Introduction

Precise point position (PPP) (Malys and Jensen 1990; Zumberge et al. 1997) has demonstrated its usefulness and power in many scientific research and civilian applications (Tim and Wright 2012; Guo et al. 2018; Liu et al. 2020). However, the traditional float PPP generally requires a convergence time of 30 min to achieve centimeter-level positioning accuracy, which is unacceptable for some time-sensitive applications. In order to shorten the convergence time and improve the positioning accuracy, PPP ambiguity resolution (PPP-AR) has been developed in recent years (Ge et al. 2008; Collins et al. 2008; Laurichesse et al. 2009). The results show that GPS PPP-AR can improve the 3D position accuracy by 54% and shorten the convergence time to about 20 min (Geng et al. 2012; Li et al. 2013).

Once the major systems, i.e., GPS, GLONASS, Galileo and BDS are fully deployed, more than 120 navigation satellites can be used for high precision positioning. The International GNSS Service (IGS) has initiated the multi-GNSS Experiment (MGEX) and built a global tracking network with more than 299 stations to collect observation data of these four systems (Rizos et al. 2013; Montenbruck et al. 2017). Several MGEX analysis centers have carried out multi-GNSS precision satellite orbit determination and clock estimation with those observation data. The emergence of these precision products makes multi-GNSS precision point positioning possible. Related studies have demonstrated that the PPP accuracy and convergence time can be significantly improved with the increasing number of satellites (Li et al. 2015; Guo et al. 2016). In addition, the time to first fix (TTFF) of PPP-AR can also be shortened to about 10 min with multi-GNSS observations (Li et al. 2018).

Although the TTFF of PPP-AR can be significantly shortened by applying multi-constellation data, it is still too long for some real-time applications. Since the atmospheric delays greatly influence the initialization time, the concept of regional augmentation PPP (RA-PPP) was proposed to shorten further the convergence time (Li et al. 2011). RA-PPP implements rapid ambiguity resolution by providing the atmospheric corrections generated from regional network stations to shorten the time of ambiguity resolution to several epochs according to the previous studies (Zhang et al. 2011; Li et al. 2014; Nadarajah et al. 2018; Li et al. 2020). Currently, RA-PPP can have the equivalent performance to the network real-time kinematic (NRTK) and become an important method to achieve high-accuracy positioning.

We develop a multi-GNSS PPP method augmented by precise atmospheric corrections to achieve instantaneous ambiguity resolution. The performance in terms of atmospheric corrections accuracy, instantaneous ambiguity resolution, as well as positioning accuracy, is evaluated with multi-GNSS observations from the Australian Regional GPS Network, South Pacific Regional GNSS Network and the Hong Kong CORS.

We introduce first the methods for deriving precise atmospheric corrections, interpolating corrections and achieving instantaneous ambiguity resolution (IAR). Then, the detailed processing strategies of the experiment are introduced. In the following, the results and analysis of multi-constellation RA-PPP are given and the conclusions are provided.

Method

In order to derive precise atmospheric corrections, the PPP-AR needs to be achieved on all stations of a regional reference network. The multi-GNSS uncalibrated phase delays (UPDs) estimation of this strategy employs the approach proposed by Li et al. (2018). After the ambiguities are fixed, the method of extracting and interpolating precise atmospheric corrections is explained. Finally, the algorithms of instantaneous ambiguity resolution at the users are presented.

Extracting precise atmospheric correction

For multi-GNSS data, the equation of pseudorange and phase observation can be written as following:

$$P_{r,j}^{i,S} = \rho_{r}^{i,S} + c \cdot t_{r}^{S} - c \cdot t^{i,S} + I_{r,j}^{i,S} + T_{d,r}^{i,S} + T_{w,r}^{i,S} + \left( {d_{r,j}^{S} - d_{j}^{i,S} } \right) + e_{r,j}^{i,S}$$
(1)
$$L_{r,j}^{i,S} = \rho_{r}^{i,S} + c \cdot t_{r}^{S} - c \cdot t^{i,S} - I_{r,j}^{i,S} + T_{d,r}^{i,S} + T_{w,r}^{i,S} + \lambda_{j}^{S} \cdot N_{r,j}^{i,S} + \lambda_{j}^{S} \cdot \left( {b_{r,j}^{S} - b_{j}^{i,Ss} } \right) + \varepsilon_{r,j}^{i,S}$$
(2)

