Adjustment theory: an introduction

By Peter Teunissen. 2000

197 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Civil and Structural Engineering, Oceanography

View Paper: https://www.semanticscholar.org/paper/6c7682c5ea33a7b85086261ab3cf829eca6a308d

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Citations in the corpus

Least-squares variance component estimation (2008, methodology): “The solution for x1 is then identical to the solution one would get when solving E{BT2 y} = BT2 A1x1 in a least-squares sense, but now with the weight matrix W2 = (BT2 W −1 B2)−1 (e.g., Teunissen, 2000).”
Weighted total least squares formulated by standard least squares theory (2012, unknown): “The covariance matrices of the least-squares estimates x̂ , ŷ, and ê are Qx̂ = (AT Q−1 y A) −1 , Qŷ = PAQy, and Qê = P⊥ A Qy , respectively (Teunissen, 2000).”
Non-negative least-squares variance component estimation with application to GPS time series (2016, background): “In this case, the covariance matrix of the constrained non-negative VCs is Qσ̂ = N−1P⊥C , where P⊥C = I − C(Ct N−1C)−1Ct N−1 is an orthogonal projector (see Teunissen 2000).”
Data-snooping procedure applied to errors-in-variables models (2013, unknown): “The formulation of the hypotheses is as follows (Teunissen, 2000b; Leick, 1998; Koch, 1999) 2 20 0 0ˆ:H versus 2 2 0 0ˆ:aH .”
Review and principles of PPP-RTK methods (2015, unknown): “…procedure, namely tz̃r = ˆ̃zr − ˇ̃zr , r = 2, . . . , n (116) According to the least-squares conditional adjustment, the unbiased estimators ˆ̃ar,GF and d ˆ̃tr,GF are corrected by the above misclosures to provide the best linear unbiased estimators (BLUEs) ˇ̃ar,GF and d ˇ̃tr,GF (Teunissen 2000).”
Assessing receiver noise using GPS short baseline time series (2006, background): “The above minimization problem is equivalent to the following maximization problem (Teunissen 2000): xk¼i ¼ argmax xj PAj y 2 Q 1y with Aj ¼ P?”
Least-squares variance component estimation: theory and GPS applications (2007, background): “Following chapter 5 of Adjustment Theory, our starting point is the representation (Teunissen, 2000a) E{y} = Ax; B x = c; D{y} = Qy (D.”
Application of least squares variance component estimation to errors-in-variables models (2013, unknown): “…and of the total residuals in the model ỹ = Ãx+ẽ were, respectively, { ŷ = PÃ ỹ = Ãx̂ = (A − Ẽ A)x̂ ê = P⊥ Ã ỹ = ỹ − ŷ = y − Ax̂ , (12) where PÃ = Ã ( Ãt Q−1ỹ Ã )−1 Ãt Q−1ỹ and P ⊥ Ã = I − Ã ( Ãt Q−1ỹ Ã )−1 Ãt Q−1ỹ are two orthogonal projectors (Teunissen 2000).”