# The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution

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Journal of Geodesy https://doi.org/10.1007/s00190-019-01245-x ORIGINAL ARTICLE The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Yibin Yao1,2 · Wenjie Peng1 · Chaoqian Xu1 · Junbo Shi1 · Shuyang Cheng3 · Chenhao Ouyang1 Received: 18 August 2018 / Accepted: 27 February 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Aiming at shortening convergence time and improving positioning accuracy, multi-GNSS precise point positioning (PPP) ambiguity resolution (PPP-AR) has been an important issue in the past decade. In this paper, a mixed (or inter-system) GPS and BDS PPP-AR model with inter-system biases considered is proposed. Datasets from the IGS MGEX network are utilized in the study to evaluate the proposed model. As a critical correction in multi-GNSS PPP-AR, the inter-system bias (ISB) can be treated as a fixed constant or unknown estimate. The effects of various ISB processing methods on other key corrections for PPP-AR, such as fractional cycle bias (FCB) and inter-system phase bias (ISPB), are analyzed. Experimental results indicate that fixing or estimating ISB approaches will not affect GPS FCB estimations. However, various ISB dealing methods will have a significant influence on some BDS FCB and ISPB estimations at some stations because of the limited BDS tracking satellites over long periods of observation. Regardless of the presence of unstable FCB products on some BDS satellites, narrow-lane FCBs on other satellites are time-continuous, and their daily changes are within the range of 0.3 cycle. And in aspect of the time to first fix (TTFF), fixing ISB is superior to estimating it. The performance of the mixed GPS and BDS PPP-AR is evaluated. Experimental results indicate that compared with the intra-system PPP-AR, the mixed method has no superiority when ISB is estimated. While it has a slight improvement in TTFF, i.e., from 969.64 to 897.96 s, however, the total fixed rate decreases from 86.5 to 85.56% when ISB is fixed as a constant. In addition, the mixed PPP-AR shows significant improvement over the intra-system PPP-AR under circumstances with limited satellite visibility. Keywords GPS · BDS · PPP-AR · ISB · Intra-system · Inter-system · Mixed 1 Introduction single system such as the Global Positioning System (GPS) (Bisnath and Gao 2009), which restricts the application and Precise point positioning (PPP) aims to provide decimeter- development of PPP. level to centimeter-level positioning accuracy with a single With the development of multi-system Global Naviga- Global Navigation Satellite System receiver (Leick et al. tion Satellite System (GNSS), which will provide sufficient 2015; Zumberge et al. 1997). As two of many factors, the navigation satellites at a certain time and enhance geometric number of visible satellites and the satellite geometry can strength, more scholars have begun to study different com- affect the positioning accuracy (Cai and Gao 2007). Usu- binations of GNSS systems in PPP. Cai and Gao (2013), ally, 30 or more minutes of initialization are needed with a Odijk et al. (2015) and Afifi and El-Rabbany (2015) noted that positioning accuracy and convergence time can, in most cases, be improved by additional GNSS observations com- * Chaoqian Xu cqxu@whu.edu.cn pared with a single GNSS system. However, most errors in PPP can only be modified by empirical models, which 1 School of Geodesy and Geomatics, Wuhan University, 129 destroy the integer characteristic of the phase ambiguities; Luoyu Road, Wuhan 430079, China thus, the ambiguities are treated as real values. Due to the 2 Key Laboratory of Geospace Environment and Geodesy, difficulty of ambiguity resolution, the positioning accuracy Ministry of Education, Wuhan University, 129 Luoyu Road, of PPP in a short time is still relatively low compared with Wuhan 430079, China that of relative positioning. 3 School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia 13 Vol.:(0123456789)

