Klau Geomatics has released processing that brings precise point positioning (PPP) and post processed kinematic (PPK) together in an optimized solution.
The autonomous solution can work anywhere without any other user inputs, such as base station data and radio/GSM links, the company added.
According to Klau Geomatics, the solution works on its own to achieve high accuracy, regardless of the location of the user. Accurate datum and tectonic plate motion corrections, specific to different countries and regions, are automatically applied to deliver the most accurate solutions.
In addition, Klau Geomatics’ NRT technology gives users — specifically those in the drone inspection industry — the ability to attain absolute accuracy to analyze change over time on 3D assets, the company said. The precise corrections are applied to data from custom-tuned KlauPPK GNSS receivers in the KlauPPK post processing software to enable centimeter-level accuracy anywhere in the world without the need for RTK, CORS or local base station data.
Finally, Klau Geomatics’ hybrid terrestrial multi station and PPP algorithm are offering even more refined accuracy in areas such as the U.S., Europe, Japan, New Zealand and Australia. Data from as many as 15 reliable long-range CORS stations, where available, are applied to the processing. The enhanced PPP solution achieves 1-3cm XYZ absolute accuracy in many parts of the world.
The same KlauPPK software workflow applies, to synchronise camera events, apply lever arm corrections, manage coordinate systems and geoids, apply site localizations, capture ground points and more. Instead of choosing a base station, which should be within 20 miles of the site, or setting up an RTK radio link, users with an active KlauPPK subscription can process a high accuracy trajectory, anywhere, without any other inputs, Klau Geomatics said.
Spaceopal has launched NAVCAST, a GNSS precise point positioning (PPP) service featuring high-accuracy positioning enhancement for end users worldwide. NAVCAST aims to actively support and to accelerate widespread adoption of Galileo.
NAVCAST provides Galileo and GPS real time orbit and clock corrections based on an algorithm RETICLE (REal-TIme CLock Estimation), developed by the German Aerospace Centre (DLR e.V.).
Galileo and GPS observations, from more than 100 receivers of the worldwide IGS network, are used to estimate the current corrections which are broadcast to registered users relaying on the standard NTRIP protocol.
NAVCAST corrections improve the user error down to the centimeter level, making it attractive for a large number of applications, the company said.
Users can appraise the accuracy levels and convergence times achievable using NAVCAST (Galileo + GPS) corrections combined with a precise point positioning (PPP) engine, on the Spaceopal website. The underlying PPP engine (dual-frequency, ionosphere-free observations) estimates the local troposphere delays and fixes the carrier-phase integer ambiguities.
NAVCAST can be considered as proof of concept and Spaceopal’s contribution to high-accuracy GNSS services. NAVCAST corrections, which are broadcast over the Internet, could be in future via satellite constellation (such as MEO satellites).
From November 2010 until end of June 2017, Spaceopal was the prime contractor responsible for Galileo operations under the Galileo Full Operational Capability (FOC) Operations Framework contract, the company said. Spaceopal GmbH will continue to operate the Galileo satellite fleet under the Galileo Service Operator (GSOp) contract. Spaceopal is actively supporting the completion of the system to expand the services up to full operational capability by 2020.
Multi-GNSS will open up PPP to a much wider range of applications.
By Francesco Basile, Terry Moore, Chris Hill, Gary McGraw and Andrew Johnson
INNOVATION INSIGHTS by Richard Langley
ARE WE THERE? In a multi-GNSS world, that is. We’ve asked that question from time to time in this column over the years. So, are we there yet? That depends. One definition of “multi” is more than one. In this sense, we were in a multi-GNSS world as long ago as 1996. In that year, we had two fully populated constellations of satellites: GPS and GLONASS. Unfortunately, the full GLONASS constellation was short-lived. Russia’s economic difficulties following the dissolution of the Soviet Union hurt GLONASS, and by 2002 the constellation had dropped to as few as seven satellites. But GLONASS was reborn, and by Dec. 8, 2011, a full 24-satellite constellation was again operational.
But another meaning of “multi” is many, implying more than two. In the late 1990s, the first satellites to host transponders for satellite-based augmentation systems were launched. So, by the mid-2000s, even though GLONASS was still undergoing its rejuvenation, we were already in a three-constellation world. And receivers then on the market provided the necessary raw measurement data to yield positioning solutions from this system of systems with potentially more continuity and greater accuracy than those obtained using GPS alone.
And so in July 2008, we featured the article “The Future is Now: GPS + GLONASS + SBAS = GNSS.” And then in June 2010, we had “GPS, GLONASS, and More: Multiple Constellation Processing in the International GNSS Service.” In the introduction to that article, we asked that same question: Are we there yet? We concluded that, for early adopters of GPS plus GLONASS data and products, we were. With Galileo test satellites in orbit and an early version of the BeiDou system operational, it was already clear that by the end of the current decade, it wouldn’t just be the early adopters who would be benefiting from multi-GNSS but virtually all users of satellite-based positioning and navigation.
Although we aren’t quite there with fully operational Galileo and BeiDou constellations, we are getting pretty close. And so researchers are looking hard at how to make the best use of multiple-constellation observations in a variety of positioning and navigation scenarios. In this month’s column, a team of such researchers examines the potential benefit of combining GPS and Galileo observations for improving precise point positioning in urban environments, following the advice we read in the Book of Ecclesiastes: “Two are better than one.”
Over the years, precise point positioning (PPP) has been applied to many real-time applications that require sub-decimeter-level accuracy over a wide area or on a global scale. It is currently a standard in scenarios characterized by open-sky conditions, where a receiver is likely to have continuous track of GNSS satellites. On the other hand, PPP’s typically long convergence time means the technique has not been widely used in constrained and transient signal environments associated with urban areas. Analysis with both simulated and real data has shown that, once Galileo reaches final operational status, the PPP convergence time will be cut by more than half when using both GPS and Galileo observations. Accordingly, multi-GNSS will open up PPP to a much wider range of applications.
To begin, we assessed the positioning performance of GPS and Galileo signals, alone or used together, in open-sky conditions. A Simulink-based software simulator was used to simulate 24-hour-long observation sessions from 10 static (fixed location) receivers spread worldwide, which were then processed with the POINT software (developed by the University of Nottingham and three other British universities) in static (receiver assumed fixed) PPP mode with an elevation cutoff angle of 10° and with carrier-phase ambiguities estimated as real or floating-point values. For each station, the simulator was run 55 times to provide a sufficient number of data points to characterize the general behavior of the processing algorithms; therefore, a total of 550 points were considered.
For better GPS-Galileo interoperability, PPP results based on the ionosphere-free (IF) combination between GPS L1 and L5 and Galileo E1 and E5a observables were considered.
The metrics used to define the positioning performance are the errors in the north, east and down components of the position once all of a daily file has been processed and the time these errors take to converge below 10 centimeters.
The open-sky condition always guarantees excellent geometry and signal continuity even considering only one constellation.
PPP Results. TABLE 1 shows the root mean square (RMS) of the errors and convergence times of the three components of position for the different configurations for the 550 points considered. Both single- and dual-constellation systems are able to provide a sub-decimeter-level accuracy after a few tens of minutes. On average, positioning with Galileo E1-E5a IF performs better that GPS L1-L5 IF: the Galileo solution is more accurate and converges faster than the GPS solution.
TABLE 1. Comparison between GPS-only, Galileo-only and GPS plus Galileo PPP results. RMS of the positioning errors and convergence times for the stations considered.
The reason for this behavior is the assumed lower noise on Galileo pseudoranges. It is well known that the quality of the pseudoranges affects the convergence time of the PPP solution.
For this reason, one would expect some improvements by employing the Galileo Alternative BOC (AltBOC) modulated E5 signal. Thanks to its very large signal bandwidth of at least 51 MHz, Galileo E5 is characterized by excellent rejection properties of both long-range and short-range multipath. However, as shown in Table 1, when comparing the PPP solutions obtained using the Galileo E1-E5 IF and E1-E5a IF combinations, they have nearly the same performance. The reason for this apparent contradiction can be found in the use of the IF combination with E1. Given that E1 represents the dominant source of error in the IF combinations, its noise is amplified by a factor of 2.34 in the IF combination with E5 and by a factor of 2.26 when combined with E5a. Also, the smaller errors (with respect to E1) in E5a are amplified by 1.26, while the one in E5 is amplified by 1.34. Therefore, depending on the noise level in the Galileo pseudoranges, there might be instances where the noise in the E1-E5 IF combination is close to the one in the E1-E5a IF combination.
The number and the geometry of the observed satellites also affect the convergence time. For this reason, when using the two systems together, the time the vertical errors take to go below 10 centimeters was reduced by 50 percent with respect to the GPS-only case and by 18 percent with respect to the Galileo-only case.
URBAN ENVIRONMENTS
The poor signal visibility and continuity associated with urban environments, together with the slow (re)convergence time of PPP, usually make the technique unsuitable for land navigation in cities. However, as demonstrated in the previous section, using a dual-constellation not only improves the visibility conditions, but also reduces the PPP convergence time. Therefore, it might be possible to extend the applicability of PPP to land navigation in certain urban areas.
To assess the positioning performance of two-constellation GNSS in these constrained environments, we analyzed the signal availability and geometry of five different simulated sites in the neighborhood of the University College London (UCL) campus. We adopted building boundaries, which determine the minimum elevation angles above which GNSS signals can be received due to building obstruction. FIGURES 1 and 2 illustrate the location and the building boundaries for each site. FIGURE 3 shows the junction (site B) between Gower Street (site A) and University Street (site C).
FIGURE 1. Locations of the urban sites that are considered in the analysis.FIGURE 2. Building obstruction masks controlling satellite visibility for each site.FIGURE 3. Google Map image showing the junction (site B) between Gower Street (site A) and University Street (site C) in the midst of the University College London main campus.
When processing data from multi-constellation GNSS, the differences between the system time of the different constellations need to be considered. For this reason, when GPS and Galileo are used simultaneously for precise positioning, the Kalman filter state vector (in general) includes the three position components, the receiver clock offset, and the GPS-Galileo Time Offset (GGTO) — whether or not a predicted value might be available in a navigation message from one of the constellations. On the other hand, in PPP processing, the multi-constellation precise products used are based on the same system time, and therefore, in theory, it is not necessary to estimate the GGTO. However, existing intersystem biases may affect the PPP performance, and so it is advisable to estimate them in the Kalman filter.
Traditionally in PPP, the state vector also includes the residual zenith wet tropospheric delay and the carrier-phase ambiguities. Therefore, the minimum number of satellites required for GPS plus Galileo PPP is six. The geometry conditions are also an important factor for assessing the GNSS positioning performance. For land navigation, the horizontal dilution of precision (HDOP), which provides information about the achievable horizontal precision (and, assuming a bias-free solution, accuracy), is particularly relevant. For many land applications, such as precision agriculture and urban positioning, horizontal accuracy is more critical than vertical accuracy. Assuming that the ranging error in the carrier phase is 20 centimeters, to have decimeter-level horizontal accuracy HDOP needs to be no larger than 5. In most cases, HDOP values as small as 2 are desired.
TABLE 2 gives an overview of the visibility and geometry conditions at the selected sites. A dual-constellation (GPS and Galileo) receiver placed at one of the two road junctions will always, or almost always, see at least six satellites with an HDOP better than 5. At sites A and C, these minimum requirements for signal availability and geometry are met for more than 75 percent of the day. Obstructions due to high buildings, such as at site E, allows us to have at least six satellites for only 13 percent of the time.
TABLE 2. Percentage of epochs in 24 hours for which dual-constellation GNSS meets the minimum visibility (number of satellites, N) and geometry requirements (horizontal dilution of precision, HDOP).
From our preliminary study, it seems clear that high-accuracy positioning in urban environments is possible, but only in some areas where buildings are relatively short, providing good signal availability and geometry. Things can slightly improve by considering additional systems, such as GLONASS and BeiDou, and by exploiting the non-line-of-sight (reflected) signals. However, it is well known that an additional obstacle for PPP in urban environments is signal discontinuity. Indeed, when a GNSS receiver loses lock on the carrier, the positioning filter needs to be reinitialized, meaning that further tens of minutes are required before reconvergence.
To test the reconvergence time of PPP in transient signal environments, a pedestrian carrying a multi-GNSS receiver was simulated to be walking along the path in FIGURE 4. The receiver was simulated to be located for the first half hour of the simulation in the front yard of UCL’s Wilkins Building (where the simulation begins and ends), before starting to move. This is to allow the initial convergence of the PPP filter.
FIGURE 4. The measured trajectory of the simulated pedestrian kinematic test.
FIGURE 5 shows the visibility for a given GNSS satellite. Only the epochs when the receiver is moving are considered. Therefore, the first 30 minutes, when the receiver is static, are not included in the plot. Data gaps due to building obstructions are visible, with the largest being about 12 minutes and the average less than 2 minutes. As a consequence, the carrier-phase ambiguities need to be estimated all over again; and, as previously mentioned, this process usually requires tens of minutes before reconvergence.
FIGURE 5. Satellite availability during the kinematic test.
FIGURE 6 shows the HDOP and the number of visible satellites for the kinematic test, while FIGURE 7 shows the RMS, over 50 simulations, of the horizontal components of the positioning error when GPS L1 and L2 and Galileo E1 and E5, linearly combined into the IF combination, are processed in kinematic PPP mode with the POINT software. At the beginning of the kinematic test, when the HDOP is well below 5, the horizontal error is at the centimeter level, while, after 33 minutes from the beginning of the simulation, building obstructions don’t permit a converged solution below the 20-centimeter accuracy level.
FIGURE 6. Horizontal dilution of precision and number of visible satellites for the kinematic test.FIGURE 7. RMS of the position errors for the kinematic test.
This short example clearly demonstrates that two-constellation PPP has, in theory, the potential to precisely navigate ground vehicles in some urban environments; however, it is too sensitive to signal discontinuity. Slow solution reconvergence to the few decimeter/centimeter level still represents the main limitation to the use of PPP for high-accuracy applications in cities. Nonetheless, GPS plus Galileo PPP easily enables sub-meter-level horizontal accuracy for most of the simulations we have carried out. After signal loss, it only took a few tens of seconds to have a horizontal accuracy of better than a meter.
SMOOTHED CORRECTIONS
As an alternative to ambiguity-fixing methods aimed to improve the (re)convergence time, we propose a method that mitigates the effect of the ionosphere and which thereby reduces the reconvergence time of the PPP solution after initial convergence has been achieved. In this new approach, while the two-frequency carrier phases are linearly combined in the traditional IF combination, the uncombined pseudoranges are corrected by a pre-smoothed ionospheric delay (via a Hatch filter), computed using the geometry-free combination of two-frequency pseudoranges.
Once the Hatch filter has converged, ideally we have IF pseudoranges with lower noise than the traditional ones. Therefore, in case the PPP filter needs to restart, we can obtain a quicker reconvergence thanks to the lower noise on the ionosphere-corrected pseudoranges. Indeed, provided that the signal gap is not very large, the ionosphere smoothing filter doesn’t need to be restarted from the raw values.
It is possible to predict the ionospheric delay computed from two-frequency carrier-phase measurements using a linear fitting model from previous measurements within a sliding time window. As an example, high-rate data recorded on July 25, 2017, from station DAEJ in Daejeon, Republic of Korea, were used to analyze the ionosphere prediction error.
In FIGURES 8 and 9, the RMS of the prediction errors for different time windows have been plotted against the data gap length. The prediction error depends on both the time latency of the observation and the elevation angle of the satellite. It increases with the data gap length, but larger time windows can damp the divergence of the error. A time window of 120 seconds was used both for satellites above and below 30° elevation angle. In this case, the error for a 5-minute prediction is about 4 centimeters for a satellite above 30° and 7 centimeters for satellites with a low elevation angle. These values are much smaller than the noise in the pseudorange measurements and can, therefore, be neglected.
FIGURE 8. RMS of the prediction errors vs. data gap length for satellite elevation angles greater than 30°.FIGURE 9. RMS of the prediction errors vs. data gap length for satellite elevation angles less than than 30°.
Multi-Frequency Combinations. The method introduced in the previous section allows users to be free from the constraint of IF observables and, therefore, to look for multi-frequency combinations aimed to minimize the noise on the pseudoranges. The next-generation GNSS satellites will broadcast open signals over three frequencies. The triple-frequency, geometry-preserving combination aimed to reduce the noise, instead of mitigating the ionosphere, can be used for positioning purposes.
TABLE 3 summarizes the assumed values for the ratios ni between the noise on different GPS and Galileo pseudoranges and the ones on L1/ E1. FIGURE 10 shows a color map of the noise amplification factor associated with different linear combinations between GPS L1, L2 and L5. The x-axis is α3, the coefficient multiplying the pseudorange on L5 in the combination, while the y-axis is the ionosphere amplification factor of the triple-frequency combination with respect to L1, q. The noise for this combination can be as little as 0.57 times the noise on L1, while the corresponding ionosphere amplification factor is 1.49. Once the smoothed ionosphere correction has converged, we can potentially have an IF pseudorange 81 percent less noisy than the L1-L2 IF, and, therefore, a much faster reconvergence.
