Improved Dual Frequency PPP Model Using GPS and ...

J. Geod. Sci. 2017; 7:1–8

Research Article

Open Access

A. Afifi* and A. El-Rabbany

Improved Dual Frequency PPP Model Using GPS and BeiDou Observations DOI 10.1515/jogs-2017-0001 Received February 19, 2016; accepted December 11, 2016

Abstract: This paper introduces a new dual-frequency precise point positioning (PPP) model, which combines GPS and BeiDou observations. Combining GPS and BeiDou observations in a PPP model offers more visible satellites to the user, which is expected to enhance the satellite geometry and the overall PPP solution in comparison with GPSonly PPP solution. However, combining different GNSS constellations introduces additional biases, which require rigorous modelling, including GNSS time offset and hardware delays. In this research, ionosphere-free linear combination PPP model is developed. The additional biases, which result from combining the GPS and BeiDou observables, are lumped into a new unknown parameter identified as the inter-system bias. Natural Resources Canada’s GPSPace PPP software is modified to enable a combined GPS/BeiDou PPP solution and to handle the newly introduced biases. A total of four data sets at four IGS stations are processed to verify the developed PPP model. Precise satellite orbit and clock products from the IGS-MGEX network are used to correct both of the GPS and BeiDou measurements. It is shown that a sub-decimeter positioning accuracy level and 25% reduction in the solution convergence time can be achieved with combining GPS and BeiDou observables in a PPP model, in comparison with the GPS-only PPP solution. Keywords: BeiDou, Global Positioning System, Precise Point Positioning

1 Introduction The precise point positioning (PPP) technique allows a user with a standalone dual-frequency global navigation satellite system (GNSS) receiver to determine his or her position at the decimeter level accuracy. PPP accuracy and

*Corresponding Author: A. Afifi: Ryerson University Toronto, Canada, E-mail: [email protected] A. El-Rabbany: Ryerson University Toronto, Canada

convergence time are influenced by the ability to mitigate all potential error sources in the system. These errors can be categorized into three classes: satellite related errors, signal propagation related errors, and receiver/antenna configuration errors (El-Rabbany, 2006). A number of comprehensive studies have been published on the accuracy and convergence time of un-differenced combined GPS PPP model (see for example: Zumberge et al, 1997; Kouba and Héroux, 2001; Colombo et al., 2004; Ge at el., 2008; Collins at al., 2010; Afifi and El-Rabbany, 2015; Afifi and ElRabbany, 2016; Abd Rabbou and El-Rabbany, 2015; Tegedor et al., 2015; Li, et al., 2013; El-Mowafy, et al., 2016; Melgard et al., 2013). Currently, there are four operational global navigation satellite systems (GNSS), namely the US GPS, the Russian GLONASS, the European Galileo system, and the Chinese BeiDou system. GPS satellites transmit signals on three different frequencies, which are referenced to the GPS time frame (GPST). GLONASS transmits three signals on different main frequencies. However, unlike the other GNSS systems, each satellite transmits different frequencies based on the frequency division multiple accesses (FDMA). GLONASS transmits its signals using the GLONASS Time system (GLONASST). The Galileo system transmits six signals on different frequencies using the Galileo time system (GST). Unlike the GLONASS satellite system, Galileo and GPS have partial frequency overlaps, which simplify the dual-system integration. The BeiDou navigation satellite system, being developed independently by China, is moving steadily forward towards completing the constellation by 2020 (BeiDou, 2015). The BeiDou system transmits three signals on different frequencies using the BeiDou time frame (BDT) (HofmannWellenhof et al., 2008; IAC, 2015; ESA, 2015; BeiDou, 2015). A drawback of a single GNSS system such as GPS is the limited number of visible satellites in urban areas. More recently, Afifi and El-Rabbany (2014) showed that combining the un-differenced GPS and Galileo observations in a PPP model improved the solution convergence time by about 25%, in comparison with the GPS-only counterpart. In addition, they showed that the inter-system bias is largely constant over a one-hour observation time span, which they used in their analysis, with a magni-

© 2017 A. Afifi and A. El-Rabbany, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