Y. Yao et al. In the past decade, ambiguity resolution-enabled PPP, positioning stability in multi-GNSS PPP is improved by fix- i.e., PPP-AR, is expected to improve positioning accuracy ing ISB as a constant, a new way that fixing ISB as a constant with short convergence time. The key to PPP-AR is to sepa- may affect the ambiguity resolution. rate the biases from float ambiguities and restore the integer This paper aims to resolve the above-mentioned issues. The characteristics of phase ambiguity. Three well-known meth- second section gives the specific component of bias products ods, the fractional cycle bias (FCB) method (Ge et al. 2008), and the specific model derivation of inter-system PPP-AR. The integer recovery clock (IRC) method (Laurichesse and Mer- third section gives the PPP-AR processing strategy, analyzes cier 2009) and decoupled satellite clock (DSC) method (Col- the effect of different processing strategies of ISB on prod- lins et al. 2008), can be utilized for PPP-AR. Teunissen and ucts for ambiguity resolution and evaluates the corresponding Khodabandeh (2015) have verified the equivalence of these inter-system GPS/BDS PPP-AR. Finally, some conclusions methods. However, PPP still suffers from a long initializa- and recommendations for future work are given. tion time of 30 min (or 1800 s) to first fix the ambiguities (Geng et al. 2011). A well-known approach to accelerate PPP-AR is to use precise ionosphere corrections (Geng et al. 2 Models for PPP‑AR 2010). Unfortunately, such precise ionosphere products nor- mally demand a dense network of reference stations with a In this section, the basic PPP models are listed, and the com- distance of several tens of kilometers (Li et al. 2014; Zou bination of ISB is derived. The key models of the intra-sys- et al. 2015) and are not available in a global context. tem and mixed GPS and BDS PPP-AR and the corresponding In terms of multi-GNSS PPP-AR, Landau et al. (2013) mathematic model of the products used for PPP-AR are given. used Trimble’s RTX system to provide the GPS + GLO- NASS ambiguity-fixed PPP service, and 1104 s was required 2.1 Basic PPP models to achieve a horizontal accuracy of better than 4 cm (90%). Liu et al. (2017) estimated the GPS and BDS FCB and car- The Uncombined PPP (U-PPP) model is available and is cho- ried out PPP-AR, and the percentage of fixing within 600 s sen to analyze multi-GNSS issues (Zhang et al. 2011). Assum- increased from 17.6 to 53.3% when compared with GPS ing that the antenna phase center offset and variance, phase alone. Odijk et al. (2017) developed an Australian multi- windup, solid tide, earth rotation and relativistic effects are GNSS PPP-RTK processing platform, which helped users corrected in advance, while the ionosphere second-order delay to obtain the precise location information in a short time. is negligible, the simplified GPS and BDS pseudo-range and Rather, Li et al. (2018) evaluated the multi-GNSS PPP-AR phase observation equations can be expressed as follows: systematically, the GPS, GLONASS, BDS and GALILEO four-system solution is optimal, the E/N/U positioning accu- Ps,G r,i = s,G r s,G + c( trG − ts,G ) + Trs,G + ̃ G Ĩr,i + G P,i (1) i racy is 1.84, 1.11 and 1.53 cm, respectively, and the time to first fix (TTFF) is reduced to 806.4 s. In addition, Nadarajah s,G s,G et al. (2018) analyzed GPS, BDS and Galileo PPP-RTK in Lr,i = s,G r + c( trG − ts,G ) + Trs,G − ̃ G Ĩr,i + BG i + G L,i i different scenarios, from large-scale to small-scale to single- (2) frequency PPP-RTK with cheap receivers, and the initiali- zation time is reduced to 900 s by applying single-receiver Ps,C r,i = s,C r s,C + (c trG + ISBGC ) − c ts,C + Trs,C + ̃ C Ĩr,i + CP,i i ambiguity resolution. All the above studies are limited to (3) intra-system studies that the separated reference satellite is s,C s,C chosen for the ambiguity resolution in the separated GNSS. Lr,i = s,C r + (c trG + ISBGC ) − c ts,C + Trs,C − ̃ iC Ĩr,i + BCi + CP,i (4) The studies about mixed (or inter-system) ambiguity resolu- where superscripts G and C stand for GPS and BDS, respec- tion that only one reference satellite is chosen for ambigu- tively; r and s stand for the receiver and satellite, respec- ity resolution among different