Effect of spatial correlation on the performances of modernized GPS and Galileo in relative positioning
Abstract
In the context of processing GNSS (Global Navigation Satellite System) data, it is known that the estimation of the ionospheric delays decreases the strength of the observation model and makes significant the time required to fix the ambiguities namely in case of long baselines. However, considering the double-differenced (DD) ionospheric delays as stochastic quantities, the redundancy in this case increases and leads to the reduction of time of fixing the ambiguities. The approach developed in the present paper makes two considerations: 1) the DD ionospheric delays are assumed as stochastic quantities and, 2) the spatial correlation of errors is accounted for based on a simple model of correlation. A simulation is made and aims to study the effect of these two mentioned considerations on the performances of the three multifrequency GNSSs; modernized GPS, Galileo and BDS which are not yet in full capability. For each GNSS, dual-frequency combinations of frequencies as well as triple-frequency combination are investigated in the simulation. The performances studied include: the time to fix the ambiguities with high success rate and the precision of coordinates in static relative positioning with varying baseline length. A method is developed to derive what we call the spatial correlation model which approximately gives the covariance between the individual errors belonging to two stations. Furthermore, the stochastic models that follow from accounting and neglecting the spatial correlation are developed. The LAMBDA (Least-squares Ambiguity Decorrelation Adjustment) method is implemented for ambiguity decorrelation. The results show that the time to fix the ambiguities caused by accounting the spatial correlation is less than the time of fix without the spatial correlation. Also, a slight superiority of Galileo in terms of performances is seen compared to the other GNSS. For all the dualfrequency combinations investigated, when processing a baseline length of 500 km with accounted spatial correlation, the time needed to successfully fix the ambiguities lies between 5 and 9 min, whereas it becomes only between 2.5 and 3 min for all the triple-frequency combinations, this is with a sampling time of 5 s. In addition, for all different combinations, the coordinates precision is less than 8 mm even for 500 km. We think that these high performances result from: 1) the precise codes of future GNSS signals, 2) the high redundancy in the observations equation and, 3) taking into account the spatial correlation in the definition of the stochastic model.
Keyword : modernized GPS, Galileo, BDS, spatial correlation, ambiguity resolution
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Arora, B. S. (2012). Evaluation of ambiguity success rates based on multi-frequency GPS and Galileo (Master Thesis). Curtin University.
Cai, C., He, C., Santerre, R., Pan, L., Cui, X., & Zhu, J. (2016). A comparative analysis of measurement noise and multipath for four constellations: GPS, BeiDou, GLONASS and Galileo. Survey Review, 48(349), 287–295. https://doi.org/10.1179/1752270615Y.0000000032
Chang, X.-W, Yang, X., & Zhou, T. (2005). MLAMBDA: A modified LAMBDA method for integer least-squares estimation. Journal of Geodesy, 79(9), 552–565. https://doi.org/10.1007/s00190-005-0004-x
Datta-Barua, S., Walter, T., Blanch, J., & Enge, P. (2008). Bounding higher-order ionosphere errors for the dual-frequency GPS user. Radio Science, 43, RS5010. https://doi.org/10.1029/2007RS003772
Deprez, C., & Warnant, R. (2016). Combining multi-GNSS for precise positioning [Conference presentation]. NAVITEC, Noordwijk, the Netherlands.
Feng, Y. (2008). GNSS three carrier ambiguity resolution using ionosphere-reduced virtual signals. Journal of Geodesy, 82, 847–862. https://doi.org/10.1007/s00190-008-0209-x
Jazaeri, S., Amiri-Simkooei, A., & Ali, S. M. (2014). On lattice reduction algorithms for solving weighted integer least squares problems: comparative study. GPS Solutions, 18, 105–114. https://doi.org/10.1007/s10291-013-0314-z
Liu, G. C., & Lachappelle, G. (2002, January 28–30). Ionosphere weighted GPS cycle ambiguity resolution. [Conference presentation]. National Technical Meeting, Institute of Navigation, San Diego, CA.
Liu, G. C. (2001). Ionosphere weighted global positioning system carrier phase ambiguity resolution (Master Thesis). Department of Geomatics Engineering, University of Calgary, Calgary, Alberta.
Nardo, A., Li, B., & Teunissen, P. J. G. (2015). Partial ambiguity resolution for ground and space-based applications in a GPS + Galileo scenario: A simulation study. Advances in Space Research, 57(1), 30–45. https://doi.org/10.1016/j.asr.2015.09.002
Ning, Y., Yuan, Y., Huang, Z., Chai, Y., & Tan, B. (2016). A long baseline three carrier ambiguity resolution with a new ionospheric constraint. International Journal of Geo information, 5(11), 198. https://doi.org/10.3390/ijgi5110198
Odijk, D., Arora, B. S., & Teunissen, P. J. G. (2014). Predicting the success rate of long-baseline GPS+Galileo (partial) ambiguity resolution. The Journal of Navigation, 67(3), 385–401. https://doi.org/10.1017/S037346331400006X
Schäcke, K. (2013). On the Kronecker Product. https://www.math.uwaterloo.ca/~hwolkowi/henry/reports/kronthesisschaecke04.pdf
Teunissen, P. J. G. (1993). Least-squares estimation of the integer GPS ambiguities. [Conference presentation]. IAG General Meeting, Invited Lecture, Section IV: Theory and Methodology, Beijing, China.
Teunissen, P. J. G. (1994). A new method for fast carrier phase ambiguity estimation. In IEEE Plans (pp. 562–573). Las Vegas, Nevada. IEEE. https://doi.org/10.1109/PLANS.1994.303362
Teunissen, P. J. G. (1995). The least squares ambiguity decorrelation adjustment: A method for fast GPS integer estimation. Journal of Geodesy, 70, 65–82. https://doi.org/10.1007/BF00863419
Teunissen, P. J. G., & Odijk, D. (1997, September 16–19). Ambiguity dilution of precision: Definition properties and application. In Proceedings of ION GPS-97 (pp. 891–899). Kansas City, USA.
Teunissen, P. J. G. (1998). Success probability of integer GPS ambiguity rounding and bootstrapping. Journal of Geodesy, 72, 606–612. https://doi.org/10.1007/s001900050199
Teunissen, P. J. G., Joosten, P., & Tiberius, C. (2002, September 24–27). A comparison of TCAR, CIR and LAMBDA GNSS ambiguity resolution. In Proceedings of ION GPS 2002 (pp. 2799–2808). Portland, OR.
Verhagen, S., & Li, B. (2012). LAMBDA software package – Matlab implementation, version 3.0. Delft University of Technology, Curtin University.
Zhang, H., & Ding, F. (2013). On the Kronecker products and their applications. Journal of Applied Mathematics, 2013, 296185. https://doi.org/10.1155/2013/296185