PPP-RTK considering the ionosphere uncertainty with cross-validation

With the high-precision products of satellite orbit and clock,uncalibrated phase delay,and the atmosphere delay corrections,Precise Point Positioning(PPP)based on a Real-Time Kinematic(RTK)network is possible to rapidly achieve centimeter-level positioning accuracy.In the ionosphere-weighted PPP–RTK model,not only the a priori value of ionosphere but also its precision afect the convergence and accuracy of positioning.This study proposes a method to determine the precision of the interpolated slant ionospheric delay by cross-validation.The new method takes the high temporal and spatial variation into consideration.A distance-dependent function is built to represent the stochastic model of the slant ionospheric delay derived from each reference station,and an error model is built for each reference station on a fve-minute piecewise basis.The user can interpolate ionospheric delay correction and the corresponding precision with an error function related to the distance and time of each reference station.With the European Reference Frame(EUREF)Permanent GNSS(Global Navigation Satellite Systems)network(EPN),and SONEL(Système d’Observation du Niveau des Eaux Littorales)GNSS stations covering most of Europe,the efectiveness of our wide-area ionosphere constraint method for PPP-RTK is validated,compared with the method with a fxed ionosphere precision threshold.It is shown that although the Root Mean Square(RMS)of the interpolated ionosphere error is within 5 cm in most of the areas,it exceeds 10 cm for some areas with sparse reference stations during some periods of time.The convergence time of the 90th percentile is 4.0 and 20.5 min for horizontal and vertical directions using Global Positioning System(GPS)kinematic solution,respectively,with the proposed method.This convergence is faster than those with the fxed ionosphere precision values of 1,8,and 30 cm.The improvement with respect to the latter three solutions ranges from 10 to 60%.After integrating the Galileo navigation satellite system(Galileo),the convergence time of the 90th percentile for combined kinematic solutions is 2.0 and 9.0 min,with an improvement of 50.0%and 56.1%for horizontal and vertical directions,respectively,compared with the GPS-only solution.The average convergence time of GPS PPP-RTK for horizontal and vertical directions are 2.0 and 5.0 min,and those of GPS+Galileo PPP-RTK are 1.4 and 3.0 min,respectively.