The Effect of Galileo on Carrier Phase Ambiguity Resolution

Transcription

1 The Effect of Galileo on Carrier Phase Ambiguity Resolution Paul Alves Department of Geomatics Engineering The University of Calgary BIOGRAPHY Paul Alves is a graduate student at the Department of Geomatics Engineering of the University of Calgary. He received a B.Sc. in Geomatics Engineering in May, 2000, and is continuing his studies, towards a Ph.D. in Geomatics Engineering, in the field of positioning of navigation at the University of Calgary. assumed to have perfectly circular orbits. The GPS constellation simulated was determined used almanac data for a 29 satellite constellation. The Galileo GNSS simulated was determined by compiling the following papers: Lucas and Ludwig (1999), Tytgat and Owen (1999), Ryan and Lachapelle (2000), and Oehler et al. (2000). Figure 1 shows the constellation simulated. ABSTRACT In order to achieve the highest level of positioning accuracy using GPS, one must first determine the carrier phase integer ambiguities. This will also be true with the Global Navigation Satellite System (GNSS) proposed by the European Union, Galileo. This paper discusses the effect of the proposed satellite system, GPS, and an integrated solution of the two systems on ambiguity resolution in kinematic mode. This paper will examine the effectiveness of ambiguity resolution with Galileo, GPS, and a combined solution. The addition of the Galileo constellation will increase the efficiency and reliability of ambiguity resolution when combined with GPS. The addition of 30 satellites transmitting with L-Band frequencies will decrease the time required for a user to wait before determining the correct integer ambiguities and therefore decrease the time before achieving the best possible position accuracy. INTRODUCTION This project consists of two distinct topics of discussion: the creation of the simulated Galileo and GPS observations and errors, and the carrier phase processing of the simulated observations. It is important to separate these two processes because the processing of the data should not influence the methods used to create the data. This would invalidate the comparison of the three systems. SIMULATION CONSTELLATIONS Figure 1: GPS and Galileo Constellation Simulated The constellation used in these simulations consists of 30 medium Earth orbit satellites. The satellites orbit in three orbital planes inclined at 54 degrees with an altitude of 23,000km. No information has been found as to the spacing of the satellites on the planes; the simulations assume that the satellites are equally spaced with 10 satellites on each orbital plane. No information has been found as to the phase of the three Galileo planes with respect to the six GPS orbital planes. This may change the solution of the combined constellation slightly. The carrier frequencies and chip rates are listed in Table 1. The E1 and E2 signals will be combined to produce a single measurement (Hein 2000). Idealized models of the two Global Navigation Satellite Systems (GNSS) were used. The two constellations are Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 1

2 Table 1: Selected Galileo Frequencies and Chip Lengths Frequency (MHz) Chip Rate (MHz) E E E1/E E E SOFTWARE SIMULATOR The University of Calgary has developed a software simulator as a quick and cheap way to generate observations for any satellite constellation with any set of signal parameters. It can generate errors for receiver noise, multipath, ionospheric delay, tropospheric delay, broadcast orbit error, and selective availability. These errors can be increased or decreased to simulate any type of environment. Dr. N. Luo, a graduate of the Department of Geomatics Engineering, the University of Calgary, developed the simulator described below. Luo s Ph.D. thesis (Luo 2000) contains a detailed description of the error generation process. These environmental conditions will have different effects on the ability to determine the correct carrier phase integer ambiguity. The effects can be characterized into two groups of errors: spatially correlated errors and spatially uncorrelated errors. The magnitudes of the errors are listed in Table 2. They are consistent with typical error values as determined using GPS. There is no indication of selective availability being applied to Galileo therefore it has not been applied to this simulation. Table 2: Magnitude of Simulated Error Sources Error source Magnitude of error Troposphere 2.3 ppm (DD) Orbit 0.1 ppm (DD) Multipath 3 m Selective Availability Off The Ionosphere error applied in the simulation was varied to examine the effects of different levels of error. The six Ionosphere level chosen are listed in Table 3. Table 3: Levels of Ionosphere Error Discription Error Level Double Difference Very low 2 ppm Low 3 ppm Medium 8 ppm High 13 ppm Very high 20 ppm Extreme 40 ppm The orbital errors for Galileo are assumed to be similar to that of GPS in terms of magnitude and update rate. This may be slightly pessimistic for Galileo because of the larger orbital radius. This would provide an orbit that is less affected by gravity perturbations. The orbit error is correlated over time with discontinuities added to simulate updates in the broadcast ephemeris. The multipath error is generated assuming a flat horizontal reflector under the antenna. The chip rates of the different signals were considered in generating the multipath error. The chip rates used are listed in Table 1 above. The Ionosphere error was generated using a global ionosphere map from the Center for Orbit Determination in Europe (CODE), an analysis center of the International GPS Service (IGS). This map defines the global trend of the Ionospheric error in a set of coefficients of a spherical harmonic expansion. The spherical harmonic expansion is then used to define a grid of vertical total electron count (TEC) values on a two dimensional Ionospheric shell above the area of interest. These grid values are interpolated to the pierce point of the observation. The interpolated values are then multiplied by a elevation mapping function to give the final Ionospheric error. The Tropospheric delay is calculated using a modified Hopfield model with a Black and Eisner mapping function. The spatial correlation of the Troposphere has been considered and the daily temporal variations have been added. The receiver noise is a function of the signal characteristics, such as chip rate. The C/A code receiver noise levels used are listed in Table 4. The carrier phase receiver noise is assumed to be one percent of the wavelengths. The frequencies used are listed in Table 1. Table 4: C/A Code Receiver Noise for the Simulated Signals GNSS Signal Receiver Noise (m) GPS L L L Galileo E1/E E E AMBIGUITY RESOLUTION The ambiguity resolution method for a multi-frequency multi-gnss system has many variations. The integer characteristic of the traditional double difference ambiguity will not be preserved. This is because of the difference of the wavelengths in the ambiguity terms. Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 2

