1 Introduction

The regional constellation of BeiDou Navigation Satellite System (BDS) has been officially operation since the end of 2012. It can provide service of navigation, timing and positioning in the Asia-Pacific Region. At the same time, IGS MGEX analysis centers and iGMAS analysis centers focus on BDS orbit and satellites clock determination [1–3]. The MGEX analysis centers such as WHU, GFZ and ESA can provide BDS precise orbit and satellites clock which lead to more researches focus on BDS precise positioning. The positioning accuracy of BDS PPP can reach a few centimeters [1–4], but more convergence time is needed and the positioning accuracy is less than GPS PPP [1, 2, 4, 5]. The reason is that BDS orbit and clock products have a lower accuracy than GPS products, due to the small network of monitor stations and the accuracy antenna model and solar radiation pressure model are unknown.

To improve performance of a GNSS mainly depends on increasing the number of navigation satellites, optimizing the spatial geometric configuration and improving of the accuracy of the observation signals [5]. At the same time, the fusion of multiple GNSSs can significantly increase the number of observed satellites and optimize the spatial geometry which can shorten convergence time of positioning. Li et al. [3] presented that the combined BDS/GPS PPP which based on un-differenced dual-frequency observations can shorten convergence time but it did not improve positioning results [4]. Zhao et al. [1] presented that when BDS and GPS data were combined to perform static PPP, the results were slightly worse compared to GPS only solutions. This could be caused by the multipath of some BDS GEO satellites and the process noise of intersystem biases for BDS. For kinematic PPP, however, the BDS/GPS combinations significantly improved the accuracy of GPS only PPP solutions. However, the above study focused on daily solutions [1]. The combined GPS/BDS positioning accuracy with short-time observations (such as 30–60 min) is unknown and PPP ambiguities are estimated as float values. Considering the PPP ambiguity resolution (PPP-AR) can improve positioning accuracy of short time.

With the development of PPP-AR in recent years, PPP ambiguity-fixed solution can improve positioning accuracy and shorten PPP convergence time [6–10]. But the convergence period to ambiguity-fixed solution is over several tens of minutes. The reason for this long period is that the slowly change geometry of satellites and other un-modeled error in observations lead to a long period before ambiguity resolution can be attempted [10, 11]. Cai and Gao [12] presented that combined GPS/GLONASS PPP can improve positioning reliable and reduce convergence time [12]. Jokinen et al. [13] and Li et al. [14] showed that combined GPS/GLONASS PPP can improve accuracy of GPS float ambiguity and reduce the time to first-fixed solution [13, 14]. Above studies focused on GPS/GLONASS PPP, however, combined GPS/BDS PPP was little studied. The impact of adding BDS observations to improve GPS PPP ambiguity resolution is worthy of investigation.

This contribution focuses on Integrating BDS and GPS to accelerate convergence and initialization time of PPP. Firstly the PDOP value and visible satellites are analysis between single system and combined BDS/GPS in the Asia-Pacific Region. Secondly, the PPP convergence time of different system combination is investigated and then the positioning accuracy of BDS, GPS and combined BDS/GPS PPP in hourly solutions is compared. Finally, GPS PPP ambiguity resolution was conduct and leaving BDS PPP ambiguities as real values, for the reason that BDS orbit and clock products have a lower accuracy than GPS products and the accuracy antenna model of BDS can’t acquire. The impact of GPS PPP ambiguity resolution for combined BDS/GPS PPP hourly solutions is also investigated

2 Combined BDS/GPS PPP Model

2.1 Function Model

For combined BDS/GPS PPP, in order to eliminate the ionosphere effect, the ionosphere-free observations are used. Considering hardware delays in observations, for a satellite j of any GNSS s (GPS or BDS) observed by receiver r, the ionosphere-free pseudo range and carrier phase observations can be expressed as follows