TABLE 3. Assumed noise, ni, on GPS and Galileo pseudoranges, i, and their ionospheric delay, q, with respect to L1/ E1.FIGURE 10. Geometry-preserving surface in the space q-α3-n (ionosphere amplification factor – L5 pseudorange multiplier – noise amplification factor) for GPS L1-L2-L5 combinations.
Similar conclusions can be drawn by considering Galileo signals. Using triple-frequency combinations with E1, E5a and E5b, we can obtain 81 percent less noise than E1-E5a IF, while a reduction of the noise in the IF pseudorange up to 90 percent was observed using E5 alone. Triple-frequency combinations involving E5 don’t bring such large improvements with respect to using E5 alone. Indeed, a maximum of 16 percent less noise can be registered when combining E1, E5a and E5 with respect to the E5 uncombined case. TABLE 4 illustrates the minimum noise amplification factor for each triple-frequency combination and its ionosphere amplification factor.
TABLE 4. Minimum noise achievable through GPS and Galileo triple-frequency pseudorange combinations and their ionospheric delay with respect to L1/ E1.
The noise associated with the ionosphere-corrected multi-frequency pseudorange combination is as large as meter level before converging to centimeter level. For this reason, a proper weighting method, which considers the varying noise on the ionosphere correction, needs to be employed.
To test the benefit of the new approach for the reconvergence time, three hours of simulated GPS and Galileo data from a static site in La Misere, Seychelles, were processed with the POINT software in kinematic mode. After 90 minutes, the PPP filter was forced to restart to simulate reconvergence. The multipath time constant was set to 5 seconds, which is a typical value for kinematic multipath. The performance of the traditional L1- L2 IF combination was compared with the triple-frequency pseudorange combination, corrected by the smoothed ionosphere delay coming from the Hatch filter.
FIGURE 11 shows the precision (RMS error over 50 simulations) of the horizontal components after filter restart. The new approach has much faster reconvergence than the traditional PPP method based on the IF combination. Indeed, while the traditional method takes about 11 minutes to have a horizontal error below 10 centimeters, using the low-noise combination, this accuracy is achieved after 171 seconds. Even better performance can be achieved considering the Galileo E5 signal (see FIGURE 12).
FIGURE 11. RMS error of the horizontal position components of static site using GPS data after filter restart.FIGURE 12. RMS error of the horizontal position components of static site using Galileo data after filter restart.
The E1-E5 IF combination requires 10 minutes for the horizontal convergence, while using E5 with the Hatch filter we have the horizontal solution converged in about 30 seconds. It is worth noticing that in the presence of static multipath, the proposed weighting method may lead to an overly optimistic weighting of the pseudorange measurements in the PPP filter and to a slower reconvergence of the positioning solution. Indeed, the long correlation time in the static multipath, of the order of a few minutes, makes it hard to filter out by the Hatch filter, hence the corrected measurements have larger errors than expected.
The effect of static multipath in the new configuration is visible in FIGURE 13, where the reconvergence of the horizontal component for the L1-L2 IF combination is compared with the new approach. In this case, the time constant of the simulated multipath was set to 1 minute. In this scenario, the triple-frequency low-noise combination corrected by the smoothed ionosphere combination quickly converges below 20 centimeters; however, it takes significantly longer than the L1-L2 IF combination to reach the 10-centimeter accuracy level.
FIGURE 13. RMS error of horizontal position component of static site using GPS data after filter restart with 1-minute multipath time constant.
Also, the new method was tested with the kinematic simulation as in the previous section. Here, the GPS triple-frequency combined pseudorange and Galileo E5 pseudorange (both corrected with the smoothed ionosphere) are processed in kinematic PPP mode with the POINT software. FIGURE 14 compares the RMS of the horizontal errors with the IF configuration. Less than a minute after the receiver lost lock on the satellites, the solution reconverged below the 20-centimeter level, while it took less than 30 seconds to go below 50 centimeters.
FIGURE 14. RMS error of the horizontal position components of kinematic trajectory using GPS and Galileo data and the smoothed ionosphere approach after filter restart.
CONCLUSIONS
In this article, we described a comparison that we carried out between GPS-only, Galileo-only and GPS plus Galileo PPP. Results based on simulated open-sky conditions demonstrated that Galileo performs better than GPS thanks to an assumed lower E1-E5a IF noise with respect to L1-L5. Two-constellation PPP enables faster (re)convergence compared to the single constellation case.
An analysis of GNSS signal availability, continuity and satellite geometry was also performed to study the feasibility of PPP in urban environments. Preliminary results, based on simulations, showed that dual-constellation (GPS plus Galileo) PPP is possible in urban areas with relatively short buildings in which a satellite minimum availability requirement is met most of the time. However, signal discontinuity still represents the major problem for traditional PPP in urban environments, due to long reconvergence times.
Finally, we proposed a new PPP configuration based on triple-frequency combinations, intended to minimize the noise on the pseudorange and corrected by a smoothed ionospheric delay. This configuration seems to provide faster reconvergence than the traditional PPP with the IF combination if applied to kinematic scenarios. In static applications, the very slow varying multipath error makes the proposed weighting method, based on the error in the smoothed ionosphere correction, overly optimistic. In such cases, the IF combination reconverges more quickly to high-accuracy levels better than 20 centimeters.
ACKNOWLEDGMENTS
The research described in this article was sponsored through a studentship agreement between the University of Nottingham and Rockwell Collins UK Limited. The article is based on the paper “Multi-Frequency Precise Point Positioning Using GPS and Galileo Data with Smoothed Ionospheric Corrections” presented at the 2018 IEEE/ION Position, Location and Navigation Symposium, held in Monterey, California, April 23–26, 2018. All figures attributed to the authors unless otherwise specified.
FRANCESCO BASILE is a postgraduate research student at the Nottingham Geospatial Institute of the University of Nottingham in the United Kingdom. He received his M.Sc. in space and astronautic engineering from the University of Rome – La Sapienza and his B.Sc. in aerospace engineering from the University of Naples – Federico II, both in Italy.
TERRY MOORE is the director of the Nottingham Geospatial Institute where he is the Professor of Satellite Navigation. He is a fellow and the president of the Royal Institute of Navigation (RIN) and also a fellow and a member of council of the Institute of Navigation (ION).
CHRIS HILL is an associate professor in the Faculty of Engineering at the University of Nottingham and a member of the Nottingham Geospatial Institute research group. He holds a Ph.D. in satellite laser ranging and he is a fellow of the RIN.
GARY MCGRAW is a technical fellow with the Rockwell Collins Advanced Technology Center in Cedar Rapids, Iowa. McGraw is a fellow of the ION and is a senior member of the IEEE.
ANDREW JOHNSON is a chief engineer at Rockwell Collions UK in Winnersh, Berkshire, United Kingdom. Johnson has a B.Sc. in electronic and electrical engineering from the University of Surrey in Guildford, United Kingdom.
FURTHER READING
Authors’ Conference Paper
“Multi-Frequency Precise Point Positioning Using GPS and Galileo Data with Smoothed Ionospheric Corrections” by F. Basile, T. Moore, C. Hill, G. McGraw and A. Johnson in Proceedings of PLANS 2018, the Institute of Electrical and Electronics Engineers / Institute of Navigation Position, Location and Navigation Symposium, Monterey, California, April 23–26, 2018, pp. 1388–1398, doi: 10.1109/PLANS.2018.8373531.
“Multi-Constellation GNSS Performance Evaluation for Urban Canyons Using Large Virtual Reality City Models” by L. Wang, P.D. Groves and M.K. Ziebart in Journal of Navigation, Vol. 65, No. 3, July 2012, pp. 459–476, doi: 10.1017/S0373463312000082.
“Potential Benefits of GPS/GLONASS/GALILEO Integration in an Urban Canyon – Hong Kong” by S. Ji, W. Chen, X. Ding, Y. Chen, C. Zhao and C. Hu in Journal of Navigation, Vol. 63, No. 4, October 2010, pp. 681–693, doi: 10.1017/S0373463310000081.
Multi-Constellation Use in Aviation Applications
“Assessment of Alternative Positioning Solution Architectures for Dual Frequency Multi-Constellation GNSS/SBAS” by G. McGraw, B.A. Schnaufer, P.Y. Hwang and M.J. Armatys in Proceedings of ION GNSS+ 2013, the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville, Tennessee, Sept. 16–20, 2013, pp. 223–232.
“Undifferenced GPS Ambiguity Resolution Using the Decoupled Clock Model and Ambiguity Datum Fixing” by P. Collins, S. Bisnath, F. Lahaye, and P. Héroux in Navigation, Vol. 57, No. 2, Summer 2010, pp. 123–135, doi: 10.1002/j.2161-4296.2010.tb01772.x.
“Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and Its Application to PPP and Satellite Precise Orbit Determination” by D. Laurichesse, F. Mercier, J.-P. Berthias, P. Broca and L. Cerri in Navigation, Vol. 56, No. 2, Summer 2009, pp. 135–149, doi: 10.1002/j.2161-4296.2009.tb01750.x.
Hatch Filter
“Combinations of Observations” by A. Hauschild, Chapter 20 in Springer Handbook of Global Navigation Satellite Systems, edited by P.J.G. Teunissen and O. Montenbruck, published by Springer International Publishing AG, Cham, Switzerland, 2017.
“The Synergism of GPS Code and Carrier Measurements” by R. Hatch in Proceedings of the Third International Geodetic Symposium on Satellite Doppler Positioning, Las Cruces, New Mexico, Feb. 8–12, 1982, Vol. II, pp. 1213–1232.
Dilution of Precision
“Dilution of Precision” by R.B. Langley in GPS World, Vol. 10, No. 5, May 1999, pp. 52–59.
Kalman Filtering
“Least-Squares Estimation and Kalman Filtering” by S. Verhagen and P.J.G. Teunissen, Chapter 22 in Springer Handbook of Global Navigation Satellite Systems, edited by P.J.G. Teunissen and O. Montenbruck, published by Springer International Publishing AG, Cham, Switzerland, 2017.
The technique of precise point positioning (PPP) is making inroads in the positioning industry. However, one issue hampering its more widespread adoption is the convergence time required for the carrier-phase ambiguities to be fully resolved so that the 10-centimeter-accuracy threshold can be surpassed. By using a multi-system, multi-carrier-frequency approach, instantaneous centimeter-level PPP can be achieved.
Innovation Insights with Richard Langley
CARRIER PHASE. It’s one of the two main measurement types or observables used by all GNSS receivers. Fundamentally, it is the instantaneous phase of a GNSS signal’s carrier, an electromagnetic wave of fixed amplitude and frequency (when transmitted), which is (optionally) modulated by a ranging code and a navigation message. It’s measured in radians, degrees or cycles and can be converted to a biased measure of the range between the receiver and satellite antennas by multiplying the value in cycles by the wavelength of the carrier in meters. The other GNSS observable is the phase of the ranging code. Initially measured in code chips or units of time, it is converted to a biased measure of the receiver-satellite range by multiplying it by the speed of light. This value is then typically called the code measurement or the pseudorange. The carrier phase is much more precise than the pseudorange by something like a factor of 100. So, while pseudoranges can be measured to a precision of tens of centimeters, carrier phases can be measured to millimeters or better.
Most GNSS receivers use pseudorange measurements to determine their position. In fact, this is the standard approach to satellite-based positioning that was introduced by GPS in the 1970s. While carrier-phase measurements, or rather their time-rate-of-change, are used for precise velocity determination, it wasn’t originally recognized that carrier-phase measurements could be used for position determination, too. The problem with the carrier phase as a measure of the range is that it has an initially unknown and potentially huge bias. This is because when a receiver starts tracking a signal’s carrier, it doesn’t know the exact number of cycles of the carrier wave stretching all the way from the satellite to the receiver. Hence, carrier-phase measurements are ambiguous as a result of this initial bias. If this ambiguity can be resolved, then carrier-phase measurements can be used for very precise positioning — positioning at the centimeter level or even better.
Over the years, various techniques have been developed to use carrier-phase measurements for positioning, most notably in differential positioning where one or more reference stations are used to position a user receiver or rover. But the technique of precise point positioning, which only requires direct uncombined measurements from the user receiver, is being actively developed and is making inroads in the positioning industry. However, one continuing issue hampering its more widespread adoption is the convergence time required for the carrier-phase ambiguities to be fully resolved so that the 10-centimeter-accuracy threshold can be surpassed. Research by the authors of this month’s article shows that by using a multi-system, multi-carrier-frequency approach, instantaneous centimeter-level PPP can be achieved. They call their technique Optimal Estimation using Uncombined Four-frequency Signals or OEUFS for short. Those of us who remember a smattering of our high-school French will agree that it is quite an eggceptional technique.
Instantaneous centimeter-level positioning used to be synonymous with the single-baseline real-time kinematic (RTK) technique. The rover was constrained to be within a few kilometers of the base station to ensure that errors would remain spatially correlated. Modeling error sources using a regional network of stations later allowed users to retain this level of accuracy within the area of network coverage. A global network of reference stations enabled the determination of precise satellite orbit and clock products, paving the way for precise point positioning (PPP).
Global centimeter-level accuracy can be achieved with PPP, at the cost of a long convergence time, often measured in hours. An additional layer of corrections, including satellite code (pseudorange) and carrier-phase biases, has enabled PPP with ambiguity resolution (PPP-AR). While an improvement in convergence time can be obtained, PPP-AR still cannot compete with RTK or network RTK in terms of time to first fix. Only by providing precise atmospheric information to PPP users, in the form of zenith tropospheric and slant ionospheric delays, can instantaneous centimeter-level accuracy be obtained. This approach led to a unification of PPP and RTK, often referred to as PPP-RTK. This scalable approach has allowed PPP users to obtain accurate positioning globally, while achieving rapid convergence when located within the regional reference network boundaries.
The modernization of GNSS includes satellites transmitting signals on multiple frequencies. The 12 GPS Block IIF satellites currently in orbit already broadcast the L5 signal, and all Galileo and BeiDou satellites launched so far have triple-frequency capabilities. In November 2017, the BeiDou constellation began a new phase of its development with the launch of the Beidou-3S satellites offering new signals compatible with the GPS L1/L5 bands. In March 2018, the European Union decided to open its Commercial Service (CS), offering at no cost the signal and correction stream for the “CS high accuracy” service. As a result, the E6 signal is now available on 14 satellites and can be tracked by modern GNSS receivers. FIGURE 1 depicts the frequency plan of the open GNSS signals, including these last evolutions, as of May 2018.
FIGURE 1. GNSS open signals (as of May 2018). (Image: authors)
With three or more frequencies, a series of widelane ambiguities can be resolved in a cascading scheme. These unambiguous widelane signals can be used to form an ionosphere-free phase measurement with lower noise than code measurements, but typically still at the decimeter level. The availability of the Galileo E6 signal provides a significant step forward for PPP-AR, permitting instantaneous convergence. As a result of frequency separation, unambiguous widelane signals have low noise characteristics, which further benefits the resolution of the whole set of ambiguities. The strategy used in our study is a generalization of the widelaning technique, based on uncombined observations, which we describe as Optimal Estimation using Uncombined Four-frequency Signals (OEUFS).
We explain how instantaneous centimeter-level PPP is achieved by first analyzing the precision of the ambiguity and range parameters in the single-satellite case. The network estimation of the uncombined Galileo phase biases is then described, followed by epoch-by-epoch and 5-minute PPP solutions based on OEUFS.
SINGLE-SATELLITE PROCESSING
To get a first grasp of the benefits of using four frequencies, we first look into single-satellite data. The aim of this analysis is twofold: first, to evaluate the ability of fixing linear combinations of ambiguities and, second, to determine the resulting precision of the unbiased range estimate once these ambiguities are fixed.
Uncombined observations on four Galileo frequencies (E1, E5a, E5b and E6) are used to model an ionosphere-free range, a slant ionospheric delay, and four carrier-phase ambiguities. It should be noted that measurements on a fifth frequency (E5) are available but, due to the proximity of E5 with respect to E5a and E5b, its impact was found to be almost negligible. We will, therefore, restrict ourselves to the four-frequency case. Only two code observations are included in the model — in this case E1 and E5a — since adding other frequencies would require the estimation of differential code biases. Thus, for single-epoch processing, additional code measurements would not usefully contribute to the solution. Observable standard deviations are set to 3 millimeters and 30 centimeters for carrier phase and code, respectively. An analysis using a zero-length baseline revealed that weak correlations do exist between signals, and multipath effects could further increase this correlation. Although taking into consideration correlations among observations would lead to a more realistic covariance matrix, these correlations were neglected in producing the results shown in this article. This is justified by the fact that correlation coefficients are usually not available, especially for real-time processing.
The above-mentioned model was inverted in a least-squares adjustment to perform covariance analysis. While the Least‐squares AMBiguity Decorrelation Adjustment (LAMBDA) method can be used for the identification of optimal linear combinations of ambiguities, the classic widelane ambiguities were found to perform equally well and were used in our work to simplify the exposition. When no ambiguities are fixed, the quality of the solution is driven by the noise on the code observations. TABLE 1 shows that, in this case, the receiver-satellite range parameter can be estimated with a precision of 0.776 meters. This value can be translated into a 3D-position precision by using the position dilution of precision (PDOP) factor. As a rule of thumb, if the PDOP for all satellites in view is equal to 1, the resulting 3D precision should be around 78 centimeters.