where the subscripts \(i\), j, r refer to satellite, frequency and receiver, respectively; s represents the constellation type (GPS, GLONASS, Galileo and BDS); \(P_{r,j}^{i,S}\) and \(L_{r,j}^{i,S}\) are the pseudorange and phase observations at frequency j, respectively; \(\rho_{r}^{i,S}\) denotes the geometric distance from the satellite to receiver; \(c\) is the speed of light in vacuum; \(t_{r}^{S}\) is the clock bias of receiver and \(t^{i,S}\) is that of satellite; \(I_{r,j}^{i,S}\) represents the ionospheric delays at different frequencies; \(T_{d,r}^{i,S}\) refers to the dry component of slant tropospheric delay while \(T_{w,r}^{i,S}\) is the remaining wet delay; \(\lambda_{j}^{S}\) is the wavelength of frequency \(j\); \(N_{r,j}^{i,S}\) refers to the integer ambiguity; \(d_{r,j}^{S}\) and \(d_{j}^{i,S}\) are the pseudorange code biases of receiver and satellites while \(b_{r,j}^{S}\) and \(b_{j}^{i,S}\) are the carrier phase delays, respectively; \(e_{r,j}^{i,S}\) and \(\varepsilon_{r,j}^{i,S}\) denote the sum of measurement noise and multipath error for the code and carrier phase observations. Other corrections such as the phase center offsets (PCO) and variations (PCV), phase windup, relativistic delays, the earth tide as well as the ocean tide loading can be precisely applied according to existing models (Wu et al. 1993; Zumberge et al. 1997; Schaer et al. 1999; Dach et al. 2006; Kouba 2001).

The ionospheric-free combination is widely used to estimate the PPP parameters in order to eliminate the first-order ionospheric delays of the code and carrier phase observations. The pseudorange (\(P_{r,IF}^{i,S}\)) and phase (\(L_{r,IF}^{i,S}\)) IF observations can be expressed based on the dual-frequency raw observations, as

$$P_{r,IF}^{i,S} = \rho_{r}^{i,S} + c \cdot t_{r}^{S} - c \cdot t^{i,S} + T_{d,r}^{i,S} + T_{w,r}^{i,S} + \left( {d_{r,IF}^{S} - d_{IF}^{i,S} } \right) + e_{r,IF}^{i,S}$$
(3)
$$L_{r,IF}^{i,S} = \rho_{r}^{i,S} + c \cdot t_{r}^{S} - c \cdot t^{i,S} + T_{d,r}^{i,S} + T_{w,r}^{i,S} + \lambda_{IF}^{S} \cdot N_{r,IF}^{i,S} + \lambda_{IF}^{S} \cdot \left( {b_{r,IF}^{S} - b_{IF}^{i,S} } \right) + \varepsilon_{r,IF}^{i,S}$$
(4)

where \(\lambda_{IF}^{S}\) represents the wavelength of IF carrier phase; \(N_{r,IF}^{i,S}\) denotes the IF phase ambiguity; \(d_{r,IF}^{S}\) and \(d_{IF}^{i,S}\) are the IF code hardware delays while \(b_{r,IF}^{S}\) and \(b_{IF}^{i,S}\) are the phase hardware delays of receiver and satellite, respectively. The IF code hardware delays are different for each system in one receiver; the differences are called inter-system biases (ISB) (Khodabandeh et al. 2016). An individual receiver clock parameter is set for each system to eliminate the code ISB. In addition, the phase hardware delays are also different, and their differences can be absorbed by IF phase ambiguities.

For PPP-AR, the float IF ambiguity is usually expressed as the combination of the wide-lane (WL) and narrow-lane (NL) ambiguity:

$$\left\{ \begin{aligned} N_{{r,IF}}^{{i,G}} = \frac{{f_{1}^{{i,G}} f_{2}^{{i,G}} }}{{f_{1}^{{i,G^{2} }} - f_{2}^{{i,G^{2} }} }}B_{{r,wl}}^{{i,G}} + \frac{{f_{1}^{{i,G}} }}{{f_{1}^{{i,G}} + f_{2}^{{i,G}} }}B_{{r,nl}}^{{i,G}} \hfill \\ N_{{r,IF}}^{{i,R}} = \frac{{f_{1}^{{i,R}} f_{2}^{{i,R}} }}{{f_{1}^{{i,R^{2} }} - f_{2}^{{i,R^{2} }} }}B_{{r,wl}}^{{i,R}} + \frac{{f_{1}^{{i,R}} }}{{f_{1}^{{i,R}} + f_{2}^{{i,R}} }}B_{{r,nl}}^{{i,R}} \hfill \\ N_{{r,IF}}^{{i,E}} = \frac{{f_{1}^{{i,E}} f_{2}^{{i,E}} }}{{f_{1}^{{i,E^{2} }} - f_{2}^{{i,E^{2} }} }}B_{{r,wl}}^{{i,E}} + \frac{{f_{1}^{{i,E}} }}{{f_{1}^{{i,E}} + f_{2}^{{i,E}} }}B_{{r,nl}}^{{i,E}} \hfill \\ N_{{r,IF}}^{{i,C}} = \frac{{f_{1}^{{i,C}} f_{2}^{{i,C}} }}{{f_{1}^{{i,C^{2} }} - f_{2}^{{i,C^{2} }} }}B_{{r,wl}}^{{i,R}} + \frac{{f_{1}^{{i,C}} }}{{f_{1}^{{i,C}} + f_{2}^{{i,C}} }}B_{{r,nl}}^{{i,C}} \hfill \\ \end{aligned} \right.$$
(5)

