Sequential best integer-equivariant estimation for geodetic network solutions

The key to high precision parameter estimation (e.g., positioning) in global navigation satellite system (GNSS) applications is to take the integer nature of the carrier-phase ambiguities into account. The class of integer estimators, like integer bootstrapping (BS) or integer least-squares (ILS), fixes the ambiguities to integer values, which can also decrease the precision of the estimates of the non-ambiguity parameters, if the probability of wrong fixing is not sufficiently small. The best integer-equivariant (BIE) estimator is optimal in the sense of minimizing the mean-squared error (MSE) of both the integer and real valued parameters, regardless of the precision of the float solution. However, like ILS, the BIE estimator comprises a search in the integer space of ambiguities, whose complexity grows exponentially with the number of ambiguities, which is not feasible for large-scale network solutions. To overcome this problem, a sequential BIE (SBIE) algorithm is proposed, which shows close to optimal performance while being part of the class with complexity of linear order. Numerical simulations are used to verify the performance of the SBIE algorithm.