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2 | A. Afifi and A. El-Rabbany tude ranging from 30 to 60 nanoseconds depending on the GNSS receiver type. Abd Rabbou and El-Rabbany (2015) showed that combining different constellations, namely GPS, GLONASS, and Galileo, improves the PPP solution in comparison with the GPS-only PPP solution. They also showed that the receiver inter-frequency bias is stable over the observation time span with a root-mean-square that varies between 2.5 cm and 3 cm. Tegedor et al., (2015), showed that adding Galileo and BeiDou constellations to the Fugro G2 service, which is a global PPP service including GPS and GLONASS constellations, will only enhance the overall solution. Li, et al. (2013) combined the GPS and BeiDou systems in a PPP model using an extended Kalman filter. In Li, et al. (2013) the precise satellite clock corrections are obtained from the IGS and Technical University Munich (TUM, Munich, Germany) for GPS and BeiDou observations, respectively. It should be noted that the BeiDou orbit and clock products from TUM are only based on six stations, and have a lower accuracy than IGS products, which result in a longer time for convergence and less positioning accuracy of the BeiDou PPP solution (Li, et al., 2013). El-Mowafy, et al., (2016) studied the effect of different PPP biases including satellite and receiver hardware biases, differential code biases, differential phase biases, initial fractional phase biases, intersystem receiver time biases, and system time scale offsets on the PPP model. Melgard et al. (2013) showed that combining multi-constellation (GPS, GLONASS, Galileo, and BeiDou) in a PPP solution improves the positioning accuracy, especially when the system biases are calibrated. In this paper a dual-frequency PPP model is developed, which combines the GPS and BeiDou observations in static mode. The developed GPS/BeiDou PPP model rigorously accounts for the system time offset and hardware delays. These additional biases are lumped together into a new unknown parameter, which is referred to as the intersystem bias, in the PPP mathematical model. The GPS receiver differential hardware delays are lumped into the GPS receiver clock error in all the developed PPP models. The hydrostatic component of the tropospheric zenith path delay is modelled through the Hopfield model, while the wet component is considered as an additional unknown parameter (Hopfield, 1972; Hofmann-Wellenhof et al., 2008). All remaining errors and biases are accounted for using existing models as shown, for example, in Kouba (2009). The GPS-only PPP solution is also obtained and is used as a reference for verifying the results. In the developed models, the GPS L1/L2 and BeiDou B1/B2 signals are used to develop dual-frequency ionosphere-free linear combinations. A sequential least-squares estimation technique is used to get the best estimates for the positioning,

inter-system bias, and hardware delay parameters. The inter-system bias parameter was found to be essentially constant over the observation time span (one hour) as well as receiver-dependent. The positioning results of the developed combined PPP models showed a sub-decimeter accuracy level and 25% convergence time improvement in comparison with the GPS-only PPP results.

2 Un-differenced PPP models GNSS observations are affected by errors and biases, which can be categorized as satellite-related errors, signal propagation-related errors and receiver/antenna-related errors (El-Rabbany, 2006; Hofmann-Wellenhof et al., 2008; Leick, 1995). GNSS errors attributed to the satellites include satellite clock errors, orbital errors, satellite hardware delay, satellite antenna phase centre variation, and satellite initial phase bias. Errors attributed to signal propagation include the delays of the GNSS signal as it passes through the ionospheric and tropospheric layers. Errors attributed to receiver/antenna configuration include, the receiver clock errors, multipath error, receiver noise, receiver hardware delay, receiver initial phase bias, and receiver antenna phase center variations. In addition to the above errors and biases, combining GPS and BeiDou observation in a PPP model introduces additional errors such as the system time difference due to the fact that each system uses a different time frame. The GPS system uses the GPS time system, which is referenced to the coordinated universal time (UTC) as maintained by the US Naval Observatory (USNO). On the other hand, the BeiDou satellite system uses the BeiDou Time (BDT), which is a continuous time scale referenced to UTC starting on January 1, 2006 (Hofmann-Wellenhof et al., 2008). Equations (1) to (4) show the ionosphere-free linear combination of GPS and BeiDou observations (Afifi and ElRabbany, 2015). P G IF =ρ G + c[dt rG − dt s ] + c[α G d P1 − β G d P2 ]r + c[α G d P1 − β G d P2 ]s + T G + ϵ PG IF