where \({B}_{r, wl}^{i}\) and \({B}_{r, nl}^{i}\) refer to float multi-constellation WL and NL ambiguities, respectively. The float WL ambiguity can be derived with the Hatch-Melbourne-Wübbena (HMW) (Hatch 1982; Melbourne 1985; Wübbena 1985) combination as:

$$\left\{ \begin{gathered} B_{{r,wl}}^{{i,G}} = \left( {\frac{{L_{{r,1}}^{{i,G}} }}{{\lambda _{1}^{{i,S}} }} - \frac{{L_{{r,2}}^{{i,G}} }}{{\lambda _{2}^{{i,S}} }} - \frac{{f_{1}^{{i,G}} P_{{r,1}}^{{i,G}} + f_{2}^{{i,G}} P_{{r,2}}^{{i,G}} }}{{\left( {f_{1}^{{i,G}} + f_{2}^{{i,G}} } \right)\lambda _{{wl}}^{{i,G}} }}} \right) = N_{{r,wl}}^{{i,G}} + b_{{r,wl}}^{G} - b_{{wl}}^{{i,G}} \hfill \\ B_{{r,wl}}^{{i,R}} = \left( {\frac{{L_{{r,1}}^{{i,R}} }}{{\lambda _{1}^{{i,R}} }} - \frac{{L_{{r,2}}^{{i,R}} }}{{\lambda _{2}^{{i,R}} }} - \frac{{f_{1}^{{i,R}} P_{{r,1}}^{{i,R}} + f_{2}^{{i,R}} P_{{r,2}}^{{i,R}} }}{{\left( {f_{1}^{{i,R}} + f_{2}^{{i,R}} } \right)\lambda _{{wl}}^{{i,R}} }}} \right) = N_{{r,wl}}^{{i,R}} + b_{{r,wl}}^{R} - b_{{wl}}^{{i,R}} \hfill \\ B_{{r,wl}}^{{i,E}} = \left( {\frac{{L_{{r,1}}^{{i,E}} }}{{\lambda _{1}^{{i,S}} }} - \frac{{L_{{r,2}}^{{i,E}} }}{{\lambda _{2}^{{i,S}} }} - \frac{{f_{1}^{{i,E}} P_{{r,1}}^{{i,E}} + f_{2}^{{i,E}} P_{{r,2}}^{{i,E}} }}{{\left( {f_{1}^{{i,E}} + f_{2}^{{i,E}} } \right)\lambda _{{wl}}^{{i,E}} }}} \right) = N_{{r,wl}}^{{i,E}} + b_{{r,wl}}^{E} - b_{{wl}}^{{i,E}} \hfill \\ B_{{r,wl}}^{{i,C}} = \left( {\frac{{L_{{r,1}}^{{i,C}} }}{{\lambda _{1}^{{i,C}} }} - \frac{{L_{{r,2}}^{{i,C}} }}{{\lambda _{2}^{{i,C}} }} - \frac{{f_{1}^{{i,C}} P_{{r,1}}^{{i,C}} + f_{2}^{{i,C}} P_{{r,2}}^{{i,C}} }}{{\left( {f_{1}^{{i,C}} + f_{2}^{{i,C}} } \right)\lambda _{{wl}}^{{i,C}} }}} \right) = N_{{r,wl}}^{{i,C}} + b_{{r,wl}}^{C} - b_{{wl}}^{{i,C}} \hfill \\ \end{gathered} \right.$$
(6)

where \({N}_{r, wl}^{i}\) represents integer WL ambiguity with the corresponding wavelength \({\lambda }_{wl}^{i}\), \({b}_{r, wl}\) and \({b}_{wl}^{i}\) denote receiver and satellite WL UPDs of each satellite system, respectively. The satellite WL UPDs can be corrected with UPD products. However, the WL ambiguities are still not close to the integer because of the receiver UPDs. In order to separate the receiver UPDs from the WL ambiguities, the WL ambiguity with the highest elevation is selected as a reference and fixed to the nearest integer for each system, respectively. The float WL ambiguity can be fixed with a round strategy by providing satellite WL UPDs (Li et al. 2018). After the WL ambiguities are fixed, the float NL ambiguities can be derived from IF ambiguities according to (5) as:

