Inter-System PPP Ambiguity Resolution Between GPS and Beidou for Rapid Initialization

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 ambiguity 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 between, 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; Wübbena 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 period 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-Wübbena (Melbourne, 1985; Wübbena, 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

 Lk  ˆ k i,m  Ni,m =  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  ˆ k ˆ 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 ˆs ˆk Nei,m = Ni,m + bi,m − bm (4) where ˆ k ˆ k ˆ k and Ni,m, Ni,1 and Ni,3 denote the wide-lane, narrow- lane and ionosphere-free ambiguity estimates, respec-  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 resolved wide-lane ambiguity comprising FCB corrections ˆs ˆk fs,1 bi,m and bm. Note that only after resolving the wide-lane and gs = is the ratio between the L1 and L2 fre- fs,2 ambiguity (Equation 3a) can the narrow-lane ambiguity quencies of GNSS ‘s’ which can be either ‘G’ for GPS or ˆk be computed (Equation 3b). Satellite FCB products bm ‘C’ for BeiDou; Lk and Lk are the carrier-phase obser- i,1 i,2 and ˆbk are estimated with a reference network, which vations on L1 and L2 in the unit of length, respectively, n k k will be detailed in section 2.1. whereas Pi,1 and Pi,2 are the pseudorange observations; c c c λs,1 = , λs,m = and λs,n = are ˆk ˆk fs,1 fs,1 − fs,2 fs,1 + fs,2 Second, bm and bn will be delivered to PPP users the wavelengths of L1, wide-lane and narrow-lane car- along with precise orbit and clock products to enable rier waves, respectively, where c is the speed of light in PPP-AR, which is also carried out using Equation3. In k k k k ˆ vacuum; Ni,3, Ni,m, Ni,1 and Ni,2 are the ionosphere- particular, wide-lane FCBs bk assist in fixing wide-lane k m free, wide-lane, L1 and L2 ambiguities, respectively; ρi ambiguities to integers with Equation 3a. The resul- denotes the non-dispersive delays containing the geo- ˘ k tant integer wide-lane ambiguity Ni,m, in combination metric distance between station i and satellite k, the with ˆbk and ˆbk, will be introduced into Equation 3b to receiver clock and the troposphere delay; bs and bk m n i,m m recover the integer property of narrow-lane ambiguity denote the uncalibrated phase delays (UPDs) in the Nˆ k . Successful narrow-lane AR to achieve N˘ k com- unit of cycles on the Melbourne-Wübbena observable i,1 i,1 pletes an ambiguity-fixed PPP solution. specific to station i and satellite k, respectively, whereas s k bi,3 and b3 for the ionosphere-free observable (Ge et al., 2008); UPDs are usually presumed to originate in station and satellite hardware biases; note that ionosphere- free pseudorange observations are ignored in Equation 2.1 FCB estimation for intra-system PPP-AR 1 for brevity.

The implementation of PPP-AR is generally divided The key issue for the FCB estimation is to separate the k k into two steps (Ge et al., 2008). First, the fractional- satellite FCBs (i.e., bm and bn) from the nominal integer ˘ k ˘ k cycle biases (FCBs), which are the fractional parts of ambiguities (i.e., Ni,m and Ni,1) and the station FCBs s s UPDs and which are critical to reconstructing resolv- (i.e., bi,m and bi,n) (Equation3). In case of intra-system able PPP ambiguities, are computed using the wide- PPP-AR where all satellites from a particular GNSS lane and narrow-lane ambiguities derived from Equa- refer to the same station UPDs, we can thus remove 4 Jianghui Geng et al. them by forming single-difference ambiguities between AR (e.g. Odijk and Teunissen, 2013). Regarding PPP, satellites, which is, according to Equation3, the hurdle is that station UPDs can no longer be re- moved through single differencing between satellites be-  Lk − Ll  Nˆ kl = i,m i,m longing to different GNSS. In contrast to Equation5,  i,m λ  s,m we have inter-GNSS single-difference ambiguities   kl kl  =N˘ − b  q  i,m m Lk L (5)  ˆ kq i,m i,m ˘ kq GC kq  Ni,m = − = Ni,m + bi,m − bm (8a)  ˆ kl ˆ kl λs,m kl  λG,m λC,m  λs,nN =λs,1N − Ne   i,1 i,3 g + 1 i,m   s q q  λ Nˆ k − λ Nˆ = λ Nˆ k − λ Nˆ −  G,n i,1 C,n i,1 G,1 i,3 C,1 i,3  =λ N˘ kl − bkl   s,n i,1 n   λG,m k λC,m q  Nei,m + Nei,m (8b) where  gG + 1 gC + 1  kl k l  Nei,m = Nei,m − Nei,m where satellite k is from GPS and q is from BeiDou.   ˆ kl ˆ k ˆ l Equation 8b can be further developed into Ni,∗ = Ni,∗ − Ni,∗ (6)  kl k l ˆ kq ˆ q ˆ k ˆ q  b∗ = b∗ − b∗ λG,nNi,1 + (λG,n − λC,n) Ni,1 = λG,1Ni,3 − λC,1Ni,3− and l is a second intra-system satellite with respect to λG,m kq λG,m λC,m q Nei,m − − Nei,m satellite k; ∗ is a wildcard representing “1”, “m” or “n” gG + 1 gG + 1 gC + 1 throughout. and afterwards

ˆ kq ˆ k ˆ q ˆ q We can then round the single-diﬀerence ambigui- λG,nNi,1 = λG,1Ni,3 − λC,1Ni,3 − (λG,n − λC,n) Ni,1− ˆ kl ˆ kl ties (Ni,m and Ni,1) to their nearest integers with the λG,m kq λG,m λC,m q goal of identifying their fractional parts as satellite-pair Ne − − Ne g + 1 i,m g + 1 g + 1 i,m FCBs. In particular, we have the single-diﬀerence FCB G G C estimates at station i ˘ kq GC kq = λG,n Ni,1 + bi,n − bn h i ˆkl ˆ kl ˆ kl  bm = Ni,m − Ni,m (9) h i (7) ˆkl ˆ kl ˆ kl  bn = Ni,1 − Ni,1 We note that

( GC G C where [·] denotes an integer rounding operation (Ge bi,m = bi,m − bi,m 6= 0 (10) et al., 2008). Note that rigorous FCB estimation should GC G C bi,n = bi,n − bi,n 6= 0 be performed over a network of reference stations, rather than at only station i. Teunissen and Khodabandeh which are the wide-lane and narrow-lane ISPBs for sta- (2015) pointed out that the FCB estimates from Equa- tion i between GPS and BeiDou, respectively. Equa- tion7 might be biased, and Geng et al.(2012) demon- tion9 is used to compute inter-GPS/BeiDou single- ˆ kq strated that this deficiency can be mitigated by the ap- difference narrow-lane ambiguities (Ni,1). On the right plication of double-difference AR on the reference net- ˆ k hand side of this equation, we then need to know Ni,3 work in advance. Single-difference FCBs can be trans- ˆ q kq and Ni,3 which are directly PPP output, and also Nei,m lated into undifferenced (pseudo-absolute) products by which is provided by resolving Equation 8a. However, assigning one satellite FCB to zero. It is such undif- q Nei,m cannot be derived since we are unable to resolve ferenced FCBs that are disseminated to PPP users to undifferenced wide-lane ambiguities with Equation 8a; ˆ kl ˆ kl convert their float Ni,m and Ni,1 estimates into resolv- ˆ q Ni,1 cannot be obtained either due to its dependence able ambiguities using Equation5. q on Nei,m (see Equation 3b).