(1)

P B IF =ρ B + c[dt rG − GB − dt s ] + c[α B d B1 − β B d B2 ]r + c[α B d B1 − β B d B2 ]s + T B + ϵ B IF

(2)

Φ G IF =ρ G + c[dt rG − dt s ] + c[α G δ L1 − β G δ L2 ]r + [α G δ L1 − β G δ L2 ]s + T G + N G IF + ϕ r0GIF + ϕ0s G + ϵ ΦG IF

IF

(3)

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Improved Dual Frequency PPP Model Using GPS and BeiDou Observations

Φ B IF =ρ B + c[dt rG − GB − dt s ] + c[α B δ B1 − β B δ B2 ]r s

+ c[α B δ B1 − β B δ B2 ] + T B + N B IF + ϕ r0BIF +

ϕ0s BIF

+ ϵ ΦB IF

(4)

where the subscripts G and B refer to the GPS and BeiDou satellite systems, respectively; P G IF and P B IF are the ionosphere-free pseudoranges in meters for GPS and BeiDou, respectively; Φ G IF and Φ B IF are the ionosphere-free carrier phase measurements in meters for GPS and BeiDou, respectively; GB is the GPS to BeiDou time offset; ρ is the true geometric range from receiver at reception time to satellite at transmission time in meters; dt r , dt s are the clock errors in seconds for the receiver at signal reception time and the satellite at signal transmission time, respectively; d P1r , d P2r , d B1r , d B2r are frequency-dependent code hardware delays for the receiver at reception time in seconds; d sP1 , d sP2 , d sB1 , d sB2 are frequency-dependent code hardware delays for the satellite at transmission time in seconds; δ L1r , δ L2r , δ B1r , δ B2r are frequency-dependent carrier-phase hardware delays for the receiver at reception time in seconds; δ sL1 , δ sL2 , δ sB1 , δ sB2 are frequencydependent carrier-phase hardware delays for the satellite at transmission time in seconds; T is the tropospheric delay in meters; N G IF , N B IF are the ionosphere-free linear combinations of the ambiguity parameters for both GPS and BeiDou carrier-phase measurements in meters, respectively; ϕ r0GIF , ϕ0s G , ϕ r0BIF , ϕ0s B are ionosphere-free IF IF linear combinations of frequency-dependent initial fractional phase biases in the receiver and satellite channels for both GPS and BeiDou in meters, respectively; c is the speed of light in vacuum in meters per second; ϵ P IF , ϵ ΦG IF , ϵ B IF , ϵ ΦB IF are the ionosphere-free linear combinations of the relevant noise and un-modeled errors in meters; α G , β G , α B , β B are the ionosphere-free linear combination coefficients for both GPS and BeiDou, which are given, respec2 2 f22 f B1 f B2 f12 tively, by: α G = f 2 −f 2 , β G = f 2 −f 2 , α B = f 2 −f 2 , β B = f 2 −f 2 . 1

2

1

2

B1

B2

B1

B2

Where f1 and f2 are GPS L1 and L2 signals frequencies; f B1 and f B2 are BeiDou B1 and B2 signals frequencies. N G IF = α G λ1 N1 − β G λ2 N2

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observations and are referenced to the GPS time frame (Montenbruck et al., 2014). IGS precise GPS satellite clock corrections include the effect of the ionosphere-free linear combination of the satellite hardware delays of the L1/L2 P(Y) code, while the BeiDou counterpart includes the effect of the ionosphere-free linear combination of the satellite hardware delays of the BeiDou B1/B2 code (Montenbruck et al., 2014; Wang et al., 2016). By applying the precise clock products for both of the GPS and BeiDou observations, Equations (1) to (4) will take the following form: P G IF =ρ G + c[dt rG − dt sprec ] + c[α G d P1 − β G d P2 ]r + T G + ϵ PG IF

(7)

P B IF =ρ B + c[dt rG − GB − dt sprec ] + c[α B d B1 − β B d B2 ]r + T B + ϵ B IF

(8)