$$ \begin{aligned} P_{IF,r}^{s,j} & = \text{ }\rho_{r}^{s,j} + c \left(dt_{r}^{s} - dt^{s,j} \right) + M_{r}^{s,j} d_{trop,r} \text{ } + c \left(b_{IFP,r}^{s} - b_{IFP}^{s,j} \right)\quad + \varepsilon \left(P_{IF,r}^{s,j} \right) \\\Phi _{IF,r}^{s,j} & = \rho_{r}^{s,j} + c \left(dt_{r}^{s} - dt^{s,j} \right) + M_{r}^{s,j} d_{trop,r} + c \left(b_{IF\phi ,r}^{s} - b_{IF\phi }^{s,j} \right) + \lambda_{IF}^{s} N_{IF,r}^{s,j} + \varepsilon \left(\Phi _{IF,r}^{s,j} \right) \\ \end{aligned} $$
(6.1)

where \( \Phi_{IF,r}^{s,j} \) and \( P_{IF,r}^{s,j} \) are ionosphere-free carrier phase and pseudo range observations, the subscript IF for ionosphere-free combination, r for station number, superscript s for GNSS, j for a given satellite, ρ s,j r as geometric distance between satellite and receiver, c for the speed of light in vacuum, dt s r as the receiver clock errors, dt s,j as the satellites clock errors, \( b_{IFP,r}^{s} \), \( b_{IF\phi ,r}^{s} \), \( b_{IFP}^{s,j} \) and \( b_{IF\phi }^{s,j} \) as receiver-dependent or satellite-dependent pseudo range and phase hardware delay, \( d_{trop,r} \) for tropospheric zenith delay (ZTD), \( M_{r}^{s,j} \) for the mapping function, \( N_{IF,r}^{s,j} \) as the ionosphere-free ambiguity, \( \varepsilon \left(\Phi _{IF,r}^{s,j} \right) \), \( \varepsilon \left(P_{IF,r}^{s,j} \right) \) for the carrier phase and pseudo range measurement noise and other errors, \( \lambda_{IF}^{s} \) is the wavelength of ionosphere-free combination.

Generally, the satellites clock errors can be corrected using IGS precise clock products \( \tilde{d}t^{s,j} = dt^{s,j} + b_{IFP}^{s,j} \) (which include satellite-dependent hardware delay) and the receiver-dependent hardware delay is grouped into receiver clock, then the Eq. (6.1) can be expressed as

$$ \begin{aligned} P_{IF,r}^{s,j} & = \text{ }\rho_{r}^{s,j} + c\tilde{d}t_{r}^{s} + M_{r}^{s,j} d_{trop,r} \quad + \varepsilon \left(P_{IF,r}^{s,j} \right) \\ \Phi _{IF,r}^{s,j} & = \rho_{r}^{s,j} + c\tilde{d}t_{r}^{s} + M_{r}^{s,j} d_{trop,r} + \lambda_{IF}^{s} B_{IF,r}^{s,j} + \varepsilon \left(\Phi _{IF,r}^{s,j} \right) \\ \end{aligned} $$
(6.2)

where \( \lambda_{IF} B_{IF,r}^{s,j} = c\left[ \left(b_{IF\phi ,r}^{s} - b_{IFP,r}^{s} \right) - \left(b_{IF\phi }^{s,j} - b_{IFP}^{s,j} \right)\right] + \lambda_{IF}^{s} N_{IF,r}^{s,j} \), as float ambiguity which include receiver and satellite-dependent hardware delay, \( \tilde{d}t_{r}^{s} = dt_{r}^{s} + b_{IFP,r}^{s} \) as the comprehensive receiver clock errors and receiver-dependent hardware delay.

It should be noted that BDS precise orbit and satellites clock products provided by MGEX are based on the ITRF reference frame, which is the same as used in single GPS data processing. It doesn’t need to transform coordinate reference frame between two systems. Due to different time system between BDS and GPS, there will have two receiver clock offsets in combined GPS/BDS PPP model, as follows

$$ dt_{r}^{G} = t_{r} - t_{sys}^{G} \quad dt_{r}^{C} = t_{r} - t_{sys}^{C} $$
(6.3)

