# An assessment of the interoperability of PPP-AR network products

- Garrett Seepersad
^{1}Email authorView ORCID ID profile and - Sunil Bisnath
^{1}

**15**:4

https://doi.org/10.1186/s41445-017-0009-9

© The Author(s) 2017

**Received: **9 April 2017

**Accepted: **30 November 2017

**Published: **15 December 2017

## Abstract

Integer ambiguity resolution of carrier-phase measurements from a single receiver can be implemented by applying additional satellite corrections (products) to mitigate unmodelled satellite equipment delays. Interoperability of different PPP-AR products would allow the PPP user to transform independently generated PPP-AR products to obtain multiple fixed solutions of comparable precision and accuracy with limited changes required to user PPP measurement processing software. The ability to provide multiple solutions would increase the reliability of the solution for, e.g., real-time processing; if there were an outage in the generation of one set of PPP-AR products, the user could instantly switch streams to a different provider.

There are currently three main public providers of real-time products that enable PPP-AR. These include School of Geodesy and Geomatics at Wuhan University (SGG-WHU), Natural Resources Canada (NRCan) and Centre National d’Etudes Spatiales (CNES). The presented research examines the PPP-AR products generated from the FCB (Fractional Cycle Bias) model and IRC (Integer Recovery Clock) model that have been transformed into the DC (Decoupled Clock) format and applied within the PPP user solution. Interoperability of the different PPP-AR products is a challenging task due to the public availability of different quality of products, limited literature documenting the conventions adopted within the network solution of the providers and unclear definitions of the corrections. The novelty of the research is in the analysis of using the transformed products. The convergence time (time to first fix and time to a pre-defined performance level), position precision (repeatability), position accuracy and solution outliers are examined. Equivalent performance was noted utilizing the different methods. Of the four solutions, FCB products had the highest accuracy. This is attributed to the products being generated using final IGS orbit and clock products. To confirm this, FCBs generated using GRG orbit and clock products were also examined and comparable performance was observed between the FCBs and IRC (GRG) products. The least accurate solution was obtained using the IRC (CNT) products, which was due to the products being archived real time products.

## Introduction

Precise Point Positioning (PPP) requires a relatively long initialization period of at least a few tens of minutes for the carrier-phase ambiguities to converge to constant values and for the solution to reach its optimal precision. The carrier-phase signals are approximately two orders of magnitude more precise than the primary pseudorange (code) signals. However, measurements of the carrier-phases are ambiguous, relative to those of the pseudoranges by an unknown number of integer cycles. In RTK, the integer nature of the carrier-phase ambiguities is uncovered by explicitly differencing simultaneous observations from multiple stations visible to the same satellites. Differencing of simultaneous observations can be thought of as an optimal correction method (Collins and Bisnath 2011) as the error sources are not modelled. Ambiguity resolution in PPP (PPP-AR) requires the equipment delays within the GPS measurements to be mitigated, which would allow for resolution of the integer nature of the carrier-phase measurements (Laurichesse and Mercier 2007; Collins 2008; Mervart et al. 2008; Ge et al. 2008; Teunissen et al. 2010; Bertiger et al. 2010; Geng et al. 2012; Lannes and Prieur 2013). Resolution of these ambiguities convert the carrier-phases into precise pseudorange measurements, with measurement noise at the centimetre-to-millimetre level compared to the metre-to-decimetre-level of the direct pseudoranges (Collins et al. 2010). If the ambiguities could be isolated and estimated as integers, then there would be more information that could be exploited to accelerate convergence to give cm-level horizontal accuracy within an hour of data collection. Collins et al. (2008) and Laurichesse et al. (2009) saw improvements in hourly position estimates by 2 cm and Geng et al. (2010) saw noticeable hourly improvements from 1.5, 3.8 and 2.8 cm to 0.5, 0.5, 1.4 cm for north, east and up, respectively.