3 This can be overcome in many ways. The carrier phase double difference observation equation is i i j j Φ = ρ + λn λn ( λn λn ) E (1) a b a b + Where ρ is the true double difference range, λ is the carrier phase wavelength, N i a is the ambiguity from station a to satellite I, and E is the remaining double difference observation s error. If the wavelength is common between observations then it can be factored out of the equation to give Φ = ρ + λ N + E (2) In the case of a GPS/Galileo combined solution the wavelengths for the observations are not common and cannot be factored. i i i i j j j j Φ = ρ + λ N λ N ( λ N λ N ) + E (3) a b Where λ i and λ j are the wavelengths of the two GNSS. The easiest solution is to have separate base satellites for each GNSS. This ensures that the double differenced observations always produce integer ambiguities. This is the approach used for the GPS/Galileo ambiguity results below. FLOAT AMBIGUITY ESTIMATION The real number (float) ambiguities are estimated to define a search range for the ambiguity search method. The float solution is described below. The ambiguities and position estimates are determined using sequential least squares. The velocity estimates were not included but process noise is added to the position variances between epochs to simulate a kinematic scenario. The carrier phase observations alone cannot be used to instantaneously determine a valid position solution because there are not enough observations to fully observe all of the estimation states. Using sequential least squares, the carrier phase observations can be used to estimate the position once enough observations are collected to observe all of the estimation states, namely position and carrier phase ambiguities. To assist with the convergence of the float solution, code observations are used to observe the kinematic position. The Lambda method (Jonge et. al., 1996) was used to search for the correct integer ambiguity. After an ambiguity set is deemed correct by the ratio test (Eirckson 1992) the statistics are recorded and the filters are reset. This allows for an average of the results from many fixed solutions of the float solution to be combined over 24 hours. a b The ratio test is the division of the sum of squared residuals using the best and second best set of ambiguities. This test statistic is approximated by a Fisher distribution. FREQUENCY COMBINATIONS The introduction of many broadcast frequencies results in many different frequency combinations that can be generated. The common GPS wide-lane combination is generated by the difference between the L1 and L2 measurements. Frequency combinations are used to reduce the effect of errors on the measurements. By differencing two measurements of different frequencies from the same satellite many common errors can be reduced. This reduction is commonly referenced to the wavelength of the resulting beat frequency, where the larger the wavelength, the greater the reduction of errors. The wavelengths of the wide-lane combinations for GPS and Galileo are listed in Tables 5 and 6, respectively. Table 5: GPS Wide-lane Wavelengths L1 L2 L3 L1 X L m X -- L m 5.81 m X Table 6: Galileo Wide-lane Wavelengths E1/E2 E5 E6 E1/E2 X E m X -- E m 3.91 m X The simulation includes results using the L1 and L2 and E1/E2 and E5 wide-lane frequency combinations to illustrate the advantages of a dual frequency system. PERFORMANCE MEASURES There are many criteria that can be used to quantitatively analyze the effectiveness of an ambiguity resolution method or in this case, the effectiveness of a GNSS. There are two main concerns for users with respect to the ambiguity resolution process they are the number of epochs required to estimate the integer ambiguities and if those ambiguities are correct. The first dictates how long a user must wait before they can achieve the greatest possible position accuracy. This is dependant on the magnitude of the differential errors and number of available satellites. As the errors increase, the time required before determining the correct integer ambiguity set increases. Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 3