$$\left\{\begin{array}{c}{B}_{r,nl}^{i,G}=\frac{{f}_{1}^{i,G}+{f}_{2}^{i,G}}{{f}_{1}^{i,G}}{N}_{r,IF}^{i,G}-\frac{{f}_{2}^{i,G}}{{f}_{1}^{i,G}-{f}_{2}^{i,G}}{N}_{r,wl}^{i,G}={N}_{r,nl}^{i,G}+{b}_{r,nl}^{G}-{b}_{nl}^{i,G}\\ {B}_{r,nl}^{i,R}=\frac{{f}_{1}^{i,R}+{f}_{2}^{i,R}}{{f}_{1}^{i,R}}{N}_{r,IF}^{i,R}-\frac{{f}_{2}^{i,R}}{{f}_{1}^{i,R}-{f}_{2}^{i,R}}{N}_{r,wl}^{i,R}={N}_{r,nl}^{i,R}+{b}_{r,nl}^{R}-{b}_{nl}^{i,R}\\ \begin{array}{c}{B}_{r,nl}^{i,E}=\frac{{f}_{1}^{i,E}+{f}_{2}^{i,E}}{{f}_{1}^{i,E}}{N}_{r,IF}^{i,E}-\frac{{f}_{2}^{i,E}}{{f}_{1}^{i,E}-{f}_{2}^{i,E}}{N}_{r,wl}^{i,E}={N}_{r,nl}^{i,E}+{b}_{r,nl}^{E}-{b}_{nl}^{i,E}\\ {B}_{r,nl}^{i,C}=\frac{{f}_{1}^{i,C}+{f}_{2}^{i,C}}{{f}_{1}^{i,C}}{N}_{r,IF}^{i,C}-\frac{{f}_{2}^{i,C}}{{f}_{1}^{i,C}-{f}_{2}^{i,C}}{N}_{r,wl}^{i,C}={N}_{r,nl}^{i,C}+{b}_{r,nl}^{C}-{b}_{nl}^{i,C}\end{array}\end{array}\right.$$
(7)

where \(N_{r, nl}^{i}\) represents integer NL ambiguity, \(d_{r, nl}\) and \(d_{nl}^{i}\) denote receiver and satellite NL UPDs of GPS, GLONASS, Galileo and BDS system, respectively. According to the method of a previous study, the NL ambiguities can also be fixed with satellite NL UPD products.

With precise satellite orbits, clocks and UPDs, one can achieve PPP-AR on all reference stations with fixed station coordinates, and the IF ambiguities can be recovered by the fixed WL and NL ambiguities. Then, the slant tropospheric wet delay of each satellite can be derived from the IF phase observations as:

$$\left\{ {\begin{array}{*{20}c} {T_{{w,r}}^{{i,G}} = L_{{r,IF}}^{{i,G}} - \left( {\rho _{r}^{{i,G}} + c \cdot t_{r}^{G} - c \cdot t^{{i,G}} + T_{{d,r}}^{{i,G}} + \lambda _{{IF}}^{G} \cdot N_{{r,IF}}^{{i,G}} + \lambda _{{IF}}^{G} \cdot \left( {b_{{r,IF}}^{G} - b_{{IF}}^{{i,G}} } \right)} \right)} \\ {T_{{w,r}}^{{i,R}} = L_{{r,IF}}^{{i,R}} - \left( {\rho _{r}^{{i,R}} + c \cdot t_{r}^{R} - c \cdot t^{{i,R}} + T_{{d,r}}^{{i,R}} + \lambda _{{IF}}^{R} \cdot N_{{r,IF}}^{{i,R}} + \lambda _{{IF}}^{R} \cdot \left( {b_{{r,IF}}^{R} - b_{{IF}}^{{i,R}} } \right)} \right)} \\ {\begin{array}{*{20}c} {T_{{w,r}}^{{i,E}} = L_{{r,IF}}^{{i,E}} - \left( {\rho _{r}^{{i,E}} + c \cdot t_{r}^{E} - c \cdot t^{{i,E}} + T_{{d,r}}^{{i,E}} + \lambda _{{IF}}^{E} \cdot N_{{r,IF}}^{{i,E}} + \lambda _{{IF}}^{E} \cdot \left( {b_{{r,IF}}^{E} - b_{{IF}}^{{i,E}} } \right)} \right)} \\ {T_{{w,r}}^{{i,C}} = L_{{r,IF}}^{{i,C}} - \left( {\rho _{r}^{{i,C}} + c \cdot t_{r}^{C} - c \cdot t^{{i,C}} + T_{{d,r}}^{{i,C}} + \lambda _{{IF}}^{C} \cdot N_{{r,IF}}^{{i,C}} + \lambda _{{IF}}^{C} \cdot \left( {b_{{r,IF}}^{C} - b_{{IF}}^{{i,C}} } \right)} \right)} \\ \end{array} } \\ \end{array} } \right.$$
(8)

In (2), the receiver clocks are estimated epoch-wise, the L1 and L2 integer ambiguities can be derived from WL and NL ambiguities, and the L1 and L2 UPDs can also be obtained with WL and NL UPDs. After the slant tropospheric wet delay is obtained, all the parameters are accurately known except for the slant ionospheric delay. Thus, the ionospheric corrections of each satellite can be straightforwardly derived as follows:

$$\left\{ {\begin{array}{*{20}c} {I_{{r,j}}^{{i,G}} = \rho _{r}^{{i,G}} + c \cdot t_{r}^{G} - c \cdot t^{{i,G}} + T_{{d,r}}^{{i,G}} + T_{{w,r}}^{{i,G}} + \lambda _{j}^{G} \cdot N_{{r,j}}^{{i,G}} + \lambda _{j}^{G} \cdot \left( {b_{{r,j}}^{G} - b_{j}^{{i,G}} } \right) - L_{{r,j}}^{{i,G}} } \\ {I_{{r,j}}^{{i,R}} = \rho _{r}^{{i,R}} + c \cdot t_{r}^{R} - c \cdot t^{{i,R}} + T_{{d,r}}^{{i,R}} + T_{{w,r}}^{{i,R}} + \lambda _{j}^{R} \cdot N_{{r,j}}^{{i,R}} + \lambda _{j}^{R} \cdot \left( {b_{{r,j}}^{R} - b_{j}^{{i,R}} } \right) - L_{{r,j}}^{{i,R}} } \\ {\begin{array}{*{20}c} {I_{{r,j}}^{{i,E}} = \rho _{r}^{{i,E}} + c \cdot t_{r}^{E} - c \cdot t^{{i,E}} + T_{{d,r}}^{{i,E}} + T_{{w,r}}^{{i,E}} + \lambda _{j}^{E} \cdot N_{{r,j}}^{{i,E}} + \lambda _{j}^{E} \cdot \left( {b_{{r,j}}^{E} - b_{j}^{{i,E}} } \right) - L_{{r,j}}^{{i,E}} } \\ {I_{{r,j}}^{{i,C}} = \rho _{r}^{{i,C}} + c \cdot t_{r}^{C} - c \cdot t^{{i,C}} + T_{{d,r}}^{{i,C}} + T_{{w,r}}^{{i,C}} + \lambda _{j}^{C} \cdot N_{{r,j}}^{{i,C}} + \lambda _{j}^{C} \cdot \left( {b_{{r,j}}^{C} - b_{j}^{{i,C}} } \right) - L_{{r,j}}^{{i,C}} } \\ \end{array} } \\ \end{array} } \right.$$
(9)

The above method ensures that the atmospheric delays can be derived at the undifferenced level so that their physical characteristics can be preserved. Different from the method used NRTK systems, which combines all errors into the generated virtual observations, the undifferenced corrections can be broadcasted separately from orbit and clock products. Since the atmospheric delays are stable over a short period of time, the broadcast interval of undifferenced atmospheric products can be extended to reduce the communication burden compared to the NRTK model. This method also has a less calculation load on the server since the atmospheric delays can be generated station by station, especially when the number of reference network stations is large. In addition, this method eliminates the need for users to establish communication with the main reference station. The corrections can be broadcasted through satellites or internet, which removes restrictions on the number of users and the reference network scale.

Interpolation of atmospheric corrections

To this end, the precise atmospheric corrections can be generated epoch-wise and provided to the user. We will now introduce the method of interpolating atmospheric corrections. Over the past few years, several methods have been developed to represent the distance-dependent biases. The typical methods include linear interpolation method (LIM) (Wanninger 1995; Wubbena et al. 1996), lower-order surface model (LSM) (Fotopoulos 2000), distance-based linear interpolation (DIM) (Gao et al. 1997), linear combination method (LCM) (Han 1997) and least squares collocation method (LSCM) (Raquet, 1997; Van der Marel, 1998). The performance of different interpolation methods has been assessed and the results have shown that LIM gave better horizontal accuracy (Dai et al. 2003; Al-Shaery et al. 2011). However, LIM is not suitable for interpolation of undifferenced (UD) corrections, because they contain some biases which are not influenced by positions. Therefore, it is necessary to add a constant parameter in the model to sensitize the function to spatially unrelated biases (Wang et al. 2020). Except for LIM, the performances of LSCM and LCM are comparable. The accuracy of LSCM depends upon the accuracy of the covariance matrix, which makes it difficult to apply in practice since it is very difficult to calculate precise covariance matrices. Thus, we choose LCM to interpolate the atmospheric corrections. But it should be modified to adapt the undifferenced model of PPP (Li et al. 2011). The modified linear combination method (MLCM) can be expressed as:

$$\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}1\\ {X}_{u}-{X}_{1}\\ {Y}_{u}-{Y}_{1}\end{array}& \begin{array}{c}1\\ {X}_{u}-{X}_{2}\\ {Y}_{u}-{Y}_{2}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\cdots \\ \cdots \\ \cdots \end{array}& \begin{array}{c}1\\ {X}_{u}-{X}_{n}\\ {Y}_{u}-{Y}_{n}\end{array}\end{array}\end{array}\right]\bullet \left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ \begin{array}{c}\vdots \\ {a}_{n}\end{array}\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$$
(10)
$$\sum_{i=1}^{n}{a}_{i}=1$$
(11)
$$\sum_{i=1}^{n}{\alpha }_{i}^{2}=min$$
(12)

where the subscript \(u\) represents the user stations and \(n\) represents the number of reference stations; \(X\) and \(Y\) are the station coordinates in the local horizontal plane system; \({a}_{i}\) refers to the interpolation coefficient. Given approximate coordinates of the user station and precise coordinates of reference stations, the interpolated atmospheric corrections can be obtained.