TABLE 1. Precision of parameters in the Galileo four-frequency (E1, E5a, E5b, E6) single-satellite case.
Even though the range is not very precise, forming the E5a-E5b widelane ambiguity from the estimated uncombined ambiguities gives a precision of 0.034 cycles, which can be reliably fixed due to the very long wavelength of the signal (9.77 meters). Adding this constraint to the system allows us to estimate the E5b-E6 widelane ambiguity with a standard deviation of 0.041 cycles (although it could also have been fixed initially). Interestingly, fixing both extra-widelane ambiguities does not significantly improve the precision of the range information derived from a single satellite. Nevertheless, due to correlations among ambiguity parameters, a precision of 0.183 cycles is now obtained for the E1-E5a widelane, an improvement of approximately 35 percent over the initial estimate.
While the E1-E5a ambiguity is not sufficiently precise for reliable instantaneous fixing based on single-satellite data from one epoch, using the geometric information from several satellites will enable single-epoch ambiguity resolution for three widelane ambiguities per satellite, as we show in the following sections. Assuming for the moment that ambiguity resolution was indeed successful on all three widelanes, Table 1 indicates that the range parameter can now be estimated with a standard deviation of 19 centimeters, a substantial improvement over the initial 78-centimeter precision. Recalling the PDOP factor introduced above, instantaneous 3D position precision at the 20-centimeter mark should then be possible with good geometry.
Including all available measurements in the model necessarily leads to the best performance. Still, TABLE 2 presents the conditional precision of parameters in three-frequency configurations. The precision for the widelane ambiguity is conditioned on first fixing the extra-widelane ambiguity, while that for the range assumes fixed extra-widelane and widelane ambiguities. The table highlights that frequency spacing plays a key role in the system performance. After fixing two widelane ambiguities, the Galileo E1-E5a-E5b configuration provides a range with a standard deviation of approximately 42 centimeters. The E1-E5a-E6 configuration is the best option, with a precision of the range parameter equal to the four-frequency case. In other words, the contribution of the E5b signal is almost negligible once the E5a-E6 ambiguity, having a wavelength of 2.93 meters, is resolved. For comparison purposes, the values for GPS are included and show that Galileo has the potential for significantly more precise instantaneous positioning.
TABLE 2. Conditional precision of parameters for three-frequency single-satellite configurations.
NETWORK SOLUTION
To demonstrate the concept of four-frequency ambiguity resolution for PPP, a phase-bias network solution for the Galileo constellation must be generated. Our solution is based on the precise satellite orbit and clock corrections produced by the Centre National d’Études Spatiales (CNES) as a part of the International GNSS Service (IGS) Multi-GNSS Experiment (MGEX). These products contain satellite clock corrections at a 30-second interval, as well as widelane biases allowing for GPS ambiguity resolution in the L1 and L2 frequency bands. For this reason, the following analysis considers both GPS and Galileo constellations.
Consistent processing of multi-frequency and multi-modulation signals requires code-bias corrections. The differential code-bias products from the German Aerospace Center (DLR), including the Galileo E6 signals, are used. Ambiguity resolution for Galileo can only be enabled with corresponding phase biases for all frequencies. To this date, the main contributors to the IGS for E6-compatible receivers are Natural Resources Canada (NRCan), CNES and Geoscience Australia. Since a global network of ground receivers tracking all four Galileo frequencies is not yet available, our solution is computed from a regional, but wide-area, network in Australia. The network consists of six reference stations with multi-system, multi-frequency receivers as depicted with red triangles in FIGURE 2. (Station CEDU is not included in the network solution because it is used later as a rover for PPP testing.) Measurements collected at a 30-second interval are retrieved from the Crustal Dynamics Data Information System (CDDIS) data archive. For the purpose of our demonstration, data from April 1, 2018, from 13:45:00 to 14:35:00 GPS Time is selected. During this period, five Galileo satellites were continuously tracked by the Australian stations, allowing the computation of a Galileo-only solution.
The phase-bias solution is a generalization in the multi-frequency case of the well-known widelane/narrowlane GPS scheme. The first step consists of resolving all integer ambiguities in the network. As we deal with four frequencies, it is required to fix four ambiguities, or their combinations, per satellite-station pass. The first three combinations used for this study are the widelanes defined from E5a-E1, E5b-E1 and E6-E1. Their ambiguities are solved, as for the dual-frequency GPS case, thanks to the Melbourne-Wübbena combination. Then, one remaining integer ambiguity (here, E1) is solved by forming the ionosphere-free phase combination between E1 and E5a (with the corresponding widelane ambiguity already resolved as an integer value). The second step aims at recovering the uncombined phase biases from the estimated linear combinations of biases. By a simple system inversion, it is possible to reconstruct the phase biases on each frequency.
FIGURE 2. Stations used to generate the Galileo phase-bias solution are represented by red triangles, while the PPP user is represented by a black square. (Image: authors)
FIGURE 3 shows the estimated biases for each frequency over the study period. The values were shifted by an integer number of the carrier wavelength for plotting purposes. The uncombined biases obtained are relatively stable, although they vary by a few centimeters over this one-hour period. These fluctuations are correlated among frequencies due to the transformation from linear combinations to uncombined biases. It should be understood that the resulting biases are not true phase biases, but rather biases to be applied to the carrier-phase observations.
FIGURE 3. Estimated Galileo phase biases for the four frequency bands over the study period. (Image: authors)
PRECISE POINT POSITIONING
We assessed the impact of using four frequencies transmitted by Galileo (E1, E5a, E5b and E6) on positioning performance by using station CEDU in Australia (see Figure 2). It is equipped with a multi-frequency receiver collecting multi-GNSS observations at 30-second intervals. Position estimates are derived from the PPP methodology using the satellite orbit and clock corrections, along with the carrier-phase and code biases, described in the previous section.
We computed three different solutions:
a GPS-only solution;
a Galileo-only solution; and
a GPS and Galileo combined solution.
For all solutions, all error sources affecting observations are modeled, including relativistic and wind-up effects, solid Earth tides and ocean loading. The a priori tropospheric zenith delay (TZD) is computed using the Vienna Mapping Function 1 (VMF1) grids, while a priori ionospheric delays are obtained from a global ionospheric map (GIM) generated at the Center for Orbit Determination in Europe (CODE). The eccentricity between the satellite antenna phase centers and the satellite center of mass is obtained from the latest version of the IGS ANTEX file, which includes frequency-dependent phase-center offsets and variations for Galileo. Since there are no Galileo-specific ground-antenna calibrations available, GPS values are used as approximations.
In all cases, we processed uncombined observations corresponding to the OEUFS strategy. For GPS, the L1C and L2W carrier-phase observations are used, along with the C1W and C2W code observations. For Galileo, the L1C, L5Q, L6C and L7Q carrier phases are used, with identical modulations for code measurements. Note that this signal identification uses the RINEX 3 conventions where, for Galileo, the L5 and L7 signals correspond to those in the E5a and E5b bands, respectively. Carrier-phase observations are given a standard deviation of 2 millimeters at zenith, while code observations are deweighted by a factor of 100. An elevation-angle-dependent weighting strategy also assigns lesser weight to satellites closer to the local horizon. Therefore, the value of 3 millimeters used in the single-satellite analysis above corresponds to a satellite tracked at an elevation angle of approximately 40 degrees.
The PPP filter includes states for the three position components, one receiver clock parameter per satellite system, inter-frequency code biases, one phase-bias parameter per frequency, a residual TZD, a residual slant ionospheric delay per satellite and carrier-phase ambiguities. To confirm the theoretical analysis from a previous section, the empirical single-epoch ambiguity-fixing success rate is first evaluated using a bootstrapping algorithm. The full vector of estimated float ambiguities is first decorrelated using the LAMBDA method, and all ambiguities having a success rate larger than 99 percent are fixed to integers. FIGURE 4 shows the number of fixed ambiguities for each solution.
FIGURE 4. Number of fixed ambiguities using a bootstrapping approach for independent, single-epoch, solutions. Number of frequencies in parentheses. (Image: authors)
Not surprisingly, the dual-frequency GPS solution is incapable of reliably fixing ambiguities within a single epoch. During this time period, five Galileo satellites are tracked. If we first consider all four frequencies from Galileo, and use the ambiguities on one satellite to provide the datum, then a total of 16 ambiguities are being estimated in the PPP filter, 12 of which are considered widelanes. Figure 4 confirms that using correlations introduced by the geometry allows instantaneous fixing of all widelane ambiguities for Galileo for most epochs. Adding GPS to the Galileo solution makes Galileo widelane fixing more reliable, but does not allow fixing of additional ambiguities. The three-frequency (E1, E5a and E6) Galileo configuration also enables instantaneous fixing of all eight widelane ambiguities, since the inclusion of E5b brings minimal additional information.
In all subsequent solutions, ambiguity estimation is performed using a more sophisticated method referred to as the best integer equivariant (BIE) approach. Because it is expected that not all ambiguities can be fixed simultaneously, a partial ambiguity resolution scheme is required. The BIE method fulfills this criterion by computing a weighted average of integer vectors. The outcome is a constrained ambiguity vector whose entries take either integer or float values. The key point of this approach is that the BIE float estimates can be improved by the averaging process with respect to the least-squares float estimates. Furthermore, by exploiting the correlations contained in the ambiguity covariance matrix, this method can effectively fix linear combinations of ambiguities. Therefore, we are not explicitly forming widelane ambiguities, but rather optimal linear combinations of ambiguities are fixed through the BIE averaging process. This strategy is implemented using the LAMBDA method to decorrelate ambiguities. Even though the BIE estimates are independent of the decorrelation, this step improves the computational efficiency of the approach.
As we explained in the previous sections, positioning with fixed widelane ambiguities can potentially allow for instantaneous precise positioning. FIGURE 5 demonstrates the epoch-by-epoch position estimates for the three solutions. As the strategy implies, the filter is entirely reset between epochs, and each point in the time series is independently determined. As expected, instantaneous ambiguity resolution with GPS alone is not feasible. Although the external information provided by the GIM is beneficial in reducing the errors, the root-mean-square (RMS) error is at the decimeter level for all components (see TABLE 3).
FIGURE 5. Instantaneous (epoch by epoch) PPP-AR solutions for GPS only, Galileo only and GPS and Galileo combined. Number of frequencies in parentheses. (Image: authors)TABLE 3. RMS errors for each instantaneous PPP-AR solution (meters).
The Galileo-only solution offers a substantial improvement in the horizontal components. These results are explained by the ambiguity-resolved widelane signals providing precise range estimates. It should be noted that only five Galileo satellites are visible during this period with a PDOP slightly exceeding a value of 3. When the full constellation of satellites will be in orbit, even better results could be obtained from a Galileo-only solution. The three-frequency (E1, E5a, E6) Galileo solution offers almost identical position estimates and is not shown here for conciseness. Combining GPS and Galileo yields the best solution with centimeter-level instantaneous positioning (refer to Table 3). For several epochs, a fully converged position is even obtained within a single epoch.
While the RMS errors of the combined GPS + Galileo solution is at the centimeter level, individual epochs can still exhibit decimeter-level errors. To demonstrate the convergence capabilities of the OEUFS strategy, we computed 5-minute PPP sessions. Even though the station is stationary, we added a large amount of process noise to the position states to simulate kinematic processing. FIGURE 6 shows the results of all 10 sessions: horizontal convergence to a few centimeters could be achieved within two epochs in all but one session.
FIGURE 6. Independent 5-minute kinematic PPP solutions using GPS and Galileo. Each trace represents a different session. (Image: authors)
CONCLUSION
We have shown that GNSS modernization is a key component for reducing the convergence time of PPP solutions. Combining multiple constellations strengthens the geometry, and using multiple frequencies allows for improved ambiguity resolution performance. In particular, tracking of the E6 Galileo commercial service signal turns out to be particularly beneficial in terms of instantaneous positioning capabilities. We demonstrated that ambiguities can be instantaneously resolved on Galileo satellites, leading to a range estimate approximately four times better than that provided using code measurements. With good satellite geometry, these frequencies can enable instantaneous 3D positioning with an accuracy of around 20 centimeters. Combining Galileo and GPS allows for single-epoch centimeter-level PPP solutions and full convergence within a few epochs.
We expect that the robustness and accuracy of the OEUFS strategy will improve in the future, with an increasing number of multi-frequency satellites and ground stations. Specifically, the additional frequencies provided by BeiDou and the Quasi-Zenith Satellite System will enhance the geometry of the solution and will further expedite convergence. Within a few years, instantaneous PPP might very well become a practical alternative to RTK for a wide range of applications.
ACKNOWLEDGMENTS
The authors acknowledge Geoscience Australia for making publicly available modernized GNSS data, as well as Paul Collins from NRCan for the review of our manuscript and technical advice. This article is published as NRCan Contribution 20180102.
MANUFACTURER
All of the stations used for the tests described in this article have PolaRx5 reference receivers manufactured by Septentrio (www.septentrio.com).
DENIS LAURICHESSE is a member of the Navigation Systems Department at CNES in Toulouse, France. He has been in charge of the DIOGENE GPS orbital navigation filter, and is now involved in navigation algorithms for GNSS. He is in charge of the CNES IGS real-time analysis center. Laurichesse was the co-recipient of the 2009 Institute of Navigation Burka Award for his work on phase ambiguity resolution.
SIMON BANVILLE is a senior geodetic engineer with the Canadian Geodetic Survey of NRCan, Ottawa, Canada, working on PPP. He obtained his Ph.D. degree in 2014 from the Department of Geodesy and Geomatics Engineering at the University of New Brunswick, under the supervision of Richard B. Langley. He is the recipient of the Institute of Navigation 2014 Parkinson Award.
“Precise Point Positioning” by J. Kouba, F. Lahaye and P. Tétreault, Chapter 25 in Springer Handbook of Global Navigation Satellite Systems, edited by P.J.G. Teunissen and O. Montenbruck, published by Springer International Publishing AG, Cham, Switzerland, 2017.
• Multi-GNSS Experiment
“The Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS) – Achievements, Prospects and Challenges” by O. Montenbruck, P. Steigenberger, L. Prange, Z. Deng, Q. Zhao, F. Perosanz, I. Romero, C. Noll, A. Stürze, G. Weber, R. Schmid, K. MacLeod and S. Schaer in Advances in Space Research, Vol. 59, No. 7, April 2017, pp. 1671–1697, doi: 10.1016/j.asr.2017.01.011.
“Zero-difference GPS Ambiguity Resolution at CNES–CLS IGS Analysis Center by S. Loyer, F. Perosanz, F. Mercier, H. Capdeville, and J.C. Marty in Journal of Geodesy, Vol. 86, No. 11, Nov. 2012, pp. 991–1003, doi: 10.1007/s00190-012-0559-2.
“Undifferenced GPS Ambiguity Resolution Using the Decoupled Clock Model and Ambiguity Datum Fixing” by P. Collins, S. Bisnath, F. Lahaye and P. Héroux in Navigation, Vol. 57, No. 2, Summer 2010, pp. 123–135, doi: 10.1002/j.2161-4296.2010.tb01772.x.
“Carrier Phase Integer Ambiguity Resolution” by P.J.G. Teunissen, Chapter 23 in Springer Handbook of Global Navigation Satellite Systems, edited by P.J.G. Teunissen and O. Montenbruck, published by Springer International Publishing AG, Cham, Switzerland, 2017.
“Theory of Integer Equivariant Estimation with Application to GNSS” by P.J.G. Teunissen in Journal of Geodesy, Vol. 77, No. 7-8, Oct. 2003, pp. 402–410, doi: 10.1007/s00190-003-0344-3.
“In partnership with Ligado Networks, we have demonstrated delivery of TerraStar X technology over both satellite and cellular networks to position vehicles with 5-centimeter (2-inch) accuracy in under a minute,” Hexagon stated in a press release. “Combining TerraStar X technology with multiple delivery channels is a significant step towards the future of Autonomous X, where cars, UAVs, industrial vehicles, trains and more will operate safely, securely, reliably and efficiently.
TerraStar X technology is built on the latest precise point positioning algorithms. According to the company, it leverages existing Hexagon capabilities in ground network infrastructure, correction data generation and data packaging for delivery.
By eliminating convergence time while providing high-accuracy global positioning, TerraStar X will form the future of Hexagon’s correction services for safety-of-life applications and Autonomous X.
When combined with automotive-grade GNSS receivers available through Hexagon Positioning Intelligence, the technology allows automotive customers to evaluate positioning performance in real time using data delivered over the cellular network or the L-band frequency using Ligado’s SkyTerra satellite in North America.
Trial networks for customer evaluation are available in California, Arizona and Michigan over satellite or cellular network, and in Germany using cellular delivery. The infrastructure is scalable, enabling timely geographic expansion to accommodate automotive development programs globally.
Commercial solutions designed for the automotive market will be available in 2019.
