Recursive Least Squares with Additive Parameters: Application to Precise Point Positioning

Abstract

There are two different but equivalent forms to solve a least squares (LS) adjustment problem, namely batch LS (BLS) and recursive LS (RLS). The batch form is usually not an appropriate approach when measurements are sequentially received over time. When the unknown parameters are constant and hence converge slowly over time to their final estimates, the RLS algorithm can be used. For time-varying parameters, the Kalman filter (KF) algorithm can be used. To properly reflect the time variations of the parameters, this method requires the appropriate choice of filter parameters, and hence tuning is an important stage. In some geodetic applications, as time progresses, new observations and parameters are added to the system of equations. For such applications, an RLS algorithm with additive parameters can be developed. The method differs from the KF in the sense that there is no need to define and manipulate the dynamic model and the noise structure of the parameters involved. The implementation of the method in both linear and nonlinear models is explained. As an application of the proposed method, the algorithm is implemented in the global positioning system (GPS) precise point positioning (PPP), either in static or kinematic mode. Generally, the PPP adjustment is performed via a sequential filter: either the LS sequential filter, or the discrete KF. We propose an efficient alternative based on the RLS with additive parameters, which is applicable to PPP. The performance of the method is investigated with regard to the repeatability and accuracy using a few 24-h data sets of four International GNSS Service (IGS) stations. Finally, the efficacy of the proposed method is investigated in the kinematic mode using two experiments: a stationary experiment and a real kinematic test. The results indicate that the proposed method can be employed as an appropriate alternative to the KF algorithm.