Instantaneous ambiguity resolution

Given the interpolated precise atmospheric corrections, the dual-frequency raw observations at user stations can be corrected and used for IAR. In this contribution, WL and IF observations are applied for ambiguity resolution. First, the WL phase observations \({L}_{r, WL}^{i}\)are formed with dual-frequency phase observations to resolve WL ambiguities:

$$\left\{ {\begin{array}{*{20}c} {L_{{r,WL}}^{{i,G}} = \frac{{f_{1}^{{i,G}} }}{{f_{1}^{{i,G}} - f_{2}^{{i,G}} }} \cdot L_{{r,1}}^{{i,G}} - \frac{{f_{2}^{{i,G}} }}{{f_{1}^{{i,G}} - f_{2}^{{i,G}} }} \cdot L_{{r,1}}^{{i,G}} } \\ {L_{{r,WL}}^{{i,R}} = \frac{{f_{1}^{{i,R}} }}{{f_{1}^{{i,G}} - f_{2}^{{i,G}} }} \cdot L_{{r,1}}^{{i,R}} - \frac{{f_{2}^{{i,R}} }}{{f_{1}^{{i,R}} - f_{2}^{{i,R}} }} \cdot L_{{r,1}}^{{i,R}} } \\ {\begin{array}{*{20}c} {L_{{r,WL}}^{{i,E}} = \frac{{f_{1}^{{i,E}} }}{{f_{1}^{{i,G}} - f_{2}^{{i,G}} }} \cdot L_{{r,1}}^{{i,E}} - \frac{{f_{2}^{{i,E}} }}{{f_{1}^{{i,E}} - f_{2}^{{i,E}} }} \cdot L_{{r,1}}^{{i,E}} } \\ {L_{{r,WL}}^{{i,C}} = \frac{{f_{1}^{{i,E}} }}{{f_{1}^{{i,G}} - f_{2}^{{i,G}} }} \cdot L_{{r,1}}^{{i,C}} - \frac{{f_{2}^{{i,C}} }}{{f_{1}^{{i,C}} - f_{2}^{{i,C}} }} \cdot L_{{r,1}}^{{i,C}} } \\ \end{array} } \\ \end{array} } \right.$$
(13)

where \({f}_{1}\) and \({f}_{2}\) are frequencies of L1 and L2 carrier phase. Since the phase observations are corrected with the interpolated atmospheric corrections, the WL observations are not affected by ionospheric and tropospheric delays and are easy to be fixed into integers. Afterward, the ambiguities in the IF solutions are replaced by NL and fixed WL ambiguities. After the NL ambiguities resolution is carried out, the IF solutions with fixed ambiguities are reconstructed by fixed WL and NL ambiguities to obtain the final positioning results. The LAMBDA method is used in the above procedure of ambiguity resolution (Teunissen 1995).

Processing strategy

The multi-constellation observations data of GPS, Galileo and BDS from the Australian Regional GPS Network and South Pacific Regional GNSS Network, as well as precise satellite orbit and clock correction provided by IGS, are used to validate the proposed method. We used 132 reference stations in Australia for UPD estimation, and the distribution of these stations is shown in Fig. 1. The Geostationary Earth Orbit (GEO) satellites of BDS are excluded in the process because the accuracy of their observations and orbit products are poor. In order to validate the performance of GLONASS, four-system observation data from Hong Kong CORS is also used. Due to the FDMA signal model of GLONASS, the inter-frequency bias (IFB) exists among different GLONASS satellites. Thus, stations with homogeneous receivers are used for GLONASS UPD estimation and ambiguity resolution to eliminate the effect of IFB. The distribution of the Hong Kong stations is shown in Fig. 2.

Fig. 1
figure 1

Distribution of selected stations in Australia for UPD estimation

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Fig. 2
figure 2

Distribution of selected stations of Hong Kong CORS

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Dual-frequency observations, i.e., L1 and L2 for GPS, E1 and E5a for Galileo, B1 and B2 for BDS, are used in the process of UPD estimation, atmospheric correction extraction as well as IAR. The sampling interval of observations is 30 s, and the elevation-dependent weight for observations is applied. The coordinates are fixed when estimating UPDs and deriving atmospheric delays. In addition, to validate the performance of multi-GNSS RA-PPP, different system combinations as well as cutoff elevation angles are set in the process of achieving IAR. Table 1 lists the specific processing strategies for UPD estimation and atmospheric correction extraction while Table 2 gives IAR strategies.

Table 1 Processing strategy for UPD estimation and atmospheric correction extraction
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Table 2 Processing strategy for IAR
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Results

The experimental results are divided into two parts. The first part verifies the reliability of atmospheric correction extraction and interpolation methods through the interpolation accuracy of ionospheric and tropospheric delays, whereas the second part shows the contribution of multi-constellation fusion to the PPP instantaneous ambiguity resolution in terms of both positioning accuracy and reliability.

Interpolated atmospheric corrections

In order to validate the proposed method, the reference stations BURA, WAGN and HYDN are chosen as regional augmentation stations, and station KELN is taken as the user station. After PPP-AR is carried out at the reference stations, the atmospheric corrections for each satellite can be extracted and interpolated at the user station. Meanwhile, the obtained interpolated values are compared with those at user stations, extracted in the same way as for the reference stations, to evaluate the accuracy of the interpolated corrections (Fig. 3).

