Effect of spatial correlation on the performances of modernized GPS and Galileo in relative positioning
In the context of processing GNSS (Global Navigation Satellite System) data, it is known that the estimation of the ionospheric delays decreases the strength of the observation model and makes significant the time required to fix the ambiguities namely in case of long baselines. However, considering the double-differenced (DD) ionospheric delays as stochastic quantities, the redundancy in this case increases and leads to the reduction of time of fixing the ambiguities. The approach developed in the present paper makes two considerations: 1) the DD ionospheric delays are assumed as stochastic quantities and, 2) the spatial correlation of errors is accounted for based on a simple model of correlation. A simulation is made and aims to study the effect of these two mentioned considerations on the performances of the three multifrequency GNSSs; modernized GPS, Galileo and BDS which are not yet in full capability. For each GNSS, dual-frequency combinations of frequencies as well as triple-frequency combination are investigated in the simulation. The performances studied include: the time to fix the ambiguities with high success rate and the precision of coordinates in static relative positioning with varying baseline length. A method is developed to derive what we call the spatial correlation model which approximately gives the covariance between the individual errors belonging to two stations. Furthermore, the stochastic models that follow from accounting and neglecting the spatial correlation are developed. The LAMBDA (Least-squares Ambiguity Decorrelation Adjustment) method is implemented for ambiguity decorrelation. The results show that the time to fix the ambiguities caused by accounting the spatial correlation is less than the time of fix without the spatial correlation. Also, a slight superiority of Galileo in terms of performances is seen compared to the other GNSS. For all the dualfrequency combinations investigated, when processing a baseline length of 500 km with accounted spatial correlation, the time needed to successfully fix the ambiguities lies between 5 and 9 min, whereas it becomes only between 2.5 and 3 min for all the triple-frequency combinations, this is with a sampling time of 5 s. In addition, for all different combinations, the coordinates precision is less than 8 mm even for 500 km. We think that these high performances result from: 1) the precise codes of future GNSS signals, 2) the high redundancy in the observations equation and, 3) taking into account the spatial correlation in the definition of the stochastic model.
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