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.