RECURSIVE LEAST SQUARES WITH REAL TIME STOCHASTIC MODELING: APPLICATION TO GPS RELATIVE POSITIONING

Abstract

Geodetic data processing is usually performed by the least squares (LS) adjustment method. There are two different forms for the LS adjustment, namely the batch form and recursive form. The former is not an appropriate method for real time applications in which new observations are added to the system over time. For such cases, the recursive solution is more suitable than the batch form. The LS method is also implemented in GPS data processing via two different forms. The mathematical model including both functional and stochastic models should be properly defined for both forms of the LS method. Proper choice of the stochastic model plays an important role to achieve high-precision GPS positioning. The noise characteristics of the GPS observables have been already investigated using the least squares variance component estimation (LS-VCE) in a batch form by the authors. In this contribution, we introduce a recursive procedure that provides a proper stochastic modeling for the GPS observables using the LS-VCE. It is referred to as the recursive LS-VCE (RLS-VCE) method, which is applied to the geometry-based observation model (GBOM). In this method, the (co)variances parameters can be estimated recursively when the new group of observations is added. Therefore, it can easily be implemented in real time GPS data processing. The efficacy of the method is evaluated using a real GPS data set collected by the Trimble R7 receiver over a zero baseline. The results show that the proposed method has an appropriate performance so that the estimated (co)variance parameters of the GPS observables are consistent with the batch estimates. However, using the RLS-VCE method, one can estimate the (co)variance parameters of the GPS observables when a new observation group is added. This method can thus be introduced as a reliable method for application to the real time GPS data processing.