Despite this difficulty, we note that the absolute val- q ˆ q 2.2 ISPB estimation for inter-system PPP-AR ues of the coefficients for Nei,m and Ni,1 in Equation9 satisfy that  λG,m λC,m  − < 0.82 cm It has been known that it is the ISPBs originating in gG + 1 gC + 1 (11) stations that can inhibit inter-GNSS double-difference  |λG,n − λC,n| < 0.14 cm Inter-GPS/BeiDou PPP-AR 5

˘ kq ˘ kq Both are far smaller than λG,n = 10.695 cm, hence parts have been assimilated into Nij,m and Nij,1, re- q ˆ q implying that minor errors within Nei,m and Ni,1, if spectively. For any given baseline, we will have a good ˆ kq number of inter-GPS/BeiDou double-difference ambi- present, will have limited impact on Ni,1. We therefore introduce the following approximations guities, and differential ISPBs between the two relevant stations can thus be computed by averaging all q  L GC GC  q ˆ q i,m estimates of bij,m and bij,n. The resultant ISPBs can  Nei,m ≈ Ni,m =  λC,m afterwards be converted into undifferenced values (e.g., (12) GC GC bi,m and bi,n ) by choosing a reference station ISPB.  ˆ q gC + 1 ˆ q 1 ˆ q  Ni,1 ≈ Ni,3 − Ni,m gC gC − 1 which refer to Equation3. Contrasting Equations 3a and4, we find that the approximation errors of Equa- C q 2.3 FCB estimation for inter-system PPP-AR tion 12 consist in the difference between bi,m − bm and ˆC ˆq bi,m − bm where the former contributes to the estimation of the latter. The impact of this approximation Once the ISPBs for a given station are known, we can on Nˆ kq will be inspected in section 2.4. With Equa- i,1 correct for them in Equations 8a and9 such that tions 8a and9 along with the substitution of Equation 12, we can calculate Nˆ kq and Nˆ kq, similar to Equation  i,m i,1 ˆ kq ˆGC ˘ kq kq  Ni,m − bi,m = Ni,m − bm 5, aiming at estimating station ISPBs. (16) ˆ kq ˆGC ˘ kq kq  Ni,1 − bi,n = Ni,1 − bn GC GC We first presume that ISPBs bi,m and bi,n should remain the same for all pairs of GPS/BeiDou satellites Then akin to Equation7, the FCBs for inter-system observed at station i. From Equations 8a and9, we can satellite pairs can be derived from form double-difference ambiguities between stations i h i kq kq ˆkq ˆ kq ˆGC ˆ kq ˆGC and j such that satellite FCBs bm and bn cancel com-  bm = Ni,m − bi,m − Ni,m − bi,m pletely, leading to h i (17) ˆkq ˆ kq ˆGC ˆ kq ˆGC  bn = Ni,1 − bi,n − Ni,1 − bi,n ( ˆ kq ˘ kq GC Nij,m = Nij,m + bij,m (13) ˆ kq ˘ kq GC which can also further be decomposed into undiffer- Nij,1 = Nij,1 + bij,n ˆk ˆk enced bm and bn, for example. where  ˘ kq ˘ kq ˘ kq  Nij,∗ = Ni,∗ − Nj,∗ Different from intra-system PPP-AR where only satel-  ˆ kq ˆ kq ˆ kq lite FCBs are required to retrieve the integer nature Nij,∗ = Ni,∗ − Nj,∗ (14)  of single-difference ambiguities between satellites, inter-  GC GC GC bij,∗ = bi,∗ − bj,∗ system PPP-AR however further asks for station-specific Note that for intra-system double-difference ambigui- ISPBs to pave the way for the recovery of integer inter- GC GC GNSS ambiguities. ties, the ISPB terms bij,m and bij,n will be absent, and ˆ kq ˆ kq Nij,m and Nij,1 have the integer property naturally. In case of inter-system AR, however, we have to correct GC GC ˆ kq for bij,m and bij,n before being able to resolve Nij,m ˆ kq 2.4 Remarks on the approximation error due to and Nij,1. Equation 12 Similar to Equation7, for all stations collecting GPS and BeiDou data simultaneously, we can calculate dif- q ferential ISPBs between stations i and j using Since the undifferenced integer Nei,m in Equation9 cannot be easily obtained, we suggest that it be replaced h i ˆGC ˆ kq ˆ kq ˆ q  bij,m = Nij,m − Nij,m by Ni,m as demonstrated in Equation 12. Contrasting h i (15) Equations 3a and4, we perceive that this replacement ˆGC ˆ kq ˆ kq  bij,n = Nij,1 − Nij,1 ˆC ˆq is in nature realized by approximating bi,m − bm with GC GC bC − bq . As a result, the bias in the unit of cycles Here we have presumed that bij,m and bij,n are both i,m m ˆ kq merely the fractional parts of ISPBs and their integer on the narrow-lane inter-GPS/BeiDou ambiguity Ni,1 6 Jianghui Geng et al. arising from this approximation is, according to Equa- (e.g., Montenbruck et al., 2015), we preferred the prod- tions9 and 12, ucts derived from the CODE (Center for Orbit Deter- mination in Europe) IGS MGEX (International GNSS kq 1 1 ˆC ˆq C q Service Multi-GNSS Experiment) endeavor throughout. ξi,1 = − bi,m − bm − bi,m + bm gG − 1 gC − 1 BeiDou GEOs (Geosynchronous Earth Orbiters) were all excluded due to their even worse orbit performance. ˆC ˆq C q ≈ 0.119 × bi,m − bm − bi,m + bm (18) The data processing schemes of this study are delin- eated in Figure2 and details therein for each processing step can refer to section2. Specifically, CODE orbits and Earth rotation parameters were fixed in all process- ˆC ˆq Since bi,m − bm can be taken as the least-squares ing. 5 s satellite clocks were computed in a simulated C q real-time mode by fixing the coordinates of the 19 sta- estimate from all contributing bi,m − bm in a network, tions in Figure1. For all processing, intra-frequency dif- we can use the uncertainty statistics of ˆbC − ˆbq to i,m m ferential code biases (e.g., P1-C1) and BeiDou IGSO/MEO roughly quantify the average approximation error of elevation dependent pseudorange biases were mitigated ˆC ˆq C q replacing bi,m − bm with bi,m − bm . For simplicity, in advance (Wanninger and Beer, 2015). We estimated we presume that ˆbC and ˆbq are independent variables zenith troposphere delays (ZTDs) every two hours with i,m m the global mapping function (Boehm et al., 2006). A and have identical uncertainty of σ0. Then the variance √ random walk process noise of 2 cm/ h was used to of ˆbC − ˆbq equates 2σ2. Similarly, the variance of i,m m 0 constrain the variation between neighboring ZTD esti- ˆqr ˆq ˆr satellite-pair FCB bm = bm − bm (r denotes a second mates. Receiver clocks were computed purely as white BeiDou satellite), which is easily derived in the FCB es- noise parameters. We corrected for absolute phase cen- 2 timation, is also 2σ0 in theory. Therefore, in this study, ter offsets and variations for both satellite and receiver we use the uncertainties of single-difference wide-lane antennas (Schmid et al., 2016), phase wind-up effects qr FCB estimates (σm ) specific to BeiDou satellite pairs (Wu et al., 1993) and station displacements according to indirectly gauge the approximation error formulated to Petit and Luzum(2010). We presumed that BeiDou in Equation 18, that is B1/B2 shared the GPS L1/L2 receiver antenna corrections. The carrier-phase precision was presumed to ˆkq qr ξi,1 = 0.119σm (19) be 2 cm while the pseudorange 2 m. A cut-off angle of 7◦ for usable observations and an elevation dependent weighting strategy were applied. For PPP-AR, in particular, we estimated epoch-wise positions without applying any constraints between epochs.

