Journal of Geodesy manuscript No. (will be inserted by the editor) Inter-system PPP ambiguity resolution between GPS and BeiDou for rapid initialization Jianghui Geng · Xiaotao Li · Qile Zhao · Guangcai Li Received: date / Accepted: date Abstract Rapid ambiguity resolution in precise point resolvable ambiguity is contributed during the transi- positioning (PPP-AR) has constantly been a difficulty tion from intra- to inter-GPS/BeiDou PPP-AR while preventing efficient initializations of user solutions. A both actually have the same model strength. Moreover, successful initialization normally requires a few tens of we provide a preliminary theoretical framework to im- minutes if only GPS data are processed, but can be plement inter-GNSS or tightly-coupled GNSS models accelerated significantly by integrating a second GNSS which can be extended to other multi-GNSS analysis. to both enhance the satellite geometry for faster am- biguity convergence and double the ambiguity quan- tity for higher partial-AR success rates. However, each Keywords GPS · BeiDou · Precise point positioning · GNSS asks for its own reference satellite to form re- Inter-system ambiguity resolution · Inter-system bias solvable ambiguities, namely intra-system PPP-AR. We propose to estimate station-specific inter-system phase biases (ISPBs) and then form resolvable ambiguities be- tween, instead of within, GNSS (i.e., inter-system PPP- 1 Introduction AR) aiming at providing one more ambiguity candidate for more efficient partial AR. We use 24 days of 5 s GPS/BeiDou data from 47 stations in China spanning an area of roughly 2000×2000 km to carry out both Integer ambiguity resolution at a single GNSS (Global intra- and inter-GPS/BeiDou PPP-AR. We find that Navigation Satellite System) station secures centimeter- about 85% of ISPBs vary minimally within 0.05 cycles level precise point positioning (PPP) accuracy (e.g., Ge from day to day, favoring precise predictions for real- et al., 2008; Zumberge et al., 1997). It is also of great time PPP-AR, despite the rare subdaily ISPB anoma- importance to signifying a successful PPP initialization lies of up to 0.1 cycles and abrupt jumps of up to 0.3 cy- which can hardly be known objectively when keeping cles at a few stations. From hourly kinematic solutions, float ambiguities. Unfortunately, rapid PPP ambiguity we find that 42.3% of them can be initialized success- resolution (i.e., PPP-AR) has constantly been a difficult fully within 5 minutes in case of inter-GPS/BeiDou problem due largely to the low precision of pseudorange PPP-AR in contrast to only 29.7% in case of intra- measurements and the slow change of satellite geome- GPS/BeiDou. The mean initialization time is therefore try. We usually need a few tens of minutes of continuous reduced appreciably from 649 s to 586 s. This 10% im- observations to ensure correct GPS-only PPP-AR (e.g., provement, though minor, is reasonable and still en- Geng et al., 2011). couraging on account of the fact that only one extra A couple of approaches have been developed to achieve J. Geng · X. Li · Q. Zhao · G. Li GNSS Research Center, Wuhan University, Wuhan, China faster PPP-AR. One widely known strategy is to intro- Collaborative Innovation Center of Geospatial Technology, duce ionosphere corrections with an accuracy of a few Wuhan, China centimeters to speed up (re-)convergences to ambiguity- E-mail: [email protected] fixed solutions (e.g., Geng et al., 2010; W¨ubbena et al., 2 Jianghui Geng et al. 2005, among others). However, such high-precision iono- the presence of inter-system phase biases (ISPBs) be- sphere products cannot be easily attained over wide tween GPS and Galileo at the station end which ham- areas, hence confining rapid PPP-AR to a regional- per the formation of resolvable double-difference am- only service. In this case, multi-frequency and multi- biguities between diverse receiver types. They further constellation GNSS data can be a more preferable so- found, and Paziewski and Wielgosz(2015) later con- lution. Geng and Bock(2013) demonstrated the po- firmed, with several hours of data, that ISPBs are in tential of simulated triple-frequency GPS data in favor general quite stable over time with a standard devia- of rapid PPP-AR gained within 2 minutes with 78% tion of a few thousandth cycle. Odijk et al.(2017) ap- of all epochs resolved. Li et al.(2017) showed that plied ISPB corrections to single-frequency RTK based PPP-AR can be achieved on average from 33.6 min- on GPS, Galileo, QZSS and NAVIC L5 signals, and utes for GPS-only solutions to 24.6 minutes for inte- found that inter-system AR, in contrast to intra-system grated GPS/BeiDou solutions over the Asian-Oceanic AR, can improve the availability of ambiguity-fixed epochs region. For central Europe, Geng and Shi(2017) rein- dramatically from 29% to 96% in case of incomplete forced that integrated GPS/GLONASS PPP-AR can GNSS constellations. While all attempts above were as- be accomplished within about 6 minutes thanks to the sociated with overlap frequencies, Gao et al.