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This paper introduces a new precise point positioning (PPP) model, which combines single-fre- quency GPS/Galileo observations in between-satellite single-difference (BSSD) mode. In the absence of multipath, all receiver-related errors and biases are cancelled out when forming BSSD for a specific constellation. This leaves the satellite originating errors and atmospheric delays un- modelled. Combining GPS and Galileo observables introduces additional biases that have to be modelled, including the GPS to Galileo time offset (GGTO) and the inter-system bias. This paper models all PPP errors rigorously to improve the single-frequency GPS/Galileo PPP solution. GPSPace PPP software of Natural Resources Canada (NRCan) is modified to enable a GPS/Galileo PPP solution and to handle the newly introduced biases. A total of 12 data sets representing the GPS/Galileo measurements of six IGS-MEGX stations are processed to verify the newly developed PPP model. Precise satellite orbit and clock corrections from IGS-MEGX networks are used for both GPS and Galileo measurements. It is shown that sub-decimeter level accuracy is possible with single-frequency GPS/Galileo PPP. In addition, the PPP solution convergence time is improved from approximately 100 minutes for the un-differenced single-frequency GPS/Galileo solution to approximately 65 minutes for the BSSD counterpart when a single reference satellite is used. Moreover, an improvement in the PPP solution convergence time of 35% and 15% is obtained when one and two reference satellites are used, respectively.

The concept of precise point positioning (PPP) was first introduced by [

The Galileo satellite system offers additional visible satellites to the user, which is expected to enhance the satellite geometry and the overall PPP solution when combined with GPS [

Generally, combining multi-constellation observations in a PPP solution improves the positioning accuracy, especially when the system biases are calibrated, as shown in [

This paper introduces a new PPP model, which combines single-frequency GPS and Galileo observables in BSSD mode. Precise corrections from the International GNSS Service multi-GNSS experiment (IGS-MEGX) network are used to account for GPS and Galileo satellite orbit and clock errors [

GNSS observations are affected by random and systematic errors, which must be accounted for to ensure that precise positioning solution is obtained. The positioning accuracy of a PPP model depends on the ability to mitigate errors and biases. These errors can be categorized into three classes, namely satellite related errors, signal propagation related errors, and receiver/antenna associated errors. The main GNSS errors include the satellite/receiver clock errors, satellite/receiver hardware delays, ionospheric and tropospheric delays, and multipath [

In addition to the above errors and biases, combining GPS and Galileo observations in a PPP model introduces additional errors such as GGTO due to the fact that each system uses a different time frame. The GPS system uses the GPS time system, which is referenced to coordinated universal time (UTC) as maintained by the US Naval Observatory (USNO). On the other hand, the Galileo satellite system has its own time frame, namely the Galileo system time (GST), which is a continuous atomic time scale with a nominal constant offset with respect to the international atomic time (TAI) [

where, the subscript G refers to the GPS satellite system and the subscript E refers to the Galileo satellite system; t_{G} is the true signal reception time; τ_{G} and τ_{E} are signal propagation times for both GPS and Galileo, respectively; P_{G} and P_{E} are the GPS and Galileo pseudorange measurements, respectively; Φ_{G} and Φ_{E} are the GPS and Galileo carrier-phase measurements, respectively; _{G} to the satellite at transmission times _{rG}(t_{G}) is the receiver clock error at reception time t_{G}; _{rG}(t_{G}) and d_{rE}(t_{G}) are frequency-dependent code hardware delays in the receiver at reception time t_{G} for GPS and Galileo, respectively; _{rG}(t_{G}) and _{G} for GPS and Galileo, respectively; δ_{rE}(t_{G}) and _{G} and λ_{E} are the wavelengths of carrier frequencies for GPS and Galileo signals, respectively; Φ_{r}(t_{0}), Φ^{s}(t_{0}) are frequency-dependent initial fractional phases in the receiver and satellite channels, respectively; t_{0} is the receiver (or satellite) initial time; N_{G} and N_{E} are the integer numbers of cycles for GPS and Galileo carrier phase measurements, respectively; GGTO is the GPS to Galileo time offset; c is the speed of light in vacuum; and ε_{P}, ε_{Φ} are the relevant noise and unmodeled errors.

As indicated earlier, precise orbit and satellite clock corrections of IGS-MGEX network are used for both GPS and Galileo observations. Clock corrections from the two networks are referred to the GPS time. In addition, they include the ionosphere-free linear combination of the satellite hardware delays of L1/L2 P(Y) code for GPS and the ionosphere-free linear combination of the satellite hardware delays of E1/E5a pilot code for Galileo [

where _{ }are the precise satellites clock corrections for both GPS and Galileo satellites, respectively, which are obtained from IGS-MGEX;_{G} and β_{E} are the ionosphere-free linear combination coefficients, which equal 1.546 and 1.261 for GPS and Galileo, respectively;

It should be pointed out that in both of our GPS-only and GPS/Galileo PPP models, the GPS receiver hardware delay is lumped to the receiver clock error as explained above. This strategy maintains the consistency of the estimated receiver clock error for both of the GPS-only and the GPS/Galileo PPP solutions [