3 Data processing ISPBs devoted to inter-GPS/BeiDou PPP-AR are specific to stations. We collected all Melbourne-Wübbena and PPP ambiguity estimates that corresponded to mean We processed 24 days (March 5–28, 2017) of 5 s dual- satellite elevations of over 10◦ from all 66 stations in frequency GPS/BeiDou data from 66 stations within Figure1. Baselines of less than 1,000 km were all picked China collected from NBASS (National BDS Augmen- and inter-GPS/BeiDou double-difference ambiguities were tation Service System) and CMONOC (Crustal Move- formed to identify their fractional parts according to ment Observation Network of China) spanning an area Equation 15, which were in turn injected into an aver- of about 2000×2000 km (Figure1). The stations were aging operation to have the differential ISPB estimates. equipped with three types of receivers, i.e., 39 ComNav Through a network adjustment by fixing the ISPBs of a K508, 15 Trimble NetR9 and 12 Unicore UR380, 19 given station to zero, we obtained undifferenced ISPBs of which with an inter-station distance of over 300 km which would facilitate inter-GPS/BeiDou PPP-AR at a were dedicated to satellite clock estimation while the re- single station. Note that for each station, we estimated mainder (i.e., 21 ComNav, 14 Trimble and 12 Unicore only one wide-lane and one narrow-lane ISPB per day. receivers) contributed to PPP-AR. Since the predicted BeiDou IGSO/MEO (Inclined Geosynchronous Satel- lite Orbiter/Middle Earth Orbiter) orbits can at present Once obtaining ISPBs, we corrected for them at only reach a nominal accuracy of tens of centimeters all 19 reference stations to calculate inter-GPS/BeiDou Inter-GPS/BeiDou PPP-AR 7

100˚ 110˚ 120˚ 130˚

40˚ 40˚

30˚ 30˚

Reference Station ComNav Trimble 20˚ Unicore 20˚ 0 km 500 km

100˚ 110˚ 120˚ 130˚

Fig. 1 Station distribution spanning an area of about 2000×2000 km within China. We used 19 stations denoted as black open stars to estimate 5 s satellite clocks in a simulated real-time mode for both GPS and BeiDou. Another 47 stations (21 ComNav K508, 14 Trimble NetR9 and 12 Unicore UR380 receivers coded with blue, red and green open circles, respectively) roughly encircled by the 19 reference stations were used to carry out PPP-AR. satellite FCBs according to Equation 17 to enable PPP- case of inter-GPS/BeiDou PPP-AR. GNSS data at the AR across GPS and BeiDou. At each reference sta- 47 PPP stations were divided into 24,656 hourly pieces tion, we chose a reference GPS satellite to constitute in total for PPP-AR tests. Wide-lane AR was accom- single-difference ambiguities with respect to the remain- plished by means of integer rounding on the FCB and ing GPS and BeiDou satellites. Any two undifferenced ISPB corrected Melbourne-Wübbena combination ob- ambiguities with mean satellite elevations of over 10◦ servations with a threshold of 0.2 cycles according to and a common observation period of more than one Dong and Bock(1989). In contrast, narrow-lane AR hour were eligible to form an inter-GPS/BeiDou am- was carried out through the LAMBDA (Least-squares biguity. Single-difference FCBs between satellites were AMBiguity Decorrelation Adjustment) method (Teu- then calculated by identifying the fractional parts, and nissen, 1995) and then validated through a ratio test later akin to the ISPB computation above, we converted with a threshold of 3 (Euler and Schaffrin, 1990). Note such FCBs into undifferenced quantities. Regarding the that PPP-AR was only attempted on ambiguities cor- high temporal stability of FCBs (Geng et al., 2011), we responding to mean satellite elevations of over 10◦. Par- estimated them every 30 s without imposing any con- tial AR was applied to only narrow-lane ambiguities if a straints on neighboring estimates. For comparison, we full AR was impossible (Teunissen et al., 1999). We re- also computed the legacy intra-system FCBs for each quired that at most two ambiguities could be declined GNSS, in which case no ISPBs need to be accounted while at least five ambiguities had to be reserved to for. start an AR trial (Geng and Shi, 2017). An AR that passes the ratio test does not necessarily mean a cor- Conventional intra-GPS/BeiDou PPP-AR can be rect screening of integer candidates, which frequently attempted after corrections of satellite FCBs, while a takes place when float ambiguities have not converged further correction of station ISPBs is also mandated in to high enough precisions. Therefore, we preferred re- attempting PPP-AR at each epoch to avoid any catas- 8 Jianghui Geng et al.