where \( dt_{r}^{G} \), \( dt_{r}^{C} \) as GPS and BDS receiver clock offset, t r is time of receiver clock, \( t_{sys}^{G} \text{ } \), \( t_{sys}^{C} \) as GPS and BDS system time. Instead of estimating BDS receiver clock offsets, it is preferable to introduce a system time difference parameter as it can reflect the difference between GPS and BDS system times [5, 12]. The BDS receiver clock offset can be expressed as

$$ \begin{aligned} dt_{r}^{C} & = t_{r} - t_{sys}^{G} - t_{sys}^{C} + t_{sys}^{G} \\ & = dt_{r}^{G} + dt_{sys} \text{ } \\ \end{aligned} $$
(6.4)

where \( dt_{sys} = t_{sys}^{G} - t_{sys}^{C} \) denotes system time difference between GPS and BDS. Considering receiver-dependent hardware delay, the BDS receiver clock offset can be expressed as

$$ \begin{aligned} \tilde{d}t_{r}^{C} & = \tilde{d}t_{r}^{G} + \tilde{d}t_{sys} \\ \tilde{d}t_{sys} & = dt_{sys} + b_{IFP,r}^{C} - b_{IFP,r}^{G} \\ \end{aligned} $$
(6.5)

Introduce Eq. (6.5) into Eq. (6.2) and linearize the equation, the Combined BDS/GPS PPP Model can be expressed as

$$ \begin{aligned}\Delta P_{IF,r}^{G,j} & = \text{ }{\varvec{\upmu}}_{r}^{G,j}\Delta {\mathbf{r}} + c\tilde{d}t_{r}^{G} \quad + M_{r}^{G,j} d_{trop,r} \quad + \varepsilon \left(P_{IF,r}^{G,j} \right) \\ {\Delta \varPhi }_{IF,r}^{G,j} & = {\varvec{\upmu}}_{r}^{G,j}\Delta {\mathbf{r}} + c\tilde{d}t_{r}^{G} \quad + M_{r}^{G,j} d_{trop,r} + \lambda_{IF}^{G} B_{IF,r}^{G,j} + \varepsilon \left(\Phi _{IF,r}^{G,j} \right) \\\Delta P_{IF,r}^{C,j} & = \text{ }{\varvec{\upmu}}_{r}^{C,j}\Delta {\mathbf{r}} + c \left(\tilde{d}t_{r}^{G} + \tilde{d}t_{sys} \right) + M_{r}^{C,j} d_{trop,r} \quad + \varepsilon \left(P_{IF,r}^{C,j} \right) \\ {\Delta \varPhi }_{IF,r}^{C,j} & = {\varvec{\upmu}}_{r}^{C,j}\Delta {\mathbf{r}} + c \left(\tilde{d}t_{r}^{G} + \tilde{d}t_{sys} \right) + M_{r}^{C,j} d_{trop,r} + \lambda_{IF}^{C} B_{IF,r}^{C,j} + \varepsilon \left(\Phi _{IF,r}^{C,j} \right) \\ \end{aligned} $$
(6.6)

where \( {\varvec{\upmu}}_{r}^{G,j} \), \( {\varvec{\upmu}}_{r}^{C,j} \) denotes GPS and BDS unit direction vector, \( \Delta {\mathbf{r}} \) denotes the three-dimensional coordinate correction. The parameters to be estimated in the equation are \( {\mathbf{{\rm{X}}}} = [\text{ }\varDelta {\mathbf{r}},\text{ }\tilde{d}t_{r}^{G} ,\text{ }\tilde{d}t_{sys} ,\text{ }d_{trop,r} ,\text{ }B_{IF,r}^{G,j} ,\text{ }B_{IF,r}^{C,j} \text{ }] \). The unknown vector \( {\mathbf{{\rm{X}}}} \) includes three coordinate parameters, a receiver clock offset, a system time difference parameter, a wet zenith tropospheric delay and real-value ambiguity parameters and the Extended Kalman Filter (EKF) can be utilized in parameters estimation.

2.2 Stochastic Model and Parameter Estimation Method

Both the fusion of function model and stochastic model in combined GPS/BDS PPP are important. When BDS and GPS data are combined to perform PPP, the initial weight ratio between GPS and BDS observations are set as 1:4, due to the GPS orbit and clock products have better quality compared with BDS ones. Whereas, the weight ratio between pseudo range and phase observations are set as 1:10,000 and the elevation-dependent weighting is also suggested in single GNSS observations.