Equations (5) to (9) can be simplified for the pseudorange and carrier phase observables after applying the corrections for the satellite clock errors, the hydrostatic component of the tropospheric zenith path delay, the correction to the ionospheric delay, the satellite differential code biases, and the other remaining biases. As stated earlier, the global ionosphere maps (GIM) are used to account for the ionospheric delay [

where _{w} is the wet component of the tropospheric zenith path;

Differencing the observations between satellites cancels out most receiver-related errors, including receiver clock error, receiver hardware delay for the same constellation, and non-zero initial phase bias [

where

If, however, a Galileo satellite is used as a reference in a tight combination, we obtain the following set of BSSD equations:

where,

Finally, the per-constellation BSSD equations take the form:

where,

It should be noticed from the above equations that the modified receiver clock error (i.e., the common term

The sequential least-squares estimation technique is used to obtain the best estimates, in the least squares sense, of the unknown parameters. The noise terms in the above observations equations are modeled stochastically using an exponential function, as described in [^{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 [

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 ration 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.

Considering the un-differenced mode, the matrix A and the vector Δu at a particular epoch are given by:

where n_{G} refers to the number of visible GPS satellites; n_{E} refers to the number of visible Galileo satellites; _{0}, y_{0} and z_{0} are the approximate receiver coordinates; _{w}, the inter-system bias ISB, and the non-integer ambiguity parameters_{E} − 1. In other words, we need a minimum of two Galileo satellites in order to contribute to the solution.

When a GPS satellite is selected as a reference to form the BSSD for both GPS and Galileo observations, the design matrix A and the vector of corrections Δu take the form:

where 1_{G} refers to the GPS reference satellite. All other parameters are as defined above. The advantage of the above system (42) is that the number of unknown parameters is reduced by two (i.e., becomes

where 1_{E} refers to the Galileo reference satellite. All other parameters are as defined above. Similar to the above BSSD scenario, the redundancy remains unchanged and equals

When two reference satellites are selected to form the BSSD, i.e., per-constellation BSSD, the design matrix A and the vector of corrections Δu take the form:

The major advantage of the above per-constellation (or loose combination) system is that the modified receiver clock error and the inter-system bias are cancelled out. In addition, the total number of unknown parameters is reduced by 4 to become

To verify the developed GPS/Galileo PPP model, GPS and Galileo measurements at six well-distributed stations (

The sampling interval for all data sets is 30 seconds, while the time span used in the analysis is three hours, which is selected at different times of the day to ensure that the four Galileo satellites are visible at each station. GPSPace PPP software of Natural Resources Canada (NRCan) was modified to enable a GPS/Galileo PPP solution as described above. The positioning results for stations CONZ, and CUT0 (January 1, 2014) and stations DLF1, and UNB3 (July 8, 2014) are presented below. Similar results are obtained from the other stations. However, a summary of the convergence times is presented below for all stations.

The single-frequency GPS/Galileo PPP solution is implemented through combining the GPS L1 signal with the Galileo E1 signal. As mentioned earlier, three different scenarios are considered when processing the data sets with the BSSD model, namely (1) a GPS satellite is selected as a reference satellite for both GPS and Galileo observables; (2) a Galileo satellite is selected as a reference satellite for both GPS and Galileo observables; and (3) two reference satellites are selected: a GPS reference satellite for the GPS observables and a Galileo satellite for the Galileo observables. To assess the PPP solution accuracy of the developed single-frequency model, un-differenced dual-frequency ionosphere-free linear combination of GPS/Galileo PPP is used as a reference.

part of the mathematical model. As shown in [

The results of the third BSSD model scenario, i.e., when a GPS satellite and a Galileo satellite are selected as references for the GPS and Galileo observables, respectively, are shown in

discussed above, this is likely attributed to the relatively weaker adjustment model of the loose combination.

The inter-system bias for the various receivers is obtained as a by-product of the PPP solution of the un-dif- ferenced and tight combination scenarios.

As shown in

A new PPP model, which combines single-frequency GPS and Galileo system observations in BSSD mode, has been introduced in this paper. Three scenarios have been considered when forming BSSD; namely a GPS satellite is selected as a reference, a Galileo satellite is selected as a reference, and two satellites, one GPS and one Galileo, are selected as references. It has been shown that a sub-decimetre level positioning accuracy can be obtained with both of the un-differenced and BSSD single-frequency GPS/Galileo PPP models. However, the PPP solution of the un-differenced model takes about 100 minutes to converge to a decimetre level positioning accuracy. The convergence time of the single-frequency GPS/Galileo PPP solution is improved by 35% and 15% when BSSDs with tight and loose combinations are used, respectively. The moderate improvement in the solution convergence time obtained with the loose combination is likely attributed to its relatively weaker adjustment model in comparison with the tight combination.

The values of the ISB have been obtained for various days and receiver types. Almost identical results have been obtained with both of the un-differenced and BSSD (tight combination) modes. It has been found that the values of the ISB are largely stable over the observation time spans. However, differences of up to 3 m have been observed, which suggest that the ISB is receiver/firmware dependent.

This research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Government of Ontario, and Ryerson University. The authors would like to thank the International GNSS service (IGS) network for providing the satellites precise products.