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

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


The realization and evaluation of mixed GPS/BDS PPP ambiguity
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

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 pared with a single GNSS system. However, most errors
 in PPP can only be modified by empirical models, which
 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
 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.
 School of Civil and Environmental Engineering, University
 of New South Wales, Sydney, NSW 2052, Australia

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution
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
 = s,G
 + c( trG − ts,G ) + Trs,G + ̃ G Ĩr,i + G
 P,i (1)
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
 + c( trG − ts,G ) + Trs,G − ̃ G Ĩr,i + BG
 + G

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
 = s,C
 + (c trG + ISBGC ) − c ts,C + Trs,C + ̃ C Ĩr,i + CP,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
 + (c trG + ISBGC ) − c ts,C + Trs,C − ̃ iC Ĩr,i + BCi + CP,i
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,

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
 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
 expressed as:
K21 = K2 − K1 (5)
 G_1S G_1
 Ñ WL = Ñ WL − Ñ WL (12)

 = 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
 ))∕ 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
 lite phase instrumental delays on frequency fi ; K21 refers Ñ NL = (f1G Ñ 1G_1G − f2G Ñ 2G_1G ) (f1G − f2G ) − f2G NWL
 (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
 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.

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
 + 2 ̃ C Kr,21
 ∕ Ci − kr,i
 G G
 − Kr,i + 2 ̃ G Kr,21
 ∕ G
 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
 = kr,i G
 − Kr,i + 2 ̃ iG Kr,21
 ∕ 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

 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

 ⎡ 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
 ⎥ 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

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

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

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)]

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

 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

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.

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

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

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)

Y. Yao et al.

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