Simultaneous estimation of GLONASS

GPS Solutions (2018) 22:19 https://doi.org/10.1007/s10291-017-0685-7

ORIGINAL ARTICLE

Simultaneous estimation of GLONASS pseudorange inter‑frequency biases in precise point positioning using undifferenced and uncombined observations Feng Zhou1,2,3 · Danan Dong1,2 · Maorong Ge3 · Pan Li3,4 · Jens Wickert3,5 · Harald Schuh3,5 Received: 11 July 2017 / Accepted: 15 November 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017

Abstract GLONASS precise point positioning (PPP) performance is affected by the inter-frequency biases (IFBs) due to the application of frequency division multiple access technique. In this contribution, the impact of GLONASS pseudorange IFBs on convergence performance and positioning accuracy of GLONASS-only and GPS + GLONASS PPP based on undifferenced and uncombined observation models is investigated. Through a re-parameterization process, the following four pseudorange IFB handling schemes were proposed: neglecting IFBs, modeling IFBs as a linear or quadratic polynomial function of frequency number, and estimating IFBs for each GLONASS satellite. One week of GNSS observation data from 132 International GNSS Service stations was selected to investigate the contribution of simultaneous estimation of GLONASS pseudorange IFBs on GLONASS-only and combined GPS + GLONASS PPP in both static and kinematic modes. The results show that considering IFBs can speed up the convergence of PPP using GLONASS observations by more than 20%. Apart from GLONASS-only kinematic PPP, the positioning accuracy of GLONASS-only and GPS + GLONASS PPP is comparable among the four schemes. Overall, the scheme of estimating IFBs for each GLONASS satellite outperforms the other schemes in both convergence time reduction and positioning accuracy improvement, which indicates that the GLONASS IFBs may not strictly obey a linear or quadratic function relationship with the frequency number. Keywords GLONASS · Pseudorange inter-frequency biases (IFBs) · Precise point positioning (PPP) · Undifferenced and uncombined model Abbreviations PPP Precise point positioning GNSS Global Navigation Satellite System * Feng Zhou [email protected] * Pan Li [email protected] 1

Engineering Center of SHMEC for Space Information and GNSS, East China Normal University, No. 500 Dongchuan Road, Shanghai 200241, China

2

Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, No. 500 Dongchuan Road, Shanghai 200241, China

3

German Research Centre for Geosciences GFZ, Telegrafenberg, 14473 Potsdam, Germany

4

School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

5

Technische Universität Berlin, 10623 Berlin, Germany

GPS Global Positioning System BDS BeiDou Navigation Satellite System IFB Inter-frequency bias UCD Uncalibrated code delay UPD Uncalibrated phase delay DCB Differential code bias PRN Pseudo-random noise IGS International GNSS service MGEX Multi-GNSS experiment CODE Center for orbit determination in Europe ESA European Space Agency ISB Inter-system bias ISFB Inter-system and inter-frequency bias DOY Day of year PANDA Positioning and navigation data analyst GDOP Geometric dilution of precision

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Introduction Over the past decades, Global Positioning System (GPS), known as the first one of the Global Navigation Satellite Systems (GNSSs), has achieved great success not only in geoscientific applications, such as geodesy, geophysics, remote sensing of atmosphere and ionosphere, but also in engineering services, i.e., surveying, navigation, and timing (Bock and Melgar 2016). As the second operational GNSS, Russia’s GLONASS has been revitalized since October 2011 and is now fully operational with 24 satellites in orbit (https://www.glonass-iac.ru/en/GLONASS). Moreover, the number of globally distributed stations with GLONASS tracking capability is increasing (Fritsche et al. 2014). Thus, increasing studies started to explore the performance of GLONASS precise positioning (Cai and Gao 2013a), tropospheric (Zhou et al. 2017) and ionospheric studies (Zhang et al. 2017). Unlike GPS, GLONASS adopts frequency division multiple access technique to distinguish the signals from individual satellites (Wanninger and Wallstab-Freitag 2007), which leads to different frequencies and inter-frequency biases (IFBs) for both pseudorange and carrier phase observations. Many studies have analyzed the features of carrier phase IFBs and demonstrated that carrier phase IFBs can be simply modeled as a linear function of the frequency number (Wanninger 2012; Al-Shaery et al. 2013). However, only a few studies analyzed the characteristics of pseudorange IFBs, and the results indicated that pseudorange IFBs can even reach up to several meters (Yamanda et al. 2010). It has been demonstrated that pseudorange IFBs are dependent not only on receiver types, but also on antenna types, domes, and firmware versions (Shi et al. 2013; Aggrey and Bisnath 2016). Although GLONASS pseudorange IFBs pose little threat on short-baseline double-differenced ambiguity resolution (Wang 2000), they indeed degrade the performance of GLONASS precise point positioning (PPP). PPP using GLONASS observations usually does not consider the IFBs. Hence, the pseudorange observations are often assigned very small weight to reduce the effect of IFBs (Cai and Gao 2013b). This significantly reduces the contribution of the pseudorange observations to the PPP solution, especially in the initialization stage, which affects both the positioning accuracy and convergence speed. Hence, the characteristic analysis and precise modeling of GLONASS pseudorange IFBs are still essential and critical. In addition, the GLONASS IFBs prevent GLONASS PPP ambiguity resolution based on Hatch (1982), Melbourne (1985) and Wübbena (1985) observables (Geng and Bock 2016) even though PPP ambiguity resolution is not the focus of this study.

