner constraint solutions and to those quantities that remain invariant with respect to a change in minimal constraints. Whenever the goal is to perform quality control on the observations, minimal or inner constraint solutions are especially relevant. Statistical testing is important for judging the quality of observations or the outcome of an adjustment. ...

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