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Session E3: All-source Intelligent PNT Method

Assessment of a Heuristic Search Approach for PPP Integer Ambiguity Resolution.
Lotfi Massarweh, Department of Geoscience and Remote Sensing, Delft University of Technology, & Peter J.G. Teunissen, Delft University of Technology, Delft; GNSS Research Centre, Curtin University of Technology, & Department of Infrastructure, University of Melbourne
Date/Time: Thursday, Sep. 22, 11:48 a.m.

When seeking a precise positioning solution [1] at centimeter level with Global Navigation Satellite Systems (GNSS), it is common to use both pseudo-range and carrier-phase measurements made available over different signal frequencies. The former ones are unambiguous and quite noisy, while the phase observables are precise at the millimeter level, but dependent on integer phase ambiguities that need to be resolved first before one can exploit the ultra-precise range-like phase measurements. In the context of Precise Point Positioning (PPP), this is possible after providing satellite phase bias corrections [2] as generally computed from a network of ground-based stations with well-known coordinates.
This Integer Ambiguity Resolution (IAR) process involves, when aiming at the highest success-rate, the solution of an Integer Least-Squares (ILS) problem [3]. This is nowadays easily achievable with the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method [4] developed in the 1990s at the Delft University of Technology. Once the integer-constrained ambiguities are correctly resolved, the estimation of the real-valued parameters can be further improved, often resulting in a very high precision ambiguity-fixed PPP solution in the coordinate domain.
With the recent growth in terms of constellations and signal frequencies, a much larger number of phase observables generally need to be processed. With that, also the dimensionality of the ambiguity resolution problem increases. This causes an exponential increase in complexity, which is also partially due to the high correlation among the ambiguities. Hence, in some cases, it may then be required to tackle a high-dimensional IAR problem in order to improve the accuracy of perhaps only a few parameters of interest. Some alternative approaches exist, which deal with the integer ambiguities directly in the coordinate domain. The Ambiguity Function Methods (AFM, [5][6]), or analogous techniques as the Modified Ambiguity Function Approach (MAFA, [7]), are some few notable examples of approaches where the search is performed in the real-valued parameter space.
In this contribution, a new conditioning strategy to tackle and alleviate the high-dimensionality of the ambiguity resolution problem is introduced, together with an assessment of its associated heuristic minimizers. Given that only a specific subset of real-valued parameters might be actually of interest for GNSS users, e.g. positioning coordinates, a parameter-split approach is also investigated, such that dimensionality of the search problem is further reduced. In this way, the search process takes place only in the domain of relevant parameters, thus within a low dimensional real-valued vector space. When some parameters are further constrained, e.g. as in the baseline-constrained model [8][9], our new approach would also lead to several additional advantages, as it represents a mathematically rigorous formulation of the constrained problem [10][11] independently of the specific type of constraints been considered. Even if not investigated in this research work, once adding new constraint conditions, some further advantages can be found as briefly described in this contribution.
In the scope of this research work, we present three heuristic strategies for tackling the search within the coordinate domain, thereby discussing and demonstrating their advantages and disadvantages. In addition to a more generic Monte-Carlo method, we will discuss the adoption of Genetic Algorithms (GA, [12][13]), along with a more systematic grid-shells expansion approach. The latter is proposed to assure a good approximation to the optimal fixed solution, still maintaining a suitable computational efficiency. By means of numerical examples, the main results following our assessment will be presented and discussed, thus paving the way for future more in-depth research works focalized on search-based methods in the parameter domain.
REFERENCES
[1] Kouba, J., Lahaye, F., Tétreault, P. (2017). Precise Point Positioning. In: Teunissen P.J.G., Montenbruck O. (eds) Springer Handbook of Global Navigation Satellite Systems. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-319-42928-1_25
[2] Khodabandeh, A., Teunissen, P.J.G. (2015). An analytical study of PPP-RTK corrections: precision, correlation and user-impact. J Geod 89, 1109–1132. https://doi.org/10.1007/s00190-015-0838-9
[3] Teunissen, P.J.G. (1993). Least-squares estimation of the integer GPS ambiguities. In Invited lecture, section IV theory and methodology, IAG general meeting, Beijing, China (pp. 1-16).
[4] Teunissen, P.J.G. (1995). The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. Journal of Geodesy 70, 65–82. https://doi.org/10.1007/BF00863419
[5] Counselman, C.C., Gourevitch, S.A. (1981). Miniature Interferometer Terminals for Earth Surveying: Ambiguity And Multipath with Global Positioning System. In IEEE Transactions on Geoscience and Remote Sensing, vol. GE-19, no. 4, pp. 244-252. https://doi.org/10.1109/TGRS.1981.350379
[6] Remondi, B.W. (1984). Using the Global Positioning System (GPS) phase observable for relative geodesy: modeling, processing, and results. Ph.D. dissertation, Center for Space Research, The University of Texas at Austin, Austin, Texas, 360 p.
[7] Cellmer, S., Wielgosz, P., Rzepecka, Z. (2010). Modified ambiguity function approach for GPS carrier phase positioning. J Geod 84, 267–275. https://doi.org/10.1007/s00190-009-0364-8
[8] Teunissen, P.J.G. (2010). Integer least-squares theory for the GNSS compass. J Geod 84, 433–447 (2010). https://doi.org/10.1007/s00190-010-0380-8
[9] Ma, L., Lu, L., Zhu, F., et al. (2021). Baseline length constraint approaches for enhancing GNSS ambiguity resolution: comparative study. GPS Solut 25, 40. https://doi.org/10.1007/s10291-020-01071-1
[10] Giorgi, G., Teunissen, P.J.G., Verhagen, S., Buist, P.J. (2012). Integer Ambiguity Resolution with Nonlinear Geometrical Constraints. In: Sneeuw N., Novák P., Crespi M., Sansò F. (eds) VII Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 137. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22078-4_6
[11] Liu, N., Zhang, Q., Yang, Y. (2018). A dynamic single epoch ambiguity resolutions algorithm with straight trajectory constraints. Adv. Space Res. 62 (4), 855–865. https://doi.org/10.1016/j.asr.2018.05.028
[12] Mirjalili, S. (2019). Genetic Algorithm. In: Evolutionary Algorithms and Neural Networks. Studies in Computational Intelligence, vol 780. Springer, Cham. https://doi.org/10.1007/978-3-319-93025-1_4
[13] Katoch, S., Chauhan, S.S. & Kumar, V. (2021). A review on genetic algorithm: past, present, and future. Multimed Tools Appl 80, 8091–8126. https://doi.org/10.1007/s11042-020-10139-6



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