Double-parameter iterative Tikhonov regularization of weak single-epoch GNSS mathematical models

Abstract

In the spirit of Tikhonov regularization, the double-parameter iterative regularization is developed to mitigate the weakness of single-epoch GNSS models. We propose a simultaneous double-parameter iterative regularization of the least-squares (LS) estimators of parameters of interest in the GNSS model to improve their accuracy properties so that variance-covariance (vc) matrices will describe their good scale. Regularization parameters (RP) are stored in the regularization operator, which plays the standardizing role. Thus, this approach considers the heteroscedasticity of unambiguous and ambiguous model parameters in the regularization principle. We used the quality-based mean squared error (mse) matrix trace minimization criterion to find the optimal RP values simultaneously. Against the unconstrained LS estimation, two variants of iterative regularization of unconstrained LS estimation are tested. The first is the double-parameter iterative Tikhonov regularization. In turn, the second one is its one-parameter counterpart. The numerical example is based on a simulation to guarantee a wide range of geometric positioning construction and provide a variety of measurement circumstances. The double-parameter iterative regularization mitigates the weakness of the single-epoch model more effectively by considering the heteroscedasticity of model parameters in the regularization principle. At the cost of losing the regularized LS estimator’s unbiased localization, the vc-matrix describes float solutions of better precision. They are less dispersed around the actual parameter values at the cost of bias. Thus, higher accuracy in the sense of mse is provided. The regularized estimator is, therefore, well-scaled with biased localization. It also provides the more peaked probability density function (PDF) of float ambiguity estimates, obtaining a higher success rate (SR) of correct integer-least squares (ILS) ambiguity resolution (AR). Therefore, the improved ILS estimator performs well in the ambiguity domain with regularized bias when processing a single-epoch data set, allowing precise GNSS positioning.