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Shi et al. (2013) estimated pseudorange IFBs using GPS + GLONASS ionospheric-free observables from 133 receivers of five manufacturers and analyzed their characteristics. Their results demonstrated that the IFBs of some receivers (i.e., Trimble NetR9 and Leica GRX1200GGPRO) showed a linear function of frequency number, while some (such as TPS NET-G3A and JAVAD TRE_G3TH DELTA) showed quadratic polynomial function. The IFBs in their study were modeled for each satellite. However, the IFBs cannot be estimated by a single station using their method. Inspired by Banville et al. (2013), GLONASS pseudorange IFBs were modeled as a linear function of frequency number in PPP model based on undifferenced and uncombined observations (Liu et al. 2017). Although the frequency response of some receiver types might not be perfectly modeled using a linear fit, estimating the slope of pseudorange IFBs will at least remove first-order effects (Banville et al. 2013). Besides, Song et al. (2014) and Chen et al. (2017) have demonstrated the impact of GLONASS pseudorange IFBs on satellite clock estimation and concluded that the influence of IFBs on the satellite clock offset can lead to poor accuracy for combined GPS + GLONASS PPP. In recent years, PPP using undifferenced and uncombined observation models provides an alternative solution, with the advantages of being more flexible when processing the future multi-frequency GNSS observations, avoiding noise amplification and being able to extract ionospheric delays. PPP based on undifferenced and uncombined observation models has attracted strong attention from the GNSS community. Until now, only limited research has been carried out on GLONASS single-system or combined PPP with undifferenced and uncombined observation models (Liu et al. 2017; Xiang et al. 2017). New challenges arise for processing the undifferenced and uncombined observations, for instance, hardware biases at the receiver and satellite end need to be properly handled (e.g., GLONASS pseudorange IFBs). We address the modeling and assessment of GLONASS pseudorange IFBs for GLONASS-only and GPS + GLONASS PPP with undifferenced and uncombined observation models. First, through a re-parameterization process, four different GLONASS pseudorange IFBs handling schemes, which are neglecting IFBs, modeling IFBs as a linear or quadratic polynomial function of frequency number, and estimating IFBs for each GLONASS satellite, respectively, are proposed for PPP using undifferenced and uncombined observation models. Then, experimental data and processing strategies are described. Finally, in terms of the convergence time and positioning accuracy, positioning performance of static and kinematic GLONASS-only and GPS + GLONASS PPP in different schemes are given and analyzed.

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GPS + GLONASS PPP models with undifferenced and uncombined observations The basic undifferenced equations for pseudorange and carrier phase observations are first derived. Then, the handling schemes of receiver uncalibrated code delays (UCDs) for GPS-, GLONASS-only, and GPS + GLONASS PPP are introduced.

i,S and 𝜉r,j are the sum of measurement noise and multipath

error for pseudorange and carrier phase observations. The satellite and receiver antenna phase center offsets and variations, relativistic effects, Sagnac effect, slant hydrostatic delay, tidal loadings, and phase wind-up must be considered in modeling (Kouba 2015), although they are not included in (1). For convenience, the following notations are defined as i,S 2

General observation models For a specific satellite-receiver link, the undifferenced observation equations of the dual-frequency pseudorange P and carrier phase L can generally be expressed as (Leick et al. 2015)

i,S 2

⎧ 𝛼 S = i,S(fm ) i,S , 𝛽 S = − i,S(fn ) i,S (fm )2 −(fn )2 ⎪ mn i,S(fm )2 −(fni,S)2 mn i,S i,S i,S − dr,n ⎨ DCBPm Pn = dm − dni,S , DCBr,Pm Pn = dr,m ⎪ di,S = 𝛼 S ⋅ di,S + 𝛽 S ⋅ di,S , di,S = 𝛼 S ⋅ di,S + 𝛽 S ⋅ di,S ⎩ r,IF n mn r,m mn r,n mn m mn IF mn

mn

(2) S where f i,S is the signal frequency ( m, n = 1, 2; m ≠ n); 𝛼mn S and 𝛽mn are frequency-dependent factors, which are inde-

� � ⎧ i,S i,S i,S i,S S i,S + d − d + 𝜀i,S + c ⋅ dt − c ⋅ dt + mf (e) ⋅ Z + 𝛾 ⋅ I ⎪ Pr,j = 𝜌i,S r w w r j r,j j r,j r,1 ⎪ i,S i,S i,S L = 𝜆 ⋅ 𝛷 ⎨ r,j j r,j � � ⎪ i,S i,S i,S i,S i,S i,S S i,S i,S + 𝜉r,j ⋅ I + 𝜆 ⋅ N + b − b = 𝜌 + c ⋅ dt − c ⋅ dt + mf (e) ⋅ Z − 𝛾 r w w r ⎪ j r,j r,j j j r,1 ⎩ where indices i , r and j ( j = 1, 2 ) refer to the satellite, receiver and carrier frequency band, respectively; superscript S denotes different satellite system, G for GPS and R for GLONASS; 𝜆i,S is the carrier wavelength on the frej

i,S is the original carrier phase measurequency band j ; 𝛷r,j

ment in cycles; 𝜌i,S r denotes the geometric distance between the satellite i and receiver r ; c is the speed of light in the vacuum; dtr and dti,S are the clock offsets of the receiver and satellite in seconds; e is the elevation angle of the satellite i ; mfw (e) is the wet mapping functions that can be retrieved using the Global Mapping Function (Boehm et al. 2006); Zw i,S is the zenith wet delay; Ir,1 is the slant ionospheric delay on i,S S the frequency f1 ; 𝛾j is the frequency-dependent multiplier ( ( )2 ) i,S i,S S , which is independent of the satelfactor 𝛾j = f1 ∕fj i,S lite pseudo-random noise (PRN) code; dr,j and dji,S are the

frequency-dependent UCDs at the receiver and satellite end, i,S is the integer phase ambiguity in cycles; respectively; Nr,j and bi,S are the frequency-dependent uncalibrated phase bi,S r,j j delays (UPDs) at the receiver and satellite end in cycles; 𝜀i,S r,j

(1)

pendent of the satellite PRN code; DCBi,S and DCBi,S P P r,P m n

m Pn

are frequency-dependent satellite and receiver differential and Pi,S , code bias (DCB) between pseudorange Pi,S r,m r,n respectively. Since the satellite clock offset dti,S is linearly dependent with the satellite UCD ( dji,S ) in (1), the satellite UCD cannot be directly isolated from the satellite clock offset without additional constraints. By convention, the IGS precise satellite clock products are estimated with the L1/L2 ionosphericfree observations. Therefore, the precise satellite clock cori,S ) is the sum of dti,S and a specific linear rection ( dtIF 12

function of the satellite UCDs ( d1i,S and d2i,S ) as (Kouba and Héroux 2001) ( S ) i,S S c ⋅ dtIF = c ⋅ dti,S + 𝛼12 ⋅ d1i,S + 𝛽12 ⋅ d2i,S 12 (3) i,S = c ⋅ dti,S + dIF 12

Substituting (3) into (1) and applying the IGS precise satellite orbit and clock products, the linearized observation model can be written as

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i,S i,S i,S i,S i,S S ⎧ pr,1 = 𝐮i,S r ⋅ 𝐱 + c ⋅ dtr + dr,1 + Ir,1 − 𝛽12 ⋅ DCBP1 P2 + mfw (e) ⋅ Zw + 𝜀r,1 ⎪ pi,S = 𝐮i,S ⋅ 𝐱 + c ⋅ dt + di,S + 𝛾 S ⋅ I i,S +𝛼 S ⋅ DCBi,S + mf (e) ⋅ Z + 𝜀i,S r w r P1 P2 2 12 r,2 r,1 r,2 � ⎪ r,2 � w ⎨ li,S = 𝐮i,S ⋅ 𝐱 + c ⋅ dt − I i,S + di,S + mf (e) ⋅ Z + 𝜆i,S ⋅ N i,S + bi,S − bi,S + 𝜉 i,S r w w r IF12 r,1 r,1 r,1 1 1 ⎪ r,1 � �r,1 ⎪ li,S = 𝐮i,S ⋅ 𝐱 + c ⋅ dt − 𝛾 S ⋅ I i,S + di,S + mf (e) ⋅ Z + 𝜆i,S ⋅ N i,S + bi,S − bi,S + 𝜉 i,S r w w ⎩ r,2 r IF12 2 r,1 2 r,2 r,2 2 r,2 i,S where pi,S r and lr denote observed minus computed values of pseudorange and carrier phase observables; 𝐮i,S r is the unit vector of the component from the receiver to the satellite; 𝐱 is the vector of the receiver position increments relative to the a priori position.

