Adjustment of observations
Adjustment of observations is a model for the solution of an overdetermined system of equations based on the principle of least squares. It is used extensively in the disciplines of surveying, geodesy, photogrammetry (the field of geomatics, collectively).
There are three forms of adjustment of observations: parametric, conditional, and combined. In parametric adjustment, one can find an observation equation h(X)=Y relating observations Y explicitly in terms of parameters X (leading to the A-model below). In conditional adjustment, there exists a condition equation g(Y)=0 involving only observations Y (leading to the B-model below), with no parameters X at all. Finally, in a combined adjustment, both parameters X and observations Y are involved implicitly in a mixed-model equation f(X,Y)=0. Clearly, parametric and conditional adjustments correspond to the more general combined case when f(X,Y)=h(X)-Y and f(X,Y)=g(Y), respectively. Yet the special cases warrant simpler solutions, as detailed below. Often in the literature, Y may be denoted L.
The equalities above only hold for the estimated parameters and observations , thus . In contrast, measured observations and approximate parameters produce a nonzero misclosure:
and the second one,
The linearized model then reads:
where are estimated parameter corrections to the a priori values, and are post-fit observation residuals. In the parametric adjustment, the second design matrix is an identity, B=-I, and the misclosure vector can be interpreted as the pre-fit residuals, , so the system simplifies to:
which is in the form of ordinary least squares. For the more general cases, Lagrange multipliers are introduced to relate the two Jacobian matrices and transform the constrained least squares problem into an unconstrained one (albeit a larger one). In any case, their manipulation leads to the and vectors as well as the respective parameters and observations a posteriori covariance matrices.
Given the matrices and vectors above, their solution is found via standard least-squares methods; e.g., forming the normal matrix and applying Cholesky decomposition, applying the QR factorization directly to the Jacobian matrix, iterative methods for very large systems, etc.
If rank deficiency is encountered, it can often be rectified by the inclusion of additional equations imposing constraints on the parameters and/or observations.
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- Parametric adjustment is similar to most of regression analysis and coincides with the Gauss-Markov model
- Combined adjustment, also known as the Gauss-Helmert model, is an instance of the errors-in-variables models
- The use of a priori parameter covariance matrix is akin to Tikhonov regularization
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (June 2014)|
Lecture notes and Technical reports:
- Nico Sneeuw and Friedhelm Krum, "Adjustment theory", Geodätisches Institut, Universität Stuttgart, 2014
- Krakiwsky, "A synthesis of recent advances in the method of least squares", Lecture Notes #42, Department of Geodesy and Geomatics Engineering, University of New Brunswick, 1975
- Cross, P.A. "Advanced least squares applied to position-fixing", University of East London, School of Surveying, Working Paper No. 6, ISSN 0260-9142, January 1994. First edition April 1983, Reprinted with corrections January 1990. (Original Working Papers, North East London Polytechnic, Dept. of Surveying, 205 pp., 1983.)
- Snow, Kyle B., Applications of Parameter Estimation and Hypothesis Testing to GPS Network Adjustments, Division of Geodetic Science, Ohio State University, 2002
- Edward M. Mikhail, Friedrich E. Ackermann, "Observations and least squares", University Press of America, 1982
- Wolf, Paul R. (1995). "Survey Measurement Adjustments by Least Squares". The Surveying Handbook: 383–413. doi:10.1007/978-1-4615-2067-2_16.
- Paul Wolf and Bon DeWitt, "Elements of Photogrammetry with Applications in GIS", McGraw-Hill, 2000
- Karl-Rudolf Koch, "Parameter Estimation and Hypothesis Testing in Linear Models", 2a ed., Springer, 2000
- P.J.G. Teunissen, "Adjustment theory, an introduction", Delft Academic Press, 2000
- Edward M. Mikhail, James S. Bethel, J. Chris McGlone, "Introduction to Modern Photogrammetry", Wiley, 2001
- Harvey, Bruce R., "Practical least squares and statistics for surveyors", Monograph 13, Third Edition, School of Surveying and Spatial Information Systems, University of New South Wales, 2006
- Huaan Fan, "Theory of Errors and Least Squares Adjustment", Royal Institute of Technology (KTH), Division of Geodesy and Geoinformatics, Stockholm, Sweden, 2010
- Gielsdorf, F.; Hillmann, T. (2011). "Mathematics and Statistics". "Springer Handbook of Geographic Information". p. 7. doi:10.1007/978-3-540-72680-7_2. ISBN 978-3-540-72678-4.
- Charles D. Ghilani, "Adjustment Computations: Spatial Data Analysis", John Wiley & Sons, 2011
- Charles D. Ghilani and Paul R. Wolf, "Elementary Surveying: An Introduction to Geomatics", 13th Edition, Prentice Hall, 2011
- Erik Grafarend and Joseph Awange, "Applications of Linear and Nonlinear Models: Fixed Effects, Random Effects, and Total Least Squares", Springer, 2012