IGS orbits and ERPs

Satellite clock Satellite Station ISPB estimation clocks estimation

Satellite FCB Satellite Station estimation FCBs ISPBs

Real-time PPP-AR

Fig. 2 Flowchart of GPS/BeiDou data processing in this study. Rectangles filled in green denote the data processing modules whereas squashed rectangles denote the data and products. Arrows indicate the input/output and processing sequences. Dashed arrows and rectangle contours mean optional data, processing and data flows. “ERP”, “FCB”, “DCB” and “ISPB” represent the Earth rotation parameter, fractional-cycle bias, differential code bias and inter-system phase bias, respectively. trophic consequence of holding presumably but incor- ISPB estimates we have achieved in the left panels, and rectly resolved ambiguities. In this study, a successful more importantly, confirm our assumption in section initialization at a particular epoch requires that, for 2.2 that all GPS/BeiDou satellite pairs observed at a all subsequent epochs, narrow-lane PPP-AR should all given station should have a common ISPB. Actually, succeed, and the resulting positions should differ from for all pairs between the 66 stations, over 96% of such truth benchmarks by less than 4 cm and 10 cm for the fractional-part residuals are within ±0.15 cycles with horizontal and vertical components, respectively. standard deviations of about 0.06 cycles for both wide- lane and narrow-lane ambiguities. However, one point worthy of attention is that there exist non-zero differential ISPBs between stations equipped with identical 4 Results receivers, antennas, domes and firmware versions. For example, both stations CHAZ and CHKH used Com- Nav receivers along with HXCCGX601A antenna and 4.1 Inter-system phase biases (ISPBs) between GPS version 55.0 firmware, and both WUHN and JSYC were and BeiDou equipped with Trimble receivers, TRM59900.00 antennas, SCIS dome and version 4.85 firmware, but their differential ISPBs surprisingly differ clearly from zero, which can reach 0.3 cycles as shown by b and b in Figure3 shows differential daily ISPB estimates and the m n Figure3. This fact seemingly contradicts what was re- fractional parts of all contributing inter-GPS/BeiDou ported by Odijk and Teunissen(2013) and Paziewski double-difference ambiguities after removal of those ISPBs and Wielgosz(2015) in which ISPBs cancel in case of over the 24 days for six representative station pairs. homogeneous stations. This intriguing issue will be in- From the top two station pairs, i.e., CHAZ–CHKH and vestigated in section 5.1 with zero and ultra-short base- WUHN–JSYC, daily wide-lane and narrow-lane ISPBs lines. can be quite stable over time with standard deviations of around 0.02 cycles. The ISPB variations from day to day are far less than 0.05 cycles, which favors the pre- In spite of the favorable results above, there are ex- diction of ISPBs for real-time inter-GPS/BeiDou PPP- ceptions breaking our assumption of temporally stable AR. The fractional-part residuals of inter-GPS/BeiDou ISPBs. As exposed by CHKH–QXGZ in Figure3, a sud- double-difference ambiguities over the 24 days appear den jump of about 0.3 cycles for wide-lane and 0.2 cy- fairly small where almost all of them fall within ±0.15 cycles for narrow-lane ISPBs occurred on day 70, though cles for CHAZ–CHKH and about 95% for WUHN–JSYC. the ISPBs before and after this jump are still as stable These statistics play as a clear indication of high-precision as those within the two top-left panels. We find that, Inter-GPS/BeiDou PPP-AR 9

Wide-lane ISPBs or fractional parts Narrow-lane ISPBs or fractional parts

0.4 ComNav bm =0.34 0.02 σ =0.034 100.0% ±0.15 ± 0.2 ComNav b =0.25 0.02 0.2 n ± -0.2 0.0 σ =0.045 99.0% ±0.15 0.2 0.2

0.4 CHAZ-CHKH -0.2

0.4 Trimble bm =0.10 0.02 σ =0.078 94.7% ±0.15 ± 0.2 Trimble b =0.24 0.02 0.2 n ± -0.2 0.0 σ =0.060 96.9% ±0.15 0.2 0.2

0.4 WUHN-JSYC -0.2

0.4 ComNav bm =0.58 0.01 σ =0.053 99.1% ±0.15 ± 0.2 ComNav b =-0.43 0.03 0.2 n ± -0.2 0.0 σ =0.063 96.0% ±0.15 0.2 0.2

0.4 CHKH-QXGZ -0.2

0.4 ComNav bm =-0.13 0.10 σ =0.066 96.3% ±0.15 ± 0.2 ComNav b =0.25 0.02 0.2 n ± -0.2 0.0 σ =0.064 96.0% ±0.15 0.2 0.2

0.4 CHKH-ZKXM -0.2

0.4 ComNav bm =0.63 0.05 σ =0.065 96.3% ±0.15 ± 0.2 Unicore b =-0.44 0.07 0.2 n ± -0.2 0.0 σ =0.059 97.3% ±0.15 0.2 0.2

0.4 CHSP-CHTZ -0.2

0.4 Trimble bm =0.14 0.05 σ =0.107 85.3% ±0.15 Fractional parts of double-difference ambiguities after removal ISPBs [cycle] ± 0.2 Unicore bn =-0.20 0.03 Inter-system phase biases for wide-lane and narrow-lane ambiguity resolution [cycle] 0.2 ± -0.2 0.0 σ =0.054 98.9% ±0.15 0.2 0.2