Handling schemes of receiver UCDs Note that the satellite UCDs can only be mitigated by forming L1/L2 ionospheric-free combination, whereas they cannot be canceled in any other combination. To account for this, satellite DCB products provided by the Center for Orbit Determination in Europe (CODE) or the Multi-GNSS Experiment (MGEX) can be used for compensation in pseudorange observations in (4) according to Guo et al. (2015). For the carrier phase observations(of (4), ) the ionospherici,S , the satellite and free combination of satellite UCD dIF 12

receiver UPDs cannot be canceled and will be absorbed into ambiguities. Since the IFBs exist in both carrier phase and pseudorange observations, GLONASS PPP ambiguity resolution is still difficult on a global scale, the ambiguities are estimated as a lumped term and treated as float values in this study. i,S i,S The receiver UCDs dr,1 and dr,2 cannot be determined in) ( an absolute sense, so actually their difference DCBi,S r,P P m n

is estimated instead ( (Guo ) et al. 2015). Note that both the i,S ionospheric delay Ir,1 and DCBs are frequency dependent.

This implies that not all parameters can be unbiasedly estimable due to rank deficiency, but only combinations ( ) of i,S and them. It is well known that the ionospheric delay Ir,1 ( ) receiver DCB DCBi,S are perfectly correlated, and they r,P P m n

are estimated as lumped terms in general. Receiver UCDs for GPS

In our model, we assume that receiver UCDs are identical for code division multiple access signals (i.e., GPS, BDS, or Galileo) for all the satellites at each frequency. The reparameterization of (4) is conducted as

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(4)

G c ⋅ d̄trG = c ⋅ dtr + dr,IF 12 ̄I i,G = I i,G + 𝛽 G ⋅ DCBG r,P1 P2 r,1 r,1 ( 12 ) i,G i,G i,G G G − dr,IF + 𝛽12 ⋅ DCBG + bG − bi,G = 𝜆G ⋅ Nr,1 N̄ r,1 + dIF r,P1 P2 r,1 1 1 12 12 ( ) i,G i,G i,G i,G G G G G G N̄ r,2 = 𝜆G ⋅ N + b − b + d − d + 𝛾 ⋅ 𝛽 ⋅ DCB r,P P r,IF IF 2 r,2 2 12 r,2 2 12

12

1 2

(5)

i,G where d̄trG , Īr,1 , and N̄ ri,G are the lumped receiver clock offset, ionospheric delay, and ambiguity terms, respectively. The parameter vector can be expressed as

̄G ,𝐍 ̄ G ]T 𝐗 = [𝐱, c ⋅ d̄trG , Zw , 𝐈̄ G ,𝐍 r,1 r,1 r,2

(6)

where 𝐗 denotes the parameter vector of the model. Receiver UCDs for GLONASS In our model we assume that receiver UCDs are different for frequency division multiple access signals (i.e., GLONASS) for all the satellites at each frequency unless the satellites share the same frequency number. If there are no special comments in this study, we use pseudorange IFBs to indicate the differences of receiver UCDs at L1/L2 frequencies. Four different schemes are proposed: 1. Neglecting pseudorange IFBs. If neglecting IFBs, the positioning model of GLONASS is the same as that of GPS. However, the IFBs cannot be completely absorbed by the parameters of receiver clock offset and slant ionospheric delays. The remaining frequency-dependent IFBs will be reflected in the pseudorange residuals. 2. Estimating pseudorange IFBs using the linear function of frequency number. The IFBs are modeled as follows i,R R dr,j = dr,j + 𝜅 i,R ⋅ ΔRr,j (7) where 𝜅 i,R denotes the frequency number (from − 7 to R 6) of the satellite i ; dr,j is the IFB for the satellite whose frequency number is 0; ΔRr,j denotes the part of IFB that is dependent on the frequency number. The re-parameterization of (4) is given as follows

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⎧ R R ⎪ c ⋅ d̄tr = c ⋅ dtr + dr,IF12 R R R ⎪ Δr,12 = Δr,2 − 𝛾2 ⋅ ΔRr,1 ⎪ ̄ i,R i,R R R R i,R ⎪ Ir,1 = Ir,1 + 𝛽�12 ⋅ DCBr,P1 P2 + 𝜅� ⋅ Δr,1 ⎪ N̄ i,R = 𝜆i,R ⋅ N i,R + bi,R − bi,R + di,R − dR IF12 r,IF12 1 r,1 r,1 1 ⎨ r,1 R ⎪ + 𝜅 i,R ⋅ ΔRr,1 + 𝛽12 ⋅ DCBRr,P P � 1 2 � ⎪ i,R i,R i,R i,R i,R R ⎪ N̄ r,2 + dIF = 𝜆i,R ⋅ N + b − b − dr,IF 2 r,2 r,2 2 12 12 ⎪ R + 𝛾2R ⋅ 𝜅 i,R ⋅ ΔRr,1 + 𝛾2R ⋅ 𝛽12 ⋅ DCBRr,P P ⎪ 1 2 ⎩

(8)

1 2

The parameter vector 𝐗 is expressed as

The parameter vector 𝐗 is expressed as

]T [ ̄R ,𝐍 ̄R 𝐗 = 𝐱, c ⋅ d̄trR , ΔRr,12 , Zw , 𝐈̄ Rr,1 , 𝐍 r,1 r,2

̄R ,𝐍 ̄ R ]T 𝐗 = [𝐱, c ⋅ d̄trR , 𝚯Rr,12 , Zw , 𝐈̄ Rr,1 , 𝐍 r,1 r,2 (9)