Integer ambiguity resolution of measurements from a single receiver can be implemented by applying additional satellite products, where the fractional component, representing the satellite equipment delay, has been separated from the integer ambiguities in a network solution. One method of deriving such products is to estimate the satellite equipment delay by averaging the fractional parts of the steady-state float ambiguity estimates (Ge et al. 2008), and the other is to estimate the pseudorange and carrier-phase clocks independently by fixing the undifferenced ambiguities to integers in advance (Collins 2008; Laurichesse et al. 2009). The initial application of ambiguity resolution to PPP was made by Ge et al. (2008) using the Uncalibrated Phase Delay (UPD) model, now called Fractional Cycle Bias (FCB) model (Geng et al. 2010, 2012). The FCB method estimates combined pseudorange/carrier-phase satellite equipment delays removing the pseudorange satellite equipment delays from common clock estimates, such as those provided by the IGS (International GNSS Service). Common clock implies the estimated satellite clock correction is applied to both pseudorange and carrier phase measurements. An alternative approach to PPP-AR was developed by Collins et al. (2008) called the Decoupled Clock (DC) model. The underlying concept of the DC model is the carrier-phase and pseudorange measurements are not synchronized with each other at the level of precision of the carrier-phase. The timing of the different observable must be considered separately, if they are to be processed together rigorously. The DC model is a reformulation of the ionosphere-free pseudorange and carrier-phase observation equations for GPS, when combined with the wide lane carrier-phase and narrow lane pseudorange observable, permits ambiguity resolution of ambiguities (Collins 2008). The Integer Recovery Clocks (IRC) model presented by Mercier and Laurichesse (2007) consists of daily averages of the wide lane biases and carrier-phase clocks. Zhang et al. (2011) presented a different approach to PPP-AR where the user is provided with the satellite equipment delay for the L1 and L2 carrier-phase signals. In Teunissen and Khodabandeh (2015), the model presented by Zhang et al. (2011) is referred to as the Common Clock (CC-1).

Similar positioning performances have been demonstrated amongst the three methods, DC (Collins et al. 2010), FCB (Ge et al. 2008; Geng et al. 2009) and IRC (Laurichesse et al. 2009). Studies such as Geng et al. (2010), Shi and Gao (2013), and Teunissen and Khodabandeh (2015) have identified the differences and the fundamental similarities between the methods. For the PPP user, the mathematical model is similar; the different PPP-AR products contain the same information and as a result would allow for a one-to-one transformations, allowing interoperability of the PPP-AR products (Teunissen and Khodabandeh 2015). The advantage of interoperability of the different PPP-AR products would be to allow the PPP user to transform independently generated PPP-AR products to obtain multiple fixed solutions of comparable precision and accuracy. The ability to provide multiple solutions would increase the reliability of the solution for, e.g., real-time processing; if there were an outage in the generation of the PPP-AR products, the user can instantly switch streams to a different provider. Interoperability also implies that the user can switch providers within the same PPP engine. The strategy of integrating interoperability of the different products would be dependent on the user application and coding infrastructure of the PPP engine. The research presented examines the PPP-AR products generated from the FCB and IRC model that have been transformed into the DC format and applied within the PPP user solution. The novelty of the research is the solution analysis using the transformed product. The convergence time (time to first fix and time to a pre-defined performance level), position precision (repeatability), position accuracy and solution outliers would be examined. The temporal and spatial behaviour of these estimated terms is examined for the various products applied to understand the unmodelled effects that introduced incorrect solution fixes.

## Overview PPP-AR techniques and products

*i*denotes the frequency dependent GPS measurements frequencies

*L*1 or

*L*2.

*ps*represents the single difference satellites, where

*p*represents the reference satellite,

*s*represents the other tracked satellite and

*u*represents the user position.

