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Table 1 GNSS single-frequency observations processed and differential code biases corrections applied

Table 1 GNSS single-frequency observations processed and differential code biases corrections applied

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However, as can be seen from Equations (10) and (11), the ionospheric parameters are affecting the constrained equations. This can be overcome by correcting these observations using the existing ionospheric models, such However, the mathematical model for the ionosphere- free described in Equations (8) and (9) presents a singular- ity model because the number of the unknowns is more than the number of equations due to the existence of the ambiguity parameters. To overcome this rank deficiency problem, additional observations should be added to con- strain the ambiguity parameters working as a priori obser- vations. Commonly, the code observations are used as a priori observations (Choy and Silcock 2011). In the cur- rent research, the a priori observation is selected as the half of code and phase difference as follows:
However, as can be seen from Equations (10) and (11), the ionospheric parameters are affecting the constrained equations. This can be overcome by correcting these observations using the existing ionospheric models, such However, the mathematical model for the ionosphere- free described in Equations (8) and (9) presents a singular- ity model because the number of the unknowns is more than the number of equations due to the existence of the ambiguity parameters. To overcome this rank deficiency problem, additional observations should be added to con- strain the ambiguity parameters working as a priori obser- vations. Commonly, the code observations are used as a priori observations (Choy and Silcock 2011). In the cur- rent research, the a priori observation is selected as the half of code and phase difference as follows:
where Hgsspir is the geometric BSSD ionosphere-free design matrix, SYpssprr and 6Yyp are the single-fre- quency BSSD and undifferenced ionosphere-free obser- vation vectors, respectively and Rgssprrand Rip are the BSSD undifferenced ionosphere-free observation var- lance—covariance (VC) matrix, respectively, considering the mathematical correlation. The error state vector for BSSD integrated system is defined as
where Hgsspir is the geometric BSSD ionosphere-free design matrix, SYpssprr and 6Yyp are the single-fre- quency BSSD and undifferenced ionosphere-free obser- vation vectors, respectively and Rgssprrand Rip are the BSSD undifferenced ionosphere-free observation var- lance—covariance (VC) matrix, respectively, considering the mathematical correlation. The error state vector for BSSD integrated system is defined as
where A;;and A>; are the float single differenced float ambiguity parameters. To verify the performance of our single-frequency GNSS PPP models, datasets from 11 globally distributed IGS- MGExX < stations are processed as shown in Fig. 1. The datasets collected at the selected stations on seven con- secutive days, i.e. 1-7 April 2014, are used for numerical analysis. The selected stations are occupied by different types of GNSS receivers as shown in Table 2. Single-fre- quency observations from GPS L1, GLONASS G1, Gali- leo El and BeiDou B1 signals are adopted in this study. The IGS antenna phase-centre offsets are used for GPS and GLONASS, while the BeiDou and Galileo antenna offsets recommended by the MGEX project are used to correct the PCOs of BeiDou and Galileo satellites (Rizos et al. 2013). To increase GNSS data sets, 6-h pos- ition solutions are analysed to represent the PPP perform- ance in a short observation time.
where A;;and A>; are the float single differenced float ambiguity parameters. To verify the performance of our single-frequency GNSS PPP models, datasets from 11 globally distributed IGS- MGExX < stations are processed as shown in Fig. 1. The datasets collected at the selected stations on seven con- secutive days, i.e. 1-7 April 2014, are used for numerical analysis. The selected stations are occupied by different types of GNSS receivers as shown in Table 2. Single-fre- quency observations from GPS L1, GLONASS G1, Gali- leo El and BeiDou B1 signals are adopted in this study. The IGS antenna phase-centre offsets are used for GPS and GLONASS, while the BeiDou and Galileo antenna offsets recommended by the MGEX project are used to correct the PCOs of BeiDou and Galileo satellites (Rizos et al. 2013). To increase GNSS data sets, 6-h pos- ition solutions are analysed to represent the PPP perform- ance in a short observation time.
Table 2 MGEX stations selected with different GNSS receivers employed CUTO, the addition of BeiDou observations significantly enhanced the positioning accuracy compared with the GPS-only due to the sufficient number of visible BeiDou satellites. For the seven-day datasets, each day is divided into four sessions. Each session is processed separately so that a total of 308 sets of results are obtained to derive a statisti- cal estimate on the positioning accuracy. To evaluate the performance of different GNSS combinations, namely, GPS only, GPS/GLONASS, GPS/Galileo, GPS/BieDou and GPS/GLONASS/Galileo/BieDou (quad-constella- tion), the positioning results for BRST and CUTO, at | April 2014, are shown herein as an example.