Since the EKF is applied for combined PPP parameters estimation, appropriate stochastic models for parameters need to be provided. The parameter of system time difference can be modeled as random walk process [5, 12], due to its stable in short time period. The other parameters can be processed the same way as single system PPP and the error corrections such as windup, solid tide et al. must be considered. A detailed and clear discussion of parameters estimation and error corrections [15, 16] can refer to Kouba and Héroux [16].

2.3 Zero Difference Integer Ambiguity Resolution

In PPP data processing, the ionosphere-free ambiguities are usually estimated as real-values, due to the existence of the receiver and satellite-dependent phase hardware which also calls uncalibrated phase delays (UPD). As we known, double-difference ambiguities can be fixed, because the UPD are canceled. If the UPD can be canceled in PPP ambiguities, the PPP ambiguity fixing can be attempted. In general, the ionosphere-free ambiguities are decomposed into wide-lane (WL) and narrow-lane (NL) ones, as follows

$$ \lambda_{IF} B_{IF,r}^{j} = \frac{{cf_{1} }}{{f_{1}^{2} - f_{2}^{2} }}B_{1,r}^{j} - \frac{{cf_{2} }}{{f_{1}^{2} - f_{2}^{2} }}B_{2,r}^{j} = \frac{{f_{2} }}{{f_{1} + f_{2} }}\lambda_{w} B_{w,r}^{j} + \lambda_{n} B_{1,r}^{j} $$
(6.7)

where

$$ \begin{aligned} B_{w,r}^{j} & = N_{w,r}^{j} + b_{w,r} - b_{w}^{j} \\ B_{1,r}^{j} & = N_{1,r}^{j} + b_{n,r} - b_{n}^{j} \\ \end{aligned} $$
(6.8)

\( B_{w,r}^{j} \) and \( B_{1,r}^{j} \) as wide-lane and narrow-lane ambiguity which include receiver and satellite-dependent UPD. \( N_{w,r}^{j} \) and \( N_{1,r}^{j} \) denote the original WL and NL integer ambiguity. \( b_{w,r} \), \( b_{n,r} \), \( b_{w}^{j} \text{ } \) and \( b_{n}^{j} \) denote receiver and satellite-dependent WL and NL UPD. Form above equation, we can see that the separation of UPD and integer ambiguity is key point in ambiguity fixing.

It is difficult to directly separate UPD and integer ambiguity, due to the linear relationship between UPD and integer ambiguity. However, the UPD have an integer part and a fractional part and the integer part will be grouped into integer ambiguity which doesn’t lose its integer property.

$$ \begin{aligned} B_{w,r}^{j} & = \tilde{N}_{w,r}^{j} + f_{w,r} - f_{w}^{j} \\ B_{1,r}^{j} & = \tilde{N}_{1,r}^{j} + f_{n,r} - f_{n}^{j} \\ \end{aligned} $$
(6.9)

\( \tilde{N}_{w,r}^{j} \) and \( \tilde{N}_{1,r}^{j} \) denotes the sum of integer ambiguity and the integer part of WL and NL UPD, f w,r , f n,r , \( f_{w}^{j} \text{ } \) and f j n denotes fractional part of receiver and satellite-dependent WL and NL UPD, which also call UPD for convenience. If fractional part of UPD can be resolved by server-end and provide to user, then the user can perform integer ambiguity fixing with single receiver.

Generally, the satellite-dependent WL UPD are stable over several days and can be estimated every day and applied to real-time PPP-AR with long update intervals [6, 7]. The fractional part of NL UPD contains not only UPD but also the bias in the estimated ambiguity, which are contaminated by inaccurate modeling of the observations. This results in the fluctuation of the NL UPD. Fortunately, fractional part of NL UPD is rather stable over a certain time span and can be estimated with short-term intervals, such as every 10–15 min. The approach proposed by Li and Zhang [14] can be applied to WL and NL UPD estimation [17] and integer ambiguity resolution in single receiver is presented in following section.