\( \Delta {\phi}_{u,i}^{ps} \) and \( \Delta {p}_{u,i}^{ps} \) represents the single differenced carrier-phase and pseudorange measurements. \( \Delta {\rho}_u^{ps} \) is the geometric range between single difference satellites and user position and tropospheric delay. \( {t}_u^{ps} \) is the first order slant ionospheric delay and *μ*
_{
i
} is the frequency dependent co-efficient (\( {\mu}_i=\frac{f_1^2}{f_i^2} \)). *dt*
^{
ps
} is the single differenced satellite clock and \( {d}_{,i}^{ps} \) is the single differenced pseudorange equipment delay. \( {z}_{u,i}^{ps} \) is the single differenced ambiguity and \( {\delta}_{,i}^{ps} \) is the single differenced carrier-phase equipment delay which are expressed in cycles and scaled by the wavelength *λ*
_{,i
}.

*dt*

^{ ps },

*δ*

^{ ps },

*d*

^{ ps }) to correct the observations as

By utilizing externally provided corrections the user’s system of observation Eqs. (5 and 6) can be solved (Teunissen and Khodabandeh 2015).

Presented in the following section is an overview of the different public providers of products that enable PPP-AR, the products and how they are applied to the PPP user equations.

### Public PPP-AR products

Currently, there are three main public providers of products that enable PPP-AR. These include School of Geodesy and Geomatics at Wuhan University (SGG-WHU) (Li et al. 2015; Wuhan University 2017) which provides global post processed FCB products, Natural Resources Canada (Collins 2008; NRCan 2015) which provides post-processed DC products and Centre national d’études spatiales (Laurichesse et al. 2009; CNES 2015) which provides post-processed and real-time IRC products.

### Fractional cycle bias (FCB) model

*IF*represents the ionospheric-free linear combination and

*WN*represents the Melbourne-Wübbena combination.

Currently SGG-WHU produces 14 sets of FCB products corresponding to different precise products such as those from COD, ESA, GFZ, GRGS, IGR, and IGS which are estimated daily (Li et al. 2015). The FCB products can be downloaded from, ftp://gnss.sgg.whu.edu.cn/product/FCB and have the prefix “SGG” and the naming convention includes the AC used for generating the FCBs. FCBs generated using the final IGS products were selected as this set of FCB products capitalized on the accuracy and reliability of the combined products IGS.

### Decoupled clock (DC) model

Where *NL* represents the narrow lane linear combination and *WL* represents the wide lane linear combination. The reformulated DC model using \( \Delta {\tilde{\tilde{p}}}_{u, NL}^{ps} \) and \( \Delta {\tilde{\tilde{\phi}}}_{u, WL}^{ps} \) rather than \( \Delta {\tilde{\tilde{\phi}}}_{u, WN}^{ps} \) was carried out to allow the PPP user to utilize the estimation of the slant ionospheric term for instantaneous re-convergence. (Collins and Bisnath 2011).

### Integer recovery clock (IRC) model

The IRC products can be downloaded from, https://igsac-cnes.cls.fr/html/products.html and have the prefix “GRG”. From this point, onwards, these products shall be referred to as IRC (GRG).

### Summary

Comparison of different public providers of products to enable PPP-AR

Fractional Cycle Bias (FCB) model | Decoupled Clock (DC) model | Integer Recovery Clock (IRC) model | ||||||
---|---|---|---|---|---|---|---|---|

Combined (GRG) | Uncombined (CNT) | |||||||

PPP-AR Products | \( d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{a}}_{q, WN}^{ps} \) \( {\tilde{a}}_{q,1}^{ps} \) | - code clock - wide lane - narrow lane | \( \delta {\tilde{t}}_{IF}^{ps} \) \( d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{\delta}}_{WN}^{ps} \) | - phase clock - code clock - wide lane clock | \( \delta {\tilde{t}}_{IF}^{ps} \) | - phase clock | \( d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{\delta}}_{,i}^{ps} \) \( {\tilde{d}}_{,i}^{ps} \) | - code clock - phase equipment delay - code equipment delay |