GNSSs (Odijk and Teunis- tively; subscript i represents the frequency fi of GNSS obser- sen 2013; Kubo et al. 2018) are only conducted in real-time vations; is the geometric range from the satellite to the kinematic (RTK), and inter-system bias (ISB) needs to be receiver; tr and ts are the redefined receiver clock bias and calibrated in advance to correct the corresponding parameter satellite clock bias, respectively (Dach et al. 2009); c is the in the client. Though Khodabandeh and Teunissen (2016) speed of light; T r is the slant tropospheric delay, and proposed the construction of an ISB lookup table to speed Tr = Md (el) ⋅ ZTH + Mw (el) ⋅ ZTw0 + Mw (el) ⋅ ΔZTW , ZTH up the ambiguity resolution, it needs frequent updates. and ZTw0 are modeled ZHD and ZWD by the empirical However, until now, no research about inter-system PPP- Saastamonien model (Saastamoinen 1973), respectively, AR, especially with non-overlapping frequencies (i.e., frequen- ΔZTW is the residual part of ZWD and Mw(el) and Md(el) cies of L1 observation on GPS and B1 observation on BDS), are the mapping function of ZWD and ZHD based on the has been conducted, and, as Choi and Yoon (2018) noted, 13

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Niell Mapping Function (NMF) (Niell, 1996), respectively, 2.2 Ambiguity resolution models with an elevation angle of el; ̃ i Ĩi is the slant ionospheric delay on frequency fi, ̃ i = − , i = 40.3 1016 m∕TECU and i f2 Usually, single-differenced (SD) wide-lane (WL) ambigu- ity is fixed first. In the U-PPP model, float WL ambiguities 2 1 i TECU is the total electron content (TEC) unit; Bi is the float ambiguity term; and P,i and L,i denote pseudo-range and can be derived from float L1 and L2 ambiguities directly: phase measurement noise and multipath effects, respectively. Ñ WL = Ñ 1 − Ñ 2 (11) Equations (5)–(9) give the specific combination of iono- In the study, a GPS satellite is chosen as the reference satel- sphere delay term Ĩ , the float ambiguity term Bi and ISB lite, and mixed GPS and BDS SD WL ambiguities can be term. expressed as: K21 = K2 − K1 (5) G_1S G_1 Ñ WL = Ñ WL − Ñ WL (12) S Ĩr,i s = s Ir,i + Kr,21 − s K21 (6) where the superscript G_1S stands for GPS and BDS satel- lite S relative to the GPS reference satellite G_1. Bi = i Ñ is = i (Nis + bi ) (7) After WL ambiguities are successfully fixed, narrow- lane (NL) ambiguities can be expressed as (Li et al. 2013): bi = (kr,i − kis − (Kr,i − Kis ) + 2 ̃ i (Kr,21 − K21 s ))∕ i (8) / Ñ NL = (f1 Ñ 1 − f2 Ñ 2 ) (f1 − f2 ) − f2 NWL ∕ (f1 − f2 ) (13) ISBGC = c trC − c trG = ISTBGC + Kr,IF C G − Kr,IF (9) where f1 and f2 are the frequencies of observations on L1 and L2, respectively. where Kr,i and Kis are the receiver and satellite code instru- The SD NL ambiguities between GPS satellites can be mental delays on frequency fi due to the receiving and expressed as: transmitting hardware; kr,i and kis are the receiver and satel- / / G G_1G lite phase instrumental delays on frequency fi ; K21 refers Ñ NL = (f1G Ñ 1G_1G − f2G Ñ 2G_1G ) (f1G − f2G ) − f2G NWL G_1G (f1 − f2G ) to differential code bias (DCB); Ii = ( 2 − 1 ) STEC , and (14) STEC is the slant TEC and given in TECUs; Ñ is is the float where the superscript G stands for the GPS satellite. ambiguity; i is the wavelength of the frequency fi ; Ni is the The SD NL ambiguities between the GPS reference integer ambiguity on frequency fi; bi is a combination of sat- satellite and the BDS satellite cannot adopt the form of ellite and receiver hardware bias, considering bi and Nis are Eq. (14) directly, and some adjustment should be applied an integrated whole, the integer part of bi will be absorbed because of the non-overlapping frequencies fiG on GPS by Nis and bi only reserves the real term of the ambiguity; and fiC on BDS. The un-differenced integer ambiguity of ISBGC is BDS-GPS raw ISB; and I STBGC is the BDS-GPS the GPS reference satellite will be treated as a datum to inter-system time bias. recover the SD WL ambiguities between GPS and BDS, A unified time datum (GPS Time) is defined for all the and the model is expressed as: GNSS systems in the precise clock products, and