# Some Generalized Equivalence Theorems for Least-Squares Adjustment

## Abstract

In 1986 an equivalence theorem had been published by B. Schaffrin and E. Grafarend that allows to check whether any form of differenced GPS data still carries the original information insofar as the Least-Squares Solution (LESS) for the parameters of interest remains the same. In its simplest form a (linearized) *Gauss-Markov Model* is compared with a corresponding *Model of Condition Equations*, after eliminating all parameters. The theorem then states that the residual vector from a least-squares adjustment comes out *identical* in both models if and only if two relations hold true, one *orthogonality* and one *rank-additivity* relation. Without the latter, we would have to admit a loss of information that usually results in a biased LESS.

In this contribution we shall generalize the existing equivalence theorem to cases that include (fixed or stochastic) constraints, random effects parameters and, in particular, the *Dynamic Linear Model* where the LESS amounts to “Kalman filtering”, with substantial benefits in some GPS ambiguity resolution problems.

## Keywords

Global Position System Nuisance Parameter Residual Vector Equivalence Theorem Dynamic Linear Model## Preview

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## References

- Baksalary, J.K. and R. Kala (1981). Linear transformations preserving best linear unbiased estimators in a general Gauss-Markov Model.
*Ann. Statist*., 9, pp. 913–916CrossRefGoogle Scholar - Baksalary, J.K. (1984). A study of the equivalence between a Gauss-Markov Model and its augmentation by nuisance parameters.
*Math. Operations-Forschung, Ser. Statistics*, 15, pp. 1–35.Google Scholar - Helmert, F.R. (1907).
*Adjustment Computations by the Method of Least-Squares (in german), 2n*^{d}Edition, Teubner: Leipzig, 578 pages.Google Scholar - Povalyaev, A.A. (1999). Filtering problem for ambiguous phase measurements,
*J. of Communications Technology and Electronics of the Russian Acad. of Sciences (English edition)*, 44, pp. 904–912.Google Scholar - Schaffrin, B. and E. Grafarend (1986). Generating classes of equivalent linear models by nuisance parameter elimination. Applications to GPS observations,
*Manus. Geodaet*., 11, pp. 262–271.Google Scholar