Since WL ambiguities have long wavelength, reaching 0.86 m, WL ambiguities can be easily and firstly fixed. The WL ambiguities can be calculated by taking the time average of the M-W combinations [18, 19] in order to reduce the effect of range noise and multipath.

$$ \begin{aligned} \left\langle {B_{w,r}^{j} } \right\rangle & = \left\langle { \left(\frac{{f_{1} }}{{f_{1} - f_{2} }}\Phi _{1,r}^{j} - \frac{{f_{2} }}{{f_{1} - f_{2} }}\Phi _{2,r}^{j} \right) - \left(\frac{{f_{1} }}{{f_{1} + f_{2} }}P_{1,r}^{j} + \frac{{f_{2} }}{{f_{1} + f_{2} }}P_{2,r}^{j} \right)} \right\rangle \\ & = \left\langle {\tilde{N}_{w,r}^{j} } \right\rangle + f_{w,r} - f_{w}^{j} \text{ } \\ \end{aligned} $$
(6.10)

where 〈*〉 denotes function of taking the time average. After correcting satellite-dependent UPD, the corrected ZD ambiguities should have very similar fractional parts and we take the mean fractional parts of all the corrected ambiguities as receiver UPD. If satellite and receiver-dependent UPD are removed, WL ambiguities can be fixed by rounding to the nearest integer value. In order to ensure accuracy of fixed WL ambiguities, the fixing decision is made according to the probability P 0, which is calculated with the following formula [20] and the minimum probability is set as 0.999.

$$ \begin{aligned} P_{0} & = 1 - \sum\limits_{i = 1}^{\infty } {\left[ {erfc \left(\frac{i - |b - n|}{\sqrt 2 \sigma } \right) - erfc \left(\frac{i + |b - n|}{\sqrt 2 \sigma } \right)} \right]} \\ erfc(x) & = \frac{2}{\sqrt \pi }\int\limits_{x}^{\infty } {e^{{ - t^{2} }} dt} \\ \end{aligned} $$
(6.11)

where, b is real-valued ambiguity, σ is its STD, n is the nearest integer of b. If WL ambiguities are successfully fixed and introduce into ionosphere free ambiguities, then the corresponding NL ambiguities can be obtained by

$$ \begin{aligned} B_{1,r}^{j} & = \frac{{f_{1} + f_{2} }}{{f_{1} }}B_{IF,r}^{j} - \frac{{f_{2} }}{{f_{1} - f_{2} }}\tilde{N}_{w,r}^{j} \\ & = \tilde{N}_{1,r}^{j} + f_{n,r} - f_{n}^{j} \\ \end{aligned} $$
(6.12)

After correcting the satellite and receiver-dependent NL UPD, the integer property of NL ambiguities can be recovered. Due to the correlation between the PPP ambiguities, the LAMBDA method [21] is applied to solve the NL ambiguities. The criterion for the ratio test is set as 2.0.

If NL ambiguities are also successfully fixed, the ionosphere-free ambiguities can be recovered by

$$ \tilde{B}_{IF,r}^{j} = \frac{{f_{2} }}{{f_{1} + f_{2} }} \left(\tilde{N}_{1,r}^{j} { + }f_{n,r} - f_{n}^{j} \right) + \frac{{f_{1} f_{2} }}{{f_{1}^{2} - f_{2}^{2} }}\tilde{N}_{w,r}^{j} $$
(6.13)

Then the ambiguity-fixed solutions can be obtained by highly weighting the \( \tilde{B}_{IF,r}^{j} \) in EKF. From Eq. (6.13), we find that the NL UPD is directly contributing to the ambiguity-fixed solutions. Hence, the accuracy of NL UPD is important to the positioning accuracy.

3 Experiments and Results Analysis

3.1 Data Collection

The experiments use observations of multi-GNSSs from 8 MGEX stations in March 11, 2014, DOY 70 and the data sampling rate is 30 s. The station distribution is showed in Fig. 6.1. The corresponding precise orbit and satellites clock products and the ‘ground truth’ are provided by GFZ, one of MGEX analysis centers. During data processing, only the dual-frequency GPS/BDS observations are used. At the same time, dual-frequency GPS observations of global distribution IGS stations are applied to estimate WL and NL UPD which proposed by Li and Zhang [17].