Fig. 3
figure 3

Illustration of the regional augmentation and user stations

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The accuracy of interpolated satellite-differenced ionospheric delays is shown in Fig. 4 with solid dots in each color representing one satellite. The statistical results of ionospheric delays of each satellite are drawn in Fig. 5. We can see from these two Figures that the BDS satellites show the best interpolation results among all three systems. The difference of interpolated ionospheric delays is smaller than 5 cm in most epochs with mean differences smaller than 1.5 cm. This is because the tracking times of BDS satellites are longer and the elevation angles are rather higher for observations from Australian stations, which means the interpolation errors are smaller than that of other systems. The accuracy of GPS and Galileo satellites is close with the mean difference of interpolated ionospheric delays less than 2.5 cm.

Fig. 4
figure 4

Differences between the interpolated and estimated ionospheric delays

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Fig. 5
figure 5

Accuracy ionospheric delays of each system

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Similar to the ionospheric delays, the results of interpolated tropospheric delays are shown in Figs. 6 and 7. The accuracy of interpolated tropospheric delays is better than that of ionospheric delays because the troposphere changes relatively slowly in space. The BDS satellites still show the best performance with mean accuracy of 0.2 cm, followed by the Galileo satellites with mean accuracy of 0.7 cm. The result for GPS satellites is 1.0 cm. The results of interpolated ionospheric and tropospheric delays are accurate enough for instantaneous ambiguity resolution and illustrate that the interpolated method is effective for reference network in the case of a 200 km separation.

Fig. 6
figure 6

Differences between the interpolated and estimated tropospheric delays

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Fig. 7
figure 7

Accuracy of tropospheric delays of each system

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In order to validate the performance of the proposed interpolation method in areas where the atmosphere undergo significant variations, we apply this method for Hong Kong Station. The results of interpolated ionospheric and tropospheric delays are shown in Figs. 8 and 9. We can see that the accuracy of the interpolated ionosphere is lower around 14:00 because the ionosphere is most active at this time of the day. The interpolation accuracy of the troposphere in Hong Kong is similar to that in Australia. Although the accuracy of the interpolated atmospheric delays in Hong Kong still meets the need for ambiguity resolution, it should be noted that the proposed method needs improvement when the ionosphere is active.

Fig. 8
figure 8

Differences between the interpolated and estimated ionospheric delays at Hong Kong station

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Fig. 9
figure 9

Differences between the interpolated and estimated tropospheric delays at Hong Kong station

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Instantaneous ambiguity resolution

With the interpolated multi-constellation atmospheric corrections, PPP with regional augmentation is carried out at user station KELN. Figure 10 and Table 3 show the float PPP results, PPP-AR NRTK and RA-PPP with GPS + Galileo + BDS observations. For the PPP float solutions, the convergence time is about 1470 s, while that of PPP-AR can be shortened to 750 s by applying the satellite UPDs. However, the ambiguity resolution can be achieved in the first epoch for RA-PPP, which is 60 s faster than for NRTK solutions. The fixing rate of the PPP-RA method is also slightly improved compared to the network RTK method.

Fig. 10
figure 10

Positioning results of PPP, PPP-AR, NRTK and RA-PPP

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Table 3 Convergence time and fixing rate of PPP, PPP-AR, NRTK and RA-PPP
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The positioning accuracy of the four solutions in different directions is shown in Table 4. The PPP results after convergence and the fixed results of other solutions are taken into consideration. With applying the atmospheric corrections, the positioning STD can reach 0.4, 0.4 and 1.0 cm on the east, north and up directions, respectively. The improvements are about 69.2%, and 20.0% on the east and north directions compared to the PPP float solution (1.3, 0.4 and 1.2 cm). The results of RA-PPP show no obvious systematic deviation having a mean differences of 0.0, 0.0 and 0.2 cm on three directions, while those of PPP solution are -1.1, -0.5 and -2.5 cm, respectively. After the ambiguities are fixed, the positioning accuracy of PPP-AR, NRTK and RA-PPP is almost equivalent, and better than 1.0 cm.

Table 4 Positioning accuracy of PPP, PPP-AR, NRTK and RA-PPP
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In order to study the effect of instantaneous ambiguity resolution, the ambiguities are fixed epoch-wise without any constraints between epochs for coordinates, receiver clocks and carrier phase ambiguity. The three different system combinations (GE, GC, GEC) and four different cases of cutoff elevation angle ranging from 10° to 40° are set during the data processing to compare the performance of multi-constellation and GPS-only RA-PPP. Figure 11 shows the percentage of IAR for different combinations under different cutoff elevation angle. We can see from the results that the percentage can be significantly improved using multi-GNSS data. The percentage of GEC combination is highest among all the combinations, followed by the GC and GE combinations. The IAR percentage can reach 100.0% for all combinations when the cutoff elevation angle is 10 degrees. As the cutoff angle increases to 20 degrees, the percentage of GPS IAR decreases to 81.7% while that of the other combinations still maintains at 100.0%. The difference of IAR percentage between multi-constellation combination and GPS-only is more significant under higher cutoff elevation angles. When the cutoff elevation angle reaches 40°, the percentage is still above 90.3% for GEC combination while that for GPS-only is 7.2%. The results for GC, GE combinations are 83.9%, 40.0%, respectively.