the cor- / / responding statement can be found in the header infor- G_1C Ñ NL = (f1C Ñ 1C − f2C Ñ 2C ) (f1C − f2C ) − (f1G Ñ 1G_1 − f2G Ñ 2G_1 ) (f1G − f2G ) mation of the precise clock files. Therefore, I STB GC is G_1C G_1 / G_1 / G − f2C (NWL + NWL ) (f1C − f2C ) + f2G NWL (f1 − f2G ) compensated in c ts,C ; thus, ISBGC can be modified as: (15) where the superscript C stands for the BDS satellite and ISBGC = Kr,IF C G − Kr,IF (10) G_1C stands for BDS satellite C relative to the GPS refer- ence satellite G_1. Considering that only GPS and BDS are discussed, ISBGC is The paper discusses the GPS and BDS inter-system simplified as ISB. In addition, ISB can be processed by two ambiguity resolutions, and the corresponding products ways, it is normally treated as a parameter that will be esti- that recover the integer characteristics of phase ambiguity mated with coordinates, GPS receiver clock bias, residual should be considered. zenith troposphere wet delay ΔZTW , ionosphere delay term Ĩ and ambiguity parameters Bi, and it also can be fixed as a SD satellite WL and NL ambiguities are used for PPP- AR usually. We can find that in a single GNSS, receiver constant (Choi and Yoon 2018). In the strict sense, all the biases kr,i , Kr,i and Kr,21 in Eq. (8) can be removed from parameters from Eqs. (1) to (4) cannot be fully separated the SD satellite float ambiguities, while they still exist due to correlation, as they will absorb a part of each other. between different GNSS systems (GPS and BDS here). Therefore, the server end and the client end are best served Thus, the receiver bias caused by the between-satellites SD by using the same processing strategy. (BSSD) in different GNSS systems should be estimated. 13

Y. Yao et al. To distinguish this bias from ISB in Eq. (10), we rename it In mixed PPP-AR, except FCB of GPS receivers, inter-system phase bias (ISPB) in the paper, and in theory, PPP-AR demands three kinds of key corrections listed in it can be expressed as: Table 1. ISPB corrections make SD satellite ambiguities ( ) ( ) between GPS and BDS available to be fixed with the help ISPBi = kC − Kr,i C r,i + 2 ̃ C Kr,21 i C ∕ Ci − kr,i G G − Kr,i + 2 ̃ G Kr,21 i G ∕ G i of the FCB products of GPS and BDS satellites. ISB is (16) usually treated as an estimated parameter with relatively With the exception of ISPB, FCB of GPS receivers and FCB large initial variance; however, it will affect the precision of GPS and BDS satellites are estimated simultaneously. The of BDS ambiguities significantly because of their correla- theoretical model can be expressed as: tion, leading to an inability of one of the SD NLs between ( ) GPS and BDS to be fixed, and specific results and analyses FCBG G = kr,i G − Kr,i + 2 ̃ iG Kr,21 G ∕ G (17) are given in the following sections. r,i i ( ) FCBSi = (KSi − kiS ) − (KiS_1 − kiS_1 ) − 2 ̃ S (K21 S S_1 − K21 ) ∕ S 3 Experiments and analyses i i (18) In this section, the processing strategy is presented first. where S_1 denotes the reference satellite relative to differ- Then, by using different ISB processing methods, the cor- ent GNSSs. responding PPP-AR products FCB and ISPB are compared Assuming that there are p GPS float ambiguities, q and analyzed. Finally, the performance of the mixed PPP- BDS float ambiguities, m reference stations including GPS AR is evaluated. observations, n reference stations including BDS observa- tions, j GPS satellites and k BDS satellites, regardless of the frequencies, the “observed minus computed” can be expressed as: 3.1 Data description G ⎡ FCBr,m ⎤ GNSS observations with a 30 s sampling rate from the ⎢ ⋮ ⎥ IGS Multi-GNSS Global EXperiment (MGEX) network ⎢ G ⎥ on DOY 20, 2017 that can receive GPS and BDS signals ⎢ FCBr,m ⎥ ⎢ ISPBGC ⎥ simultaneously are chosen for the experiments. After ⎡ Ñ 1 − N1 ⎤ ⎡ R1 G G G S1G ⎤ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ r,1 ⎥ removing stations with less than 12 h of data, 115 stations ⋮ ⋮ ⎢ ⋮ ⎥ ⎢ Ñ G − N G ⎥ ⎢ RG S G ⎥ ⎢ ISPBr,n ⎥ GC (Fig. 1) remain. In order to ensure the reliability of the ⎢ p p ⎥ = ⎢ p p ⎥ ⎢ FCBs,G ⎥ (19) ISPBs and the BDS FCBs, more than three BDS satellites ⎢ Ñ 