Φ G IF =ρ G + cdt rG − c[dt sprec + [α G δ P1 − β G δ P2 ]s ] + c[α G δ L1 − β G δ L2 ]r − c[α G δ L1 − β G δ L2 ]s + T G + N G IF + ϕ r0GIF + ϕ0s G + ϵ ΦG IF

(9)

IF

Φ B IF =ρ B + cdt rG + c[dt sprec + [α B δ B1 − β B δ B2 ]s ] + c[α B δ B1 − β B δ B2 ]r − c[α B δ B1 − β B δ B2 ]s + T B + N B IF + ϕ r0BIF + ϕ0s BIF + ϵ ΦB IF

(10)

For simplicity, the receiver and satellite hardware delays are written as: b r p = c[α G d P1 − β G d P2 ]r b r B = c[α B d B1 − β B d B2 ]r b r I = c[α G δ L1 − β G δ L2 ]r + ϕ r0GIF b r BI = c[α B δ B1 − β B δ B2 ]r + ϕ r0BIF

(5) b sp = c[α G d P1 − β G d P2 ]s

N B IF = α B λ B1 N B1 − β B λ B2 N B2

(6)

where λ1 and λ2 are the GPS L1 and L2 signals wavelengths in meters; λ B1 and λ B2 are the BeiDou B1 and B2 signals wavelengths in meters; N1 , N2 are the integer ambiguity parameters of GPS signals L1 and L2, respectively;N B1 ,N B2 are the integer ambiguity parameters of BeiDou signals B1 and B2, respectively. Precise orbit and satellite clock corrections from the IGS-MGEX network are produced for both GPS/BeiDou

b sB = c[α B d B1 − β B d B2 ]s b sΦ = c[α G δ L1 − β G δ L2 ]s + ϕ0s GIF b sBΦ = c[α B δ B1 − β B δ B2 ]s + ϕ0s BIF In the combined GPS/BeiDou un-differenced PPP model, the GPS receiver clock error is lumped with the

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4 | A. Afifi and A. El-Rabbany GPS receiver differential code biases. In order to maintain consistency in the estimation of a common receiver clock offset, this convention is used when combining the ionosphere-free linear combination of GPS L1/L2 and BeiDou B1/B2 observations. This, however, introduces an additional bias to the BeiDou ionosphere-free PPP mathematical model, which represents the difference in the receiver differential code biases of both systems. Such an additional bias is commonly known as the inter-system bias, which is referred to as ISB in this paper. In our PPP mode, the Hopfield tropospheric correction model along with the Vienna mapping function are used to account for the hydrostatic component of the tropospheric delay (Hopfield, 1972; Boehm and Schuh, 2004). Other corrections are also applied, including the effect of ocean loading (Bos and Scherneck, 2014; IERS, 2010), Earth tide (Kouba, 2009), carrier-phase windup (Leick, 2004; Wu et al., 1993), Sagnac (Kaplan and Heagarty, 2006), relativity (HofmannWellenhof et al., 2008), and satellite and receiver antenna phase-center variations (Dow et al. 2009). The noise terms are modeled stochastically using an exponential model, as described in Afifi and El-Rabbany (2014). With the above consideration, the GPS/BeiDou ionosphere-free linear combinations of both the pseudorange and carrier phase can be written as: ˜ rG − dt sprec + T G + ϵ PG P G IF = ρ G + dt IF ˜ rG − dt sprec + ISB GB + T B + ϵ B P B IF = ρ B + dt IF Φ G IF

˜ rG − dt sprec + T G + N ˜ G + ϵ ΦG = ρ G + dt IF IF

(11) (12) (13)

˜ rG − dt sprec + T B + N ˜ B IF + ISB GB + ϵ ΦB IF Φ B IF = ρ B + dt (14) ˜ rG represents the sum of the receiver clock error where dt ˜ rG = cdt rG + b r p ; ISB is the and receiver hardware delay dt ˜ G and inter system bias as follows ISB GB = b r B − b r p ; N IF ˜ N B IF are given by: ˜ G = N G + b r Φ + b r p − b sI − b sP N IF IF

(15)

˜ B IF = N B IF + b r BΦ + b r p − b sBI − b sB N

(16)

from both systems plus six parameters, while the number of equations equals double the number of visible satellites. This means that the redundancy equals nG +nB -6. In other words, at least 6 mixed satellites are needed for the solution to exist. In comparison with the GPS-only un-differenced scenario, which requires a minimum of 5 satellites for the solution to exist, the addition of BeiDou satellites increases the redundancy by nB -1. In other words, we need a minimum of two satellites from BeiDou system in order to contribute to the solution.