\( {\tilde{\delta}}_{WN}^{ps} \) | - wide lane clock | |||||||

Provided product units | \( d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{a}}_{q, WN}^{ps} \) \( {\tilde{a}}_{q,1}^{ps} \) | - seconds - cycles - cycles | \( \delta {\tilde{t}}_{IF}^{ps} \) \( \delta {\tilde{t}}_{IF}^{ps}-d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{\delta}}_{WN}^{ps} \) | - seconds - nanoseconds - nanoseconds | \( \delta {\tilde{t}}_{IF}^{ps} \) \( {\tilde{\delta}}_{WN}^{ps} \) | - seconds - cycles | \( d{\tilde{t}}_{IF}^{ps} \) \( {\tilde{\delta}}_{,i}^{ps} \) \( {\tilde{d}}_{,i}^{ps} \) | - seconds - nanoseconds - nanoseconds |

Data rate | \( d{\tilde{t}}^{ps} \) \( {\tilde{a}}_{q,1}^{ps} \) \( {\tilde{a}}_{q, WN}^{ps} \) | - 30 s - 15 min - daily | \( \delta {\tilde{t}}_{IF}^{ps} \) \( d{\tilde{t}}^{ps} \) \( {\tilde{\delta}}_{WN}^{ps} \) | - 30 s - 30 s - 30 s | \( \delta {\tilde{t}}_{IF}^{ps} \) | - 30 s | \( d{\tilde{t}}^{ps} \) \( {\tilde{\delta}}_{,i}^{ps} \) \( {\tilde{d}}_{,i}^{ps} \) | - 5 s - 5 s - 5 s |

\( {\tilde{\delta}}_{WN}^{ps} \) | - daily | |||||||

General assumptions | Constant \( {\tilde{a}}_{q,W}^{ps} \) are estimated daily by averaging arc-dependent estimates. | No constraints or smoothing applied. | \( \delta {\tilde{t}}_{IF}^{ps} \)aligned to the satellite pseudorange clocks within a narrow lane cycle. \( {\tilde{\delta}}_{WN}^{ps} \) represents a daily average | \( {\tilde{\delta}}_{WN}^{ps} \) estimated as unconstrained in the network work solution with white noise added at each epoch. | ||||

Products used | Post-processed | Post-processed | Post-processed | Archived real-time | ||||

Network solution | Global. IGS final products | Global. EMR rapid products | Global. GRG final products | Global. GRG ultra-rapid products | ||||

PPP user model | \( \Delta {p}_{u, IF}^{ps},\Delta {\phi}_{u, IF}^{ps},\Delta {p}_{u, NL}^{ps},\Delta {\phi}_{u, WL}^{ps} \) or \( \Delta {p}_{u, IF}^{ps},\Delta {\phi}_{u, IF}^{ps},\Delta {\phi}_{u, WN}^{ps} \) | User defined | ||||||

P1, P2 correction | \( {\tilde{d}}_{,i}^{ps} \) | |||||||

L1, L2 correction | \( {\tilde{\delta}}_{,i}^{ps} \) | |||||||

NL correction | \( {\tilde{a}}_{q,1}^{ps} \) | |||||||

WL correction | \( {\tilde{a}}_{q, WN}^{ps} \) | \( {\tilde{\delta}}_{WN}^{ps} \) | \( {\tilde{\delta}}_{WN}^{ps} \) |

## Product transformation

While the different strategies (FCB, DC, IRC) make different assumptions, there are fundamental similarities between them. For the PPP user, the mathematical model is similar; the different PPP-AR products contain the same information and as a result should permit one-to-one transformations between them, allowing interoperability of the PPP-AR products. The advantage of interoperability of the different PPP-AR products would be to permit the PPP user to transform independently generated PPP-AR products to obtain multiple fixed solutions of comparable precision and accuracy. The ability to provide multiple solutions would increase the reliability of the solution for, e.g., real-time processing; if there was an outage in the generation of the PPP-AR products, the user can instantly switch streams to a different provider. The following sections examine the transformation matrix used to transform the IRC and FCB products to the DC format. While the original DC format are in units of seconds and nanoseconds, the following transformed parameters are in units of meters.