1C − N1C ⎥ ⎢ RC1 RGC S1C ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⋮ ⋮ 1 should be observed at the station simultaneously. And due ⋮ ⋮ ⎥ ⎢ ⋮ ⎥ ⎢ ̃C ⎥ ⎢ C GC ⎥ ⎢ FCBs,G ⎥ to the distribution of visible BDS satellites, a total of 80 ⎣ Nq − Nq ⎦ ⎣ Rq Rq C SqC ⎦ ⎢ j s,C ⎥ with 13 stations in Asia-Pacific region are treated as ref- ⎢ FCB1 ⎥ erence stations to calculate the products for PPP-AR and ⎢ ⎥ ⎢ ⋮ s,C ⎥ the other 35 stations in Asia-Pacific region are chosen as ⎣ FCBk ⎦ roving stations to validate the feasibility of mixed GPS/ BDS PPP-AR and analyze the effect of ISB. In matrix Ri, all the elements in ith column are − 1, and the 3.2 Data processing elements of the other columns are zero. In matrix Si, one element in each line is 1, corresponding to a certain satellite, Two methods of dealing with ISB are applied to PPP-AR: and the other elements are zero. Due to the rank deficiency, one is estimating ISB epoch by epoch, and another is fixing one GPS satellite and one BDS satellite FCB should be set ISB as a constant. For BDS, given the poor precision of to zero. All the estimations FCBs and ISPBs are limited to within − 0.5 and 0.5 cycles (Yi et al. 2017). The WL prod- ucts are treated as time-invariant values and estimated as Table 1 Corrections for mixed Correction daily constants, while NL products are time-dependent and PPP-AR estimated every epoch. Least-squares estimates are used for 1 ISBs sole FCB estimation (Li et al. 2017), and the precision of the 2 FCBs on GPS left input WL and NL ambiguity is calculated based on the and BDS ambiguity precision in float PPP by the variance–covariance satellites propagation law. 3 ISPBs 13

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Fig. 1 Distribution of the 80 reference stations (blue dots) and 35 roving stations (red triangles) from the MGEX network geostationary orbit (GEO) satellite ephemeris products and and fixed as a constant derived from the estimation at the the lack of available PCO/PCV and satellite-induced code last epoch. Second, the datasets at reference stations are bias corrections for GEO satellites, only geosynchronous processed again with the fixed ISB. Third, the ambiguities orbit (IGSO) and medium earth orbit (MEO) satellites are results, whose first-hour results are not used because of poor used here. Least-squares AMBiguity Decorrelation Adjust- precision, are processed based on Eq. (19); then, both PPP- ment (LAMBDA) method (Teunissen 1995) is used for PPP- AR products based on ISB-estimated and ISB-fixed methods AR. In addition, due to a weak strength of a GNSS model can be obtained at the reference stations. Finally, the PPP- in ambiguity resolution, it is often impossible to resolve all AR products are broadcast to the roving stations to evaluate eligible NL ambiguities (Teunissen and Verhagen 2009). the mixed PPP-AR method. In this case, a partial AR (PAR) strategy (Teunissen et al. 1999) is tried, and the specific processing steps (Geng and Shi 2017; Cheng et al. 2017) are adopted in this study. The 3.3 Comparison of PPP‑AR products based detailed configurations are shown in Table 2. on different ISB processing methods The modified RTKLIB software (Takasu and Yasuda, 2010) is applied in the study. Considering the different Due to the different ISB processing methods, different ISBs methods of dealing with ISB, the products (FCB and ISPB) may lead to different FCBs and ISPBs. The primary issue for PPP-AR will differ. First, the 24-h GPS and BDS obser- of this study is whether ISB is a constant or whether it can vation datasets at all stations are processed together with be treated as a constant. ISB estimations, and ISB at every station can be obtained Table 2 Processing strategies of PPP-AR Elevation angle 7° Sampling rate 30 s Raw observation GPS/BDS: code: 0.3 m; phase: 3 mm Satellite-induced code bias BDS: model with stochastic model (Guo et al. 2016) Antenna PCO and PCV GPS: igs14_1958_plus.atx BDS: corrections by European Space Agency (ESA) (Dilssner et al. 2014) Troposphere ZWD Estimation √ (random walk process 10 mm/ h) √ Ionosphere delay Estimation (random walk process 60 mm/ h) Satellite orbit and clock errors Final precise products from GFZ Ambiguity resolution LAMBDA and PAR Estimator Kalman filter 13