3 Least Squares Estimation Technique Under the assumption that the observations are uncorrelated and the errors are normally distributed with zero mean, the covariance matrix of the un-differenced observations takes the form of a diagonal matrix. The elements along the diagonal line represent the variances of the code and carrier phase measurements, respectively. In our solution, we consider the ratio between the standard deviation of the code and carrier-phase measurements to be 100. The general linearized form for the above observation equations around the initial (approximate) vector u0 and observables l can be written in a compact form as: f(u, l) ≈ A∆u − w − r ≈ 0

(17)

where u is the vector of unknown parameters; A is the design matrix, which includes the partial derivatives of the observation equations with respect to the unknown parameters u; ∆u is the unknown vector of corrections to the approximate parameters u0 , i.e., u = u0 + ∆u; w is the misclosure vector and r is the vector of residuals. The sequential least-squares solution for the unknown parameters ∆ui at an epoch i can be obtained from (Vanicek and Krakiwsky, 1986):

When using the combined GPS/BeiDou undifferenced PPP model, the ambiguity parameters lose their integer nature, as they are contaminated by the receiver and satellite hardware delays. It should be pointed out that the number of unknown parameters in the combined PPP model equals the number of visible satellites

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Improved Dual Frequency PPP Model Using GPS and BeiDou Observations

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T −1 T −1 2ui = 2ui−1 + M−1 i−1 Ai (Cl i + Ai Mi−1 Ai ) [wi − Ai 2ui−1 ]

(18)

−1 −1 T −1 T −1 −1 M−1 i = Mi−1 − Mi−1 Ai (Cl i + Ai Mi−1 A ) Ai Mi−1

(19)

−1 −1 T −1 T −1 −1 C2u i = M−1 i = Mi−1 − Mi−1 Ai (Cl i + Ai Mi−1 A ) Ai Mi−1

(20)

where ∆ui−1 is the least-squares solution for the estimated parameters at epoch i-1; M is the matrix of the normal equations; C l and C ∆u are the covariance matrices of the observations and unknown parameters, respectively. It should be pointed out that the usual batch least-squares adjustment should be used in the first epoch, i.e., for i = 1. The batch solution for the estimated parameters and the inverse of the normal equation matrix are given, respectively, by (Vanicek and Krakiwsky, 1986): T −1 −1 T −1 2u1 = [C−1 x0 + A1 Cl1 A1 ] A1 Cl1 w1

(21)

−1 T T −1 M−1 1 = [Cx0 + A1 Cl1 A1 ]

(22)