### Fractional cycle bias

The FCB products consist of \( d{\tilde{t}}_{IF}^{ps},{\tilde{a}}_{q,1}^{ps} \) and \( {\tilde{a}}_{q, WN}^{ps} \) which has been estimated from the network solutions. The FCBs utilized within this analysis were generated using IGS final orbit and clock products. The fundamental differences between the FCB and DC is that \( {\tilde{a}}_{q,1}^{ps} \) was not determined in the DC method, but assimilated within the clock estimates. Also, \( {\tilde{a}}_{q,W}^{ps} \) are assumed constant over a 24-h period whereas in the DC method the \( {\tilde{\delta}}_{WN}^{ps} \) is neither constrained nor smoothed. The difference in symbology between the FCB and DC approaches is a result of the different representation of the products. DC products are described as a clock term, whereas the FCB products are described as a correction term to be applied to the ambiguity.

### Integer recovery clock

Where *κ* represents \( \frac{\lambda_1-{\lambda}_2}{\lambda_1+{\lambda}_2} \) and on the RHS \( d{\tilde{t}}_{IF}^{ps},{\tilde{d}}_{,1}^{ps} \) and \( {\tilde{d}}_{,2}^{ps} \) are in units of meters and \( {\tilde{\delta}}_{,1}^{ps} \) and \( {\tilde{\delta}}_{,2}^{ps} \) are in units of cycles.

### Analysis of transformed products

While the transformations in the previous section assumed the transformed satellite clocks and equipment delays were in units of meters, the results in this section are presented in terms of time, as the satellite equipment delays should be thought of as an unmodelled timing error. For comparison of the products, the transformed products were corrected for the orbital radial error as the satellite clock and orbital radial error are strongly correlated.

## Performance of transformed products

The station was analyzed in static mode. Receiver clocks were estimated epoch-by-epoch. The zenith tropospheric delays were also estimated each epoch with a random walk co-efficient of 2 cm/sqrt (hour). The station coordinates were initialized using a pseudorange only solution with an initial constraint of 10 m. The IGS absolute antenna model file was used and ocean loading coefficients were obtained from Scherneck (2013) for each of the sites processed. An elevation cut-off angle was set to 10°. The transformed IRC (GRG) and IRC (CNT) products were processed with their original satellite orbit files to maintain consistency between satellite orbits, clocks and equipment delays.

rms of final solution produced by York-PPP from 24-h datasets using data from 55 sites for DOY 25 to 31, GPS week 1829, of 2015 provided by the IGS. Satellite products were provided by NRCan, CNES and Wuhan University. All units are in millimetres

DC | IRC (GRG) | IRC (CNT) | FCB | |||||
---|---|---|---|---|---|---|---|---|

Float | Fixed | Float | Fixed | Float | Fixed | Float | Fixed | |

Northing | 5 | 5 | 5 | 4 | 7 | 6 | 5 | 4 |

Easting | 6 | 3 | 5 | 3 | 8 | 7 | 5 | 3 |

Horizontal | 8 | 6 | 7 | 5 | 11 | 9 | 7 | 5 |

Vertical | 11 | 10 | 11 | 10 | 15 | 10 | 9 | 9 |

3D | 13 | 12 | 13 | 11 | 18 | 14 | 11 | 10 |

The horizontal position error, Fig. 11 (a–d), all four solutions had an overshoot of 55–60 cm after 30 s of data processing. Float and fixed solutions were equivalent within the first 10 min as fixing was only performed after 10 min. Convergence of the float solution to the predefined threshold (5 cm) was 14 min and fixing the ambiguities improved convergence by only 1 min. Similar convergence was noted in the horizontal component for all four solutions.