Y. Yao et al. 3.3.1 ISB Table 3 RMS and bias of ISB estimations compared with ISB-III ISB type Mean RMS (ns) Max RMS (ns) Max bias (ns) The time series of ISB with coordinates fixed (ISB-I), ISB with coordinates and ambiguities fixed (ISB-II) and ISB ISB-I 0.69 6.74 11.19 fixed as a constant (ISB-III) at stations COCO, CUUT, ISB-II 0.71 6.77 11.18 JFNG and XMIS can be seen in Fig. 2. The results show that ISB has a different order of magnitude at every station, and its change during a day indicates that ISB is not a constant; be discussed further. The specific influence of these results in addition, it varies the most at station CUUT, ranging from on FCB and ISPB will be discussed next. 45.78 to 59.0 ns, and it is in the normal range compared with the corresponding results (Odijk and Teunissen 2013), while 3.3.2 FCB the other three vary within 2 ns. Tables 3 and 4 give the RMS and bias statistics of ISB-I FCB is a key parameter for recovering the integer ambigui- and ISB-II compared with ISB-III at 35 stations. The tables ties. Two kinds of FCB products are calculated based on PPP show that the difference between ISB-I and ISB-II can be ambiguities with ISB fixed and estimated. The satellites G02 negligible, the RMS and bias of ISB-I less than 0.5 ns are and C08 are chosen as the reference satellites for GPS and 69.4% and 8.3% and ISB has a significant fluctuation exceed- BDS. The difference of WL FCB caused by various ISB ing 0.5 ns. In addition, it can be found that RMS and bias dealing methods can be seen in Fig. 3. The results show that less than 2 ns are 94.4% and 91.7%, respectively. Never- the differences of all GPS satellites are less than the 0.015 theless, a 2 ns bias can reach approximately 60 cm bias in cycle, and the differences of the BDS IGSO satellites are less geometric distance. The issue of whether this 2 ns will affect than the 0.02 cycle, while on BDS MEO satellites, the dif- the positioning results and ambiguity estimations needs to ferences are relatively large, reaching up to a 0.27 cycle on Fig. 2 Time series of three different ISBs at four stations [COCO (left top), CUUT (right top), JFNG (left bottom), XMIS (right bottom)] 13

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Table 4 Difference statistics ISB Type RMS (%) Max bias (%) between ISB estimation and ISB-III (%) < 0.5 ns < 1 ns < 2 ns < 0.5 ns < 1 ns < 2 ns ISB-I 69.4 91.7 94.4 8.3 58.3 91.7 ISB-II 66.7 91.7 94.4 5.6 55.6 91.7 large errors. Nevertheless, FCB products will not affect the following tests since most of the BDS NL FCBs are stable within one hour, and only satellites over elevation angles of 15 degrees are used for PPP-AR. In addition, as shown in Fig. 5, the large values of the BDS NL FCB differences focus on MEO satellites, such as those of the BDS WL FCB differences. 3.3.3 ISPB ISPB is another necessary correction item for mixed PPP- AR. The different ISB processing strategies lead to the dif- ferent WL ISPBs, and their differences can be seen in Fig. 6. The results show that 84% of the difference values are less Fig. 3 WL FCB difference between fixed and estimated ISBs than the 0.1 cycle, and only 7% exceed the 0.3 cycle, which indicates that fixing or estimating ISB will not obviously affect WL ISPB. With the exception of a few stations, the C11. It cannot be simply stated if ISB has a direct influence differences of the corresponding NL ISPB are slight with on MEO satellites, as the limited tracking of MEO satel- respect to the remaining stations (the specific results are not lites should be responsible for this result due to the satellite presented for simplicity). Instead, the standard deviations and station distribution, meaning that ISB has a remarkable (STDs) of NL ISPB with fixed ISB at 35 roving stations are influence on MEO satellites. shown in Fig. 7. The STDs are calculated every 15 min, and Figure 4 shows the GPS satellites NL FCB with ISB-I and the results show that the NL ISPBs are not very stable in ISB-III as well as their differences. The results show