where C0x is a priori covariance matrix for the approximate values of the unknown parameters. The design matrix A and the vector of corrections to the unknown parameters ∆x of the combined un-differenced GPS/BeiDou PPP model take the following forms: )︁ (︁ )︁ (︁ )︁ ⎤ ⎡(︁ y0 −Y G1 z0 −Z 1G x0 −X 1G 1 m1G 0 0 ··· 0 0 ··· 0 1G 1G 1G f ρ ρ ρ ⎥ ⎢(︁ 0 1 )︁ (︁ 0 1 )︁ (︁ 0 1 )︁ ⎥ ⎢ x0 −X G y0 −Y G z0 −Z G 1G 1 m 0 1 · · · 0 0 · · · 0 ⎥ ⎢ ρ1G 1G 1G f ρ0 ρ0 ⎥ ⎢ 0 ⎥ ⎢ . . . . . . . . . . . . ⎢ .. .. .. .. .. .. .. .. .. .. . . .. ⎥ ⎥ ⎢(︁ )︁ (︁ )︁ (︁ )︁ ⎥ ⎢ x0 −X Gn y0 −Y Gn z0 −Z Gn 1nG ⎥ ⎢ 1 m 0 0 · · · 0 0 · · · 0 nG nG nG f ⎥ ⎢(︁ ρ0 )︁ (︁ ρ0 )︁ (︁ ρ0 )︁ n n ⎥ ⎢ x0 −X n y0 −Y G z0 −Z G 1nG G ⎥ ⎢ 1 m 0 0 · · · 1 0 · · · 0 f ⎥ ⎢(︁ ρ0nG )︁ (︁ ρ0nG )︁ (︁ ρ0nG )︁ A = ⎢ x −X1 ⎥ 1 1 y −Y z −Z 1B 0 0 ⎥ ⎢ 0 1B B B B 1 m 1 0 · · · 0 0 · · · 0 1B f ⎥ ⎢(︁ ρ0 )︁ (︁ ρ1B 0 )︁ (︁ ρ0 1 )︁ ⎥ ⎢ x −X1 1 y0 −Y B z0 −Z B 1B ⎢ 0 B 1 mf 1 0 · · · 0 1 · · · 0⎥ 1B 1B ⎥ ⎢ ρ1B ρ ρ 0 0 0 ⎥ ⎢ ⎢ .. .. .. .. .. .. .. .. .. .. . . .. ⎥ ⎥ ⎢ . . . . . . . . . . . . ⎥ ⎢(︁ )︁ (︁ )︁ (︁ )︁ ⎥ ⎢ x0 −X Bn y0 −Y Bn z0 −Z Bn 1nB 1 0 · · · 0 0 · · · 0 1 m ⎥ ⎢ ρ nB f ⎦ ⎣(︁ 0 n )︁ (︁ ρ0nB n )︁ (︁ ρ0nB n )︁ x0 −X B y0 −Y B z0 −Z B 1nB 1 m 1 0 · · · 0 0 · · · 1 1nB 1nB 1nB f ρ ρ ρ 0

0

2x×(n+6)

0

⎡

∆x ∆y ∆z ˜ rG dt

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ zpd w ⎥ ⎥ ⎢ ⎢ISB ⎥ ⎢ GB ⎥ ⎢ ˜1 ⎥ ∆x = ⎢ N ⎥ ⎢ G ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎥ ⎢ nG ⎥ ⎢ N ⎢ ˜G ⎥ ⎢ ˜1 ⎥ ⎢ NB ⎥ ⎥ ⎢ ⎢ .. ⎥ ⎣ . ⎦ ˜ nB N B

(23)

n+6

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6 | A. Afifi and A. El-Rabbany where n G refers to the number of visible GPS satellites; n B refers to the number of visible BeiDou satellites; n = n G +n B is the total number of the observed satellites for both GPS/BeiDou systems; x0 , y0 and z0 are the approximate receiver coordinates; X jG , Y jG , Z jG , j=1, 2, . . . , nG are the known GPS satellite coordinates; X kB , Y kB , Z kB , k=1, 2, . . . , nB are the known BeiDou satellite coordinates; ρ0 is the approximate receiver-satellite range. The unknown parameters in the above system are the corrections to the receiver coordinates, ∆x, ∆y, and ∆z, the wet component of the tropospheric zenith path delay zpdw , the inter-system bias ISBGB , and the non-integer ambiguity parameters N.

4 Results and discussion To verify the developed combined PPP models, GPS and BeiDou observations at four globally distributed stations (Fig. 1) were selected from the IGS tracking network (Dow et al. 2009). These stations are occupied by GNSS receivers, which are capable of simultaneously tracking both GNSS constellations. Only one hour of observations with maximum possible number of BeiDou satellites of each data set is considered in our analysis. All data sets have an interval of 30 seconds.

used to assess the performance of the newly developed PPP model. Figure 2 shows the satellite availability during the observation time window (one hour) for each GNSS system at DLF1 station.

Figure 2: DLF1 station GNSS satellite availability.

As shown in Fig. 2, the GPS system offers eight visible satellites for one hour; however the addition of BeiDou satellites increases the number of visible satellites to 14. Figure 3 shows the positioning results in the East, North, and Up directions, respectively, for the GPS-only PPP model. As can be seen, the un-differenced GPS-only PPP solution indicates that the model is capable of obtaining a sub-decimetre level accuracy. However the solution takes about 20 minutes to converge to decimetre level accuracy.

