Best Integer Equivariant (BIE) Ambiguity Resolution Based on Tikhonov Regularization for Improving the Positioning Performance in Weak GNSS Models

In complicated scenarios, due to the low precision of float solutions and poor reliability of fixed solutions, it is challenging to achieve a balance between accuracy and reliability of the integer least squares (ILS) estimation. To address this dilemma, the best integer equivariant (BIE) estimation, which makes a weighted sum of all possible candidates, has recently been attached great importance. The BIE solution approaches the float solution at a low ILS success rate, maintaining positioning reliability. As the success rate increases, it converges to the fixed solution, facilitating high-precision positioning. Furthermore, the posterior variance of BIE estimation provides the capability of reliability evaluation. However, in environments with a limited number or a deficient configuration of available satellites, there is a sharp decline in the strength of the GNSS precise positioning model. In this case, the exactness of weight allocation for integer candidates in BIE estimation will be severely compromised by unmodeled errors. When the ambiguity is incorrectly fixed, the wrongly determined optimal candidate is probably assigned an excessively high weight. Therefore, the BIE solution in a weak GNSS model always exhibits a significant positioning error consistent with the fixed solution. Moreover, the posterior variance of BIE estimation approximately resembles that of a fixed solution, losing error warning ability. Consequently, the BIE estimation may exhibit lower reliability compared to the ILS estimation employing a validation test with a loose acceptance threshold. To improve the positioning performance in weak GNSS models, a BIE ambiguity resolution (AR) method based on Tikhonov regularization is proposed in this paper. The method introduces Tikhonov regularization into the least squares (LS) estimation and the ILS ambiguity search, mitigating the serious impact of unmodeled errors on the BIE estimation under weak observation conditions. Meanwhile, the regularization factors are appropriately selected by utilizing an optimized approach established on the L-curve method. Simulation experiments and field tests have demonstrated that the method can significantly enhance the positioning accuracy and reliability in weak GNSS models. Compared to the traditional BIE estimation, the proposed method achieved accuracy improvements of 73.6% and 69.3% in the field tests with 10 km and 18 km baselines, respectively.