0.4 CHLH-JSYC -0.2 64 68 72 76 80 84 8864 68 72 76 80 84 88 Day of year 2017

Fig. 3 Time series of differential wide-lane and narrow-lane daily inter-system phase biases (ISPBs), and the fractional parts of inter-GPS/BeiDou double-difference ambiguities after removal of ISPBs on all 24 days. Six station pairs are presented representing baselines with identical or mixed receivers. Station codes are displayed at the bottom of the left panels while receiver types shown at the top left corners and the mean (bm and bn) along with the standard deviations (cycles) of ISPBs shown at the top right corners. Here ISPBs are plotted after removal of the mean. Correspondingly, the fractional parts of double-difference ambiguities after removal of these ISPBs are shown in the right panels with the standard deviations and the percentages of those falling within ±0.15 cycles exhibited at the top part of each panel. Note that the standard deviations (bm and bn) for CHKH–QXGZ are calculated after eliminating the jumps on day 70. Gaps are due to GNSS observation loss. 10 Jianghui Geng et al. for the data in this study, such jumps are all associated for CHKH–ZKXM and CHLH–JSYC, and the narrow- with GNSS observation gaps lasting for days or upgrade lane ISPBs for CHSP–CHTZ. This phenomenon may of receiver firmware. The observation gaps may relate relate to the hardware bias instability within specific to cold start events of receivers, which unfortunately receivers, which however needs further inspections on cannot be confirmed here due to loss of log files by sta- receiver architecture. Despite this problem, daily differ- tion operators. Despite those jumps, the fractional-part ential ISPBs in fact track well the first-order signatures residuals still have over 95% falling in ±0.15 cycles. A of the 2-hourly ISPBs. In most cases, the differences close look at all residuals in the right panels for the top between the daily and 2-hourly ISPBs do not exceed three station pairs reveals that most of them are fairly 0.1 cycles, thereby limiting the adverse impacts of in- around zero and seemingly normally distributed despite accurate daily ISPBs on inter-GPS/BeiDou PPP-AR. a negligible portion of outliers. At this point, we may have been holding a ques- However, this is not always the case. For the station in mind concerning whether it is the approximation pair CHKH–ZKXM, its wide-lane residual time tion errors in Equation 12 that contribute considerably series fluctuates irregularly especially during the days to the subdaily instability of ISPBs. We thus compute of 65–66 and 75–77, and clearly differs from the resid- the wide-lane FCB uncertainties for all BeiDou satellite ual distribution outlined at the top three station pairs, pairs throughout the 24 days, whose mean is 0.15 cy- though overall we still have 96.3% of residuals within cles (3σ). According to section 2.4, the resultant biases ±0.15 cycles. This phenomenon is also manifested for on inter-GPS/BeiDou narrow-lane ambiguities will be the narrow-lane residuals at CHSP–CHTZ. Even inter- mostly smaller than 0.018 cycles. This suggests that estingly, CHLH–JSYC appears to reveal almost promi- the approximation errors induced by Equation 12 are nent undulations within its wide-lane residuals where insignificant overall and matter minimally with respect only 85.3% stay within ±0.15 cycles with a standard to the notable subdaily signatures of ISPBs in Figure deviation of 0.107 cycles. This is dramatically worse 4. than those for all other station pairs in Figure3. Con- sequently, the wide-lane ISPBs for CHKH–ZKXM, the Despite those abnormal subdaily signatures, the ISPB narrow-lane ISPBs for CHSP–CHTZ and the wide-lane variations across days are in general quite small. Figure ISPBs for CHLH–JSYC in the left panels show clear 5 shows the distribution of the day-to-day variations of rise and fall of up to 0.1 cycles over the 24 days. The both wide-lane and narrow-lane ISPBs. It can be seen ISPB standard deviations are resultantly increased to that about 85% of absolute variations are smaller than 0.05–0.1 cycles, roughly doubling those of the top three 0.05 cycles, and over 95% are below 0.1 cycles. This station pairs. Such pronounced ISPB anomalies will means that in most cases ISPBs are temporally sta- jeopardize the prediction accuracy of ISPBs for inter- ble enough and we can precisely predict them for the GPS/BeiDou PPP-AR. next few days to facilitate real-time inter-GPS/BeiDou PPP-AR. However, we should also keep in mind that These abnormal residuals indicate that daily esti- there exist such risks that we predict seriously biased mates may not suffice to describe the temporal variation ISPBs for succeeding days which prevent the recovery of ISPBs. Hence, in Figure4, we re-estimate differential of integer inter-GPS/BeiDou ambiguities. ISPBs for the bottom three station pairs shown in Fig- ure3, but at a two-hour interval instead. This time we expectedly achieve smaller fractional-part residuals of inter-GPS/BeiDou double-difference ambiguities. 4.2 Positioning performance To be specific, the percentage of wide-lane residuals at CHLH–JSYC falling within ±0.15 cycles grow substan- tially from 85.3% to 94.6%, while all other percentages approach 99%. More favorably, the distribution Table1 shows the RMS of ambiguity-fixed epoch-wise of these new residuals now resembles those illustrated positions with respect to daily position estimates. These by CHAZ–CHKH, WUHN–JSYC and CHKH–QXGZ RMS statistics can be taken as a measure of the po- in Figure3, that is we no longer see any clear irregular sitioning accuracy of PPP-AR. We calculated statis- fluctuations. From the left panels of Figure4, we can see tics for four types of PPP-AR solutions where both that some ISPBs do show pronounced and sometimes GPS and BeiDou data were always processed, but dif- rapid subdaily variations, such as the wide-lane ISPBs ferent types of ambiguities were fixed. To be specific, we have four solutions where only BeiDou ambigui- Inter-GPS/BeiDou PPP-AR 11

Daily Wide-lane ISPBs 2-Hourly Wide-lane ISPBs or fractional parts Daily Narrow-lane ISPBs 2-Hourly Narrow-lane ISPBs or fractional parts

0.4 σ =0.046 99.0% ±0.15 0.4 0.2 0.2 0.0 0.2 0.0 b =-0.12 0.11 b =0.25 0.03 m ± n ± σ =0.045 98.3% ±0.15 0.4 0.2 0.2 0.0 0.4 CHKH-ZKXM 0.2

0.4 σ =0.051 98.4% ±0.15 0.4 0.2 0.2 0.0 0.2 0.0 b =0.63 0.06 b =-0.44 0.08 m ± n ± σ =0.036 99.6% ±0.15 0.4 0.2 0.2 0.0 0.4 CHSP-CHTZ 0.2

0.4 σ =0.077 94.6% ±0.15 0.4 0.2 0.2 0.0 0.2 0.0 b =0.14 0.08 b =-0.20 0.03 m ± n ± σ =0.046 99.8% ±0.15 0.4 0.2 0.2 Inter-system phase biases for ambiguity resolution [cycle] 0.0 0.4 CHLH-JSYC 0.2 64 68 72 76 80 84 8864 68 72 76 80 84 88 Day of year 2017 Fractional parts of double-difference ambiguities after removal ISPBs [cycle]

Fig. 4 Time series of 2-hourly wide-lane and narrow-lane differential ISPBs, and the fractional parts of double-difference ambiguities after removal of these ISPBs on all 24 days. Three station pairs are presented corresponding to the bottom three pairs in Figure3 and thus the meaning of similar symbols refers to the caption of Figure3. Wide-lane and narrow-lane ISPBs are intentionally offset by about 0.4 cycles from each other to avoid overlap of symbols. Blue and red curves denote the 2- hourly wide-lane and narrow-lane ISPB estimates while open squares and circles denote the daily ISPBs presented previously in Figure3.