The strength of applying ambiguity resolution was illustrated in the time to attain a steady state. All three solutions illustrated different convergence trends before attaining a steady state: DC products took 5.2 h, IRC (GRG) products took 3 h and IRC (CNT) as well as FCB products took 2 h. The ambiguity resolved solution improved the time attain a steady state as the time was reduced to 1.2 h.

The vertical position error, Fig. 11 (e–h), had an overshoot of 36, 45, 54 and 60 cm for the DC, IRC (GRG), IRC (CNT) and FCB products respectively. The same predefined threshold for the vertical component was maintained. Convergence times were 18, 16, 15 and 13 min for the DC, IRC (GRG), IRC (CNT) and FCB products, respectively. Applying ambiguity resolution did not improve the time to attain a steady state in the vertical component: 4.5, 3,5.8 and 2 h for the DC, IRC (GRG), IRC (CNT) and FCB products, respectively. Slower convergence was noted in the vertical component due the strong correlation between the atmospheric effects and the vertical component. Any unmodelled components of the atmospheric effects required time averaging or a-priori information to be provided to the user (Collins and Bisnath 2011; Shi and Gao 2014).

## Challenges of interoperability of PPP products

Interoperability of the different PPP products is a challenging task due to the public availability of different quality of products, limited literature documenting the conventions adopted within the network solution of the providers and unclear definitions of the corrections.

Presented in Table 1 was a summary of the different qualities of the products that were utilized within the study. IRC products were generated from a network of reference stations globally distributed, available in real-time and post-processed using final orbits and clocks. Similar to the IRC, the DC and FCB products were generated from a global network of solutions but post-processed. Post-processed orbits and clocks have an accuracy of ~2.5 cm and ~75 ps, respectively, whereas the predicted half of ultra-rapid orbits and clocks have an accuracy of ~5 cm and ~3 ns respectively. While it is evident in existing literature PPP-AR is possible in real-time, the solution is more sensitive as the orbit and clocks have a lower accuracy and more susceptible to outages and network failures.

Another critical component to be accounted for is the difference in the modelling of yaw maneuvers. Yaw maneuvers occur when the actual yaw angle differs from the nominal yaw angle. The nominal yaw angle is the orientation angle by which a satellite would maintain optimal solar visibility throughout its orbit, provided it could spin arbitrarily fast. The actual yaw angle is the orientation that the satellite can maintain due to its limited rate of yaw.

All satellites fail to maintain their nominal orientation when their orbits pass close to the Earth-Sun axis. These are the eclipsing orbits with turns at both orbit noon and orbit midnight. During a satellite eclipse, Block II GPS satellites behaved unpredictably because of hardware sensitivity, spinning beyond the nominal amount upon entering the sun’s shadow. The Block IIR and Block IIF generations of satellites were designed to be able to maintain their nominal attitude even during orbit noon and orbit midnight (Bar-Sever 1996; Dilssner et al. 2011). For Block IIR, the yaw maneuver is constrained by a maximum yaw rate of 0.2 deg./s (Kouba 2009) and Block IIF is constrained by a maximum yaw rate of 0.11 deg./s (Dilssner 2010). The attitude model of the GPS satellites affects the computation of measurement geometry through variations of the transmitter phase center location and carrier-phase measurement wind-up. It also affects the modelling of the solar radiation pressure force acting on the GPS satellites due to the changes in illumination geometry (Kuang et al. 2016).

The expectation is that Block IIR and Block IIF satellites can maintain their nominal attitude even when orbiting through the penumbra and the umbra. As a result of the difficulties in determining the exact moment of exiting the umbra, modelling inconsistencies between ACs have been observed (Seepersad et al. 2016). In the DC format, for example, yaw angles and instances of orbit noon and midnight are provided. The expectation is all ACs would begin to provide this information to the PPP user to increase consistency between the network and user solution. If the networked defined periods of orbit noon, orbit midnight and yaw angles are not provided, it is recommended to the PPP user not to resolve ambiguities of satellites exiting the umbra if differences in yaw modelling exists between the network and user may exist. For more details on inconsistencies between ACs and combining PPP-AR products can be found in Seepersad et al. (2016).