3. Estimating pseudorange IFBs using quadratic polynomial function of frequency number. The IFBs are modeled as follows i,R R R dr,j = dr,j + 𝜅 i,R ⋅ ΔRr,j + (𝜅 i,R )2 ⋅ 𝛺r,j (10) R denotes the part of IFB that is quadraticwhere 𝛺r,j

dependent on the frequency number. The re-parameterization of (4) is given as follows ⎧ R ̄R ⎪ c ⋅ dtr = c ⋅ dtr + dr,IF12 ⎪ ΔR = ΔR − 𝛾 R ⋅ ΔR r,2 2 r,1 ⎪ r,12 R R R = 𝛺r,2 − 𝛾2R ⋅ 𝛺r,1 ⎪ 𝛺r,12 ⎪ ̄ i,R i,R R R R R i,R 2 i,R ⎪ Ir,1 = Ir,1 + 𝛽�12 ⋅ DCBr,P1 P2 + 𝜅� ⋅ Δr,1 + (𝜅 ) ⋅ 𝛺r,1 ⎨ N̄ i,R = 𝜆i,R ⋅ N i,R + bi,R − bi,R + di,R − dR IF12 r,IF12 1 r,1 r,1 1 ⎪ r,1 R R ⎪ +𝜅 i,R ⋅ ΔRr,1 + (𝜅 i,R )2 ⋅ 𝛺r,1 + 𝛽12 ⋅ DCBRr,P P 1 2 � � ⎪ ⎪ N̄ i,R = 𝜆i,R ⋅ N i,R + bi,R − bi,R + di,R − dR IF12 r,IF12 2 r,2 r,2 2 ⎪ r,2 R R ⎪ +𝛾2R ⋅ 𝜅 i,R ⋅ ΔRr,1 + 𝛾2R ⋅ (𝜅 i,R )2 ⋅ 𝛺r,1 + 𝛾2R ⋅ 𝛽12 ⋅ DCBRr,P P 1 2 ⎩

(11)

The parameter vector 𝐗 is expressed as R ̄R ,𝐍 ̄ R ]T 𝐗 = [𝐱, c ⋅ d̄trR , ΔRr,12 , 𝛺r,12 , Zw , 𝐈̄ Rr,1 , 𝐍 r,1 r,2

R ⎧ c ⋅ d̄trR = c ⋅ dtr + dr,IF ⎪ 𝛩i,R = 𝛩i,R − 𝛾 R ⋅ 𝛩i,R12 2 r,1 ⎪ i,Rr,12 i,R r,2 i,R R ⎪ Īr,1 = Ir,1 + 𝛽12 ⋅ DCBRr,P P + 𝛩r,1 1 2 ⎪ ̄ i,R i,R i,R i,R i,R i,R R ⎨ Nr,1 = 𝜆1 ⋅ (Nr,1 + br,1 − b1 ) + dIF12 − dr,IF12 (14) i,R R R ⎪ + 𝛩r,1 + 𝛽12 ⋅ DCBr,P P 1 2 ⎪ i,R i,R i,R i,R i,R i,R R ̄ ⎪ Nr,2 = 𝜆2 ⋅ (Nr,2 + br,2 − b2 ) + dIF12 − dr,IF 12 i,R ⎪ R R + 𝛾2R ⋅ 𝛩r,1 + 𝛾2R ⋅ 𝛽12 ⋅ DCBr,P P ⎩

(12)

4. Estimating pseudorange IFBs for each GLONASS satellite. The IFBs are modeled as follows i,R i,R R dr,j = dr,j + 𝛩r,j (13) i,R denotes the part of IFB that is dependent on where 𝛩r,j

(15)

It is noted that schemes 1, 2 and 3 are all full rank i,R i,R models. However, for scheme 4, 𝛩r,j and Īr,1 are linearly dependent, the following constraint equations can be added to overcome the datum deficiency in the equation systems n ∑ k,R 𝛩r,12 =0 (16) k=1

where n is the number of GLONASS satellites in view. Receiver UCDs for GPS + GLONASS There are two strategies to handle the receiver clock offset parameter for combined GNSS PPP processing. One is that an independent receiver clock offset parameter for each GNSS system is introduced, which is applicable for both the single satellite system and the combined satellite systems. The other is to consider the receiver UCD difference among different GNSSs and then introduce the inter-system biases (ISBs) at the receiver end (Lou et al. 2016). In this study, the latter one is adopted and the receiver clock offset for GPS signals is chosen as reference. Note that there are no differences in handling the IFBs by applying the abovementioned four schemes for GLONASS-only and GPS + GLONASS PPP, except the ISB parameter included in GLONASS observation equations in the latter strategy. However, the ISB and IFB parameters in the fourth scheme are highly correlated. In order to avoid the high correlation, the ISB and IFB parameters are merged into a single parameter for every GLONASS satellite. Thus, we call it “ISFB” parameter. The re-parameterization of (4) is given as

GLONASS satellite PRNs and their frequency. The reparameterization of (4) is conducted as

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⎧ G G ⎪ c ⋅ d̄tr = c ⋅ dtr + dr,IF12 i,R i,R ⎪ ISFB = 𝛩 − 𝛾 R ⋅ 𝛩i,R + dR − dG r,IF12 r,IF12 r,1 2 ⎪ i,G r,12i,G r,2 G G ⎪ Īr,1 = Ir,1 + 𝛽12 ⋅ DCBr,P1 P2 i,R ⎪ Ī i,R = I i,R + 𝛽 R ⋅ DCBR + 𝛩r,1 r,P1 P2 r,1 ⎪ r,1 � �12 ⎪ N̄ i,G = 𝜆G ⋅ N i,G + bG − bi,G + di,G − dG + 𝛽 G ⋅ DCBG r,P1 P2 IF12 r,IF12 1 r,1 12 1 ⎪ r,1 � � r,1 ⎨ N̄ i,G = 𝜆G ⋅ N i,G + bG − bi,G + di,G − dG + 𝛾 G ⋅ 𝛽 G ⋅ DCBG r,P1 P2 IF12 r,IF12 2 r,2 2 12 2 ⎪ r,2 � � r,2 ⎪ N̄ i,R = 𝜆i,R ⋅ N i,R + bi,R − bi,R + di,R − dG IF12 r,IF12 1 r,1 r,1 1 ⎪ r,1 i,R R ⎪ + 𝛩r,1 + 𝛽12 ⋅ DCBRr,P P 1 2 � � ⎪ i,R i,R i,R i,R i,R G ⎪ N̄ r,2 = 𝜆i,R + dIF ⋅ N + b − b − dr,IF 2 r,2 r,2 2 12 12 ⎪ i,R R + 𝛾2R ⋅ 𝛩r,1 + 𝛾2R ⋅ 𝛽12 ⋅ DCBRr,P P ⎪ 1 2 ⎩ The parameter vector 𝐗 is expressed as

]T [ ̄R , 𝐍 ̄G ,𝐍 ̄G ,𝐍 ̄R ,𝐍 ̄R 𝐗 = 𝐱, c ⋅ d̄trG , 𝐈𝐒𝐅𝐁Rr,12 , Zw , 𝐈̄ G , 𝐈 r,1 r,1 r,1 r,2 r,1 r,2

(18) Note that the four schemes for GPS + GLONASS PPP are all full rank models. In scheme 4, considering that the ISB parameter has the same impact on all GLONASS satellites, we can isolate IFBs from ISFBs by subtracting the common parts of ISFB estimations for all GLONASS satellites.

