Regularized integer least-squares estimation: Tikhonov's regularization in a weak GNSS model

The strength of the GNSS precise positioning model degrades in cases of a lack of visible satellites, poor satellite geometry or uneliminated atmospheric delays. The least-squares solution to a weak GNSS model may be unreliable due to a large mean squared error (MSE). Recent studies have reported that Tikhonov's regularization can decrease the solution's MSE and improve the success rate of integer ambiguity resolution (IAR), as long as the regularization matrix (or parameter) is properly selected. However, there are two aspects that remain unclear: (i) the optimal regularization matrix to minimize the MSE and (ii) the IAR performance of the regularization method. This contribution focuses on these two issues. First, the "optimal" Tikhonov's regularization matrix is derived conditioned on an assumption of prior information of the ambiguity. Second, the regularized integer least-squares (regularized ILS) method is compared with the integer least-squares (ILS) method in view of lattice theory. Theoretical analysis shows that regularized ILS can increase the upper and lower bounds of the success rate and reduce the upper bound of the LLL reduction complexity and the upper bound of the search complexity. Experimental assessment based on real observed GPS data further demonstrates that regularized ILS (i) alleviates the LLL reduction complexity, (ii) reduces the computational complexity of determinate-region ambiguity search, and (iii) improves the ambiguity fixing success rate.