Table 1 RMS (cm) of ambiguity-fixed kinematic positions with respect to daily position estimates in the east, north and up components. We show four types of PPP-AR solutions where both GPS and BeiDou data are used but only BeiDou ambiguities are fixed, only GPS ambiguities are fixed, intra-GPS/BeiDou ambiguities are fixed or inter-GPS/BeiDou ambiguities are fixed. The last column presents the percentages of successfully initialized hourly solutions. Solutions East North Up Percentage Only BeiDou fixed 0.9 0.8 3.0 59.6% Only GPS fixed 0.8 0.7 2.5 99.0% Intra-GPS/BeiDou fixed 1.1 0.7 2.8 99.4% Inter-GPS/BeiDou fixed 1.0 0.7 2.9 99.4% ties were fixed, only GPS ambiguities were fixed, intra- initialized successfully. Summing up all ambiguity-fixed GPS/BeiDou ambiguities were fixed or inter-GPS/BeiDou epochs within each initialized hourly solutions, we find ambiguities were fixed. Due largely to the relatively that all solution types can attain a positioning accuracy poor BeiDou satellite geometry, only 59.6% of BeiDou- of around 1.0 cm in the horizontal and 2.5 cm in the only fixed solutions are successfully initialized within an vertical directions on average. One phenomenon that is hour while the other three types have over 99% to be worth noting here is that inter-GPS/BeiDou PPP-AR 12 Jianghui Geng et al.

20 Li et al.(2017) and Liu et al.(2017). More importantly, 84.4% ±0.05 84.4% ±0.05 we ﬁnd that inter-GPS/BeiDou PPP-AR asks for even 95.7% ±0.1 95.5% ±0.1 shorter time to achieve successful initializations than intra-GPS/BeiDou over all 24 days as illustrated by 15 Wide-lane Narrow-lane the clear gap between the red and black traces in the bottom panel of Figure6. On average, the largest re- 10 duction of the initialization time is 112 s on day 84 and the smallest is 34 s on day 68. Overall, 586 s of data

Percentage[%] are required to initialize inter-GPS/BeiDou PPP-AR 05 while 649 s for intra-GPS/BeiDou PPP-AR, reflecting a 9.7% improvement. This improvement appears less significant compared to those achieved from single-GNSS 00 to dual-GNSS PPP-AR above, but is reasonable and 0.1 0.0 0.1 still encouraging since only one extra resolvable ambi- Variation of ISPBs across neighbouring days [cycle] guity is contributed in case of inter-GPS/BeiDou PPP- AR. In another respect, this result demonstrates that Fig. 5 Distribution of ISPB variations (cycles) across neigh- the dual-GNSS PPP-AR efficiency can still be amelio- boring days. Bars for wide-lane and narrow-lane statistics are rated even if the number of candidate ambiguities to be staggered for comparison. The top left corner of the panel shows the percentage of wide-lane ISPB variations falling fixed grows by only one, hence reinforcing the finding within 0.05 cycles and 0.1 cycles, whereas the top right for by Geng and Shi(2017). narrow-lane ISPBs. The top two panels of Figure6 present for each day does not achieve better positioning accuracy than that the percentages of the hourly solutions whose initial- of intra-GPS/BeiDou, though the former has one more ization times are shorter than 300 s and 600 s. When ambiguity candidate to be fixed. This result is in fact only GPS or BeiDou ambiguities are fixed to integers, anticipated and it is underlaid by the fact that, fun- less than 3% of solutions can achieve successful initial- damentally, inter-GPS/BeiDou and intra-GPS/BeiDou izations within 300 s. In case of a threshold of 600 s, on PPP-AR have equal GNSS model strength. Similar re- the contrary, GPS ambiguities can be resolved more effi- sults were also reported by Gao et al.(2017) and Paziewski ciently than BeiDou counterparts, as on average 21% of and Wielgosz(2017), suggesting that we should not ex- GPS-only PPP-AR succeed while only 8.4% for BeiDou- pect too much on the improvement of positioning accu- only PPP-AR. This result that GPS outperforms Bei- racy through inter-GNSS processing. Dou can be comprehended according to the facts that the GPS satellite geometry changes faster, and the GPS satellites outnumber the BeiDou IGSO/MEO satellites at almost each epoch of data (Li et al., 2017). How- 4.3 Initialization performance ever, if we resolve both GPS and BeiDou ambiguities as attempted in the intra-GPS/BeiDou PPP-AR, the percentages of successfully initialized solutions within Although inter-GPS/BeiDou PPP-AR does not achieve 300 s and 600 s will be increased tremendously to over a higher positioning accuracy, the initialization time 25% and 65%, respectively, on average over the 24 days. may be shortened since one more resolvable ambigu- More encouragingly, resolving one more ambiguity in ity is contributed to improve the success rate of partial case of inter-GPS/BeiDou PPP-AR grows the percent- AR (Geng and Shi, 2017). The bottom panel of Fig- age of successful initializations within 300 s significantly ure6 displays the mean initialization time over all 47 from 29.7% to 42.3%. This amelioration is less pro- PPP stations on each day. As expected, the initializa- nounced when we count the successful initializations tion times are always much shorter in case of resolving within 600 s instead, but can still reach a seven percent both GPS and BeiDou ambiguities than those in case of increment. We also find that such percentage rise is resolving ambiguities of a single GNSS. In particular, achieved near uniformly on each day when contrasting the mean initialization time is more than 1,000 s for the black and red traces in the top two panels of Figure both GPS-only and BeiDou-only fixed solutions, while 6, strongly suggesting that inter-GPS/BeiDou PPP-AR only about 600 s of data are required on average for is indeed more efficient than intra-GPS/BeiDou PPP- successful PPP-AR when both GPS and BeiDou ambi- AR in achieving successfully initialized solutions using guities are resolved, which echos the similar results by less than 10 minutes of data. Inter-GPS/BeiDou PPP-AR 13

BeiDou only fixed Intra-GPS/BeiDou fixed GPS only fixed Inter-GPS/BeiDou fixed 90 75.4%

60 68.4%

30 21.0% 8.4% <600s,Percentage[%] 0 60

40 42.3% 29.7% 20 2.5% 1.3%

<300s,Percentage[%] 0

16 1414s

12 1063s

8 649s

586s

Initialization time[100s] 4 64 68 72 76 80 84 Day of year 2017

Fig. 6 Initialization time (seconds) and the percentages of the initialization times within 300 s and 600 s for all PPP stations over the 24 days. Four types of PPP-AR solutions are presented in diﬀerent colors and the details of these solution types refer to the caption of Table1. The mean initialization time and mean percentages over the 24 days are shown beside the end of each color coded trace. Note that the unit of time for the bottom panel is 100 s.