4 The percentage of correctly determined ambiguity sets within a 24-hour period is an important reliability measure. If the ambiguities are incorrectly resolved the resulting baseline solution may be biased. The percentage of correctly determined ambiguity sets is influenced by other ambiguity resolution measures. If the time to resolve ambiguities is unusually short it may cause the ambiguities to fix incorrectly. The percentage of ambiguities correctly resolved within a particular ambiguity set may also be a good measure to consider. When the number of ambiguities estimated increases the success rate of fixing all of the ambiguities correctly decreases because there are more dimensions. However, when the number of ambiguities increases the success rate of a partial set of the ambiguities increases. This measure will show how many ambiguities within an incorrect ambiguity set are correct. If one ambiguity within an ambiguity set for a GNSS were consistently resolved incorrectly then the percentage of the correctly resolved ambiguity sets would be zero even though most of the ambiguities within the sets were correctly resolved. This measure will indicate a candidate for partial ambiguity fixing. TEST SETUP To test the effectiveness of each GNSS to resolve integer ambiguities, the level of errors is steadily increased until a difference in the systems appear. The level of errors will increase as the inter-antenna distance between the reference and remote stations increase. This is tested by simulating observations for four stations. These stations are separated to test the effectiveness at different baseline distances and thus different correlated error levels. The relative station positions are shown in Figure 2. This network was simulated for a region in Southern Alberta. From these stations all of the possible baselines combinations are used. RESULTS The results presented in this section were produced using the simulated observations described above. Two different simulations will be discussed: The single frequency solution using L1 and E1/E2 and the dual frequency L1 L2 combination and the E1/E2 E5 combination. These will demonstrate the performance of the Galileo, GPS and combined constellations using the common float processing module described above. Single Frequency Results The single frequency results were processed using six different Ionosphere error levels. The error levels are described above. These varying levels of error will increase as the baseline distance between the receivers increases. Error Levels Figure 3 shows the percentage of correctly determined GPS only ambiguity sets for the various levels of Ionosphere error. These results show independent trends for each level of error as a function of baseline length. Similar trends are exhibited by the single frequency Galileo and combined GPS/Galileo systems (Figures 4 and 5, respectively). These results are as expected for an L1 only GPS receiver. For example, real-time kinematic positioning (RTK) becomes unreliable when the baseline distance is greater than 15 km, under usual conditions. It is also reasonable to assume that L1 only RTK positioning is reliable when the baseline is less than one kilometer even under severe conditions. 1 km 10 km 20 km Figure 2: Relative Station Positions Figure 3: Percentage of Correctly Resolved Ambiguity Sets for Single Frequency GPS only with Various Error Levels Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 4

5 Figure 4: Percentage of Correctly Resolved Ambiguity Sets for Single Frequency Galileo only System with Various Error Levels Figure 6: Time to Fix Ambiguities Correctly for Single Frequency GPS only with Various Error Levels Figure 5: Percentage of Correctly Resolved Ambiguity Sets for the Single Frequency Combined GPS/Galileo system with Various Error Levels Figure 7: Time to Fix Ambiguities Correctly for Single Frequency Galileo only with Various Error Levels The time required to fix ambiguities is an important characteristic for RTK users in the field. If the time to fix can be reduced then the availability of a system is increased. The time to fix is a function of the differential errors and the baseline length. Under extreme conditions the time to fix may take hours. A time to fix this extreme is not practical for real-time use. Figure 6 shows the time to fix correctly for the GPS only system. Baseline results that fixed correctly less than 4 times were omitted from the plot because at least 4 samples are required to get a reliable estimate. Again there are clear trend lines for each level of error. These results show exponential trends as a function of baseline length. As predicted, it takes longer to fix correctly under high error conditions. Figures 7 and 8 show similar trends for Galileo only and the GPS/Galileo combined system. Figure 8: Time to Fix Ambiguities Correctly for the Single Frequency Combined GPS/Galileo system with Various Error Levels Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 5