“Ligado’s expertise in satellite delivery and proactive involvement in this project enabled rapid deployment of our TerraStar X correction technology over the test area,” said Sara Masterson, positioning services segment manager with Hexagon’s Positioning Intelligence division. “Their unique spot-beam technology enables efficient delivery of the higher bandwidth correction data required for this application and adds a delivery method providing continental scale coverage.”
The geostationary Skyterra satellite operated by Ligado uses a 22-meter reflector-based antenna to deliver an L-band signal over North America. Several of the L-band DGPS/PPP service providers, including Terrastar, have used the Skyterra-1 satellite since its 2010 launch to support North American coverage.
Hexagon has been providing highly reliable, precise GNSS corrections under VERIPOS, TerraStar, Oceanix, and SmartNet brands for more than 20 years, the company said. It operates the world’s largest reference station network, consisting of more than 4,500 stations.
“Hexagon is uniquely positioned to offer end to end solutions from correction data generation through to GNSS positioning solutions in the vehicle,” said Brian Deobald, vice president, strategic product and ecosystem development, Ligado Networks. “We are excited to partner with Hexagon on this opportunity to demonstrate the delivery of TerraStar X technology, using high throughput, cost-efficient satellite connectivity to enable superior performance and reliability for autonomous driving applications.”
Ligado. This development has no relationship to the current Ligado Networks petition before the Federal Communications Commission to repurpose some of its mobile satellite systems spectrum to broadcast from ground-based transmitters. That matter is still pending, and there is currently no such signal being broadcast.
In this month’s column, we travel along the road of PPP development, examine its current status and look at where it might go in the near future
By Sunil Bisnath, John Aggrey, Garrett Seepersad and Maninder Gill
Innovation Insights with Richard Langley
PPP. It’s one of the many acronyms (or initialisms, if you prefer) associated with the uses of global navigation satellite systems. It stands for precise point positioning. But what is that? Isn’t all GNSS positioning precise? Well, it’s a matter of degree.
Take GPS, for example. The most common kind of GPS signal use, that implemented in vehicle “satnav” units; mobile phones; and hiking, golfing and fitness receivers, is to employ the L1 C/A-code pseudorange (code) measurements along with the broadcast satellite orbit and clock information to produce a point position.
Officially, this is termed use of the GPS Standard Positioning Service (SPS). It is capable of meter-level positioning accuracy under the best conditions. There is a second official service based on L1 and L2 P-code measurements and broadcast data called the Precise Positioning Service (PPS).
In principle, because the P-code provides somewhat higher precision code measurements and the use of dual-frequency data removes virtually all of the ionospheric effect, PPS is capable of slightly more precise (and accurate) positioning. But because the P-code is encrypted, PPS is only available to so-called authorized users.
While meter-level positioning accuracy is sufficient for many, if not most applications, there are many uses of GNSS such as machine control, surveying and various scientific tasks, where accuracies better than 10 centimeters or even 1 centimeter are needed. Positioning accuracies at this level can’t be provided by pseudoranges alone and the use of carrier-phase measurements is required. Phase measurements are much more precise than code measurements although they are ambiguous and this ambiguity must be estimated and possibly resolved to the correct integer value.
Traditionally, phase measurements (typically dual-frequency) made by a potentially moving user receiver have been combined with those from a reference receiver at a well-known position to produce very precise (and accurate) positions. If done in real time (through use of a radio link of some kind), this technique is referred to as real-time kinematic or RTK.
A disadvantage of RTK positioning is that it requires reference station infrastructure including a radio link (such as mobile phone communications) for real-time results. Is there another way? Yes, and that’s PPP. PPP uses the more precise phase measurements (along with code measurements initially) on at least two carrier frequencies (typically) from the user’s receiver along with precise satellite orbit and clock data derived, by a supplier, from a global network. Precision, in this case, means a horizontal position accuracy of 10 centimeters or better.
In this month’s column, we travel along the road of PPP development, examine its current status, and look at where it might go in the near future.
In a 2009 GPS World “Innovation” article co-authored by Sunil Bisnath, the performance and technical limitations at the time of the precise point positioning (PPP) GPS measurement processing technique were described and a set of questions asked about the potential of PPP, especially with regard to the real-time kinematic (RTK) measurement processing technique.
Since the 2009 article, we’ve seen a significant amount of research and development (R&D) activity in this area. Many scientific papers discuss PPP and making use of PPP — a search on Google Scholar for “GNSS PPP” delivers nearly 7,000 results, and for “GPS PPP” more than 15,000 results! Will PPP eventually overtake RTK as the de facto standard for precise (that is, few centimeter-level) positioning? Or, in light of PPP R&D developments, should we be asking different questions, such as will multiple precise GNSS positioning techniques compete or complement each other or perhaps result in a hybrid approach?
In almost a decade, have we seen much in the way of positioning performance improvement, where “performance” can refer to positioning precision, accuracy, availability and integrity? Or, to some users, has the Achilles’ heel of PPP — the initial position solution convergence period — only been reduced from, for example, 20 minutes to 19 minutes? From such a perspective, all of this PPP research might not appear to have produced much tangible benefit. Advances have been made from this research and we will explore them here. Also, aside from many researchers working diligently on their own PPP software, there are now a number of well-established PPP-based commercial services — a number that has grown and been affected by the wave of GNSS industry consolidation over the decade. Consequently, there is much more to this story.
This month’s article summarizes the current status of PPP performance and R&D, and discusses the potential future of the technique. In the first part of the article, we will present brief explanations of conventional dual-frequency PPP, recent research and implementations, and application of the evolved technique to low-cost hardware. We will conclude the article with a rather dangerous attempt at near-term extrapolation of potential upcoming developments and conceivable implications.
Conventional PPP
The concept of PPP is based on standard, single-receiver, single-frequency point positioning using pseudorange (code) measurements, but with the meter-level satellite broadcast orbit and clock information replaced with centimeter-level precise orbit and clock information, along with additional error modeling and (typically) dual-frequency code and phase measurement filtering. Back in 1995, researchers at Natural Resources Canada were able to reduce GPS horizontal positioning error from tens of meters to the few-meter level with code measurements and precise orbits and clocks in the presence of Selective Availability (SA). Subsequently, the Jet Propulsion Laboratory introduced PPP as a method to greatly reduce GPS measurement processing time for large static networks. When SA was turned off in May 2000 and GPS satellite clock estimates could then be more readily interpolated, the PPP technique became scientifically and commercially popular for certain precise applications.
Unlike static relative positioning and RTK, conventional PPP does not make use of double-differencing, which is the mathematical differencing of simultaneous code and phase measurements from reference and remote receivers to greatly reduce or eliminate many error sources. Rather, PPP applies precise satellite orbit and clock corrections estimated from a sparse global network of satellite tracking stations in a state-space version of a Hatch filter (in which the noisy, but unambiguous, code measurements are filtered with the precise, but ambiguous, phase measurements). This filtering is illustrated in FIGURE 1, where measurements are continually added in time in the range domain, and errors are modeled and filtered in the position domain, resulting in reduced position error in time.
FIGURE 1. Illustration of conventional PPP measurement and error modeling in state-space Hatch filter, resulting in reduced position error in time.
The result is the characteristic PPP initial convergence period seen in FIGURE 2, where the position solution is initialized as a sub-meter, dual-frequency code point positioning solution, quickly converging to the decimeter-level in something like 5 to 20 minutes, and a few centimeters after ~20 minutes when geodetic-grade equipment is used (at station ALGO, Algonquin Park, Canada, on Jan. 2, 2017). For static geodetic data, daily solutions are typically at the few millimeter-level of accuracy in each Cartesian component.
FIGURE 2. Conventional geodetic GPS PPP positioning performance characteristics of initial convergence period and steady state for station ALGO, Algonquin Park, Canada, on Jan. 2, 2017.
The primary benefit of conventional PPP is that with the use of state-space corrections from a sparse global network, there is the appearance of precise positioning from only a single geodetic receiver.
Therefore, baseline or network RTK limitations are removed in geographically challenging areas, such as offshore, far from population centers, in the air, in low Earth orbit, and so on, and without the need for the requisite terrestrial hardware and software infrastructure. PPP is now the de facto standard for precise positioning in remote areas or regions of low economic density, which limit or prevent the use of relative GNSS, RTK or network RTK, but allow for continuous satellite tracking. These benefits translate into the main commercial applications of offshore positioning, precision agriculture, geodetic surveys and airborne mapping, which also are not operationally bothered by initial convergence periods of tens of minutes.
For urban and suburban applications, RTK and especially network RTK allow for near-instantaneous, few-centimeter-level positioning with the use of reference stations and regional satellite (orbit and clock) and atmospheric corrections. The use of double-differencing and these local or regional corrections allows sufficient measurement error mitigation to resolve double-differenced phase ambiguities. All of this additional information is not available to conventional PPP, limiting its precise positioning performance, but which is considered in PPP enhancements.
Progress on PPP Convergence Limitations
Over the past decade or so, PPP R&D activity can be categorized as follows:
Integration of measurements from multiple GNSS constellations, transitioning from GPS PPP to GNSS PPP;
Resolution of carrier-phase ambiguities in PPP user algorithms — in an effort to increase positional accuracy and solution stability, but foremost in an effort to reduce the initial convergence period; and
Use of a priori information to reduce the initial convergence and re-convergence periods and improve solution stability, making use of available GNSS error modeling approaches.
Unlike relative positioning, which makes use of measurements from the user receiver as well as the reference receiver, PPP only relies on measurements from the user site. This situation results in weaker initial geometric strength, and so the addition of more unique measurements is welcome. To make use of measurements from all four GNSS constellations (GPS, GLONASS, Galileo and BeiDou), user-processing engines must account for differences in spatial and temporal reference systems between constellations and numerous equipment delays between frequencies and modulations. The former can be done so that any number of measurements from any number of constellations can be processed to produce one unique PPP position solution. The latter requires a great deal of calibration, especially for heterogeneous tracking networks and user equipment (antenna, receiver and receiver firmware), most notably for the current frequency division multiple access GLONASS constellation.
FIGURE 3 shows typical multi-GNSS float (non-ambiguity-fixed) horizontal positioning performance at multi-GNSS station GMSD in Nakatane, Japan, on March 24, 2017. As with all modes of GNSS data processing, more significant improvement with additional constellations can be seen in sky-obstructed situations.
FIGURE 3. Typical conventional multi-GNSS PPP float horizontal positioning accuracy for station GMSD, Nakatane, Japan, March 24, 2017 (G: GPS, R: GLONASS, E: Galileo and C: BeiDou).
Related to multi-constellation processing is triple-frequency processing afforded by the latest generation of GPS satellites and the Galileo and BeiDou constellations. More frequencies mean more measurements, although with the same satellite-to-receiver measurement geometry as dual-frequency measurements. Again, additional signals require additional equipment delay modeling, in this case especially for the processing of GPS L1, L2 and L5 observables.
For processing of four-constellation data available from 20 global stations in early 2016, FIGURE 4 shows the average reduction of float (non-ambiguity-fixed) horizontal error from dual- to triple-frequency processing of approximately 40% after the first five minutes of measurement processing. In terms of positioning, this result, for this time period with a limited number of triple-frequency measurements, means a reduction in average horizontal positioning error from 43 to 26 centimeters within the first five minutes of data collection.
FIGURE 4. Average dual- and triple-frequency static, float PPP horizontal solution accuracy for 20 global stations. Data collected from tracked GPS, GLONASS, Galileo and BeiDou satellites in early 2016.
PPP with ambiguity resolution, or PPP-AR, was seen as a potential solution to the PPP initial solution convergence “problem” analogous to AR in RTK. Various researchers put forward methods, in the form of expanded measurement models, to isolate pseudorange and carrier-phase equipment delays to estimate carrier-phase ambiguities. These methods remove receiver equipment delays through implicit or explicit between-satellite single-differencing and estimate satellite equipment delays in the network product solution either as fractional cycle phase biases or altered clock products.
FIGURE 5 illustrates the difference between a typical GPS float and fixed solution (for station CEDU, Ceduna, Australia, on June 28, 2017). Initial solution convergence time is reduced, and stable few-centimeter-level solutions are reached sooner. For lower quality data, ambiguity fixing does not provide such quick initial solution convergence. Fixing is dependent on the quality of the float solution; and, for PPP, the latter requires time to reach acceptable levels of accuracy. Therefore, depending on the application, PPP-AR may or may not be helpful.
FIGURE 5. Typical float (red) and fixed (pink) GPS PPP horizontal solution error at geodetic station CEDU, Ceduna, Australia, on June 28, 2017.
To consistently reduce the initial solution convergence period, PPP processing requires additional information, as is the case for network RTK, in which interpolated satellite orbit, ionospheric and tropospheric corrections are needed since double-differenced RTK baselines over 10 to 15 kilometers in length contain residual atmospheric errors too large to effectively and safely resolve phase integer ambiguities. For PPP, uncombining the ionospheric-free code and phase measurements from the conventional model is required, to directly estimate slant ionosphere propagation terms in the filter state.
In this form, the model can allow for very quick re-initialization of short data gaps by using the pre-gap slant ionospheric (and zenith tropospheric) estimates as down-weighted a priori estimates post-gap — making these estimates bridging parameters in the estimation filter. Expanding this approach, external atmospheric models can be used to aid with initial solution convergence.
FIGURE 6 illustrates, for a large dataset, that applying a spatially and temporally coarse global ionospheric map (GIM) to triple-frequency, four-constellation float processing can reduce one-sigma convergence time to 10 centimeters horizontal positioning error from 16 to 6 minutes. If local ionospheric (and tropospheric) corrections are available and AR is applied, PPP (sometimes now referred to as PPP-RTK) can produce RTK-like results with a few minutes of initial convergence to few-centimeter-level horizontal solutions.
FIGURE 6. Averaged horizontal error from 70 global sites in mid-2016 using four-constellation, triple-frequency processing.
PPP Processing with Low-Cost Hardware
As the impetus for low-cost, precise positioning and navigation for autonomous and semi-autonomous platforms (such as land vehicles and drones) continues to grow, there is interest in processing such low-cost data with PPP algorithms. For example, it has been shown that with access to single-frequency code and phase measurements from a smartphone, short-baseline RTK positioning is possible. It has also been shown that similar smartphone data can be processed with the PPP approach. From the origins of PPP, it may be argued that single-frequency processing and many-decimeter-level positioning performance is not “precise.” But we will avoid such semantic arguments here (but see “Insights”), and focus on the use of high-performance measurement processing algorithms to new low-cost hardware. We are currently witnessing great changes in the GNSS chip market: single-frequency chips for tens-of-dollars or less; and boards with multi-frequency chips for hundreds-of-dollars. And these chips will continue to undergo downward price pressure with increases in capability, and be further enabled for raw measurement use in a wider range of applicable technology solutions. There are now a number of low-cost, dual-frequency, multi-constellation products on the market, with additional such products as well as smartphone chips coming soon.
To process data from such products with a PPP engine, modifications are required to optimally account for single-frequency measurements in the estimation filter, optimize the measurement quality control functions for the much noisier code and phase measurements compared to data from geodetic receivers, and optimize the stochastic modeling for the much noisier code and phase measurements. The single-frequency measurement model can be modified to either make use of the Group and Phase Ionospheric Calibration linear combination (commonly referred to as GRAPHIC) or ingest data from an ionospheric model. Due to the use of low-cost antennas, as well as the low-cost chip signal processing hardware, code and phase measurements suffer from significant multipath and noise at lower signal strengths; therefore, outlier detection functions must be modified. Also, the relative weighting of code and phase measurements must be customized for more realistic low-cost data processing.
FIGURE 7 compares the carrier-to-noise-density ratio (C/N0) values from ~1.5 hours of static GPS L1 signals collected from a geodetic receiver with a geodetic antenna, a low-cost receiver chip with a patch antenna, and a tablet chip and internal antenna, as a function of elevation angle. Received signal C/N0 values can be used as a proxy for signal precision. The three datasets were collected at the same time in mid-September 2017 in Toronto, Canada, with the receivers and antennas within a few meters of each other. The shading represents the raw estimates output from each receiver, while the solid lines are moving-average filtered results.
FIGURE 7. Carrier-to-noise-density ratios of ~1.5 hour of static GPS L1 signals from a geodetic receiver with a geodetic antenna, a low-cost receiver chip with a patch antenna, and a tablet chip and internal antenna, as a function of elevation angle.
Keeping in mind the log nature of C/N0, the high measurement quality of the geodetic antenna and receiver are clear. The low-cost chip and patch antenna signal strength structure is similar, but, on average, 3.5 dB-Hz lower. And the tablet received signal strength is lower still, on average a further 4.0 dB-Hz lower, with greater degradation at higher signal elevation angles and much greater signal strength variation.
The PPP horizontal position uncertainty for these datasets is shown in FIGURE 8. Note that reference coordinates have been estimated from the datasets themselves, so potential biases, in especially the low-cost and tablet results, can make these results optimistic. Given that only single-frequency GPS code and phase measurements are being processed, initial convergence periods are short and horizontal position error reaches steady state in the decimeter range. The geodetic and the low-cost results are comparable at the 2-decimeter level, whereas the tablet results are worse, at the approximately 4-decimeter level. Initial convergence of the geodetic solution is superior to the others, driven by the higher quality of its code measurements. The grade of antenna plays a large role in the quality of these measurements, for which there are physical limitations in design and fabrication. While geodetic antennas can be used, this is not always feasible, given the mass limitations of certain platforms or the cost limitations for certain applications.