Experimental data and processing strategies In order to test and validate the proposed handling schemes, observation data of globally distributed stations from the IGS network was selected. The strategies of data processing were then described in detail.

Experimental data sets To validate the handling schemes of GLONASS pseudorange IFBs on PPP performance, observation data sampled in 30 s from 132 stations of the IGS network were selected, which covered a one-week period of DOY (Day of Year) 183–189 in 2016. These stations are globally evenly distributed and equipped with receivers from six manufacturers as shown in Table 1. Figure 1 shows the geographical distribution of the selected stations with GPS and GLONASS tracking capability.

Processing strategy GPS-, GLONASS-only, and combined GPS + GLONASS PPP in static and simulated kinematic modes were performed in our study. GPS and GLONASS precise orbit and clock products provided by ESA (European Space Agency) were held fixed. In the data processing, the elevation cutoff

13

(17)

angle was set to 7° and the elevation-dependent weighting for the observations with elevations below 30° was applied. The zenith wet delays were estimated as random-walk noise, and their spectral density values were empirically set to 10−8 m2/s (Li and Zhang 2014). The slant ionospheric delays were treated as white noise (Guo et al. 2016; Liu et al. 2017). The float phase ambiguities were estimated as constant for each continuous satellite arc. The initial standard deviation values for GPS and GLONASS carrier phase observations were both set to 0.003 m, while the measurement error ratio between pseudorange and carrier phase observations was set to 100. For combined GPS + GLONASS PPP, the system weighting ratio of GPS and GLONASS was assumed to be 1:1 (Lou et al. 2016). In static PPP mode, the position Table 1 Details of GNSS receivers of the selected IGS stations Manufacturer

Receiver type

Number of stations

JAVAD

TRE_G3TH DELTA TRE_3 DELTA TRE_G3TH SIGMA EGGDT LEGACY NETG3 NET-G3A LEGACY ODYSSEY_E GR10 GR25 GRX1200GGPRO GRX1200 + GNSS NETR5 NETR8 NETR9 POLARX3ETR POLARX4 POLARX4TR POLARXS –

21 2 1 4 2 1 15 1 2 4 9 12 1 4 4 37 1 7 3 1

JPS TPS

LEICA

TRIMBLE SEPTENTRIO

SUM

132

Page 7 of 14 19

GPS Solutions (2018) 22:19 Fig. 1 Geographical distribution of the selected 132 globally IGS tracking stations

In this study, the 24-h observation data for each station is divided into eight sessions (each session is 3 h) to evaluate the PPP performance in short observation time spans. In total, there are about 7392 convergence tests used in the experiment. The PPP performance in terms of convergence time and positioning accuracy in the horizontal and vertical components is evaluated at the 68 and 95% confidence level in static and kinematic mode. The convergence time is determined when the positioning error is lower than 0.2 m (95%) and 0.1 m (68%) in the horizontal and vertical components. Taking the first epoch for example, position errors of the first epoch in horizontal and up components for each test can be obtained, and then sort them by absolute values for each component. Take the 68 and 95% quantiles, respectively, as the position errors of the first epoch at the 68 and 95% confidence level. This statistical method, which has been adopted by Leandro et al. (2011), van Bree and Tiberius (2012), Lou et al. (2016), and de Oliveira et al. (2017), is generally used Table 2 Summary of GLONASS pseudorange IFBs handling schemes Items

Descriptions

IFB0 IFB1

Neglecting pseudorange IFBs Estimating pseudorange IFBs using the linear function of frequency number Estimating pseudorange IFBs using quadratic polynomial function of frequency number Estimating pseudorange IFBs for each GLONASS satellite

IFB2 IFB3

GLONASS‑only PPP Figure 2 displays the horizontal and vertical errors of GPSand GLONASS-only static PPP in different schemes, based on the statistics over all the tests. Table 3 presents the convergence time determined in the horizontal and vertical components. We can see that it takes 26.5 and 20.0 min for IFB0 to converge to the defined 68% level in horizontal and vertical components, respectively. The convergence time is 36.5 and 29.5 min in two components at the 95% level, respectively. The convergence performance of IFB0 is worse than that of GPS in both components mainly owing to the neglecting pseudorange IFBs for IFB0 solutions. After considering pseudorange IFBs (i.e., IFB1, IFB2, and IFB3 solutions) in GLONASS-only PPP processing, the

GPS IFB0 IFB1 IFB2 IFB3 1.4 95% level 95% level 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 36.5 min 29.5 min 0.4 0.4 0.2 0.2 23.0 min 0.0 23.5 min 0.0 68% level 68% level 0.5 0.5 0.4 0.4 0.3 0.3 26.5 min 20.0 min 0.2 0.2 0.1 0.1 13.5 min 0.0 15.5 min 0.0 0 20 40 60 80 100 0 20 40 60 80 100120

Vertical error (m)

Results and analysis

to assess convergence performance per epoch. Furthermore, the threshold values of 0.2 m (95%) and 0.1 m (68%) in the horizontal and vertical components have been suggested by Lou et al. (2016). For convenience, the four schemes mentioned above, summarized in Table 2, are marked as IFB0, IFB1, IFB2, and IFB3, respectively.

Horizontal error (m)

coordinates were considered as constants, while they were modeled as white noise in kinematic PPP mode (Li and Zhang 2014). The positioning performance was assessed with respect to either the coordinates from IGS weekly solutions, or the average values from seven consecutive daily PPP solutions with the Positioning And Navigation Data Analyst (PANDA, Liu and Ge 2003) software in static mode.