(2017) in- greatly enhanced satellite geometry in high-latitude ar- vestigated the feasibility of performing inter-system AR eas. Later, Liu et al.(2017) combined GPS, GLONASS between non-overlap frequencies from GPS and Bei- and BeiDou data over a China region and reached suc- Dou. It was stated that a single-difference ambiguity cessful PPP-AR within 10 minutes for 90% of all test with a wavelength of a few millimeters has to be taken solutions. All these studies, albeit specific to regional into account along with ISPBs before forming a re- or continental areas, corroborated the expectation that solvable inter-GPS/BeiDou double-difference ambigu- introducing more GNSS satellites can indeed contribute ity. With two medium-length baselines, they reported to faster PPP-AR. that the time to first fix could be shortened by a few seconds in case of inter-GPS/BeiDou AR, compared to Moreover, Geng and Shi(2017) found that if more intra-GPS/BeiDou AR, if only seven or fewer satellites ambiguities contribute to partial AR where only a sub- were usable. In addition, Khodabandeh and Teunissen set of candidates are fixed to initialize PPP, its suc- (2016) analyzed in theory the implications of ISPBs to cess rate will be improved considerably from 60% to inter-system PPP-AR which were formulated using un- over 80%. This explains in part why multi-GNSS PPP- differenced uncombined observables. AR can be achieved within a shorter observation pe- riod than that required by GPS-only trials. Recalling In this study, we develop an approach of forming that multi-GNSS PPP-AR is normally carried out ex- inter-system ambiguities using GPS and BeiDou data clusively within each GNSS (e.g., Geng and Shi, 2017; modulated on diverse frequencies with the intent of in- Li et al., 2017; Liu et al., 2017), namely intra-system vestigating the potential of inter-system PPP-AR in or loosely-coupled PPP-AR where each GNSS has its further speeding up initializations. The article is out- own reference satellite, we conceive of an idea that inter- lined as follows. Section2 details the methods of both system or tightly-coupled PPP-AR where oppositely all intra- and inter-system PPP-AR, and the strategy of GNSS share one reference satellite may further reduce calculating ISPBs. Section3 exhibits the GPS/BeiDou the initialization time, because the number of resolvable data and the processing schemes, whereas section4 ambiguities will grow, though merely by one. illustrates the achievement of rapid PPP-AR due to inter-GPS/BeiDou data processing. Section5 addresses Inter-system AR has been studied mostly on double- relevant issues on ISPBs and inter-GPS/BeiDou ambi- difference ambiguities formed on overlap frequencies among guities. Conclusions are drawn in section6. GNSS, such as the L5 frequency shared by GPS, Galileo, QZSS (Quasi-Zenith Satellite System) and NAVIC (NAV- igation with Indian Constellation). Julien et al.(2003) simulated Galileo data on E1 and E5a which overlap 2 Methods the GPS L1 and L5 frequencies, respectively, and found that the initialization time in case of inter-GPS/Galileo AR on a 20-km baseline could be a few seconds shorter than that of intra-GPS/Galileo AR. With real observa- For dual-frequency PPP-AR, usually the undifferenced tions, Odijk and Teunissen(2013) began to be aware of Melbourne-W¨ubbena (Melbourne, 1985; W¨ubbena, 1985) and the ionosphere-free combination observables be- Inter-GPS/BeiDou PPP-AR 3 tween station i and satellite k are both used tions1 and2, that is 8 Lk > ^ k i;m > Ni;m = 8 k k k k > λs;m gsLi;1 − Li;2 gsPi;1 + Pi;2 > > Lk = − > > i;m > k s k > gs − 1 gs + 1 > = N˘ + b − b (3a) > < i;m i;m m > k s k <> =λs;m Ni;m + bi;m − bm λ > ^ k ^ k s;m k (1) >λs;nNi;1 = λs;1Ni;3 − Nei;m > g2 1 > gs + 1 > Lk = s Lk − Lk > > i;3 g2 − 1 i;1 g2 − 1 i;2 > > s s > = λ N˘ k + bs − bk (3b) > : s;n i;1 i;n n :> k k s k =ρi + λs;1 Ni;3 + bi;3 − b3 where k ˘ k ^s ^k Nei;m = Ni;m + bi;m − bm (4) where ^ k ^ k ^ k and Ni;m, Ni;1 and Ni;3 denote the wide-lane, narrow- lane and ionosphere-free ambiguity estimates, respec- 8 λ tively; N˘ k and N˘ k denote the nominal integer wide- > λ N k =λ N k + s;m N k i;m i;1 < s;1 i;3 s;n i;1 i;m lane and narrow-lane ambiguities which contain the in- gs + 1 (2) teger parts of UPDs; bs and bs are the station FCBs :> k k k i;m i;n Ni;m =Ni;1 − Ni;2 for wide-lane and narrow-lane ambiguities, respectively, k k k whereas bm and bn are the satellite FCBs; Nei;m is the re- solved wide-lane ambiguity comprising FCB corrections ^s ^k fs;1 bi;m and bm.