FIGURE 8. Horizontal positioning error (compared to final epoch solutions) for geodetic, low-cost and tablet data processed with PPP software customized for single-frequency and less precise measurements.
Comments Regarding the Near Future
The PPP GNSS measurement processing approach was originally designed to greatly reduce computation burden in large geodetic networks of receivers by removing the need for network baseline processing. The technique found favor for applications in remote areas or regions with little terrestrial infrastructure, including the absence of GNSS reference stations. Given PPP’s characteristic use of a single receiver for precise positioning, various additional augmentations have been made to remove or reduce solution initialization and re-initialization interval to near RTK-like levels. But, to what end?
This question can be approached from multiple perspectives. From the theoretical standpoint, there is the impetus to maximize performance — millimeter-level static positioning over many hours, and few-centimeter-level kinematic positioning in a few minutes — by augmenting PPP in any way necessary. There is the academic exercise of maximizing performance without the need for local or regional reference stations – apparent single-receiver positioning, or truly wide-area augmentation. In terms of engineering problems, we can work to do more with less, that is, decimeter-level positioning with ultra-low-cost hardware, or the same with less, that is, few-centimeter-level positioning with low-cost hardware. And from the practical or commercial aspect, the great interest is for the implementation of evolved PPP methods for applications that can efficiently and effectively make use of the technology.
In terms of service providers, be it regional or global, commercial or public, there is momentum to provide enhanced correction products that are blurring the lines across the service spectrum from constellation-owner tracking to regional, terrestrial augmentation. A public GNSS constellation-owner, through its constellation tracking network, can provide PPP-like corrections and services. A global commercial provider with or without regional augmentation can provide similar services. The key is providing multi-GNSS state-space corrections for satellite orbits, satellite clocks, satellite equipment delays (fractional phase biases), zenith ionospheric delay and zenith tropospheric delay at the temporal and spatial resolution necessary for the desired positioning performance at reasonable cost, that is, subscription fees that particular markets can bear.
Given these correction products, PPP users have a greater ability to access a wide array of positioning performance levels for various new applications, be it few-decimeter-level positioning on mobile devices to few-centimeter-level positioning for autonomous or semi-autonomous land, sea and air vehicles. PPP can be used for integrity monitoring and perhaps safety-of-life applications where low-cost is a necessity and relatively precise positioning for availability and integrity purposes is required. For safety critical and high-precision applications, such as vehicle automation, PPP can be used alongside, or in combination with, RTK for robustness and independence with low-cost hardware. Such a parallel and collaborative approach would require a hybrid user processing engine and robust state-space corrections from a variety of local, regional and global sources, as we are seeing from some current geodetic hardware-based commercial services.
Near-future trends should also include more low-cost, multi-sensor integration with PPP augmentation. Optimized navigation algorithms and efficient user processing engines will be a priority as the capabilities of low-cost equipment continue to increase and low-cost integrated sensor solutions are required for mass-market applications. Analogous to meter-level point position GNSS, lower hardware costs should drive markets to volume sales, PPP-like correction services, and GNSS-based multi-sensor integration into more navigation technology solutions for various industry and consumer applications.
Clearly, the future of PPP continues to be bright.
SUNIL BISNATH is an associate professor in the Department of Earth and Space Science and Engineering at York University, Toronto, Canada. For over twenty years, he has been actively researching GNSS processing algorithms for a wide variety of positioning and navigation applications.
JOHN AGGREY is a Ph.D. candidate in the Department of Earth and Space Science and Engineering at York University. He completed his B.Sc. in geomatics at Kwame Nkrumah University of Science and Technology, Ghana, and his M.Sc. at York University. His research currently focuses on the design, development and testing of GNSS PPP software, including functional, stochastic and error mitigation models.
GARRETT SEEPERSAD is a navigation software design engineer for high-precision GNSS at u-blox AG and concurrently is completing his Ph.D. in the Department of Earth and Space Science and Engineering at York University. His Ph.D. research focuses on GNSS PPP and ambiguity resolution. He completed his B.Sc. in geomatics at the University of the West Indies in Trinidad and Tobago. He holds an M.Sc. degree in the same field from York University.
MANINDER GILL is a geomatics designer at NovAtel Inc. and concurrently is completing his M.Sc. in the Department of Earth and Space Science and Engineering at York University. His M.Sc. research focuses on GNSS PPP and improving positioning accuracy for low-cost GNSS receivers. He holds a B.Eng. degree in geomatics engineering from York University.
FURTHER READING
• Comprehensive Discussion of Technical Aspects of Precise Point Positioning
“Precise Point Positioning” by J. Kouba, F. Lahaye and P. Tétreault, Chapter 25 in Springer Handbook of Global Navigation Satellite Systems, edited by P.J.G. Teunissen and O. Montenbruck, published by Springer International Publishing AG, Cham, Switzerland, 2017.
• Earlier Precise Point Positioning Review Article
“Precise Point Positioning Using IGS Orbit and Clock Products” by J. Kouba and P. Héroux in GPS Solutions, Vol. 5, No. 2, October 2001, pp. 12–28, doi: 10.1007/PL00012883.
“Precise Point Positioning for the Efficient and Robust Analysis of GPS Data from Large Networks” by J.F. Zumberge, M.B. Heflin, D.C. Jefferson, M.M. Watkins and E.H. Webb in Journal of Geophysical Research, Vol. 102, No. B3, pp. 5005–5017, 1997, doi: 10.1029/96JB03860.
“Reduction of PPP Convergence Period Through Pseudorange Multipath and Noise Mitigation” by G. Seepersad and S. Bisnath in GPS Solutions, Vol. 19, No. 3, March 2015, pp. 369–379, doi: 10.1007/s10291-014-0395-3.
“Global and Regional Ionospheric Corrections for Faster PPP Convergence” by S. Banville, P. Collins, W. Zhang and R.B. Langley in Navigation, Vol. 61, No. 2, Summer 2014, pp. 115–124, doi: 10.1002/navi.57.
“A New Method to Accelerate PPP Convergence Time by Using a Global Zenith Troposphere Delay Estimate Model” by Y. Yao, C. Yu and Y. Hu in The Journal of Navigation, Vol. 67, No. 5, September 2014, pp. 899–910, doi: 10.1017/S0373463314000265.
“External Ionospheric Constraints for Improved PPP-AR Initialisation and a Generalised Local Augmentation Concept” by P. Collins, F. Lahaye and S. Bisnath in Proceedings of ION GNSS 2012, the 25th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville, Tennessee, Sept. 17–21, 2012, pp. 3055–3065.
“Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and Its Application to PPP and Satellite Precise Orbit Determination” by D. Laurichesse and F. Mercier, J.-P. Berthias, P. Broca and L. Cerri in Navigation, Vol. 56, No. 2, Summer 2009, pp. 135–149.
“Resolution of GPS Carrier-phase Ambiguities in Precise Point Positioning (PPP) with Daily Observations” by M. Ge, G. Gendt, M. Rothacher, C. Shi and J. Liu in Journal of Geodesy, Vol. 82, No. 7, July 2008, pp. 389–399, doi: 10.1007/s00190-007. Erratum: doi: 10.1007/s00190-007-0208-3.
“Isolating and Estimating Undifferenced GPS Integer Ambiguities” by P. Collins in Proceedings of ION NTM 2008, the 2008 National Technical Meeting of The Institute of Navigation, San Diego, California, Jan. 28–30, 2008, pp. 720–732.
• Precise Positioning Using Smartphones
“Positioning with Android: GNSS Observables” by S. Riley, H. Landau, V. Gomez, N. Mishukova, W. Lentz and A. Clare in GPS World, Vol. 29, No. 1, January 2018, pp. 18 and 27–34.
Tallysman, a manufacturer of high-performance GNSS antennas and related products, has introduced a magnetic-mount triple-band (plus L-band) GNSS antenna, TW7972, and a dual-band antenna, TW7872.
They are designed for precision agriculture, autonomous vehicles, navigation, real-time kinematic (RTK), precise point positioning (PPP), and other applications where precision matters. The ability of the TW7972 to access L-Band correction services extends its utility to a wider range of applications.
The introduction of these antennas is a continuation of Tallysman’s expansion into broader band GNSS antennas. These antennas are the first releases in a line of new enclosures that will be used for additional broadband GNSS solutions.
Photo: Tallysman
The antennas employ Tallysman’s Accutenna technology.
The TW7972 is capable of receiving GPS L1/L2/L5, GLONASS G1/G2/G5, BeiDou B1/B2, Galileo E1/E5a+b and L-band correction services (1164 MHz to 1254 MHz + 1525 MHz to 1606 MHz).
The TW7872 is capable of receiving GPS L1/L2, GLONASS G1/G2, BeiDou B1 and Galileo E1.
The precisely tuned antennas have a tight pre-filter to protect against intermodulation and saturation caused by high-level cellular 700 MHz and other signals.
The antennas provide superior multi-path signal rejection, a linear phase response, and a tight phase-center variation (PCV) at a new economical price point, Tallysman said. The antennas provide comparable or superior performance to higher priced triple- and dual-band GNSS antennas on the market.
The TW7972 and TW7872 are housed in a magnetic-mount, IP67 weather-proof enclosure with pre-tapped screw holes. The antennas can also be ordered without the magnet.
The TW3967 (28-dB gain) and the TW3972E (35-dB gain) are the embedded versions of the TW7972. The TW3867 and TW3872E are the embedded versions of the TW7872. They are available with a wide selection of connectors and custom cable lengths, and can be custom tuned by Tallysman to ensure optimum performance within the customer’s enclosure.
French Space Agency CNES has signed a cooperation agreement with the company Geoflex, granting it the right to spin off software developed by CNES that employs satellite precise point positioning (PPP) technology.
Under the agreement, CNES is granting Geoflex a license to use its patented technologies in this field with a view to offering a global commercial operational service. This partnership ties in with the agency’s strategy of spinning off its research and development results.
The agreement was signed June 28 at the Toulouse Space Show by Lionel Suchet, CNES’s director of innovation, applications and science, and Romain Legros, chairman of Geoflex.
The Geoflex team was able to draw on more than 10 years’ experience in GNSS precise positioning when they founded their start-up to pursue this project. Through this cooperation agreement with CNES, Geoflex is set to benefit from significant opportunities worldwide in real-time precise positioning, navigation and timing, serving a broad customer base employing applications such as topographic mapping, construction and civil engineering, agriculture, shipping, rail, driverless vehicles and unmanned aerial systems.
“Today’s space technologies will drive revolutionary changes in usage in the future,” Suchet said after signing the agreement. “Through their commitment to developing a global GNSS precise positioning service, CNES and Geoflex are showing that France has a key role to play in innovating and in growing our future economy. The people at this start-up are looking to shake up the status quo, so it was natural that CNES should support them.”
Examining the interoperability of precise point positioning products
By Garrett Seepersad and Sunil Bisnath
INNOVATION INSIGHTS with Richard Langley
CARRIER PHASE. We’ve all heard the term and recognize it as a more precise observable for GNSS positioning, navigation and timing than code phase, more commonly called the pseudorange. The carrier-phase measurement is the phase of the received continuous radio-frequency sinusoidal waveform that “carries” the pseudorandom noise ranging codes and the navigation messages. The underlying carrier of a satellite signal can be recovered and its phase measured at regular intervals by the receiver once it locks onto the signal.
As long as there is no interruption in the carrier tracking, the receiver can generate a continuous series of measurements of the cumulative phase or cycle count including fractional cycles. The initial value at signal lock-on is arbitrary. Ideally, it would equal the exact number of cycles (and fractional cycle) of the waveform between the antenna of the satellite and the antenna of the receiver.
If that was the case, then we could simply multiply that cycle count by the wavelength of the carrier in meters, say, and we would have the initial geometric distance (or range) to the satellite. Then we could update this value as time progresses with the receiver’s measurements and have a continuous sequence of range values, which, when corrected for satellite and receiver clock errors and other effects, would allow the receiver’s position to be accurately determined. But because we don’t know the true initial cycle count, the carrier-phase measurements are ambiguous by a constant integer amount (when measured in cycles). This characteristic of the observable is referred to as the integer ambiguity.
It was realized early in the development of GPS, that if the integer ambiguity of carrier-phase measurements could be resolved, we would have a very precise observable for positioning, navigation and timing, some two orders of magnitude more precise than the code-based pseudorange. Instead of measurement precisions of tens of centimeters, we could have precisions of tenths of millimeters.
In the early 1980s using the few test GPS satellites in orbit at the time, surveyors and geodesists developed a series of clever techniques that allowed them to make use of carrier-phase measurements to determine the baseline between pairs of receivers by estimating combinations of the ambiguities as unknowns along with the receiver relative coordinates or, for short baseline work, use a calibration procedure before starting a survey.
Now jump forward a few decades. While it is still common practice to double difference carrier-phase measurements between pairs of satellites and pairs of receivers to determine relative receiver coordinates, the technique of precise point positioning or PPP, which uses carrier-phase (and pseudorange) measurements from a single user receiver, is growing in popularity. But, the integer ambiguity problem is still with us and has to be addressed by the analysis software. The ambiguities are often estimated as real- rather than integer-valued quantities, in part because of the contribution of satellite hardware biases to the carrier-phase measurements.
However, it is possible to resolve the ambiguities to integer values by using PPP ambiguity resolution products distributed by several research organizations. In this month’s column, we take a look at the interoperability of these products for increasing the reliability and precision of position solutions and reducing the time required for a solution to converge to a required level of accuracy.
Ambiguity resolution in precise point positioning (hereafter, PPP-AR) requires that hardware delays within the GPS measurements be mitigated, which will then allow for resolution of the integer ambiguities within the carrier-phase measurements. Resolution of these ambiguities converts the carrier-phases into precise “range” measurements, with measurement noise at the centimeter-to-millimeter level compared to the meter-to-decimeter level of the C/A- and P(Y)-code pseudoranges. If the ambiguities could be isolated and estimated as integers, then that information could be exploited to accelerate PPP convergence to provide, for example, few-centimeter horizontal positioning accuracy within tens of minutes or even minutes from a cold start.
Integer ambiguity resolution of measurements from a single receiver can be implemented by applying additional satellite products, where the fractional component — representing the satellite hardware delay — has been separated from the integer ambiguities in a network solution. One method of deriving such products is to estimate the satellite hardware delay by averaging the fractional parts of steady-state real-valued or floating-point (float) ambiguity estimates, and the other is to estimate the receiver clock offset in the pseudorange and carrier-phase measurements independently by fixing the undifferenced ambiguities to integers in advance.
Similar positioning performances have been demonstrated among three approaches of different groups or agencies using the two methods: FCB (Fractional Cycle Bias), IRC (Integer Recovery Clock) and DC (Decoupled Clock). For the PPP user, the mathematical model is similar. The different PPP-AR products contain the same information and, as a result, should allow for one-to-one transformations, allowing interoperability of the PPP-AR products. The advantage of interoperability of the various products is to allow the PPP user to transform independently generated products to obtain multiple fixed solutions of comparable precision and accuracy, with no changes to the core PPP user software. An overview of the different providers and their products is presented in FIGURE 1.
FIGURE 1. Public providers of PPP-AR products. (Source: Richard Langley)
The ability to use different products would increase the reliability of a positioning solution in real-time processing, for example. If there was an outage in the generation of a particular PPP-AR product, a user could instantly switch streams to a different provider. The research presented in this article examines the PPP-AR products generated from the FCB and IRC models that have been transformed into the DC format and applied within a PPP user solution. The novelty of the research is the solution analysis using the transformed product. We examine the convergence time (time-to-first-fix and time to a pre-defined performance level), position precision (repeatability), position accuracy and solution outliers. The temporal and spatial behavior of these estimated terms is examined for the different products applied to understand the unmodeled effects responsible for incorrect solution fixes.
The Role of PPP-AR Products
The standard GPS pseudorange ( and ) and carrier-phase () observation equations are given by
(1)
where i denotes the frequency-dependent GPS measurements for frequencies L1 or L2, s represents the tracked satellite, r represents the receiver, is the geometric range between the satellite s and the user position, T is the tropospheric delay, is the first order slant ionospheric delay, γi is the frequency dependent coefficient, dts is the satellite clock and is the pseudorange hardware delay. is the ambiguity term and is the carrier-phase hardware delay, both of which are expressed in cycles and scaled by the wavelength λi . The error sources can be grouped into two main components, the geometric parameters and the timing parameters. Included in the timing parameters are the clock offsets and the hardware delay terms. Understanding the role of the hardware delays is critical in isolating the integer ambiguities.
The following equations illustrate the effects of not mitigating the hardware delay. The set of equations was simplified by combining the clock and hardware delay parameters. Processing the carrier-phase measurements with the pseudoranges (code measurements) ensures that the pseudoranges provide a reference for the carrier-phase measurements and for the clock parameters. An implication of this is the manifestation of the hardware delay present in both the estimated clock parameters and the ambiguities.