Convergence time (min)

Fig. 2 GPS- and GLONASS-only static PPP convergence performance in different schemes

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Table 3 Convergence time of GPS- and GLONASS-only static and kinematic PPP in different schemes (unit: min)

GPS Solutions (2018) 22:19 Static PPP

GPS IFB0 IFB1 IFB2 IFB3

Kinematic PPP

H (95%)

H (68%)

V (95%)

V (68%)

H (95%)

H (68%)

V (95%)

V (68%)

23.5 36.5 31.5 28.5 24.5

19.5 26.5 23.0 20.0 15.5

23.0 29.5 27.0 25.0 23.0

17.0 20.0 17.5 16.0 13.5

51.0 – – – –

42.5 80.5 69.0 65.0 55.0

50.0 – – – –

34.0 99.5 85.0 80.0 73.0

90˚N


60˚N 30˚N 0˚ 30˚S 60˚S 90˚S 90˚N 60˚N 30˚N

GPS IFB0 IFB1 IFB2 IFB3 1.4 95% level 95% level 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 51.0 min 50.0 min 0.4 0.4 0.2 0.2 0.0 0.0 68% level 68% level 0.5 0.5 0.4 0.4 0.3 0.3 80.5 min 99.5 min 0.2 0.2 0.1 0.1 34.0 min 0.0 42.5 min 0.0 0 30 60 90 120150 0 30 60 90 120150180 Convergence time (min)

Vertical error (m)

19

Fig. 4 GPS- and GLONASS-only kinematic PPP convergence performance in different schemes

0˚ 30˚S 60˚S 90˚S 180˚

120˚W 60˚W

0˚

60˚E

120˚E

180˚

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 Fig. 3 Average global GDOP of GPS (top) and GLONASS (bottom) on DOY 183, 2016 with an elevation cutoff angle of 7°

convergence performance is improved comparing to IFB0. Meanwhile, it can be seen from Table 3 that IFB2 performs better than IFB1, while IFB3 performs best among the handling schemes. Compared with IFB0, the convergence time of IFB3 is notably improved by 32.9% from 36.5 to 24.5 min and by 22.0% from 29.5 to 23.0 min in horizontal and vertical components at the 95% level, respectively. At the 68% level, the convergence time is significantly reduced by 41.5% from 26.5 to 15.5 min and by 34.2% from 20.0 to 13.5 min in horizontal and vertical components, respectively. It is noted that the convergence time of static PPP in vertical components is less than that in horizontal components. One reason may be that the phase ambiguities are more correlated with the east component than the vertical

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component, the convergence of horizontal component can be weakened through their correlation with phase ambiguities (Blewitt 1989). The other reason may relate to the north–south ground tracks of GPS and GLONASS satellites in the earth-fixed reference frame (Melbourne 1985). It is also shown in Fig. 2 and Table 3 that the convergence time of IFB3 is less than that of GPS at the 68% level, which is also confirmed by Lou et al. (2016). Figure 3 displays the average global Geometric Dilution of Precision (GDOP) of GPS and GLONASS with elevation cutoff angle of 7° on DOY 183, 2016. The GDOP values are calculated every 15 min and averaged daily to show the geometry strength in different regions around the world. It can be seen that the GDOP distribution of GPS is fairly uniform globally, while the GLONASS GDOP shows evident regional characteristics. The GLONASS GDOP is smaller in the middle- and high-latitude regions than that in the low-latitude regions. In high-latitude regions, the GDOP values of GLONASS are even smaller than GPS. The GDOP factors will actually have a direct impact on the PPP positioning performance. There are 72 stations (more than 50%) located in high-latitude whose latitude is larger than 45°.

Page 9 of 14 19

GPS Solutions (2018) 22:19 Table 4 Positioning accuracy comparison of GPS- and GLONASS-only static and kinematic PPP in different schemes (unit: cm)

Static PPP


Kinematic PPP

H (95%)

H (68%)

V (95%)

V (68%)

H (95%)

H (68%)

V (95%)

V (68%)

3.2 4.0 4.0 3.9 3.9

1.5 1.7 1.6 1.6 1.6

3.9 5.6 5.6 5.5 5.5

1.8 2.5 2.5 2.5 2.5

7.3 68.7 48.5 41.9 30.6

3.1 5.6 5.1 4.9 4.4

9.0 51.0 37.6 33.1 26.8

3.7 7.6 7.0 6.8 6.4

0.25 0.20 0.15

R

Ωr,12 (m)

0.10 0.05 0.00 −0.05 −0.10 −0.15 −0.20 −0.25 0

25

50 75 Stations

100

125

Fig. 5 The estimated second-order item of IFB2 for each station on DOY 183, 2016 (stations are numbered in alphabetical order)

Fig. 6 Pseudorange observation residuals of GPS- and GLONASS-only PPP at two selected IGS stations (WROC and RIO2) on DOY 183, 2016 (The first 3 h)

Figure 4 indicates the horizontal and vertical errors of kinematic PPP in different schemes, and Table 3 also presents the convergence time of kinematic PPP at the 95 and 68% level in the horizontal and vertical components, respectively. It is noted that GLONASS-only PPP cannot converge in the horizontal and vertical components at the 95% level. Similar to the GLONASS-only static PPP, at the 68% level, IFB0 performs worst in the convergence performance, while IFB3 performs best. Compared with IFB0, the convergence time of IFB3 is notably improved by 31.7% from 80.5 to 55.0 min and by 26.6% from 99.5 to 73.0 min in the horizontal and vertical components at the 68% level, respectively. Unlike static PPP, the convergence performance of GLONASS-only kinematic PPP is much worse than that of GPS-only, especially in the vertical component. It is reasonable that the precise orbit and clock products of GLONASS are worse than that of GPS by a factor of 2–3 (Guo et al. 2017). Also, the larger GLONASS GDOP at low-latitudes, as shown in Fig. 3 (bottom), is unfavorable.

P1 residual (m) RIO2 P2 residual (m) P1 residual (m) WROC P2 residual (m) 3 3 RMS: 0.075 m GPS RMS: 0.880 m RMS: 0.890 m 2 2 GPS RMS: 0.101 m 1 1 0 0 −1 −1 −2 −2 3 3 −3 −3 ICB0 RMS: 0.406 m RMS: 0.247 m ICB0 RMS: 0.629 m RMS: 0.521 m 2 2 1 1 0 0 −1 −1 −2 −2 3 3 −3 −3 RMS: 0.101 m ICB1 RMS: 0.485 m RMS: 0.446 m 2 2 ICB1 RMS: 0.131 m 1 1 0 0 −1 −1 −2 −2 3 3 −3 −3 RMS: 0.097 m ICB2 RMS: 0.451 m RMS: 0.437 m 2 2 ICB2 RMS: 0.124 m 1 1 0 0 −1 −1 −2 −2 3 3 −3 −3 RMS: 0.081 m ICB3 RMS: 0.245 m RMS: 0.359 m 2 2 ICB3 RMS: 0.079 m 1 1 0 0 −1 −1 −2 −2 −3 −3 00:00 01:00 02:00 00:00 01:00 02:00 03:00 00:00 01:00 02:00 00:00 01:00 02:00 03:00