6 System Comparison It is unrealistic to compare the extreme situations to determine whether one system is better or worse than another. It is more reasonable to compare situations that are likely in real-world applications and still show separation between the systems. Figures 9 and 10 show a comparison of the three systems for a medium and low level of Ionosphere error, respectively. This represents the usual performance and conditions of these systems. The GPS constellation has the least number of satellites of the systems compared. For this reason, it is expected that GPS only solution will perform the worst by taking the longest time to fix correctly and showing the lowest percent correct. The combined system performs very well, fixing very quickly and almost always correctly. Figure 10: Single Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing Time to Fix Ambiguities Correctly and Percentage of Correctly Resolved Ambiguity Sets for a Low Level of Error Dual Frequency Results The dual frequency results were processed using the same six Ionosphere error levels as the single frequency results. A detailed comparison between the dual frequency results and the single frequency results has been omitted because the advantages of the dual frequency results are obvious. The single frequency solutions begin to deteriorate with a medium level of error while the dual frequency results don t degrade until the errors become very high. Figure 9: Single Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing Time to Fix Ambiguities Correctly and Percentage of Correctly Resolved Ambiguity Sets for a Medium Level of Error The L1 L2 and E1/E2 E5 frequency combinations were used for GPS and Galileo respectively. The wavelength of the GPS combination is 0.86 meters while the wavelength of the Galileo combination is 0.80 meters. This difference is evident in the results below. The longer wavelength is less influenced by measurement errors when trying to resolve the integer ambiguities. The results show that the GPS only system performs slightly better than the Galileo results; this is due to the difference in wavelengths discussed above. The very high and extreme error level results will be compared. These results are the only ones that show a comparable separation in performance between the systems. Error Levels Figure 11 shows the percentage of correctly resolved ambiguity sets for the dual frequency solution using GPS only. These results are considerably better than those of the single frequency solutions. Again, there are distinct trend lines for each level of error as a function of baseline length. This is also apparent in Figures 12 and 13, which Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 6

7 show the percentage of correctly resolved ambiguity sets for the Galileo and combined GPS/Galileo solutions. Figure 11: Percentage of Correctly Resolved Ambiguities for Dual Frequency GPS only with Various Error Levels Figure 13: Percentage of Correctly Resolved Ambiguities for the Dual Frequency combined GPS/Galileo system with Various Error Levels The time to fix ambiguities correctly also shows distinct trend lines between the different levels of error. There appears to be an exponential trend in the extreme error levels and a slight trend in the very high error level. Figures 14, 15, and 16 show the time to fix ambiguities correctly for GPS only, Galileo only, and the combined GPS/Galileo systems, respectively. Figure 12: Percentage of Correctly Resolved Ambiguities for Dual Frequency Galileo only with Various Error Levels Figure 14: Time to Fix Ambiguities Correctly for Dual Frequency GPS only with Various Error Levels Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 7

8 Figure 15: Time to Fix Ambiguities Correctly for Dual Frequency Galileo only with Various Error Levels Figure 16: Time to Fix Ambiguities Correctly for Dual Frequency the Combined GPS/Galileo system with Various Error Levels Figure 17: Dual Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing the Time to Fix Correctly and Percentage of Correctly Resolved Ambiguities for a Very High Level of Error The combined GPS/Galileo solution fixes very quickly but is more often incorrect than the GPS or Galileo only systems. The difference in percent correct is quite small and still above 90 percent. This result shows an interesting characteristic that may happen when the number of ambiguities to be estimated increases. The number of dimensions grows making it more difficult to fix a complete ambiguity set correctly but the success rate for a partial ambiguity set increases. Figure 18 shows the percentage of correctly resolved ambiguities within an incorrect ambiguity set. The missing plot markers indicate that there were no incorrect ambiguity fixes. For the combined GPS/Galileo case, on average, at least 90 percent of the ambiguities within a fixed set are correct even if the overall ambiguity set is incorrect. This system is a good candidate for partial ambiguity fixing because of the high number of ambiguities that need to be resolved. System Comparison As discussed above, the dual frequency wavelength used for GPS is longer than that of Galileo. This will cause the GPS results to be slightly better than that of Galileo. This is apparent in Figure 17, which shows a comparison of the three systems in terms of percentage of correctly resolved ambiguities and the time required to fix ambiguities. The GPS only results are similar for the Galileo system for the percent correct but on average fix ambiguities faster. Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 8