Table 2 Statistics on partial AR for the four solution types deﬁned in Table1. Partial AR here is only for narrow-lane PPP ambiguities. nwl and nnl are the mean numbers of resolved wide-lane and narrow-lane ambiguities, respectively, when a solution is successfully initialized. Note that nwl can also be taken as the number of candidate narrow-lane ambiguities to n be resolved. The numbers in parentheses are those for BeiDou. pAR is the percentage of successfully initialized solutions nall through partial AR within all initialized ones. tsi is the averaged initialization time (min.) while tfAR is the time when all candidate ambiguities are resolved, which is in other words a full AR.

npAR Solutions nwl nnl ×100% tsi (min.) tfAR (min.) nall Only BeiDou fixed 6 5 26.6% 23.6 24.7 Only GPS fixed 7 6 64.9% 17.7 21.9 Intra-GPS/BeiDou fixed 7(5) 6(4) 74.3% 10.8 16.0 Inter-GPS/BeiDou fixed 7(5) 6(4) 75.1% 9.8 15.6

One explanation for the improvement from intra- to fore in Table2 we show the mean number of fixed inter-GPS/BeiDou PPP-AR traces back to the claim by ambiguities when a solution is initialized successfully, Geng and Shi(2017) that the efficiency of partial AR the percentage of successful initializations through par- will be improved if more ambiguities are incorporated tial AR and the mean times spent on achieving suc- into the integer-cycle resolution. Inter-GPS/BeiDou PPP- cessful initializations and full AR. We find that par- AR has one extra candidate ambiguity to be resolved tial AR in general plays a major role in PPP initializa- compared to intra-GPS/BeiDou PPP-AR, and there- tions since candidate narrow-lane ambiguities cannot be 14 Jianghui Geng et al.

all resolved in any type of solutions listed in Tables2. ] e l CUT0-CUT2 CUBB-CUT3 CUAI-SPA8 c 0.8

The solutions initialized through partial AR account for y c [ only 26.6% in case of BeiDou-only ﬁxed solutions, which )

is much lower than 64.9% in case of GPS-only ﬁxed so- 2 ( 0.6

lutions. This inferiority can also be recognized if we e i t n

make a comparison on the mean times spent on achiev- i

a 0.4 t ing successful initializations (tsi) and full AR (tfAR) be- r e tween BeiDou-only and GPS-only ﬁxed solutions (the c n

u 0.2 last two columns of Table2). For BeiDou, t is only

si d n about 1 minute shorter than t which is as minimal as a

fAR s a 4% improvement, whereas for GPS it is over 4 minutes B 0.0 P S which is an appreciable 18% amelioration. The contri- I

e bution of partial AR to rapidly initializing PPP can be n 0.2 a l more manifested if GPS and BeiDou ambiguities are re- - e d solved simultaneously, as illustrated by the fourth line i

W 0.4 of Table2. In this case, 74.3% of successfully initialized solutions are achieved through partial AR, showing a ] m [

10 percent increment compared to that for GPS-only ) 0.6 2

ﬁxed solutions. The mean initialization time declines (

s e

correspondingly by about 7 minutes (i.e., a signiﬁcant i

t GPS-BeiDou L1/B1 GPS-Galileo L1/E1 0.3 n

39% reduction). More interestingly, if we instead try i a t to ﬁx inter-GPS/BeiDou ambiguities, the mean initial- r e c

ization time can be further reduced by 1 minute (i.e., n 0.0 u

a perceivable 9% improvement) and the percentage of d n a partial-AR initialized solutions is increased by about

s 0.3 one percent. We have to acknowledge that this improve- B C S I ment on the success rate of partial AR is more or less 1 0.6 B

marginal, but is reasonable considering the fact that we / 1 resolve only one more ambiguity in inter-GPS/BeiDou E /

1 64 68 72 76 80 84 PPP-AR whose satellite geometry and observations ac- L Day of year 2017 tually stay the same as those for intra-GPS/BeiDou PPP-AR. Fig. 7 Differential wide-lane GPS/BeiDou ISPBs (cycle, top panel), and GPS-L1/Galileo-E1, GPS-L1/BeiDou-B1 inter- system code biases (ISCBs, meter, bottom panel) from March 5th to 28th, 2017 over homogeneous stations on three zero and 5 Discussions ultra-short baselines equipped with identical receivers, antennas, domes and firmware versions (refer to Table3 for the three station pairs CUT0–CUT2, CUBB–CUT3 and CUAI– SPA8). The error bars (2σ) are also plotted and color coded 5.1 Why do ISPBs differ between homogeneous along with the ISPB and ISCB estimates. Note that SPA8 stations? does not have Galileo data and is thus excluded in the bottom panel.

The top two panels of Figure3 show unexpectedly that ing the method formulated in section2, we computed ISPBs can differ significantly between homogeneous sta- the differential wide-lane ISPBs and their uncertainties tions equipped with identical receivers, antennas, domes (2σ) on each baseline for each day, as shown in the and firmware versions. To minimize any adverse im- top panel of Figure7. It can be seen that the ISPBs pact of residual atmospheric refractions and orbit er- are in general stable over the 24 days, except those for rors on ISPB estimations, we chose three representa- baseline CUBB–CUT3 with even one dropout on day tive zero and ultra-short GNSS baselines provided by 82. The day-to-day variation usually does not exceed Curtin University spanning the same observation pe- 0.1 cycles. Strikingly, we find that almost all wide-lane riod as that in this study. From Table3, each base- ISPBs are statistically far away from zero values on ac- line employs completely identical receivers, antennas, count of the error bars denoting the 2σ uncertainties. domes and firmware versions at its both stations. Us- This result confirms the finding revealed by Figure3, Inter-GPS/BeiDou PPP-AR 15

