Optimal linear combinations of triple frequency carrier phase data from future global navigation satellite systems

Abstract

With the introduction of a third frequency on GPS Block IIF satellites and the implementation of Galileo, there will be three freely available carrier phase measurements transmitted from each system. This change in satellite navigation infrastructure will enable the use of linear combinations of the original measurements that are not currently available. As a result, it is conceivable that there may be an optimal choice of combination coefficients for a given positioning campaign. This article outlines some of the motivations of using linear combinations of Global Navigation Satellite System (GNSS) data. For example, linear combinations can be used to eliminate or mitigate individual sources of error, they can be used to alleviate excessive computational burdens, and they can be used to reduce the necessary bandwidth in communication systems. Upon establishing the motivations for using linear combinations of data, the mathematical theory involved in creating linear combinations is given. The variance of the combined signal is shown to be a weighted sum of the variances of the error sources in the untransformed signals. The weights depend on the choice of combination coefficients and the nominal frequencies of the carrier signals. As a result, there are certain choices of combination coefficients that eliminate or mitigate individual sources of error. Three categories of combinations are developed: those that eliminate the ionospheric effect, those that mitigate the effects of thermal noise and multipath, and those that mitigate the tropospheric effects. The relationships between these various categories of linear combinations are shown geometrically and the concept of optimal linear combinations of data is discussed. Finally, experimental results using optimal linear combinations of data are shown. The results are obtained using a commercially available software simulator and a GNSS processing engine from the Mobile Multi-Sensor Research Group in the Geomatics Engineering Department of the University of Calgary. It is shown that there indeed exist combinations that produce approximately the same ambiguity estimation accuracy as the pure L1 (or E1) signals, but that deliver far better baseline precision results. However, like the pure L1 (or E1) signals, instantaneously resolving integer ambiguities with these combinations will be impossible for baselines longer than about 15 km depending on the existing ionospheric conditions.