L1-Norm Estimation for Mixed Additive and Multiplicative Error Models: Theory and Application to GNSS Positioning

Although Global Navigation Satellite System (GNSS) observations are verified to suffer from mixed additive and multiplicative errors, current methodologies still rely heavily on conventional additive error models (AEMs). Furthermore, existing research on mixed additive and multiplicative error models (MAMEMs) is largely confined to least squares frameworks and simulated scenarios. To address this, this study explores parameter estimation for MAMEMs using the robust L1-norm approach. Four novel algorithms are proposed: the L1-norm-based Gauss Theorem Algorithm (L1-Gauss), Linear Programming Algorithm (L1-LP), Relaxation Method (L1-RM), and Weighted Relaxation Method (L1-WRM). Among them, L1-WRM considers observation weights, while the other three algorithms treat observations as equal weights. Using real-world GNSS datasets, the proposed algorithms are compared with least squares (LS), weighted least squares (WLS) and M-robust weighted least squares (MWLS) under AEMs, as well as the bias-corrected WLS (bcWLS) estimator specifically designed for MAMEMs. Experimental results demonstrate that: Statistically significant differences exist in accuracy among the eight algorithms. In scenarios affected by multipath or non-line-of-sight (NLOS) effects, the accuracy of the bcWLS outperforms that of the WLS, and the accuracy of the L1-WRM outperforms that of the MWLS, with robust algorithms achieving superior accuracy. In scenarios without multipath or NLOS effects, the accuracy of bcWLS is virtually consistent with WLS, showing a difference in 3D Mean Error of only about 2 cm. Similarly, the accuracy of L1-WRM is virtually consistent with MWLS, with a 3D Mean Error difference of only about 8 cm; in this case, non-robust algorithms outperform robust algorithms. Regarding single-epoch execution time, the maximum computation time across the five MAMEMs algorithms is 95.5939 ms, fully satisfying the requirements for 1 Hz real-time processing. These findings validate the practical application value of the proposed algorithms in GNSS pseudorange relative positioning.