Table 3 Three zero and ultra-short baselines provided by Curtin University (saegnss2.curtin.edu.au/ldc). Note that the two stations on each baseline have identical receivers, antennas, domes and firmware versions. Baseline Receiver Antenna Dome Firmware version Length (m) CUT0–CUT2 Trimble NetR9 TRM59800.00 SCIS 5.20,16/DEC/2016 0.0 CUBB–CUT3 JAVAD TRE G3T DELTA TRM59800.00 SCIS 3.7.0a1-106-a669-loc 4.3 CUAI–SPA8 Sept PolaRxS TRM59800.00 SCIS 2.9.0 349.2 while further strengthened by precluding the possibly Table 4 Initialization times (min.) in case of different interfering atmosphere or orbit biases through zero and number of BeiDou IGSOs employed by intra- and inter- ultra-short baselines. GPS/BeiDou PPP-AR. BeiDou MEOs are excluded while all available GPS satellites are used. The first column lists the IGSOs that are used in PPP-AR and from line 2 until 7 the Odijk and Teunissen(2013) and Paziewski and Wiel- number of IGSOs is increased from 1 to 6 simply according to their pseudo-random noise numbers. tsi is the mean initializa- gosz(2015) demonstrated that the differential inter- tion time. The numbers before “/” are for inter-GPS/BeiDou system code biases (ISCBs) quite approached zero be- PPP-AR while those after for intra-GPS/BeiDou. The last tween homogeneous stations collecting GPS and Galileo two columns show the percentages of solutions whose initial- data on overlap frequencies. We hence implement the ization times are shorter than 300 s and 600 s. algorithms depicted by Odijk and Teunissen(2013) to BeiDou IGSOs tsi (min.) tsi <300 s tsi <600 s calculate differential daily ISCBs on the L1–E1 fre- C06 15.9/17.7 5.4%/2.5% 32.4%/20.9% quency for baselines CUT0–CUT2 and CUBB–CUT3. C06, C07 14.8/17.1 7.7%/3.4% 39.9%/25.3% Unfortunately, SPA8 does not deliver Galileo data and C06–C08 13.2/15.2 13.5%/6.5% 50.8%/34.8% C06–C09 12.3/14.0 19.6%/10.3% 59.3%/44.6% its baseline is thus excluded herein. The bottom panel C06–C10 11.0/12.7 29.2%/16.5% 67.8%/54.8% of Figure7 shows the ISCBs and their 2 σ uncertain- C06–C10, C13 10.2/11.5 36.6%/23.6% 73.6%/64.0% ties. As expected, the inter-GPS/Galileo code biases are very close to zero and quite stable over the 24 days, confirming the observations by Odijk and Teunissen fewer satellites (e.g., 1–4) within each GNSS contribute (2013) and Paziewski and Wielgosz(2015). We then fur- to positioning, because forming intra-GNSS double-difference ther attempt to compute the differential ISCBs between ambiguities in this case will be greatly limited, while the GPS L1 and BeiDou B1 frequencies for these two forming inter-GNSS ambiguities instead will exploit the baselines. Surprisingly, CUT0–CUT2 has a clear non- carrier-phase observations from all GNSS. In this study, zero inter-GPS/BeiDou code bias of about 0.6 m while however, we do not investigate such GNSS-difficult con- around −0.5 m for CUBB–CUT3. Moreover, the inter- ditions, but purely gain insight into the impact of the GPS/BeiDou code biases of CUBB–CUT3 again mani- number of BeiDou satellites on inter-GPS/BeiDou PPP- fests pronounced day-to-day variations of up to 0.2 m, AR efficiency. echoing the instable temporal behavior of ISPBs observed in Figure3. This finding conveys a message that inter-GNSS biases related to different frequencies are Table4 therefore exhibits the initialization times for likely to stay away from zero values even among homo- both intra- and inter-GPS/BeiDou PPP-AR in case of geneous stations. We hence contend that there should different number of BeiDou satellites. Note that we used be some station-specific inter-GNSS biases subject to all available GPS satellites, but change the number of frequency disparities that may not remain sufficiently BeiDou IGSOs while excluding all BeiDou MEOs. The stable over time and cannot cancel even between ho- IGSOs to be involved in PPP-AR were chosen simply mogeneous stations, which needs more investigation on according to their pseudo-random noise numbers, as de- receiver architectures. lineated by the first column of Table4. In general, the more BeiDou satellites are used, the faster initializations can be achieved. When only one IGSO C06 is used, in particular, the mean initialization time over all 47 stations on all 24 days can be reduced from 17.7 min- 5.2 How does the number of BeiDou satellites affect utes in case of intra-GPS/BeiDou PPP-AR to 15.9 min- inter-GPS/BeiDou PPP-AR? utes in case of inter-GPS/BeiDou PPP-AR, showing a moderate 10% improvement. This improvement tends to be slightly more significant which reaches up to 13% Odijk et al.(2017) reported that the benefit of inter- when 2–5 IGSOs contribute to PPP-AR, but returns GNSS model to AR efficiency can be more significant if to 11% when 6 IGSOs are all used. If BeiDou MEO 16 Jianghui Geng et al. data are further processed, the improvement drops to intra-GPS/BeiDou, which more clearly demonstrates 10% again as shown by Figure6. Similarly, from the the superiority of inter-GNSS processing. third column of Table4, when only C06 is accounted, only 5.4% of the inter-GPS/BeiDou solutions can be Finally, in this study, we enable inter-GPS/BeiDou initialized successfully within 5 minutes while 2.5% for PPP-AR by pre-calibrating station ISPBs. However, intra-GPS/BeiDou solutions, showing a slight improve- it is a common concern that such station-specific bi- ment. This improvement will be gradually and steadily ases cannot be always provided for PPP users before- increased to 13 percent when all IGSOs are involved. hand, and an online estimation in real time by the However, this improvement stays around 12 percent in users themselves is preferable. Unfortunately, Paziewski case of the 600 s threshold. Overall, these results sug- and Wielgosz(2015) demonstrated that the strategy gest that the number of contributing BeiDou IGSOs has of taking ISPBs as additional unknowns will under- slight impacts on the superiority of inter-GPS/BeiDou mine the GNSS model strength and thus deteriorate over intra-GPS/BeiDou PPP-AR, when we have al- the AR efficiency. Although the contribution of inter- ready had sufficient number of GPS satellites to bolster GPS/BeiDou PPP-AR to rapid initializations cannot a strong GNSS model. be overly expected, this study provides a feasible route to enabling inter-GNSS PPP-AR which may potentially benefit tightly-coupled multi-GNSS data processing on a global scale (see Garcia-Serrano et al., 2016). 6 Conclusions and outlook Acknowledgements This work is funded by National Sci- ence Foundation of China (41674033), China Earthquake In- strument Development Project (Y201707) and State Key Re- We develop a strategy to enable inter-GPS/BeiDou AR search and Development Programme (2016YFB0501802). We at a single station with the aim of speeding up PPP are grateful to IGS, NBASS (National BDS Augmentation initializations. Wide-lane and narrow-lane ISPBs which Service System), CMONOC (Crustal Movement Observation are specific to stations have to be predetermined through Network of China) and Curtin University (saegnss2.curtin.edu.au/ldc) for the GPS/BeiDou data and the high-quality orbit, clock inter-GPS/BeiDou double-difference ambiguities before and ERP products. We thank the high-performance com- we are able to align undifferenced GPS and BeiDou am- puting facility at Wuhan University where all computational biguities for inter-GNSS PPP-AR. work of this study were accomplished.

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