(2)
By not mitigating the hardware delay terms ( and), they are absorbed within the estimated ambiguity terms, rendering the integer nature of the ambiguity term inaccessible. The user observation equations do not contain sufficient information to solve for an integer-ambiguity-resolved user position. Ambiguity resolution would only become possible if information about the satellite hardware delays were provided to the user. The receiver hardware delay can be removed by single differencing (between satellites).
In the following section, we present an overview of the different public providers of products that enable PPP-AR, their 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 are Scripps Institution of Oceanography, which provides regional real-time FCB products; Natural Resources Canada (NRCan), which provides post-processed and real-time DC products; and Centre National d’Etudes Spatiales (CNES), which provides post-processed and real-time IRC products.
FCB Model. The initial application of ambiguity resolution to PPP was the Uncalibrated Phase Delay (UPD) model, now called the Fractional Cycle Bias (FCB) model. The FCB method estimates the hardware delay by averaging the fractional parts of the steady-state float ambiguity estimates to be removed from common satellite clock estimates. The FCB products consist of , and , where WN indicates the Melbourne-Wübbena (widelane ambiguity) combination and IF indicates the ionosphere-free linear combination.
DC Model. The underlying concept of the decoupled clock model is that the carrier-phase and pseudorange (code) measurements are not synchronized with each other at an equivalent level of precision. The timing of the different observables must be considered separately if they are to be processed together rigorously. The decoupled clock model is a reformulation of the ionosphere-free carrier-phase and pseudorange observation equations. When combined with the narrowlane pseudorange and the widelane phase, ambiguity resolution is possible. The DC products transmitted to the user are , and .
IRC Model. The integer recovery clocks estimate constant daily widelane pseudorange/carrier-phase hardware delays by averaging arc-dependent estimates. Using float-solution estimates of the range parameters, narrowlane ambiguity resolution is performed and the ionosphere-free satellite carrier-phase clocks are estimated. In 2014, the format of the IRC products was changed from , and to a state-space uncombined representation, such that the satellite hardware delay is provided for each observable (, ) and satellite pseudorange clock ().
Summary. The three publicly provided products to enable real-time PPP-AR are listed in TABLE 1 along with their primary characteristics. The table summarizes the various measurements used, different products transmitted and the varying data rate of the transmitted products.
TABLE 1. Comparison of different publicly provided real-time products to enable PPP-AR.
Product Transformation
While the different strategies (FCB, FC, IRC) make different assumptions, there are fundamental similarities among them. The mathematical models for the PPP user are similar, as the different products contain the same information and as a result would allow for a one-to-one transformation. The following sections examine the transformation matrix used to transform the IRC and FCB products to the DC format (see Figure 2.)
FIGURE 2. Transformation of FCB and IRC products to DC input format.
FCB. The FCB products consist of , and , which are estimated in the network solution using International GNSS Service (IGS) ultra-rapid orbit and clock products. The fundamental difference between the FCB and DC products is that is not determined in the DC method, but assimilated within the clock estimates. Also, is assumed constant over a 48-hour time period, whereas in the DC method the is neither constrained nor smoothed. Here is the transformation matrix used to transform from FCB to the DC model:
(3)
where z1 is the single-differenced L1 ambiguity and zw is the single-differenced widelane ambiguity.
IRC. The original IRC products used a decoupled-like approach, where independent clocks ( and ) were transmitted for the pseudorange and carrier-phase measurements and widelane satellite hardware delays () were estimated. A redefined model was presented in 2014, where a state-space approach was adopted such that one phase bias per phase observable ( and ) was identified and broadcast. The primary benefit of such an approach is interoperability, allowing the network and user side to implement different ambiguity resolution methods. Here is the transformation matrix used to transform from IRC to the DC model: (4)
where d12 represents .
Analysis of Transformed Products. In FIGURES 3 to 5, we present the FCB and IRC products transformed to the DC format. The presented format was selected because it represents the nature of the transmitted real-time DC products. The philosophy of the DC model refers to the satellite hardware delay as an unmodeled timing error and, as such, the satellite carrier-phase clocks in Figure 3 are in units of seconds and in Figures 4 and 5 are in units of nanoseconds. Nanoseconds were selected because of the magnitude of the relative satellite pseudorange and widelane clock error, as well as being more bandwidth efficient.
FIGURE 3. Transformed FCB and IRC satellite carrier-phase clock correction on day-of-year 28 of 2015 for PRN 10. DC was included for comparison.
Figure 3 illustrates the FCB and IRC products transformed to the DC satellite carrier-phase clock. The satellite-clock corrections presented were not differenced with respect to a reference satellite, to illustrate their differences in an absolute nature. If the clocks are differenced, in a relative nature, they are equivalent. The data gaps in the FCB products are expected because of the regional nature of the products. Unlike the DC and IRC products, the FCB pseudorange clocks illustrate different trends such as those between hours 3 and 4. The noise illustrated in the IRC clock can be removed either by filtering or by differencing with respect to another satellite clock.
In Figure 4, we present the relative satellite clock error ( − ) for the transformed FCB (upper subplot) and IRC (lower subplot) products. For the original DC product (middle subplot), a simple moving average filter was applied with a bin size of five minutes to reduce the noise and illustrate the underlying equipment delay. The relative satellite clock error represents the difference between the pseudorange and carrier-phase clocks. The distinct differences of the products are easily visible, such as the filtering present within FCB and IRC products in contrast to the DC. The underlying relative satellite clock error is also significantly different in contrast to the DC product, such that FCB and IRC have an average relative satellite clock error of -0.041 ± 0.101 nanoseconds and -0.645 ± 0.005 nanoseconds, respectively, whereas the DC has an average of 8.465 ± 1.546 nanoseconds.
FIGURE 4. Transformed FCB and IRC products to code-phase relative clock correction on day-of-year 28 of 2015 for PRN 10. DC was included for comparison. Linear trend has been removed.
Figure 5 shows the relative satellite widelane clock error for the transformed FCB (upper subplot) and IRC (lower subplot) products. For the original DC product (middle subplot), a simple moving average filter was applied with a bin size of five minutes, to reduce the noise and illustrate the underlying equipment delay. The relative satellite clock error represents the difference between the widelane clocks and phase clocks. Similar to the relative satellite clock error, the differences in the transformed relative satellite widelane clock error are noticeable. As expected, the transformed FCB has a constant widelane estimate of -0.24 nanoseconds, whereas the transformed IRC and DC have an average widelane estimate of 0.0589 ± 0.002 and 3.6704 ± 0.34 nanoseconds, respectively.
FIGURE 5. Transformed FCB and IRC products to code-phase relative widelane clock correction on day-of-year 28 of 2015 for PRN 10. DC was included for comparison. Linear trend has been removed.
Performance of Transformed Products
One of the metrics we can use to examine the performance of the transformed products is the quality of the solution in the position domain. The solutions were examined with respect to the time for convergence to a pre-defined threshold and position stability. We used five stations from the Scripps Orbit and Permanent Array Center (SOPAC) network for days 23 to 30 of 2015. These five stations were selected because of the regional nature of FCB products provided by SOPAC. We show the results for site Brand Basin (BRAN) on day-of-year 30 of 2015 as it reflects the performance of the whole dataset processed.
In FIGURES 6 to 8, we show the varying convergence periods at the site BRAN on day-of-year 30 for the “float” and “fixed” solutions using the different PPP-AR products, where fixed means the ambiguity-resolved solution and float the unresolved solution. Figure 6 uses the decoupled clock products, and the fixed solution performs as expected. After a few minutes, the solution attains the correct ambiguity candidate, and a fixed state is maintained.
FIGURE 6. Position errors for site BRAN located in Burbank, Calif., on day-of-year 30 of 2015 illustrating the difference between the float and fixed solutions using the DC products.FIGURE 7. Position errors for site BRAN located in Burbank, Calif., for day-of-year 30 of 2015 illustrating the difference between the float and fixed solutions using the IRC products.FIGURE 8. Position errors for site BRAN located in Burbank, Calif., for day-of-year 30 of 2015 illustrating the difference between the float and fixed solutions using the FCB products.
The performance of the fixed solution using the IRC products is depicted in Figure 7. Initial convergence is similar to the DC products in the northing and easting components where a fixed state is attained after a few epochs. In the up component, the solution quality deteriorates after 30 minutes. What is also easily visible is the solution sensitivity to changes in the satellite geometry. As the number of satellites changes, the fixed ambiguities change, causing datum shifts in the user solution.
Similar trends were also observed when the transformed FCB products were used, with the results presented in Figure 8. The solution deterioration is most evident in the easting component, as the incorrect integer candidate is selected.
Challenges of Interoperability
Interoperability of the various PPP-AR products is a challenging task because of the different qualities of the publicly available products, limited literature documenting the conventions adopted within the network solution of the providers, and unclear definitions of the corrections.
In TABLE 2, we summarize the various qualities of the products we used in the study, showing why it was challenging to perform a consistent comparison. IRC products were generated from a network of reference stations globally distributed and in real time. Similar to the IRC products, the DC products were generated from a global network of solutions, but post-processed, and the FCB products were based on a regional network of reference stations, but were available in real time. Post-processed orbits and clocks have an accuracy of ~2.5 centimeters and ~75 picoseconds, respectively, whereas the predicted half of ultra-rapid orbits and clocks have an accuracy of ~5 centimeters and ~3 nanoseconds, respectively. While it is evident in the existing literature that PPP-AR is possible in real time, the solution is rather sensitive to changes experienced by the PPP user solution, such as varying local conditions and satellite geometry. The sensitivity is illustrated in Figures 7 and 8 with solution jumps typically occurring when there is a change in the number of satellites.
TABLE 2. Summary of the different quality of products provided by public providers to enable PPP-AR.
The general assumption when PPP-AR products are estimated within a network is that the PPP user would follow similar conventions when using the products. Consequences of different conventions adopted may result in incorrect ambiguities being resolved. For example, if inconsistent satellite antenna conventions were adopted between the network and user, then when phase wind-up corrections are applied, fractional cycles would be introduced. The introduced fractional cycles would result in incorrect ambiguities being resolved. FIGURE 9 shows the orientation of the spacecraft body frame for GPS Block IIR/IIR-M satellites adopted in the IGS axis convention (subplot (a)) and those provided in the manufacturer specifications (subplot (b)). The difference between the manufacturer specifications and IGS axis convention is the orientation of the x- and y-axes.
FIGURE 9. Orientation of the spacecraft body frame for GPS Block IIR/IIR-M satellites as (a) adopted within the International GNSS Service axis convention, and (b) those provided in the manufacturer specifications.
Conclusions
The mathematical model for the PPP user is similar for all PPP-AR products, as the different products contain the same information and, as a result, would allow for one-to-one transformations, allowing interoperability of the PPP-AR products. Interoperability of the various 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. The ability to provide multiple solutions would increase the reliability of the solution such as in 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.
We looked at the PPP-AR products provided by three organizations and examined position solutions for a set of stations in the SOPAC network with respect to convergence time to the pre-defined threshold and position stability.
Using the decoupled clock products, we found that the fixed solutions performed as expected. After a few minutes, a solution attains the correct ambiguity candidate and a fixed state is maintained. Unlike the fixed solutions using the decoupled clock products, instantaneous convergence was not attained in the horizontal and vertical components when the transformed IRC and FCB products were used. The ambiguity-resolved solutions were sensitive to changes in the satellite geometry. As the number of satellites change, the fixed ambiguities change, causing datum shifts in the user solution.
The unstable solutions from both transformed products are attributed to the magnitude of the relative satellite code and widelane clock errors. Additional refinement of the transformation model is required as the satellite hardware delay has not been completely mitigated. Mismodeling of the hardware delay was absorbed by the ambiguity terms, causing incorrect fixed solutions.
Future Research
Future prospective research includes refinement of the proposed transformation models to include the mismodeled effects, thus providing the user with a more reliable solution. The functional model needs to be further examined to ensure that the corrections were applied consistently. Further analysis of the instability of the user solution is required, as solution jumps typically occur when there are changes in the number of satellites tracked. Also to be analyzed are the post-fit residuals, to examine the effects of mismodeling. The temporal and spatial behavior of the estimated terms will be examined for the different products used to understand the unmodeled effects that introduce incorrect solution fixes. We would also consider increasing the number of reference stations to further test the reliability of the transformed products under varying user conditions.
Acknowledgments
We acknowledge Paul Collins, Jianghui Geng and Denis Laurichesse for our valuable discussions and their suggestions. The research was funded by the Natural Sciences and Engineering Research Council of Canada. The results we have presented were derived from data and products provided by Natural Resources Canada, Scripps Institution of Oceanography, Centre National d’Etudes Spatiales and the International GNSS Service.
This article is based on the paper “Examining the Interoperability of PPP-AR Products” presented at ION GNSS+ 2015, the 28th International Technical Meeting of The Satellite Division of the Institute of Navigation held in Tampa, Fla., Sept. 14–18, 2015.
GARRETT SEEPERSAD is a Ph.D. candidate at York University, Toronto, Canada, in the Department of Earth and Space Science and Engineering. He completed his B.Sc. in geomatics at the University of the 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.
SUNIL BISNATH is an associate professor in the Department of Earth and Space Science and Engineering at York University. His research interests include geodesy and precise GNSS positioning and navigation.
Further Reading
• PPP Ambiguity Resolution Techniques
“Review and Principles of PPP-RTK Methods” by P.J.G. Teunissen and A. Khodabandeh in Journal of Geodesy, Vol. 89, No. 3, 2014, pp. 217–240, doi: 10.1007/s00190-014-0771-3.
“A Novel Un-differenced PPP-RTK Concept” by B. Zhang, P.J.G. Teunissen and D. Odijk in Journal of Navigation, Vol. 64, Supplement S1, 2011, pp. S180–S191, doi: 10.1017/S0373463311000361.
“Isolating and Estimating Undifferenced GPS Integer Ambiguities” by P. Collins in Proceedings of the 2008 National Technical Meeting of The Institute of Navigation, San Diego, Calif., January 28–30, 2008, pp. 720–732.
“Resolution of GPS Carrier-Phase Ambiguities in Precise Point Positioning (PPP) with Daily Observations” by M. Ge, G. Gendt, M. Rothacher, C. Shi and J. Liu in Journal of Geodesy, Vol. 82, No. 7, 2008, pp. 389–399, doi: 10.1007/s00190-007-0187-4.
“Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and Its Application to PPP” by D. Laurichesse and F. Mercier in Proceedings of ION GNSS 2007, the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation, Fort Worth, Texas, Sept. 25–28, 2007, pp. 839–848.
Improved Convergence for GNSS Precise Point Positioning by S. Banville, Ph.D. dissertation, Department of Geodesy and Geomatics Engineering, Technical Report No. 294, University of New Brunswick, Fredericton, New Brunswick, Canada. Recipient of The Institute of Navigation 2014 Bradford W. Parkinson Award.
Real-time single-frequency precise point positioning for cars and trains
By Peter de Bakker and Christian Tiberius
INNOVATION INSIGHTS with Richard Langley
“IT’S GETTING BETTER ALL THE TIME.”This refrain from the Beatle’s song could well describe precise point positioning or PPP. PPP is a positioning technique that relies on GNSS carrier-phase measurements (in addition to code or pseudorange measurements) from a user’s receiver along with satellite orbit and clock data much more precise (and accurate) than that included in broadcast satellite navigation messages to achieve accuracies down to the centimeter level. It also requires a more sophisticated model of the measurements compared to that used in most consumer GNSS equipment and even some professional devices, including accounting for residual tropospheric propagation delay, carrier-phase windup, and even solid Earth tides.
PPP has been around for more than a decade and ongoing research has gradually improved its capabilities. Until recently, it has been used primarily with dual-frequency GPS observables. However, the technique is not restricted to GPS. It works equally well with observables from other constellations including GLONASS, Galileo and BeiDou. As long as precise orbit and clock products are available (typically from the International GNSS Service or its participating analysis centers), then PPP positioning solutions are possible. And, single-frequency PPP is also possible. The primary advantage of dual-frequency PPP is that the ionospheric propagation delay is almost completely removed by linearly combining the measurements on the two frequencies, taking advantage of the dispersive nature of signal propagation through the ionosphere. But, if good predictions of the ionospheric delay at, say, the L1 GPS frequency are available, then it is possible to do single-frequency PPP. While not as accurate as dual-frequency PPP, the technique is considerably more accurate than typical pseudorange point positioning (the so-called Standard Positioning Service).
PPP is also traditionally a post-processing technique. That is, data is collected but it is not processed until some later convenient time when the necessary precise products are available. Such an approach is useful for many applications but clearly not for navigation, which requires real-time positioning. But in the past few years, a number of commercial and non-commercial entities have started streaming real-time satellite orbit and clock corrections over the Internet and various radio links, making real-time PPP a reality.
In this month’s Innovation column, we bring together, perhaps for the first time, single-frequency and real-time PPP. Our authors describe a series of experiments they have conducted on roadways and a railway achieving sub-meter horizontal positioning at a 95 percent confidence interval. Such accuracies may already be sufficient for freeway lane and railway track guidance. But we might expect even better accuracies in the future. After all, PPP is getting better all the time.
