Theory of integer equivariant estimation with application to GNSS

Abstract

Abstract. Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called 'fixed' baseline estimator is known to be superior to its 'float' counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the 'float' baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its 'float' and its 'fixed' counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a GaussMarkov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss-Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove-restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have.

Best prediction in linear models with mixed integer/real unknowns: theory and application

“…As was pointed out in Teunissen (2003), the expression for theâ BIEE is identical to its Bayesian counterpart as given in Betti et al (1993) and Gundlich and Koch (2002); also see Teunissen (2001) and Gundlich and Teunissen (2004). This equivalence nicely bridges the gap between the present nonBayesian approach and the Bayesian approach.…”

“…This equivalence nicely bridges the gap between the present nonBayesian approach and the Bayesian approach. Despite this similarity, however, there are important differences in the probabilistic evaluation of the solution, as described in Teunissen (2003).…”

Distributional theory for the DIA method

“…For instance, although the weights p i (t) are binary in case of the DIA estimator, smoother weighting functions of the misclosures will provide for a larger class of estimators that may contain estimators with better performances for certain defined criteria. This in analogy with integer (I) and integer-equivariant (IE) estimation, for which the latter provides a larger class containing the optimal BIE estimator (Teunissen 2003b). And just like the nonBayesian BIE estimator was shown to have a Bayesian counterpart (Teunissen 2003b;Betti et al 1993), the nonBayesian DIA estimator with smoother weights may find its counterpart in methods of Bayesian and information-theoretic multimodel inference (Burnham and Anderson 2002).…”

A New Method For DGPS Ambiguity Resolution?

“…Optimality refers here to the maximization of the probability of correct integer estimation (the success rate) or to the minimization of the mean squared error. The three optimal estimators, one for each class, have been identified in [4][5][6].…”