## Conclusions

Interoperability of PPP-AR products is important, as it can increase the reliability of the user solution while offering similar performance, in regards to precision and accuracy. Interoperability of the products is possible for the PPP user, as the mathematical model, to enable an ambiguity resolved solution is similar. The different PPP-AR products contain the same information and would allow for a one-to-one transformation, allowing interoperability of the PPP-AR products. The PPP user will be able to transform independently generated PPP-AR products to seamlessly integrate within their PPP user solution. The seamless integration of the transformed products will allow the PPP user to have multiple solutions, which will increase the reliability of the solution, for e.g., real-time processing. During real-time PPP processing, if there was an outage in the generation of the PPP-AR products, the user can instantly switch streams to a different provider.

The three main public providers of products that enable PPP-AR were examined, which included School of Geodesy and Geomatics at Wuhan University (SGG-WHU), Natural Resources Canada (NRCan) and Centre National d’Etudes Spatiales (CNES). Equivalent performance was noted utilizing the different methods. Of the four solutions, FCB products had the highest accuracy. This is attributed to the products being generated using final IGS orbit and clock products. To confirm this, FCBs generated using GRG orbit and clock products were also examined and comparable performance was observed between the FCBs and IRC (GRG) products. The least accurate solution was obtained using the IRC (CNT) products, which was due to the products being archived real time products.

As the results indicated, interoperability of PPP-AR products is feasible. While feasible, there were challenges when processing the different PPP-AR products. These challenges were due to the same conventions not being followed between the network and user solution, for e.g. different satellite antenna convention. When different satellite antenna convention was used, fractional cycles was introduced when carrier-phase wind-up correction was applied. Another critical component to be accounted for is the difference in the modelling of yaw maneuvers. Difficulties in determining the exact moment when an eclipsing satellite exits the umbra, results in modelling inconsistencies between ACs. If network-defined periods of orbit noon, orbit midnight and yaw angles are not provided, it is recommended that the PPP user not attempt to resolve ambiguities of satellites exiting the umbra if differences in yaw modelling exists between the network and user.

### Future work

Future work will consist of introducing a more advanced AR and validation technique; GPS L5; multi-constellation float and analysis of the post-fit residuals to examine the effects of mismodelling. The temporal and spatial behaviour of these estimated terms will be examined for the different products applied to understand the unmodelled effects that introduced incorrect solution fixes. The number of reference stations examined will also be increased to further test the reliability of the transformed products under varying user conditions.

## Declarations

### Authors informations

Garrett Seepersad is a Ph.D. candidate at York University, Toronto, Canada, in the Department of Earth and Space Science and Engineering. He has completed his B.Sc. in Geomatics at the University of West Indies and his M.Sc. in Geomatics Engineering at York University. His area of research currently focuses on the development and testing of PPP functional, stochastic and error mitigation models.

Dr. Sunil Bisnath is an Associate Professor in the Department of Earth and Space Science and Engineering at York University in Toronto, Canada. His research interests include GNSS processing algorithm development for positioning and navigation applications.

### Funding

Funding was provided by Natural Sciences and Engineering Research Council of Canada (NSERC). NSERC aims to make Canada a country of discoverers and innovators for the benefit of all Canadians. The agency supports university students in their advanced studies, promotes and supports discovery research, and fosters innovation by encouraging Canadian companies to participate and invest in postsecondary research projects. NSERC researchers are on the vanguard of science, building on Canada’s long tradition of scientific excellence.

### Authors’ contributions

Both authors contributed equally in the writing and preparation of the manuscript. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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