The single-frequency precise point positioning (SF-PPP) method, developed at Delft University of Technology, was previously demonstrated to provide lane-level position accuracy on a freeway in post-processing mode. Important applications of SF-PPP are lane-level traffic state estimation and lane-level specific driver advice for next-generation car navigation. For a functional system, as well as for advanced experiments in this field, the computed positions have to be available in real time. Therefore, a new real-time implementation of the SF-PPP method was developed as part of the Dutch Dynamic Lane Guidance project. In this article, we outline aspects of the real-time implementation, and we present experimental results from this new implementation collected on a busy freeway in the Netherlands and in a parking lot, as well as results from a railway experiment.
In these experiments, a test vehicle was equipped with a low-end, automotive-type single-frequency receiver with a patch antenna to collect raw GPS observations. A 3G mobile communications link was used to obtain data-correction streams over the Internet using the Ntrip protocol. The SF-PPP processing was performed on a laptop computer onboard the vehicle, in real time. Various forms of ground-truth positions were used to assess the real-time SF-PPP positioning accuracy. For some of our tests, the vehicle was also equipped with high-end GPS antennas and receivers to provide ground truth. The position solutions obtained with the SF-PPP algorithm have been compared to (post-processed) network-RTK solutions using the Netherlands Positioning Service (NETPOS). Additional validation was performed by means of a 5-centimeter-accuracy road-infrastructure map from Rijkswaterstaat, the Dutch Ministry of Infrastructure and the Environment, and by a centimeter-level a priori ground survey.
The new real-time SF-PPP software was tested successfully with performance comparable to our previous post-processing software, and meeting the required accuracy for freeway lane identification. Statistics on the performance are provided, as well as their dependence on a number of external parameters including the number of available satellites.
Precise corrections from both the German Aerospace Center (Deutsches Zentrum für Luft- und Raumfahrt or DLR) and the International GNSS Service (IGS) were used. Delays in the correction streams vary between providers and can increase further in the event of a time-out of the mobile link. The influence of these delays is considered, and an optimal approach for dealing with outages is discussed.
PPP Model and Corrections
The GNSS positioning model is non-linear. The observations are non-linear functions of the unknown parameters plus noise.
To solve for the unknown parameters (including the receiver position coordinates), through least squares estimation, the model must be linearized around an approximate solution.
In our SF-PPP model, the primary observations are, from each satellite, the pseudorange measurement and the carrier-phase measurement. The unknown parameters are the receiver position vector and the receiver clock offset, both of which are involved in the linearization, and also the ambiguity, associated with the carrier-phase measurement, for which the model is already linear.
In the context of PPP, it is important to note that in addition to the linearization around the initial approximate values, the computed observations contain a number of a priori model values for parameters which are not estimated, including:
The precise satellite position and clock offset (including the relativistic effect): The GPS satellite positions and clock offsets are computed from the broadcast products (navigation message) and corrected with real-time data streams via Ntrip. The correction streams of DLR and IGS were used at different times as detailed in Table 1. In post-processing older files, the satellite orbits and clocks are taken from sp3 files, but to keep the processing as close as possible to the real-time functionality, these are first converted to corrections to the broadcast products.
The (neutral) troposphere delay: The troposphere delay is modeled with the a priori Saastamoinen model using the Ifadis mapping function and parameters from the 1976 U.S. Standard Atmosphere.
The ionosphere delay and satellite differential code bias: The ionosphere delay is computed a priori using the one-day predicted Global Ionosphere Maps (GIMs) from the Center for Orbit Determination in Europe (CODE), together with the corresponding differential code biases.
The carrier-phase observations are corrected for the phase wind-up at the receiver and satellite. The user orientation is estimated from the vehicle velocity vector.
TABLE 1. Four SF-PPP field tests.
Besides the primary observations, the ambiguity estimate from the previous epoch can be added to the current epoch as an additional observation per satellite, because it is assumed to be constant in the absence of a cycle slip.
Observations from different epochs are assumed to be uncorrelated, and consequently the ambiguity estimates from previous epochs are uncorrelated to the current observations. Observations to different satellites are also assumed to be uncorrelated.
The carrier-phase ambiguities are the only parameters propagated from a previous epoch to the current epoch. The receiver position coordinates (and receiver clock offset) are estimated each epoch anew — no vehicle dynamics model is involved.
The computed positions are finally corrected for solid Earth tides with an efficient numerical model. Computed positions result in the International Terrestrial Reference Frame (ITRF) 2008 at the epoch of the observations.
In parallel with the positioning filter, statistical hypothesis testing is used to detect errors in the observations or propagated ambiguities (such as those caused by excessive multipath or a cycle slip), based on the detection, identification and adaptation (DIA) procedure. First, an overall model test is run at each epoch to test the validity of the model and observations. If the test is rejected, data snooping is applied to determine which observation is most likely to have caused the problem. If one of the pseudorange measurements is identified, it is removed from the model. If either a carrier-phase measurement or ambiguity is identified, the ambiguity for that satellite is reset; that is, the propagated ambiguity is removed.
Experiments
Four field tests that we have carried out are considered here.
In October 2012, more than 100 laps were driven over a 5-kilometer stretch of the A13 freeway between Delft and Rotterdam. The data collected were reprocessed to validate the new real-time software implementation (but obviously carried out in post-processing mode).
The first real-time tests were performed in December 2014 and later in May 2015 on the same stretch of the A13 freeway.
In May 2015, a third dataset was collected on a recently constructed and nicely outlined parking lot in Delft.
In July 2015, a train carriage was equipped with a GPS receiver and data were collected on a train trip from the center of The Netherlands to the far southern part — a distance of more than 200 kilometers.
Details of the four field tests are collected in Table 1.
Ground Truth
In our earlier experiments, the ground truth for the vehicle positions was computed with measurements from high-end equipment onboard the same vehicle. Both the antenna of the SF-PPP receiver and the high-end antennas were rigidly connected to a wooden beam on the roof rack of the van (positions of the two high-end antennas at both ends of the beam were obtained through network RTK GPS). As our results from this experiment show, the performance, and especially the precision, is very good, but a moderate bias of 17 centimeters in the cross-track direction was observed (see FIGURE 1 and TABLE 2). The suspect cause of this bias was the antenna location, close to the side of the vehicle and not attached to the metal roof itself.
FIGURE 1. 2D histogram of SF-PPP position errors (with respect to the network RTK GPS solution) in horizontal directions for the 2012 test on the A13 freeway, expressed in local east and north directions (left), and in cross-track and along-track directions (right). The color indicates the number of samples in each bin.TABLE 2. Statistics of the position errors in each direction, for the 100 laps on the A13 freeway.
Therefore, during more recent experiments, the test vehicle was only equipped with a patch antenna for the low-end, automotive-type GPS receiver, and attached directly to the roof of the car, in the middle of the centerline of the vehicle. In this case, the metal roof acts as a ground plane for the antenna, improving the gain and not acting as a source of multipath. However, this setup also has complications for the accuracy assessment. Thus, instead of computing accurate ground truth from the measurements from high-end equipment directly near the test receiver, a number of other ways were used to determine the ground truth.
During the first real-time test on the A13 freeway, a 5-centimeter accurate road infrastructure map from Rijkswaterstaat was used as previously mentioned. This comparison was done both visually and numerically.
For our next experiment, we selected a recently constructed parking lot with a simple, neat rectangular layout. By surveying the corners of the rectangle and using the repetitive pattern, a schematic drawing of the parking lot was made, and used to evaluate the positioning performance in a visual manner. The car was first driven over the lined-up parking spaces in a lengthwise manner, circling round at each end of the parking lot, and changing lanes once each lap at the same point. Then the car was driven along the edges of the rows of parking spaces to and fro over the parking lot.
SF-PPP positions were obtained live in the vehicle while driving. The raw (single-frequency) observations of this experiment were also post-processed with the RTKLib software package using the nearby permanent DLF1 station at the TU Delft GNSS observatory on a very short baseline (less than 1 kilometer). The ambiguity-fixed results could then be used to also numerically assess the SF-PPP positioning performance.
For the test on the train, again the network RTK GPS solution provided the ground truth positions. Two antennas were mounted along the centerline of the carriage at a fixed offset from each other: a patch antenna for the single-frequencyreceiver and a geodetic antenna for the ground truth. With this known offset, and the direction of motion, the ground truth position for the single-frequency receiver was obtained.
The ground-truth positions, either in the European Terrestrial Reference System (ETRS) 89 (from NETPOS or our own survey) or in the local national reference frame Rijksdriehoeksmeting (National Triangulation System) / Normaal Amsterdams Peil (Amsterdam Ordnance Datum) or RD/NAP, have been transformed into ITRF2008, to allow for comparison with the SF-PPP positions.
Computational Performance, Data Rates
The real-time software was used under the 64-bit Windows 8.1 operating system on a moderately fast laptop with i5-4200U CPU running at 1.60 GHz. The software consists of uncompiled Matlab R2014b scripts and functions using timer objects to repeatedly read in new observations, corrections and ephemerides, and to update the position computation. The software can run with data arriving at about 20 Hz in the current state on this platform, but was used with 5-Hz data because of limitations of the receiver to provide raw data and to prevent any overrun. It should be noted that only a few obvious potential computational bottlenecks were targeted; the software was not optimized for efficiency.
The RT SF-PPP implementation relies on a 3G mobile Internet connection for a number of data products. The ionosphere map, which is a predicted product (24 hours ahead), comes as a 200-kilobyte file (and 5 kilobytes for the associated differential code biases), which covers the globe and is valid for 24 hours. The file contains 13 maps at 2-hour intervals, between which interpolation in time is required.
Spatial interpolation is also required for the ionosphere pierce point of each satellite signal, between the grid points in the map (at intervals of 5 degrees in longitude and 2.5 degrees in latitude). The satellite orbit/position corrections (every 60 seconds) and satellite clock corrections (every 10 seconds) are retrieved over the Internet using the Ntrip protocol by means of the Bundesamt für Kartographie und Geodäsie (BKG) Ntrip Client (BNC), which passes these on to Matlab.
The data-rate used by this correction stream is about 1 kilobit per second. The corrections are applied to the broadcast ephemerides (in quasi-Keplerian-element form), which are therefore also required. These satellite ephemerides can be extracted by the GPS receiver itself (from the GPS navigation message), but in our implementation are also collected via Ntrip for convenience only, with a bandwidth consumption of 6 kilobits per second. Note that, much like the software implementation itself, the data stream has not been optimized for any particular bandwidth limitation. For instance, orbit and clock corrections are needed only for those satellites in view, and hence transmitting the data for all satellites of the constellation is not needed.
Results
In this section, we present the results of our tests, followed in the next section with a discussion of important common factors affecting accuracy and continuity of RT SF-PPP.
Road-Test A13 Freeway (100 Laps). Under different conditions, we collected a large amount of data with a van, driving repeatedly the same 5-kilometer stretch of road on the A13 freeway from Rotterdam to Delft. The test amounted to almost a full day of driving.
2D histograms of the results are shown in Figure 1 with corresponding statistics in TABLE 2. Note a small bias in the cross-track direction. The total number of position solutions was 2.0× 105.
Road-Test A13 Freeway (Real Time). The results of the real-time freeway road test are shown in FIGURE 2. The different lanes used by the vehicle are clearly visible in the figure. The number of GPS satellites is indicated by the color bar. Shown is the Delft-Zuid / TU Delft exit of the A13 freeway, roughly a 300 × 300 meter area, taken from the Digitaal Topografisch Bestand (DTB) of Rijkswaterstaat. Note that only the cross-track performance can be assessed in this manner, but fortunately this is exactly the performance aspect that is most interesting for the target application of lane identification. Note also that if the vehicle was not driving exactly in the middle of the lane, which to some extent is unavoidable, this effect cannot be separated from the positioning errors.
FIGURE 2. SF-PPP solution displayed on a 5-centimeter accurate road infrastructure map, on Dec. 18, 2014.
The 95-percent error southbound and northbound is 0.65 meters and 0.58 meters respectively, in the cross-track direction.
Road-Test Parking Lot.FIGURE 3 shows an aerial photograph (left) and schematic drawing (right) of the 3M company parking lot in Delft showing measured positions and driven tracks. The lines in red and yellow represent the measured tracks while driving the same loop over the parking lot again and again (more than 60 times in total), and the purple lines show the track while driving around and following the parking space boundaries with the left front wheel of the test vehicle (4 laps). These lines show both the SF-PPP position error and the driver error. The white parking spaces are each 2 meters wide.
FIGURE 3. Aerial photograph, from Google Earth, (left) and schematic drawing (right) of the parking lot in Delft showing measured positions and driven tracks.
The position errors in local north, east and up directions for part of the first dynamic session, of about 4.5 laps, of the 3M parking lot experiment (lane change 1) are shown in the upper panel of FIGURE 4. We see a clear periodic signal as well as a bias in each direction. The driving direction gives an approximation of the heading (shown in the bottom panel), which confirms that the periodic signal coincides with the driven laps.
FIGURE 4. Position errors (top) in local north, east and up directions and heading (bottom) for part of the first dynamic session, about 4.5 laps, of the 3M parking lot experiment (lane change 1).
The figure shows that the errors in the position solution are on the order of 0.2 meters, and consist of a bias in each of the three directions and a periodic signal with a period equal to the lap-time (confirmed by the driving direction of the vehicle). Since the bias does not depend on the orientation of the vehicle, and given the slow variation over time, the most likely cause is a residual ionosphere error or errors in the satellite products. The repeating pattern, on the other hand, is most probably related to multipath or near-field effects related to the vehicle antenna.
Rail-Test Amersfoort to Simpelveld. The train carriage with the GPS antennas installed was pulled by a 1955-built diesel-electric locomotive. A trip of more than 200 kilometers was made, over the main Intercity Network of Nederlandse Spoorwegen (NS) / ProRail (Dutch Railways). Only the last 20 kilometers were on a local line to a historic railway station.
The overhead power line (about 1 meter above the GPS antennas) and portals seem to have no impact on the SF-PPP positioning performance. An example of the positioning accuracy is shown in FIGURE 5. The figure shows position error scatter for an almost 20-kilometer stretch of nearly straight east-west track through rural and forest areas (Weert to Roermond). The time span of the data is 10 minutes, and the data rate was 5 Hz. SF-PPP positions were compared with NETPOS network RTK GPS solutions. Generally, eight satellites were received and used in the SF-PPP solution. The corresponding error statistics are presented in TABLE 3.
FIGURE 5. Position error scatter for an almost 20-kilometer stretch of nearly straight east-west track through rural and forest areas (Weert to Roermond); 10 minutes of data at 5 Hz.TABLE 3. Statistics of the position errors, over 2994 epochs, in along- and cross-track directions, for the position scatter shown in Figure 5.
A heavy steel-construction bridge along the route at the River Lek near Culemborg, 15 kilometers south of Utrecht, was found to degrade positioning performance considerably. The heavy steel construction of the bridge hampers reception of GPS satellite signals. The positioning performance on the bridge is shown in FIGURE 6. The computed SF-PPP trajectory overlaid on a Google Earth aerial photograph is shown on the left.
FIGURE 6. Positioning performance on the Lek Bridge. Left: measured trajectory overlaid on a Google Earth aerial photograph. The number of satellites available is indicated by the color bar. Right top: SF-PPP positions in local east-north directions. Right bottom: Absolute cross-track offset of position solution with respect to a straight line, as a function of time.
From the positions, one can clearly see the train driving straight on the right-hand track (going south) on the ramp onto the bridge, and on the ramp down from the bridge. However, on the bridge itself, position solutions show considerably larger variations of up to 8 meters. The image shows a 250-meter stretch of the track. Also, the number of satellites available, and used in the position solution, drops considerably (indicated by the color bar) while the train is on the bridge. On the right of the figure at the top, the SF-PPP positions in local east-north coordinates are shown along with a straight line between the first and last epochs, representing the assumed straight track. The plot at bottom right shows the absolute cross-track offset of the position solutions with respect to the straight line, as a function of time, over 250 5-Hz epochs.
Analysis
Two factors significantly affect the performance of our tests: the number of satellites available and the continuity and latency of the corrections.
Number of Satellites.As can be expected, the SF-PPP position accuracy depends to a large extent on the number of satellites used to compute the solution. For the third test, the road-test in the 3M parking lot, the three-dimensional position error (SF-PPP versus RTK GPS) is shown as a boxplot in FIGURE 7 in which various accuracy measures are plotted as a function of the number of satellites for the second and longest dynamic part of the test (lane change 2), consisting of about 12,000 epochs of data. During this session, the available number of satellites varied between 10 and 12. This number was reduced artificially by increasing the elevation mask angle to 15 and to 30 degrees. The red lines show the medians, the boxes show the 25th and 75th percentiles, the dashed lines cover all data points not considered outliers, and outliers are plotted with red plus signs. The graph shows a clear improvement going from six to seven or more satellites.
FIGURE 7. Boxplot of 3D position error vs. the number of satellites for the second and longest dynamic part of the 3M parking lot test (lane change 2).