GPS time (hh:mm)

GPS time (hh:mm)

13

19

Page 10 of 14


GPS IFB0 IFB1 IFB2 IFB3 1.4 95% level 95% level 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 23.5 min 23.0 min 0.4 0.4 0.2 0.2 13.0 min 0.0 13.5 min 0.0 68% level 68% level 0.5 0.5 0.4 0.4 0.3 0.3 19.5 min 17.0 min 0.2 0.2 0.1 0.1 12.5 min 11.0 min 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100120

Vertical error (m)


The positioning accuracy of static and kinematic PPP in different schemes is summarized in Table 4. For static PPP, the results of the last epoch for each test are used for statistics, which has been adopted by Li et al. (2016) and Pan et al. (2014). Since GLONASS-only PPP does not converge at 95% level, the positioning accuracy in kinematic mode is calculated through the same converge period, here two hours later for each test is chosen. From the table, it is noted that the positioning accuracy of GLONASS-only static PPP is comparable with each other. Therefore, we conclude that considering GLONASS pseudorange IFBs makes very small improvement on positioning accuracy, the improvement of which is below 6% from 1.7 to 1.6 cm. This is mainly due to the carrier phase observations will play a dominant role in positioning after convergence. For kinematic PPP, it can be seen that IFB2 performs better than IFB1 and IFB3 perform best. Compared with IFB0, the positioning accuracy of IFB3 is significantly improved by 55.5% from 68.7 to 30.6 cm and by 47.5% from 51.0 to 26.8 cm in horizontal and vertical components at the 95% level, respectively. At the 68% level, the positioning accuracy is obviously improved by 21.4% from 5.6 to 4.4 cm and by 15.8% from 7.6 to 6.4 cm in horizontal and vertical components, respectively. Similar to convergence performance, the positioning accuracy of GLONASS-only kinematic PPP is much lower than that of GPS-only, especially in the vertical component.

Convergence time (min) Fig. 7 GPS-only and GPS + GLONASS static PPP convergence performance in different schemes Table 5 Convergence time of GPS-only and GPS + GLONASS static and kinematic PPP in different schemes (unit: min)

13

The estimated second-order item of IFB2 for each station on DOY 183, 2016 is shown in Fig. 5. There are 59 (about R 45%) stations, of which absolute values of 𝛺r,12 are larger than 0.01 m. This means that the second-order corrections of GLONASS satellite whose frequency number is − 7 can even reach 0.49 m. Hence, the positioning performance that IFB2 outperforms IFB1 is expectable. Since observation residuals contain measurement noises and other unmodeled errors, they can be used as an important indicator to evaluate the positioning model. Figure 6 shows the pseudorange observation residuals at two selected IGS stations (WROC and RIO2) in GPS- and GLONASSonly PPP processing on DOY 183, 2016 (The first 3 h). In the figure, different colors represent different satellites. Neglecting GLONASS IFB (the panel of IFB0), larger systematic biases can be seen, compared to the pseudorange residuals of considering IFB. The RMS statistics are also displayed in each panel. Overall, the statistical results clearly demonstrate that IFB3 has the smallest pseudorange residuals for GLONASS, suggesting that the IFBs in GLONASS pseudorange observations have been properly handled in the IFB3 model. It is apparent that the GPS pseudorange residuals at RIO2 are much noisier than GLONASS, which may due to the poorer quality for GPS observations on that day.

GPS + GLONASS PPP Figure 7 shows the horizontal and vertical errors of GPSonly and GPS + GLONASS static PPP in different schemes. Table 5 presents the convergence time in the horizontal and vertical components. It can be seen that the convergence performance of GPS + GLONASS PPP is better than that of GPS-only PPP, no matter considering GLONASS pseudorange IFBs or not. Similar to GLONASS-only static PPP, the convergence performance with considering GLONASS pseudorange IFBs is better than that without considering IFBs. Compared with the convergence performance of GLONASS-only static PPP, the improved percentage of GPS + GLONASS with different handling schemes is smaller. Compared with IFB0, the convergence time of IFB3 is improved by 28.9% from 19.0 to 13.5 min and by 21.9% from 16.0 to 12.5 min in horizontal and vertical components at the 95% level, respectively. At the 68% level, the

Static PPP


Kinematic PPP

H (95%)

H (68%)

V (95%)

V (68%)

H (95%)

H (68%)

V (95%)

V (68%)

23.5 19.0 17.0 15.0 13.5

19.5 17.5 15.5 14.0 13.0

23.0 16.0 14.5 14.0 12.5

17.0 13.5 12.5 11.5 11.0

51.0 27.0 24.5 21.5 19.5

42.5 24.0 21.0 17.0 16.5

50.0 21.0 18.5 19.5 17.5

34.0 18.0 16.0 15.0 14.0

Page 11 of 14 19

GPS IFB0 IFB1 IFB2 IFB3 1.4 95% level 95% level 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 51.0 min 50.0 min 0.4 0.4 0.2 0.2 16.5 min 0.0 19.5 min 0.0 68% level 68% level 0.5 0.5 0.4 0.4 0.3 0.3 42.5 min 34.0 min 0.2 0.2 0.1 0.1 14.0 min 0.0 17.5 min 0.0 0 30 60 90 120150 0 30 60 90 120150180

Vertical error (m)



Convergence time (min) Fig. 8 GPS-only and GPS + GLONASS kinematic PPP convergence performance in different schemes

convergence time is reduced by 25.7% from 17.5 to 13.0 min and by 18.5% from 13.5 to 11.0 min in horizontal and vertical components, respectively. Note that the improvement in the convergence performance is more obvious in the horizontal component than in the vertical component. Figure 8 illustrates the horizontal and vertical errors of kinematic PPP in different schemes. The convergence time of kinematic PPP at the 95 and 68% level in the horizontal and vertical components, respectively, are presented in Table 5. It can be seen that, by adding GLONASS observations, the convergence time can be shortened by more than 50% in both horizontal and vertical components when considering GLONASS pseudorange IFBs. IFB0 performs worst in the convergence performance at the 95 and 68% level, while IFB3 performs best. Compared with IFB0, Table 6 Positioning accuracy comparison of GPS-only and GPS + GLONASS static and kinematic PPP in different schemes (unit: cm)