Fig. 11
figure 11

Percentage of IAR for different combinations with different cutoff elevation angles from 10° to 40° at KELN

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The usable number of satellites of each combination under different cutoff elevation angles are drawn in Fig. 12, which can explain the difference above. The usable satellites are less than 4 in many epochs for GPS-only solutions when the cutoff elevation angle is 40 degrees; this means the PPP cannot be achieved. However, the number of GEC combinations is still larger than 10 under 40 degrees, which can ensure the successful ambiguity resolution. It is worth mentioning that the usable number of GPS + BDS satellite combinations are larger than 7 in most of the situation; thus, the performance of GC IAR solution is fairly good. The reason is that the altitude angle of BDS satellites in this area is generally higher; this is also consistent with the conclusion mentioned above that interpolated atmospheric corrections of BDS is better than other systems.

Fig. 12
figure 12

Number of satellites of each combination with different cutoff elevation angles

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The results in Hong Kong are basically consistent with those in Australia. Atmospheric corrections derived from HKST, HKWS and T430 are used to interpolate for HKSS; the average distance between the reference station and user station is about 20 km. The percentage of IAR for different combinations with different cutoff elevation angles are presented in Fig. 13. The IAR percentage can be up to 99.5% for GREC combination, while the percentage for GPS-only is 88.3% when the cutoff elevation angle is 10°. The performance of IAR can also be improved with other combinations, the percentages of GC, GE and GR combinations are 99.4%, 99.2% and 92.7%, respectively. When the cutoff elevation angle reaches 30°, the percentage is still above 99.0% for GREC combination while that for GPS-only is 42.9%. The results for GC, GE and GR combinations are 95.1%, 93.9% and 53.7%, respectively. The IAR results of both Australia and the Hong Kong area indicate that multi-constellation fusion can significantly improve RA-PPP reliability.

Fig. 13
figure 13

Percentage of IAR for different combinations with different cutoff elevation angles from 10° to 30° at HKSC

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The positioning accuracy can benefit from the multi-constellation combinations too. Figure 14 gives the RMS of the positioning results for different observation combinations with 10° cutoff elevation angle at KELN. The positioning accuracy are better than 1.0 cm for all four solutions. The GEC and GE combination shows better performances than GC and GPS-only solutions. The positioning accuracy for GEC and GE combination is 0.3, 0.2 and 0.4 cm on the east, north and up directions, respectively, while those of GPS-only solutions are 0.4, 0.8 and 0.5 cm. The accuracy improves by 25.0%, 75.0% and 20.0%, respectively.

Fig. 14
figure 14

Positioning accuracy of different combinations with 10° cutoff elevation angle

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Conclusion

We developed a multi-constellation PPP method augmented by precise atmospheric corrections to achieve IAR. In the proposed method, multi-constellation GNSS precise atmospheric corrections are derived from PPP fixed solutions at reference stations and provided to user stations for correcting dual-frequency raw observations. With corrected observations of GPS, GLONASS, Galileo and BDS, the IAR can be achieved at user stations.

This method is validated with observation data from the Australian Regional GPS Network, South Pacific Regional GNSS Network and the Hong Kong CORS. With multi-GNSS UPD products, PPP-AR is carried out at the reference stations and regional atmospheric corrections are derived. The accuracy of interpolated ionospheric corrections is smaller than 2.5 cm, while interpolated tropospheric corrections are smaller than 1.0 cm. The interpolated atmospheric corrections are accurate enough for achieving IAR. However, the interpolation method still needs improvement under practical circumstances especially when the ionosphere is active. This will be a subject of our future research.

The ambiguity resolution can be achieved in one epoch with regionally computed atmospheric corrections, and the convergence time of positioning is significantly shortened compared to the PPP float and PPP-AR solution. RA-PPP can improve the positioning accuracy too; the positioning error is 0.4, 0.4 and 1.0 cm in the east, north and up directions, respectively. The improvements are about 69.2.4% and 20.0% on the east and up directions compared to the PPP float solution. RA-PPP also provides an advantage over NRTK for reference network in the case of a 200 km separation; the time to first ambiguity resolution can be shortened from 90 to 30 s.

The improvement of multi-constellation RA-PPP in terms of positioning accuracy and AR reliability has also been validated. The benefit of multi-constellation fusion is significant for a high cutoff elevation angle. The percentage of IAR can still be above 90.0% for multi-constellation fusion solutions, while the result of GPS-only solutions is only 7.2% when the cutoff elevation angle reaches 40°. As for positioning accuracy, the positioning error for the multi-constellation combination is 0.3, 0.2 and 0.4 cm on the east, north and up directions, respectively, while the results are 0.4, 0.8 and 0.5 cm for GPS-only solutions. Compared to GPS-only results, the accuracy for multi-constellation solutions is improved by 25.0%, 75.0% and 20.0%, respectively.