PPP Correction-Stream Outages.To determine the optimal approach to an interruption in the correction data stream, we studied the variation of the corrections over time. Suppose we lose reception of the correction stream at epoch 0, and we keep using the last-received corrections (simply hold onto them). Then the change in values can be interpreted as the additional error introduced in the positioning algorithm by the outage on the mobile link. The effect is not catastrophic. Only after about 200 seconds do the additional satellite clock errors grow to the decimeter level. The position errors remain even smaller.
However, one might wonder whether this can be improved further by performing a linear extrapolation of the corrections, for example, using a number of previous epochs. We looked at what would happen in this case if 5 minutes of previous data are used. For the clock errors, there is no real benefit — the errors only grow larger. But the position errors do remain smaller during the first 5 minutes of extrapolation. After that time, the errors are larger than those without the linear extrapolation (just holding onto the last corrections). The effect of increasing the order of the polynomial extrapolation was also considered. The polynomials of different order outperform each other at different extrapolation times, and also the number of previous epochs used for the polynomial estimation impacts this. Further optimization to reduce the satellite position errors might well be possible, but may be of marginal value, since, the extrapolated clock error is dominant and polynomial extrapolation does not improve this. Simply using the most recent corrections is thus a straightforward and acceptable approach.
Conclusions
In this article, we outlined a real-time implementation of single-frequency GPS precise point positioning. With a fairly low-cost GPS receiver and reception of a modest correction data stream, it is possible to achieve sub-meter horizontal positioning accuracy, in real-time, live in the vehicle (95-percent error of better than 1 meter). Actual results were shown from four field tests: two tests using a vehicle on a freeway, a vehicle test in a parking lot, and one test on a train.
The number of satellites used in the position solution has a big effect on the positioning performance; seven or more satellites yields a good position accuracy. And up to 5 minutes outage of the satellite position and clock corrections does not seem to pose a serious threat to SF-PPP positioning performance.
Acknowledgments
The Dynamic Lane Guidance project under which the first road test was carried out was funded by the Ministry of Infrastructure and Environment, the Province of Noord-Brabant and the Eindhoven Regional Government in the context of Brabant in-car III. This project was carried out in close cooperation with colleagues in the Transport and Planning Department at TU Delft.
We acknowledge the provision of the Real-Time Clock Estimation (RETICLE) satellite clock products by André Hauschild at DLR for several of our field tests. We are also grateful for the use of the IGS Real-Time Service. Also, we acknowledge the provision of the NETPOS network RTK GPS service as ground truth by Lennard Huisman of Kadaster, the Dutch Land Registry and Mapping Agency. Colleague Hans van der Marel analyzed the NETPOS RTK-GPS solution of the train test. Colleagues of the TU Delft Railway Engineering Department offered the opportunity to carry out the test on the train trip from Amersfoort to Simpelveld.
Manufacturers
The vehicle receivers used for the tests were u-blox AG TIM LP and 7P modules in evaluation kits fed by a Tri-M Technologies Inc. Big Brother SM-66 or Taoglas Dominator AA.161 antenna. A Trimble Navigation R7 receiver with a Zephyr Geodetic antenna was used to establish ground truth for some tests.
PETER DE BAKKER is a researcher in the Faculty of Civil Engineering and Geosciences at Delft University of Technology (TU Delft). He recently finished his Ph.D. dissertation on user algorithms for GNSS precise point positioning, and is working on localization for automotive applications, including autonomous vehicles.
CHRISTIAN TIBERIUS is an associate professor in the Faculty of Civil Engineering and Geosciences at TU Delft. He has been involved in GNSS positioning and navigation research since 1991, currently with an emphasis on data quality control, satellite-based augmentation and precise point positioning.
Further Reading
• Earlier Work on Single-Frequency Precise Point Positioning
“Lane Identification with Real Time Single Frequency Precise Point Positioning: A Kinematic Trial” by R.J.P. Van Bree, P.J. Buist, C.C.J.M. Tiberius, B. van Arem and V.L. Knoop in Proceedings of ION GNSS 2011, the 24th International Technical Meeting of the Satellite Division of The Institute of Navigation Portland, Ore., Sept. 19–23, 2011, pp. 314–323.
“Real Time Satellite Clocks in Single Frequency Precise Point Positioning” by R.J.P. Van Bree, C.C.J.M. Tiberius and A. Hauschild in Proceedings of ION GNSS 2009, the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation, Savannah, Ga., Sept. 22–25, 2009, pp. 2400–2414.
“Single-frequency Precise Point Positioning with Optimal Filtering” by A.Q. Le and C. C. J. M. Tiberius in GPS Solutions, Vol. 11, No. 1, 2007, pp. 61–69, doi: 10.1007/s10291-006-0033-9.
• Single- vs. Dual-Frequency Precise Point Positioning
Improved Convergence for GNSS Precise Point Positioning by S. Banville, Ph.D. dissertation, Department of Geodesy and Geomatics Engineering, Technical Report No. 294, University of New Brunswick, Fredericton, New Brunswick, Canada. Recipient of The Institute of Navigation 2014 Bradford W. Parkinson Award.
“Variations of Box Plots” by R. McGill, J.W. Tukey and W.A. Larsen in The American Statistician, Vol. 32, No. 1, Feb. 1978, pp. 12–16, doi: 10.2307/2683468.
A new high-accuracy technique using one dual-frequency GNSS receiver, precise point positioning (PPP) offers the possibility of cost-effectively obtaining coordinates. This study investigates the accuracy of kinematic PPP for hydrographic applications on rivers, and shows results comparable to double-difference solutions.
By Ashraf Abdallah and Volker Schwieger
Duisburg Harbor, Germany: site of the PPP survey.
Precise Point Positioning (PPP) is a challenging surveying technique for high-accuracy results. It offers the advantage of using one dual-frequency GNSS instrument. Estimation of a PPP solution is based on the ionosphere-free linear combination for code data and carrier-phase data.
Bernese Software. Bernese software V. 5.2 is a GNSS post-processing software, using GNSS measurement data for static and kinematic surveying. It processes the data in double-difference (differential GNSS) and zero-difference (PPP solution) techniques. The software was developed at the Astronomical Institute of the University of Bern.
Bernese software contains a group of different tools or programs to complete the processing for double-difference or zero-difference mode. The estimation of the two techniques has the same processing schedule in most of the pre-processing stages. The change appears later within the parameter estimations section.
As shown in Figure 1, the processing starts with downloading the related orbits from the CODE (Center for Orbit Determination in Europe) FTP server. The orbit tools include the updating of the Earth orientation parameters to be in Bernese format, converting the satellite data to a specific format and generating the standard orbit format for Bernese software. A preprocessing program contains the smoothing of the RINEX data from outliers and cycle slips.
Figure 1. Bernese software processing schedule.
This smoothing step is following by converting the RINEX into Bernese binary format. The receiver clock is synchronized with respect to the GPS time and stored to observation files using clock synchronization tools. Using the code solution, a kinematic file is written to be inserted in the next parameter estimation procedure. For double-difference solution, a baseline is created, and this baseline is corrected from cycle slips for phase data. Parameter estimation is carried out by least-square estimation for the phase and code GNSS observations.
Kinematic PPP Solution. Bernese software provides the possibility to obtain the PPP solutions in automatic script (Bernese Protocol Engine [BPE]). The satellite orbit and clock ephemeris data from CODE center were used with intervals of 5 seconds to obtain highly accurate results. Satellite and receiver phase center offsets are considered. Tropospheric correction is applied using the Global Mapping Function (GMF) model for the hydrostatic and wet delay estimation. Regarding ionospheric correction, the estimation of the PPP solution is based on the linear ionospheric-free combination, with high-order ionospheric parameters to improve the estimation.
The ocean tidal loading correction is considered in the PPP estimation. Atmosphere tidal loading is also corrected.
Figure 2 gives the analysis flowchart. Some outputs of the PPP solution could be visualized, such as the satellite phase and code residuals. The high residuals might come from the lower elevation angles of the satellites. Moreover, the residuals appear because of the effect of the remaining observation errors, such as atmospheric delay, multipath, or even the satellite orbit and clock residuals.
Figure 2. Flowchart of analysis strategy.
Regarding kinematic PPP solution, the error values in the east, north and ellipsoidal height are calculated with respect to the double-difference solution from Bernese software. The root-mean-square (RMS) error, which refers to the double-difference solution, and the standard deviation (SD), which is related to the mean value of the PPP solution error, are calculated, and the frequency histogram is plotted.
An antenna and a receiver were mounted on the surveying vessel to collect the GNSS data with an interval of 1 second during two days.
Experimental Work. Two kinematic trajectories were observed on the Rhine River in Duisburg, Germany, as a part of the project “HydrOs — Integrated Hydrographical Positioning System.” The project was launched in cooperation with Department M5 (Geodesy) of the German Federal Institute of Hydrology (BfG) and the Institute of Engineering Geodesy at the University of Stuttgart (IIGS) .
An antenna and a receiver were mounted on the surveying vessel (inset photo, opener) to collect the GNSS data with an interval of 1 second during two days. The virtual SAPOS (SAtellitenPOSitionierungsdienst der deutschen Landesvermessung) reference station was considered as a reference station, provided from the SAPOS-NRW team. SAPOS is a continuously operating reference station (CORS) GNSS service collecting data throughout Germany.
Results and Discussions
The layout of the first trajectory, which was observed for more than three hours, is presented in Figure 3. The measurements started from the inner harbour in Duisburg. The left figure shows the overview layout, and the right figure illustrates a zoom-in of the trajectory below two bridges. The white line refers to the kinematic PPP trajectory; the cross-hatched white line shows interpolated points between two solved points from the PPP solution. Because of loss of GNSS signals from the bridges, the yellow line indicates the actual vessel trajectory below bridges.
Figure 3. Layout of the first trajectory [DOY: 2014/126], zoom-in on bottom. (Photos: Google Earth)As mentioned before, the double-difference solution of the Bernese software is considered as the reference solution for the PPP solution. The PPP residuals for phase and code observations (not using double-difference solution) are presented in Figure 4. Here the residual values in phase and code have a gap because of the loss of GNSS signals, which starts from epoch 438 to 486 [GPS week second = 199845: 200115]. Additionally, there are some cycle slips from epoch 883 to 892 [GPS week second = 202105: 202150].
Figure 4. Satellite residuals for the first trajectory [DOY: 2014/126].To assess the accuracy of the PPP solution for this hydrographic trajectory, Figure 5 illustrates the analysis results for this trajectory between the double-difference and PPP solutions. The X-axis refers to the number of observations (one epoch/5 seconds), and the Y-axis indicates the error value in meters. Figure 5.1 shows the error plot (m) in east, north and height. As shown previously, the error values have a gap in the solution because of the loss of lock below the bridges. Moreover, there are some cycle slips later on, which decrease the estimated kinematic PPP accuracy.
Figures 5.2 and 5.3 provide the error plot for the east and north and east and height directions. The blue points refer to the errors, and the red cross refers to the mean value. Table 1 summarizes the PPP results.
Table 1. Statistical results of the first trajectory [DOY: 126/2014].Five percent of the PPP errors are eliminated to get outlier-free results. The SD (95%) of the kinematic PPP solution is obviously improved to reach 5.0 cm, 1.20 and 5.0 cm in east, north and height directions, respectively.
To distinguish between the standard deviation and the standard deviation based on 95 percent of the data, Figure 5 shows additionally the histogram of SD in Figures 5.4, 5.5 and 5.6 for east, north and height respectively. Figures 5.7, 5.8 and 5.9 provide the error with 95 percent of the results. Absolutely, the error range is improved by eliminating 5 percent of the data including outliers.
Figure 5. Analysis results for the first trajectory. Standard deviations shown in plots on the left, with outliers excluded, right.
Second Data Set. The second trajectory on the Rhine River was observed [DOY: 127] for more than 5 hours (see Figure 6). Sixteen satellites were observed during the measurement time.
Figure 6. Layout of the second trajectory [DOY: 127/2014]. (Photo: Google Earth)In Figure 7, the phase and code residuals are plotted. Some outliers are reported in this graph, which refers to cycle slips during the observations.
Figure 7. Satellite residuals for the second trajectory [DOY: 127/2014].Figure 8 illustrates the PPP results for this kinematic trajectory. Figure 8.1 shows the PPP error values in the east, north and height directions with respect to the double-difference solution from Bernese software.
Figure 8. Kinematic PPP solution for the second trajectory. Standard deviations shown in plots on the left, with outliers excluded, right.
The first 40 minutes of that trajectory were realized in a quasi-static observation technique (nonmoving vessel) from GPS week second 281660: 284060. The result obtained from this solution is more accurate due to the high number of satellites, and the trajectory did not include the bridges area. Figure 8.2 and 8.3 show errors in east and north, and east and height.
As shown in Table 2, the maximum and minimum values for the error range, which are presented in detail in Figure 8.4, 8.5 and 8.6, are reported in the east, north and height directions. These figures show the frequency histogram for the PPP errors. The RMS error from the solution is 2.10 cm and 2.90 cm in east and north respectively, with an RMS error of 5.60 cm in height. The standard deviation is definitely improved after eliminating 5 percent of the PPP errors as outliers. The standard deviation for 95 percent of the results shows 1.5 cm in east and north and 3 cm in height. The error histograms for 95 percent of the data are provided in Figures 8.7, 8.8 and 8.9.
Table 2. Statistical results of the second trajectory [DOY: 127/2014].The second trajectory clearly provides a higher accuracy than the first. Its data has a higher number of satellites and lower outliers than the first. Figure 8 shows the histogram of the second trajectory is similar to the Gaussian distribution curve.
Acknowledgments
The authors would like to thank Annette Scheider for receiving the GNSS measurements through the HydrOs project, our BfG partners Harry Wirth and Marc Breitenfeld, and Bernhard Galitzki form SAPOS-NRW for providing us with the reference stations.
This article is based on a peer-reviewed paper presented at the FIG Working Week, May 2015, in Sofia, Bulgaria.
Manufacturers
A Leica 1203+ antenna and GX1230+ GNSS receiver collected the data shown here.
Ashraf Abdallah is an assistant lecturer in engineering, Aswan University, Egypt, and a Ph. D. student at the Institute of Engineering Geodesy (IIGS), Stuttgart University, Germany. He received a master’s degree from Aswan University in applications of single-frequency GNSS.
Volker Schwieger is a full professor at the University of Stuttgart and director of the IIGS. He received a Ph.D. from the University of Hannover, focusing on GPS for monitoring applications.
The new version of the precise point positioning software includes several updates and new features, including creation of GAPS Basic and Advanced user submission pages.
The GAPS Basic user submission page allows for quick and easy submission of observation files for users who frequently use GAPS’ default processing options.
The GAPS Advanced user submission page includes:
• User-selection of the orbit and clock products to be used.
• User-selection of the carrier-phase and pseudorange observables to be used.
• Optional use of GPS L2C in place of P2 for all satellites currently transmitting L2C.
• Optional use of GPS L5 in place of L2 for all satellites currently transmitting L5.
• Optional use of static-mode satellite clock interpolation (if 30s clock product is used and logging interval < 30s).
• User-selection of GDOP cut-off threshold.
• User-selection of positional convergence condition and maximum number of iterations for least-squares filter.
• Enhanced the cycle-slip detection algorithm (following Blewitt, 1990).
• A minimum of 4 satellites per epoch are required before estimation begins.
• User-selection of all available neutral atmosphere delay (NAD) prediction model and mapping function (MF) combinations.
NAD prediction models include: UNB-VMF1 (NCEP), UNB-VMF1 (CMC), VMF1 (ECMWF), UNB3m, GPT2 (1×1 deg.), ESA 2.5 and None
Mapping function options are: Vienna MF and Niell MF.
• User option to not estimate NAD.
• User option to not estimate tropospheric gradients.
• Optional use of a user-provided receiver antenna calibration file.
• NAD estimation automatically terminated if receiver rises above neutral atmosphere threshold (50,000 ft = 15,240 m).
• Added option to estimate precipitable water (if a meteorological file is submitted).
• New .ion, .cmp, .nad, and .DOP output files as well as modified formatting of the .par file.
• Improved reporting of processing parameter options and results in the HTML output.
• Added receiver clock and DOP plots.
• Added height component to kml output.
For information on the processing strategy, visit the web page. Your feedback (suggestions, bug reports, etc.) is welcome via the GAPS Development Team email: [email protected].
![Figure 3. Layout of the first trajectory [DOY: 2014/126], zoom-in on bottom. (Photo: Google Earth)](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Figure-4-W.jpg)
![Figure 4. Satellite residuals for the first trajectory [DOY: 2014/126].](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Figure-5-W.jpg)
![Table 1. Statistical results of the first trajectory [DOY: 126/2014].](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Table1.jpg)
![Figure 6. Layout of the second trajectory [DOY: 127/2014]. (Photo: Google Earth)](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Figure-7-W.jpg)
![Figure 7. Satellite residuals for the second trajectory [DOY: 127/2014].](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Figure-8-W1.jpg)
![Table 2. Statistical results of the second trajectory [DOY: 127/2014].](https://stage.globalpositioningnews.com/wp-content/uploads/2015/12/Table2.jpg)