Fig. 6.1
figure 1

Station distribution

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The PDOP value and visible satellites of GPS and combined GPS/BDS on 8 MGEX stations is analyzed. In Fig. 6.2, the left figure shows the average PDOP value of each station. The right figure shows the average visible satellites of each station. The elevation angle cutoff is set as 10°.

Fig. 6.2
figure 2

The PDOP value and observed satellites of MGEX stations

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From Fig. 6.2, we can find that when the elevation angle cutoff is set as 10°, the average PDOP value is 2.03 and the number of average visible satellites is 9 for single GPS constellation. It also can be seen that the average PDOP value reduces to 1.41 and the number of average visible satellites increases to 16 when combined GPS/BDS constellations. The improvement rate of visible satellites and PDOP is 89 and 31 %. The above analysis shows combined BDS/GPS constellations can increase visible satellites and reduce PDOP.

3.2 Comparison of Convergence Time of Different Combined System PPP

In order to assess the performance of the combined GPS/BDS PPP, the convergence time, the positioning accuracy of hourly solutions and daily solutions are compared among BDS, GPS and combined GPS/BDS. When BDS and GPS data are combined to perform PPP, the initial weight ratio between GPS and BDS observations are set as 1:4.

Figures 6.3 and 6.4 show the convergence time series of BDS, GPS and combined GPS/BDS PPP on station GMSD and NNOR. The positioning results are compared with the ‘ground truth’ which provide by GFZ network solutions. The large fluctuations of time series are caused by simulated data breaks every 2 h on all observed satellites.

Fig. 6.3
figure 3

Convergence time series of PPP with different system on GMSD station

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Fig. 6.4
figure 4

Convergence time series of PPP with different system on NNOR station

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As we can see from Figs. 6.3 and 6.4, the combined GPS/BDS PPP can accelerate convergence compares with GPS PPP and obviously faster than BDS PPP. It also can see that the convergence time of BDS PPP is longer than GPS PPP solutions. The main reason for this phenomenon is that the GPS orbit and satellites clock products have better quality compared with BDS ones.

Figure 6.5 shows the accuracy of PPP hourly solutions and daily solutions of 8 MGEX stations can also indicate this phenomenon. As can be seen from left figure, the positioning accuracy of GPS PPP and combined GPS/BDS PPP hourly solutions are generally better than 10 cm in 3D component and the accuracy of GPS/BDS PPP is better than GPS PPP. It also can be seen, the 3D positioning biases of BDS PPP hourly solutions are reaching a few decimeters. Therefore, the convergence time of BDS PPP is longer than combined GPS/BDS ones. As shown in right figure, however, the daily solutions of BDS PPP are comparative to GPS and combined GPS/BDS PPP. The positioning accuracy of BDS PPP daily solutions is within a few centimeters and slightly worse than GPS one. Whereas, the positioning accuracy of daily solutions of combined GPS/BDS PPP and GPS PPP is at the same level.

Fig. 6.5
figure 5

PPP hourly solutions (left) and daily solutions (right) of 8 MGEX stations with different system

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In order to assess the performance of convergence and positioning accuracy of combined GPS/BDS PPP, Fig. 6.6 shows the positioning results of PPP in different period of daily data set on station CUT0, JFNG and MAL2. It can be seen, when the data set is less than 2 h, the performance of convergence and positioning accuracy of combined GPS/BDS PPP are better than single system PPP solutions. When the data set is more than 2 h, the positioning accuracy of combined GPS/BDS PPP is at the same level and slightly better than BDS PPP solutions.