that the first few hours, and the largest STD reaches up to a 0.29 time-dependent GPS NL FCBs are stable in a day, most of cycle; however, the ISPBs become more and more stable them change within a 0.1 cycle during a day and the largest with time, and the stable NL ISPBs will contribute to the one does not exceed 0.3 cycle. Similar to the GPS WL FCB mixed PPP-AR. difference in Fig. 3, the results in Fig. 4 show that the GPS NL FCB differences are also small; over 99.7% are less than 3.4 Comparison of ISB‑fixed and ISB‑estimated the 0.05 cycle. We can draw the conclusion that whether or PPP‑AR not ISB is fixed, there is a negligible influence on GPS FCB estimations. According to the above analysis, it is difficult to determine The same results from the BDS satellite NL FCB are whether ISB should be fixed as a constant. More experi- given in Fig. 5. However, the BDS NL FCBs are not sta- ments should be carried out to draw this conclusion. We ble when compared with those of GPS. NL FCBs on C06, apply these PPP-AR products to obtain the coordinates. C08 and C10 vary within a 0.1 cycle during a day, regard- Three indicators are used to present the positioning results: less of a few exceptional data, while the daily changes of time to first fix (TTFF), positioning accuracy at TTFF and other satellites can reach up to the 0.6 cycle. This finding is total fixed rate. The ratio test (Frei and Beutler 1990) and mainly caused by the constellation distribution, as numerous bootstrapped success rate test (Teunissen 1998) are used data with low elevation angles participate in calculation but in the integer ambiguity validation. We employed a strict are not necessarily inaccurate. Though the error-dependent ambiguity validation to lower AR failure rate. Their thresh- weighing scheme is used during processing, it still cannot old values are 3.0 and 0.999, respectively. Since the PAR guarantee that all the gross error can be detected because method is applied in the study, we redefine the time when of the limited samples. Although a BDS satellite-induced the horizontal positioning accuracy is less than 5 cm and code bias model (Guo et al. 2016) is utilized here, the cor- the ambiguities are fixed as TTFF (Li et al. 2018), and the rections at low elevation angles are still not accurate due to 13

Y. Yao et al. Fig. 4 Time series of GPS NL FCB with fixed ISB (top) and estimated ISB (middle) as well as their differences (bottom) samples fixed incorrectly as well as non-fixed samples are their errors before and after LAMBDA, the number of unified as non-fixed. fixed ambiguities in the mixed PPP-AR is the same as that The 24-h datasets used for PPP-AR are initialized every in the intra-system PPP-AR with respect to the majority of hour. Regardless of the first hour and the periods that lack the samples, and the simple results of one sample are given ISPB results, the total number of samples is 748. All the in Fig. 8. This result indicates that one of the ambiguities results of the mixed PPP-AR are compared with those of the (C3, here) cannot be fixed, and, according to the test, it intra-system PPP-AR. cannot be avoided by simply removing it. In fact, rather The results of the intra-system PPP-AR with ISB esti- than a specific ambiguity with poor accuracy, the overall mation (Method a) and mixed or inter-system PPP-AR accuracy of BDS satellite ambiguities results in one ambi- with ISB estimation (Method b) are given in Table 5. The guity of BDS satellites having a relatively large error after results show that the positioning accuracy at TTFF in these LAMBDA. We believe that ISB should be responsible for two methods is almost identical, except for the aspects of this finding because of the strong correlation between ISB the TTFF and total fixed rate. The former method, with a and ambiguities of BDS satellites. Figure 9 shows these TTFF of 992.73 s and a fixed rate of 83.82%, outperforms correlations between ISB and satellite ambiguities on GPS the latter with 1035.63 s and 81.15% when ISB is esti- and BDS regardless of the positive and negative correla- mated as a parameter. Theoretically, in the mixed PPP-AR, tion, and the results indicate that satellite ambiguities on the accessorial ambiguity subset can accelerate the TTFF, BDS have stronger correlation compared with those on but it does not. By analyzing the specific ambiguities and GPS. 13