Table 2 MGEX stations selected with different GNSS receivers employed CUTO, the addition of BeiDou observations significantly enhanced the positioning accuracy compared with the GPS-only due to the sufficient number of visible BeiDou satellites. For the seven-day datasets, each day is divided into four sessions. Each session is processed separately so that a total of 308 sets of results are obtained to derive a statisti- cal estimate on the positioning accuracy. To evaluate the performance of different GNSS combinations, namely, GPS only, GPS/GLONASS, GPS/Galileo, GPS/BieDou and GPS/GLONASS/Galileo/BieDou (quad-constella- tion), the positioning results for BRST and CUTO, at | April 2014, are shown herein as an example.
2 The quad-GNSS satellite availability and positioning errors for Station BRST using different GNSS combinations Figures 5 and 6 show the distribution of the absolute positioning errors after 2h for the four developed single-frequency PPP models. A total of 308 datasets are used for all constellation combinations except GPS/ BeiDou where only 196 datasets are used due to the una- vailability of BeiDou satellite in some stations. The posi- tioning distribution for the ionosphere-free models gives more precise solution in comparison with the GIM- based solution in general for all the datasets. On the other hand, comparable positioning errors distributions Figures 2 and 3 show the GNSS satellite availability and positioning errors with time for the various GNSS constellation combinations at stations BRST and CUTO, respectively, using the developed BSSD-IF model. All PPP solutions are referenced to the GNSS station coordinates published by Center for Orbit Deter- mination in Europe (CODE 2015). For station BRST, it can be seen that the major contribution to the PPP sol- ution enhancement is due to the additional GLONASS observations. This is due to the good availability of GLO- NASS compared with the other constellation, which sig- nificantly affects the overall satellite geometry. On the other hand, because of their limited number of visible sat- ellites, the addition of Galileo and BeiDou systems has a marginal effect on the positioning accuracy, in compari- son with the GPS-only solution. In contrast, for Station
2 The quad-GNSS satellite availability and positioning errors for Station BRST using different GNSS combinations Figures 5 and 6 show the distribution of the absolute positioning errors after 2h for the four developed single-frequency PPP models. A total of 308 datasets are used for all constellation combinations except GPS/ BeiDou where only 196 datasets are used due to the una- vailability of BeiDou satellite in some stations. The posi- tioning distribution for the ionosphere-free models gives more precise solution in comparison with the GIM- based solution in general for all the datasets. On the other hand, comparable positioning errors distributions Figures 2 and 3 show the GNSS satellite availability and positioning errors with time for the various GNSS constellation combinations at stations BRST and CUTO, respectively, using the developed BSSD-IF model. All PPP solutions are referenced to the GNSS station coordinates published by Center for Orbit Deter- mination in Europe (CODE 2015). For station BRST, it can be seen that the major contribution to the PPP sol- ution enhancement is due to the additional GLONASS observations. This is due to the good availability of GLO- NASS compared with the other constellation, which sig- nificantly affects the overall satellite geometry. On the other hand, because of their limited number of visible sat- ellites, the addition of Galileo and BeiDou systems has a marginal effect on the positioning accuracy, in compari- son with the GPS-only solution. In contrast, for Station
3 The quad-GNSS satellite availability and positioning errors for Station CUTO using different GNSS combinations
3 The quad-GNSS satellite availability and positioning errors for Station CUTO using different GNSS combinations
4 The positioning accuracy of the four developed single-frequency PPP models for stations GRAC (left) and BRUX (right)
4 The positioning accuracy of the four developed single-frequency PPP models for stations GRAC (left) and BRUX (right)
5 Distribution of the absolute positioning errors for the different GNSS combinations using our GNSS PPP techniques, namely, (a) undifferenced GIM-based model, (b) ionosphere-free PPP model
5 Distribution of the absolute positioning errors for the different GNSS combinations using our GNSS PPP techniques, namely, (a) undifferenced GIM-based model, (b) ionosphere-free PPP model
Table 3 The statistical analysis namely the mean and the standard deviation of 3D positioning results sets for the four single- frequency PPP developed models after 2 h of data processing (unit: metre)
Table 3 The statistical analysis namely the mean and the standard deviation of 3D positioning results sets for the four single- frequency PPP developed models after 2 h of data processing (unit: metre)
6 Distribution of the absolute positioning errors for the different GNSS combinations using our developed GNSS PPP tech- niques, namely, (a) BSSD-PPP, (b) BSSD-IF-PPP
6 Distribution of the absolute positioning errors for the different GNSS combinations using our developed GNSS PPP tech- niques, namely, (a) BSSD-PPP, (b) BSSD-IF-PPP
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