Fig. 6.6
figure 6

The positioning results of PPP in different period of daily data set on station CUT0, JFNG and MAL2

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3.3 Comparison Between PPP Ambiguity-Fixed Solutions and Ambiguity-Float Solutions

The fusion of GPS and BDS to PPP can significantly increase the number of observed satellites, optimize the spatial geometry which can shorten convergence time and improve positioning accuracy with short-time observations. However, the improved performance of positioning accuracy of combined GPS/BDS PPP is not obvious. In order to further improve positioning accuracy with short-time observations, GPS PPP ambiguity fixing is attempted, leaving BDS PPP ambiguities as real values, for the reason that BDS orbit and clock products have a lower accuracy than GPS products and the accuracy antenna model of BDS can’t acquire. When attempting PPP ambiguity fixing, the WL ambiguity can be calculated by taking the time average of the M-W combinations and fixed to the nearest integer value. WL ambiguity fixing decision is made if the probability P 0 is large than 0.999. The NL ambiguity fixing is attempted by the LAMBDA method and the criterion for the ratio test is set as 2.0. The procedure of PPP ambiguity fixing can reference to Sect. 1.2.3.

Figure 6.7 shows the positioning accuracy of GPS/BDS PPP ambiguities fixed and float solutions in three component and 3D component with hourly observations.

Fig. 6.7
figure 7

The hourly solutions of combined GPS/BDS PPP with ambiguities fixed or as float values

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As can be seen from left sub-figure, when PPP ambiguity is fixed, the positioning accuracy in NEU component is improved in different degree. It can be seen more clearly in right sub-figure that the positioning accuracy is within 5 cm of combined GPS/BDS PPP with GPS ambiguity fixing and the performance is better than GPS/BDS ambiguities float solutions.

Table 6.1 statistics the positioning RMS of ambiguities fixed and floats solutions in three component and 3D component with hourly observations and its improvement rate.

Table 6.1 The RMS of PPP ambiguities fixed and floats solutions with hourly observations and its improvement rate
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The result shows, the RMS of ambiguities fixed solutions are better than 1 cm in horizontal components and 3 cm in the vertical. Comparing with float solutions, the positioning accuracy is improved by 59.1 % in N component, 87.0 % in E component and 39.1 % in U component. The improvement in E component is obvious.

4 Conclusion

Due to BDS orbit and satellites clock products have a lower accuracy than GPS products, BDS PPP need long time to convergence to centimeter-level and the positioning accuracy is less than GPS PPP solutions. Combined BDS/GPS PPP can benefit from more visible satellites and enhanced satellite geometry distribution which would accelerate the convergence speed of PPP. This contribution focuses on integrating BDS and GPS to accelerate convergence and initialization time of PPP. The experiment results with data set of 8 MGEX stations show

  1. (1)

    When the elevation angle cutoff is set as10°, the average PDOP value is 2.03 and the number of average visible satellites is 9 for GPS only. However, the average PDOP value reduces to 1.41 and the number of average visible satellites increases to 16 when combined GPS/BDS constellations. The improvement rate of visible satellites and PDOP is 89 % and 31 %. Combined BDS/GPS constellations can increase visible satellites and reduce PDOP.

  2. (2)

    Combined GPS/BDS PPP can accelerate convergence compares with GPS PPP and obviously faster than BDS PPP. For PPP hourly solutions, the positioning accuracy of GPS PPP and combined GPS/BDS PPP hourly solutions are generally better than 10 cm in 3D component and BDS PPP solutions can only reach a few decimeters. For PPP daily solutions, the positioning accuracy of BDS PPP are within a few centimeters and slightly worse than GPS PPP solutions. The positioning accuracy of combined GPS/BDS PPP and GPS PPP is at the same level.

  3. (3)

    Further improvement of positioning accuracy with short-time observations can obtain when GPS PPP ambiguities are fixed. The RMS of ambiguities fixed solutions is better than 1 cm in horizontal components and 3 cm in the vertical. Comparing with RMS of float solutions, the positioning accuracy is improved by 59.1 % in N component, 87.0 % in E component and 39.1 % in U component. The improvement in E component is obvious.

In summary, the fusion of GPS and BDS to PPP can significantly increase the number of observed satellites, optimize the spatial geometry and shorten convergence time of positioning. At the same time, GPS PPP ambiguity fixing can further improve positioning accuracy with short-time observations.