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Fig. 5 Time series of BDS NL FCBs with fixed ISB (top) and estimated ISB (middle) as well as their differences (bottom) Fig. 6 Cumulative probability distribution of WL ISPB differences at all the stations Fig. 7 STDs of NL ISPB at 35 stations every 15 min 13

Y. Yao et al. Table 5 TTFF, positioning accuracy and fixed rate of four different PPP-AR methods: the intra-system PPP-AR-estimating ISB (a), the mixed PPP-AR-estimating ISB (b), the intra-system PPP-AR-fixing ISB (c) and the mixed PPP-AR-fixing ISB (d) Methods TTFF (s) Accuracy (cm) Fixed rate (%) (Fixed/total) E N U a 992.73 1.38 1.23 5.65 83.82 (627/748) b 1035.63 1.30 1.18 5.41 81.15 (607/748) c 969.64 1.45 1.24 5.57 86.50 (647/748) d 897.96 1.40 1.22 5.74 85.56 (640/748) Fig. 9 Correlation between ISB and satellite ambiguities on f1 (top) and f2 (bottom) on GPS and BDS (TTFF corresponds to Fig. 8) constant, the mixed PPP-AR can shorten the TTFF, ranging Fig. 8 STD before and after LAMBDA and the fixed ambiguity val- from one to seven epochs, but can affect the stability of the ues removing the integer part at TTFF positioning results slightly because of the ISPB with limited accuracy compared with the intra-system PPP-AR. Considering that real-time ISB cannot be obtained accu- 3.5 Results of limited satellites rately in practical applications, ISB is fixed as a constant to solve the problem. The statistical results of the intra-system As shown above, the mixed PPP-AR has no inspiring improve- and mixed PPP-AR with fixed ISB (Methods c and d, respec- ments but increases the complexity of the algorithm compared tively) are also given in Table 5. It can be found that fix- with the intra-system PPP-AR. Though ISB can be fixed as a ing ISB improves the fixed rate and shortens the TTFF, and constant to simplify the process to some extent, it still needs to the mixed PPP-AR is superior to the intra-system PPP-AR be calibrated in advance, as does the ISPB. Regardless of the in terms of TTFF, as the TTFF is shortened by approxi- difficulties of implementation, we believe the only advantage mately 72 s, from 969.64 to 897.96 s. However, the fixed of the mixed PPP-AR is that it provides one more alternative rate decreases slightly, from 86.50 to 85.56%; not only is ambiguity pair to fix the ambiguities under circumstances with there a relatively large difference between the fixed ISB limited visible satellites and to accelerate the TTFF. Figure 11 and the actual ISB but also the inaccurate ISPB is likely shows the positioning results with limited satellites at sta- responsible for this. In addition, the positioning accuracy tions CUUT (three GPS and three BDS satellites) and NNOR in the PPP-AR-fixing ISB is slightly lower than that of the (four GPS and two BDS satellites). The results indicate that PPP-AR-estimating ISB since the fixed ISB does not match compared with the results in the intra-system PPP-AR, the the actual ISB well, and the coordinates will absorb their ambiguities can be fixed correctly in the mixed PPP-AR, and difference to some extent. the positioning accuracy at the TTFF (20.5 min or 1230 s) The specific E/N/U component time series of the intra- is improved from 2.81, 3.58 and 14.71 cm to 2.12, 0.37 and system and mixed PPP-AR-fixing ISB at stations COCO and 8.08 cm at the east, north and up components, respectively, at XMIS can be seen in Fig. 10. Similar to all the statistics in station CUUT, and from 4.41, 4.86 and 11.89 cm to 0.74, 0.6 Table 5, the results also indicate that when fixing ISB as a 13

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution Fig. 10 Four-hour time series of the positioning results on E/N/U components at stations COCO (left) and XMIS (right) Fig. 11 Time series of the positioning results on E/N/U components at stations CUUT (left) and NNOR (right) 13

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