Figure 1: Analysis stations.

The positioning results for stations DLF1 are presented below. Similar results are obtained for the other stations. However, a summary of the convergence times and the three-dimensional PPP solution standard deviations are presented below for all stations. Natural Resources Canada’s GPSPace PPP software was modified to handle data from the GPS and BeiDou systems, which enables a combined PPP solution as detailed above. In addition to the combined PPP model, we also obtained the solutions of the un-differenced ionosphere-free GPS-only which is

Figure 3: The positioning results of the GPS-only PPP model.

Figure 4 shows the combined GPS/BeiDou PPP model positioning results. As shown in Fig. 4 the convergence time of the combined GPS/BeiDou PPP solution is reduced to 15 minutes to reach the decimeter level of accuracy. Figure 5 summarizes the convergence times for the GPS-only and the combined GPS/BeiDou PPP models at all stations, which confirm the PPP solution consistency.

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Improved Dual Frequency PPP Model Using GPS and BeiDou Observations

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Figure 4: The positioning results of the GPS/BeiDou PPP model.

Figure 5: Summary of PPP solution convergence times at all stations.

To further assess the performance of the two PPP models, the solution output is sampled every 10 minutes and the standard deviation of the computed station coordinates is calculated for each sample. Figure 6 shows the position standard deviations in the East, North, and Up directions, respectively. Examining the standard deviations indicates that the performance of the combined PPP model is slightly better than the GPS PPP model. As the number of epochs, and consequently the number of measurements, increases, the performance of the two models tends to be comparable.

5 Conclusions This paper presented a new PPP model, which combines the GPS and BeiDou observations in the un-differenced mode. The developed combined GPS/BeiDou PPP model accounts for the combined effect of the different GNSS time reference and hardware delays through the introduction of a new unknown parameter, which is identified as the intersystem bias, in the combined PPP mathematical model. It has been shown that the positioning solutions of the GPS-only and the GPS/BeiDou PPP are comparable and are at the sub-decimeter level accuracy. However, the conver-

Figure 6: Summary of positioning standard deviations in East, North, and Up directions of two PPP models

gence time of the combined GPS/BeiDou PPP solution is reduced by about 25% in comparison with the GPS-only PPP. Acknowledgement: This research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors would like to thank the International GNSS service - Multi-GNSS Experiment (IGSMGEX) network.

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8 | A. Afifi and A. El-Rabbany