9 The percentage of correctly resolved ambiguities within an incorrect ambiguity set for the extreme error case shows the same trend as in the very high error case (Figure 20). When the combined GPS/Galileo system fixes ambiguities, on average, more than 90 percent of the ambiguities within the set are correct. Given that the combined GPS/Galileo constellation provides twice as many observations as the GPS or Galileo only systems, the resulting positioning results using the very slightly biased ambiguities may still provide better results than the unbiased GPS or Galileo only systems. Figure 18: Dual Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing the Percentage of Correctly Resolved Ambiguities Within an Incorrect Ambiguity Set for a Very High Level of Error Figure 19 shows a comparison of the three systems in terms of percentage of the correctly resolved ambiguities and time to fix ambiguities for the extreme error case. These results have similar conclusions to those in the very high error case. The time to fix ambiguities for the GPS only solution is slightly better than that of the Galileo only solution. This is most likely because of the difference in wavelengths. The combined GPS/Galileo system shows the fastest time to fix ambiguities correctly, however, it results in the lowest percentage of correctly resolved ambiguities between the systems. Figure 20: Dual Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing the Percentage of Correctly Resolved Ambiguities Within an Incorrect Ambiguity Set for an Extreme Level of Error CONCLUSIONS The University of Calgary has developed a software simulator capable of creating observations for any constellation with any set of signal parameters. This is a cheap and efficient way of analyzing the effects of a proposed satellite system without the need for expensive hardware. Figure 19: Dual Frequency GPS only, Galileo only, and the combined GPS/Galileo systems comparing the Time to Fix Correctly and Percentage of Correctly Resolved Ambiguities for an Extreme Level of Error This software was used to generate observations for a fictitious Galileo constellation of satellites. The system consists of 30 medium Earth orbit satellites transmitting on three frequencies. The GPS only constellation was generated using a broadcast almanac that contains 29 satellites. Using these data sets, single frequency and dual frequency results were computed. The dual frequency results were calculated using the L1 L2 and E1/E2 E5 frequency combinations. The wavelengths of these combinations are 0.86 meters and 0.80 meters respectively. Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 9

10 The single frequency Galileo only system performs better than the GPS only system. This may be due to the extra Galileo satellite or the difference in satellite distributions. The single frequency combined constellation performs significantly better than the GPS and Galileo only systems in terms of time to fix ambiguities and the percentage of correctly determined ambiguities. The dual frequency results are slightly more difficult to interpret. The GPS only solution is slightly better than the Galileo only system. This is because of the difference in wavelengths between the systems. If this trend continues for longer wavelengths, the modernized GPS system provides the longest wavelength of 5.81 meters. The combined GPS/Galileo solution fixes the fastest between the three systems. It also provides the lowest percentage of correctly fixed ambiguity sets, however most of the ambiguities within the incorrectly determined ambiguity set are correct for the combined GPS/Galileo solution. This lower level of correctly determined ambiguity sets only occurs when the errors are set to an extremely high level of error. Under normal circumstances the combined GPS/Galileo dual frequency system will fix ambiguities faster than the independent GPS and Galileo only systems and fixes correctly greater than 99 percent of the time with distances upwards of 30 kilometers. These simulations show that the best ambiguity resolution scheme is to incorporate as many measurements as possible. The use of a frequency combination alone reduces the number of available observation. The resulting estimation filter contains half of the available observations because two measurements, one for each frequency, are combined to give one observation. Jonge, P. & C. Tiberius (1996) The LAMBDA Method for Integer Ambiguity Estimation: Implementation Aspects. Publications of the Delft Geodetic Computing Centre, No. 12, August, 1996, pp Lou, N. (2000) Precise Relative Positioning of Multiple Moving Platforms Using GPS Carrier Phase Observables. UCGE Reports, Number 20147, January, 2001, Lucas, R., D. Ludwig (1999) Galileo: System Requirements and Architecture. ION GPS-99. pp Oehler, V., G. Hein, B. Eissfeller, B. Ott. (2000) GNSS- 2/Galileo End-to-End Simulations for Aviation, Urban and Maritime Applications. IAIN World Congress ION Annual Meeting, 2000, pp Parkinson, B., J. Spiker, P. Axelrad, P. Enge, P. Zarchan (1996) Global Positioning System: Theory and Applications Volume 1, American Institute of Aeronautics and Astronautics, Inc., United States of America, 1996 Ryan, S., G. Lachapelle (2000) Impact of GPS/Galileo Integration on Marine Navigation. IAIN World Congress ION Annual Meeting, 2000, pp Tytgat, L., J.I.R. Owen. (2000) Galileo The Evolution of a GNSS. IAIN World Congress ION Annual Meeting, 2000, pp ACKNOWLEDGMENTS The author would like to thank Dr. G. Lachapelle for support, advice, and encouragement throughout this project. This project was funded by the Canadian Space Agency as part of the GALA project. REFERENCES Eirckson, C. (1992) An Analysis of Ambiguity Resolution Techniques for Rapid Static GPS Surveys Using Single Frequency Data, ION GPS-92, September 16 18, 1992, Albuquerque, New Mexico, pp Hein, G. (2000) Galileo: Design Options for the European GNSS-2. Navtech Seminars Course 310 Notes, September 18, 2000, Salt Lake City, Utah. Proceedings of IONGPS 2001, Session C4, Salt Lake City, Utah, Sept 11 14, 2001 Page 10