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This paper introduces a new dual-frequency precise point positioning (PPP) model, which combines the observations of three different GNSS constellations, namely GPS, Galileo, and BeiDou. Our model is based on between-satellite single-difference (BSSD) linear combination, which cancels out some receiver-related biases, including receiver clock error and non-zero initial phase bias of the receiver oscillator. The reference satellite can be selected from any satellite system GPS, Galileo, and BeiDou when forming BSSD linear combinations. Natural Resources Canada’s GPS Pace PPP software is modified to enable a combined GPS, Galileo, and 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 Galileo measurements. It is shown that using the BSSD linear combinations improves the precision of the estimated parameters by about 25% compared with the GPS-only PPP solution. Additionally, the solution convergence time is reduced to 10 minutes for both BSSD scenarios, which represent about 50% improvement in comparison with the GPS-only PPP solution.

Global navigation satellite systems (GNSS) precise point positioning (PPP) has proven to be capable of providing positioning accuracy at the sub-decimeter and decimeter levels in static and kinematic modes, respectively. PPP accuracy and convergence time are controlled by the ability to mitigate all potential error sources in the system. Several comprehensive studies have been published on the accuracy and convergence time of undiffer- rent combined GPS/Galileo PPP model [

A drawback of a single GNSS system such as Global Positioning System (GPS) is the limited number of visible satellites in urban areas. With the addition of Galileo and BeiDou satellites, a PPP solution based on the combined GPS, Galileo, and BeiDou observations becomes more feasible. Combining the three satellite constellations offers more visible satellites to users, which in turn enhance the satellite geometry and are expected to improve the overall positioning solution [

This paper develops triple GNSS (GPS, Galileo, and BeiDou) PPP model, which rigorously accounts for the additional combination biases. These additional biases are lumped together into a new unknown parameter, which is referred to as the inter-system bias, in the PPP mathematical model. The GPS receiver differential hardware delays are lumped to 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 [

Traditionally, PPP has been carried out using dual-frequency ionosphere-free linear combinations of carrier- phase and pseudorange GPS measurements. Equations (1) to (4) show the ionosphere free linear combination of GPS, Galileo, and BeiDou observations [

where the subscripts G, E, and B refer to the GPS, Galileo, and BeiDou satellite systems, respectively;_{r}, dt^{s} are the clock errors in seconds for the receiver at signal reception time and the satellite at signal transmission time, respectively;

which are given, respectively, by:

Where f_{1} and f_{2} are GPS L_{1} and L_{2} signals frequencies; f_{E}_{1} and f_{E}_{5a} are Galileo E_{1} and E_{5a} signals frequencies; f_{B}_{1} and f_{B}_{2} are BeiDou B_{1} and B_{2} signals frequencies.

where λ_{1} and λ_{2} are the GPS L_{1} and L_{2} signals wavelengths in meters; λ_{E}_{1} and λ_{E}_{5a} are the Galileo E_{1} and E_{5a} signals wavelengths in meters; λ_{B}_{1} and λ_{B}_{2} are the BeiDou B_{1} and B_{2} signals wavelengths in meters; N_{1}, N_{2} are the integer ambiguity parameters of GPS signals L_{1} and L_{2}, respectively; N_{E}_{1}, N_{E}_{5a} are the integer ambiguity parameters of Galileo signals E_{1} and E_{5a}, respectively; N_{B}_{1}, N_{B}_{2} are the integer ambiguity parameters of BeiDou signals B_{1} and B_{2}, respectively.

Precise orbit and satellite clock corrections of IGS-MGEX networks are produced for both GPS/Galileo observations and are referred to GPS time. IGS precise GPS satellite clock correction includes the effect of the ionosphere-free linear combination of the satellite hardware delays of L_{1}/L_{2} P(Y) code, while the Galileo counterpart includes the effect of the ionosphere-free linear combination of the satellite hardware delays of the Galileo E_{1}/E_{5a} pilot code. In addition, BeiDou satellite clock correction includes the effect of the ionosphere-free linear combination of the satellite hardware delays of B_{1}/B_{2} code [

For simplicity, the receiver and satellite hardware delays will take the following forms:

In the combined GPS, Galileo and BeiDou un-differenced PPP model, the GPS receiver clock error is lumped with the 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 L_{1}/L_{2}, Galileo E_{1}/E_{5a} and BeiDou B_{1}/B_{2} observations in a PPP model. This, however, introduces an additional bias in the Galileo 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 [

where

When using the combined GPS, Galileo, and BeiDou un-differenced PPP model, the ambiguity parameters lose its integer nature as they are contaminated by 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 from any system plus seven parameters, while the number of equations equals double the number of the visible satellites. This means that the redundancy equals

As indicated earlier, In case of considering a GPS satellite is selected as a reference for all the GNSS observables [

where

Similarly, when a Galileo satellite is selected as a reference, using Equations (16) to (21) leads to:

where,

In case of selecting a BeiDou satellite as a reference, Equations (16) to (21) leads to:

where,

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. In our solution, we consider that the ratio between the standard deviation of the code and carrier-phase measurements to be 100. When forming BSSD, however, the differenced observations become mathematically correlated. This leads to a fully populated covariance matrix at a particular epoch.

The general linearized form for the above observation equations around the initial (approximate) vector u^{0} and observables l can be written in a compact form as:

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; Δu is the unknown vector of corrections to the approximate parameters u^{0}, i.e., u = u^{0} + Δu; w is the misclosure vector and r is the vector of residuals. The sequential least-squares solution for the unknown parameters Δu_{i} at an epoch i can be obtained from (Vanicek and Krakiwsky, 1986):

where Δu_{i}_{ }_{−}_{ }_{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_{Δu} 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 [

where

To verify the developed combined PPP models, GPS, Galileo, and BeiDou observations at four globally distributed 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 GPS, Galileo, 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 used to assess the performance of the newly developed PPP model.

As shown in

of visible satellites will be 14 satellites, however by combining the three satellite systems the number of visible satellites will be 19 satellites.

As mentioned earlier, the reference satellite can be considered from any constellations of the GNSS. In this paper the BSSD PPP model is formed using different reference satellite from each constellation: the first considers a GPS satellite as a reference for all GNSS observables, while the second considers a Galileo satellite as a reference and the third considered a BeiDou satellite as a reference satellite. As can be seen in

To further assess the performance of the various PPP models, the solution output is sampled every 10 minutes and the standard deviation of the computed station coordinates is calculated for each sample.

This paper presents a PPP model, which combines GPS, Galileo, and BeiDou observations in BSSD mode. Different reference satellite has been considered when forming BSSD, namely GPS reference satellite, Galileo reference satellite and BeiDou reference satellite. It has been shown that the newly developed PPP model improves the solution convergence time by about 50%, in comparison with the un-differenced GPS-only PPP model. In addition, the newly developed model improves the precision of the estimated parameters by about 25%, in comparison with the un-differenced GPS-only model. As the number of epochs increases, the performance of the various models tends to be comparable. Almost identical results are obtained through the BSSD combination when either a GPS, Galileo or BeiDou satellite is selected as a reference.

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 (IGS-MEGX) network.

Akram Afifi,Ahmed El-Rabbany, (2016) Improved Between-Satellite Single-Difference Precise Point Positioning Model Using Triple GNSS Constellations: GPS, Galileo, and BeiDou. Positioning,07,63-74. doi: 10.4236/pos.2016.72006