References Abd Rabbou, M. and A. El-Rabbany (2015). “Precise Point Positioning using Multi-Constellation GNSS Observations for Kinematic Applications.” Journal of Applied Geodesy. Volume 9, Issue 1, Pages 15–26, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: 10.1515/jag-2014-0021. Afifi, A., El-Rabbany, A. (2016). Precise Point Positioning Using Triple GNSS Constellations in Various Modes, Sensors Journal, 16(6), 779; doi:10.3390/s16060779. Afifi, A. and El-Rabbany, A. (2015). An innovative dual frequency PPP model for combined GPS/Galileo observations. Journal of Applied Geodesy, 9(1), 27-34, doi:10.1515/jag-2014-0009. Afifi, A., El-Rabbany, A. (2014). Enhanced model for precise point positioning with single and dual frequency GPS/Galileo observables. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Spatial Inf. Sci., XL-2, 173177, Toronto, Canada BeiDou GNSS (2015). BeiDou navigation satellite system, Retrieved on June 28, 2015 (http://en.beidou.gov.cn/). Boehm, J. and Schuh, H. (2004). Vienna mapping functions in VLBI analyses. Geophysical Research Letters 31(1): L01603 0160101604. doi: 10.1029/2003gl018984. Bos, M.S, and Scherneck, H.-G. (2011). Ocean tide loading provider. Retrieved December 1, 2014, from http://holt.oso.chalmers.se/ loading/. Collins, P., S. Bisnath, F. Lahaye, and P. Héroux (2010). Undifferenced GPS ambiguity resolution using the decoupled clock model and ambiguity datum fixing. NAVIGATION, 57(2), 123-135. Colombo, O.L., A.W. Sutter, and A.G. Evans (2004). Evaluation of precise, kinematic GPS point positioning. ION GNSS 17th International Technical Meeting of the Satellite Division, 21-24 Sept. 2004, Long Beach, CA. Dow, J.M., Neilan, R. E., and Rizos, C. (2009). The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. Journal of Geodesy 83(3-4), 191–198, doi: 10.1007/s00190008-0300-3. El-Mowafy A., Deo M. and C. Rizos (2016). On biases in Precise Point Positioning with multi-constellation and multi-frequency GNSS data. IOP Science, Meas. Sci. Technol., Volume 27, Number 3, 2016 035102 (10pp), http://iopscience.iop.org/0957-0233/27/3. ESA (2015). European Space Agency, Retrieved June 28, 2015, from (http://www.esa.int/ESA). Ge, M., Gendt, G., Rothacher, M., Shi, C., and Liu, J. (2008). Resolution of GPS carrier-phase ambiguities in precise point positioning (PPP) with daily observations. Journal of Geodesy, 82(7), 401. doi: 10.1007/s00190-007-0208-3. Hopfield, H.S. Tropospheric refraction effects on satellite range measurements. APL Tech. Dig. 1972, 11, 11–19. Hofmann-Wellenhof, B., H. Lichtenegger, and E. Wasle (2008). GNSS global navigation satellite systems; GPS, Glonass, Galileo & more. Springer Wien New York. 501pp. IERS (2010). International earth rotation and reference system services conventions (2010), IERS Technical Note 36, available at http://www.iers.org/IERS/EN/Publications/TechnicalNotes/ tn36.html/. Kaplan, E. and C. Heagarty (2006). Understanding GPS principles and applications, Artech House Inc.

Kouba, J. (2009). A guide to using international GNSS service (IGS) products. Available at http://igscb.jpl.nasa.gov/igscb/resource/ pubs/UsingIGSProductsVer21.pdf. Kouba, J. and P. Heroux (2001) Precise point positioning using IGS orbit and clock products. GPS Solutions, 5(2), PP 12-28. Leick, A. (2004). GPS satellite surveying, 3rd ed., John Wiley and Sons. Melgard, T., Tegedor, J., de Jong, K., Lapucha, D., and Lachapelle, G. (2013). Interchangeable integration of GPS and Galileo by using a common system clock in PPP. ION GNSS+, September 16-20, 2013, Institute of Navigation, Nashville TN. Tegedor, J., Lapucha, D., Ørpen, O., Vigen, E., Melgard, T., and Strandli, R. (2015). The New G4 Service: Multi-constellation Precise Point Positioning Including GPS, GLONASS, Galileo and BeiDou, Proceedings of the 28th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2015), Tampa, Florida, September 2015, pp. 1089-1095. Montenbruck O., Steigenberger P., Khachikyan R., Weber G., Langley R.B., Mervart L., and Hugentobler U. (2014). IGS-MGEX: preparing the ground for multi-constellation GNSS science, InsideGNSS 9(1):42-49. Li W., Teunissen P.J.G., Zhang B. and Verhagen J.S. (2013). Precise Point Positioning using GPS and Compass observations. China Satellite Navigation Conf. (CSNC). 2013, Proc. ed. J Sun et al (Berlin: Springer), pp. 367–78. Vanicek, P., and Krakiwsky, E. J. (1986). Geodesy: The concepts. 2nd edition, North-Holland, Amsterdam, The Netherlands. Wang, N., Yuan, Y., Li, Z., and Montenbruck O. (2016). Multi-GNSS differential code biases (DCBs) estimation within MGEX. IGS Workshop 2016, Sydney, Australia. DOI: 10.13140/RG.2.1.3755.6885 Wu, J. T., S, C. Wu, G.A. Hajj, W.I. Bertiger, and S. M. Lichten (1993). Effects of antenna orientation on GPS carrier phase. Man. Geodetica 18, pp. 91-98. Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, and Webb FH (1997). Precise point positioning for the eflcient and robust analysis of GPS data from large networks. J Geophys Res 102(B3):5005– 5017.

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