Evaluation of BDS and GPS RAIM availability based on data collected in June 2020

The satellite pseudo-range fault detection with the Receiver Autonomous Integrity Monitoring(RAIM)method is affected by several satellite observations and the geometric distribution of satellites.The poor geometry distribution of satellites will conceal the positioning errorcaused by the satellite pseudo-range fault,resulting in unreliable detection results.Therefore,the availability evaluation must be made before RAIM to ensure that the fault detection performance will not be affected.On June 23,2020,China successfully launched the 30 th(last)navigation satellite of BeiDou’s third-generation navigation satellite system(BDS-3),which is also the 55 th BeiDou navigation satellite.Combining all the available satellites of BDS-1,BDS-2 and BDS-3,the positioning performance of BDS can be greatly improved.In order to evaluate the RAIM availability of BeiDou Navigation Satellite System(BDS)and Global Positioning System(GPS)in China,this paper first deduces the mathematical models and their characteristics of the three RAIM availability evaluation methods.Then,the study area(N10°-70°,E60°-150°)is divided into 4536 grid points at intervals of 1°×1°in latitude and longitude,and the elevations of these grid points are taken from the global terrain data file.The Horizontal Protection Level(HPL)values of these grid points are calculated during 8-15 June 2020 using BDS and GPS ephemeris data.The RAIM availability differences between the two systems are compared and analysed.The analysis shows the Horizontal Protection Level method(HPLM)based on single-satellite pseudo-range fault is the most practical and convenient.During the 8-day observation period,the HPL values of BDS are significantly smaller than those of GPS in terms of geographic location and observation time,and the variation of HPL time series of BDS is also smaller than that of GPS,which indicates that the RAIM availability of GPS in China is not as good as that of BDS.Most importantly,in the four flight stages of the aircraft’s Oceanic/Continental lowdensity En-route,Continental En-route,Terminal En-route and Non-precision approach(NPA),BDS can completely satisfy its RAIM availability requirement,while GPS can only meet the availability requirement of the En-route(Oceanic/Continental low density)phase,and the availability of the other three phases can at least reach 99.714%.

Fast projection onto the ordered weighted ℓ_(1) norm ball

In this paper,we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weightedℓ_(1)(OWL1)norm ball.In particular,an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem.Global and local quadratic convergence results,as well as the finite termination property,of the algorithm are proved.Numerical comparisons with the two best-known methods demonstrate the efficiency of our method.In addition,we derive the generalized Jacobian of the studied projector which,we believe,is crucial for the future designing of fast second-order nonsmooth methods for solving general OWL1 norm constrained problems.