Table 7 Positioning accuracy comparison of GPS-only and GPS + GLONASS kinematic PPP during convergence period of 10 and 20 min in different schemes (unit: cm)

the convergence time of IFB3 is significantly improved by 27.8% from 27.0 to 19.5 min and by 16.7% from 21.0 to 17.5 min in horizontal and vertical components at the 95% level, respectively. At the 68% level, the convergence time of IFB3 is obviously improved by 31.3% from 24.0 to 16.5 min and by 22.2% from 18.0 to 14.0 min in horizontal and vertical components, respectively. The positioning accuracy of GPS-only and GPS + GLONASS static and kinematic PPP in different schemes is summarized in Table 6. Similar with GLONASS-only PPP, considering GLONASS pseudorange IFBs achieves very small improvement on the positioning accuracy of GPS + GLONASS static PPP, the improvement of which is below 9% from 1.2 to 1.1 cm. Unlike GLONASS-only kinematic PPP, it is noted that the positioning accuracy of GPS + GLONASS kinematic PPP is comparable with each other. Shi et al. (2013) pointed out that the accuracy of GPS + GLONASS PPP after GLONASS pseudorange IFB calibration was greatly improved during the convergence period. Table 7 illustrates positioning accuracy comparison during convergence period. It is noted that, if neglecting the GLONASS pseudorange IFBs, the horizontal accuracy of GPS + GLONASS PPP is lower than that of GPS-only during 10- and 20-min convergence period at the 95 and 68% level. With considering IFBs, the horizontal and vertical accuracy of GPS + GLONASS PPP is higher than that of GPS-only PPP during the convergence period. Note that the schemes that considering IFBs perform better than IFB0. Hence, we conclude that considering IFBs or not mainly affects the accuracy of combined GPS + GLONASS PPP during the convergence period. Meanwhile, it is also observed that IFB3 performs better than IFB1, but worse than IFB2, which may

Static PPP


Kinematic PPP

H (95%)

H (68%)

V (95%)

V (68%)

H (95%)

H (68%)

V (95%)

V (68%)

3.2 2.4 2.4 2.4 2.4

1.5 1.2 1.2 1.2 1.1

3.9 3.8 3.8 3.8 3.8

1.8 1.9 1.9 1.9 1.9

7.3 4.1 4.1 4.0 4.0

3.1 2.0 2.0 1.9 1.9

9.0 6.7 6.7 6.7 6.6

3.7 3.0 3.0 3.0 3.0

10 min


20 min

H (95%)

H (68%)

V (95%)

V (68%)

H (95%)

H (68%)

V (95%)

V (68%)

95.4 101.1 87.2 78.5 84.0

48.2 51.4 43.3 41.3 42.7

148.7 142.9 123.0 116.5 123.1

64.9 62.3 54.8 52.8 54.6

74.9 76.5 65.9 59.0 62.2

37.4 38.5 32.4 30.7 31.3

112.9 103.4 89.0 84.1 88.5

48.2 45.0 39.5 38.0 39.2

13

19

Page 12 of 14

be due to more introduced parameters in the IFB3 that the strength of the observation equations is weakened.

Conclusions In this study, we proposed a general approach to model GLONASS pseudorange IFBs for GLONASS-only and combined GPS + GLONASS PPP through a re-parameterization process. Four different GLONASS pseudorange IFBs handling schemes, which are neglecting IFBs, modeling IFBs as a linear or quadratic polynomial function of frequency number, and estimating IFBs for each GLONASS satellite, were proposed for PPP based on undifferenced and uncombined observation models. One week of observation data in July 2016 (DOY 183–189) from 132 stations of IGS network were adopted to validate the model, and preliminary positioning results have been achieved. The numerical results showed that for GLONASS-only and GPS + GLONASS PPP, considering pseudorange IFBs can accelerate the convergence in both static and kinematic PPP. The convergence time was shortened by more than 20%. Apart from GLONASS-only kinematic PPP, the positioning accuracy of GLONASS-only and GPS + GLONASS PPP is comparable among the four schemes. Besides, the accuracy of combined GPS + GLONASS PPP with considering IFBs was evidently improved during the convergence period. Compared to the static PPP, it was found that the convergence performance and positioning accuracy of GLONASS-only kinematic PPP were much worse than that of GPS-only, especially in the vertical component, which may be due to lower accuracy of GLONASS orbit and clock products (Guo et al. 2017). Also, the higher GDOP values for GLONASS at low-latitude regions are another important factor. In general, the scheme of estimating IFBs for each GLONASS satellite is superior in both the convergence time reduction and positioning accuracy improvement. These results indicate that generally, the GLONASS pseudorange IFBs may not strictly obey a linear or quadratic function relationship with the frequency number. Hence, it is recommended that this scheme can be used to handle GLONASS pseudorange IFBs in PPP processing when GLONASS observations are used. Acknowledgements Feng Zhou is financially supported by the China Scholarship Council (CSC) for his study at the German Research Centre for Geosciences (GFZ). We would like to thank the IGS for providing GNSS ground tracking data, DCB, precise orbit and clock products. The figures were generated using the public domain Generic Mapping Tools (GMT) software (Wessel et al. 2013). This work is sponsored by the National Key R&D Program of China (No. 2017YFE0100700), the National Natural Science Foundation of China (Nos. 61372086 and 41771475) and the Science and Technology Commission of Shanghai (Nos. 13511500300 and 15511101602).

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Page 13 of 14 19 Feng Zhou is currently a Ph.D. student at East China Normal University. He received his Master’s degree at China University of Mining and Technology in 2012. His current research mainly focuses on multi-constellation and multi-frequency GNSS PPP.

Danan Dong is currently a professor of East China Normal University. He obtained his Ph.D. degree from Massachusetts Institute of Technology (MIT) and had worked in Jet Propulsion Laboratory (JPL) for 18 years. His main research interest is in the study of geophysics with application of GPS technology.

Maorong Ge received his Ph. D. at the Wuhan University, Wuhan, China. He is now a senior scientist at the German Research Centre for Geosciences (GFZ), Potsdam, Germany. He has been in charge of the IGS Analysis Center at GFZ and is now leading the real-time software group. His research interests are GNSS data processing and the related a l g o r i t h m s a n d s o f t wa r e development.

Pan Li is currently a scientist at the German Research Centre for Geosciences (GFZ). He obtained his Ph. D. degree at the Wuhan University in 2016. His current research mainly focuses on GNSS PPP ambiguity resolution.

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GPS Solutions (2018) 22:19 Jens Wickert holds a joint professorship of Technische Universität Berlin and the German Research Centre for Geosciences (GFZ) on GNSS Remote Sensing, Navigation and Positioning. His main research interests are interdisciplinary GNSS applications for Earth Observation.

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Harald Schuh is currently the Director of the Department 1 “Geodesy”, German Research Centre for Geosciences (GFZ). He is also the president of the International Association of Geodesy (IAG) since 2016. His research interests are Very Long Baseline Interferometry technique and its applications.