A generalized least-squares filter designed for GNSS data processing

The Kalman filter stands as one of the most widely used methods for recursive parameter estimation. However, its standard formulation typically assumes that all state parameters avail initial values and dynamic models, an assumption that may not always hold true in certain applications, particularly in global navigation satellite system (GNSS) data processing. To address this issue, Teunissen et al. (2021) introduced a generalized Kalman filter that eliminates the need for initial values and allows linear functions of parameters to have dynamic models. This work proposes a least-squares approach to reformulate the generalized Kalman filter, enhancing its applicability to GNSS data processing when the parameter dimension varies with satellite visibility changes. The reformulated filter, named generalized least-squares filter, is equivalent to the generalized Kalman filter when all state parameters are recursively estimated. In this case, we demonstrate how both the generalized Kalman filter and the generalized least-squares filter adaptatively manage newly introduced or removed parameters. Specifically, when estimation is limited to parameters with dynamic models, the generalized least-squares filter extends the generalized Kalman filter by allowing the dimension of estimated parameters to vary over time. Moreover, we introduce a new element of least-squares smoothing, creating a comprehensive system for prediction, filtering, and smoothing. To verify, we conduct simulated kinematic and vehicle-borne kinematic GNSS positioning using the proposed generalized least-squares filter and compare the results with those from the standard Kalman filter. Our findings show that the generalized least-squares filter delivers better results, maintaining the positioning errors at the centimeter level, whereas the Kalman-filter-based positioning errors exceed several decimeters in some epochs due to improper initial values and dynamic models. Moreover, the normal equation reduction strategy employed in the generalized least-squares filter improves computational efficiency by 23% and 32% in simulated kinematic and vehicle-borne kinematic positioning, respectively. The generalized least-squares filter also allows for the flexible adjustment of smoothing window lengths, facilitating successful ambiguity resolution in several epochs. In conclusion, the proposed generalized least-squares filter offers flexibility for various GNSS data processing scenarios, ensuring both theoretical rigor and computational efficiency.