Abstract
In satellite navigation, the key to high precision is to make use of the carrier-phase measurements. The periodicity of the carrier-phase, however, leads to integer ambiguities. Often, resolving the full set of ambiguities cannot be accomplished for a given reliability constraint. In that case, it can be useful to resolve a subset of ambiguities. The selection of the subset should be based not only on the stochastic system model but also on the actual measurements from the tracking loops. This paper presents a solution to the problem of joint subset selection and ambiguity resolution. The proposed method can be interpreted as a generalized version of the class of integer aperture estimators. Two specific realizations of this new class of estimators are presented, based on different acceptance tests. Their computation requires only a single tree search, and can be efficiently implemented, e.g., in the framework of the well-known LAMBDA method. Numerical simulations with double difference measurements based on Galileo E1 signals are used to evaluate the performance of the introduced estimation schemes under a given reliability constraint. The results show a clear gain of partial fixing in terms of the probability of correct ambiguity resolution, leading to improved baseline estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
\({\varvec{z}}_{(\mathcal {I})}\) addresses all entries of \({\varvec{z}}\) with index \(i \in {\mathcal {I}}\).
References
Agrell E, Eriksson T, Vardy A, Zeger K (2002) Closest point search in lattices. IEEE Trans Inf Theory 48(8):2201–2214
Babai L (1986) On Lovász Lattice reduction and the nearest lattice point problem. Combinatorica 6(1):1–13
Blewitt G (1989) Carrier phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km. J Geophys Res Solid Earth 94(B8):10187–10203
Dai L, Eslinger D, Sharpe T (2007) Innovative algorithms to improve long range RTK reliability and availability. In: Proceedings of the 2007 National Technical Meeting of the Institute of Navigation, pp 860–872
De Jonge PJ, Tiberius CCJM (1996) The LAMBDA Method for integer ambiguity estimation: implementation aspects. Publications of the Delft Computing Centre, LGR-Series 12
Euler HJ, Schaffrin B (1991) On a measure for the discernibility between different ambiguity solutions in the static-kinematic GPS-mode. In: IAG Symposia no 107, Kinematic systems in geodesy, surveying, and remote sensing, pp 285–295
Frei E, Beutler G (1990) Rapid static positioning based on the fast ambiguity resolution approach FARA: theory and first results. Manuscripta geodaetica 15(6):325–356
Hagenauer J, Offer E, Papke L (1996) Iterative decoding of binary block and convolutional codes. IEEE Trans Inf Theory 42(2):429–445
Han S (1997) Quality-control issues relating to instantaneous ambiguity resolution for real-time GPS kinematic positioning. J Geodes 71(6):351–361
Hassibi A, Boyd S (1998) Integer parameter estimation in linear models with applications to GPS. IEEE Trans Signal Process 46(11):2938–2952
Jaldén J, Ottersten B (2005) Parallel implementation of a soft output sphere decoder. In: Signals, systems and computers, 2005. Conference Record of the Thirty-Ninth Asilomar conference on, IEEE, pp 581–585
Khanafseh S, Pervan B (2010) New approach for calculating position domain integrity risk for cycle resolution in carrier phase navigation systems. IEEE Trans Aerosp Electron Systems 46(1):296– 307
Lenstra AK, Lenstra HW, Lovász L (1982) Factoring polynomials with rational coefficients. Math Annal 261(4):515–534
Li T, Wang J (2012) Some remarks on GNSS integer ambiguity validation methods. Surv Rev 44(326):230–238
Li T, Wang J (2013) Theoretical upper bound and lower bound for integer aperture estimation fail-rate and practical implications. J Navig 66(3):321–333
Parkins A (2011) Increasing GNSS RTK availability with a new single-epoch batch partial ambiguity resolution algorithm. GPS Solut 15(4):391–402
Schnorr CP, Euchner M (1994) Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math Program 66(1–3):181–199
Studer C, Bölcskei H (2010) Soft-input soft-output single tree-search sphere decoding. IEEE Trans Inf Theory 56(10):4827–4842
Studer C, Burg A, Bölcskei H (2008) Soft-output sphere decoding: algorithms and VLSI implementation. IEEE J Sel Areas Commun 26(2):290–300
Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70(1–2):65–82
Teunissen PJG (2001) GNSS ambiguity bootstrapping: theory and application. In: Proceedings of international symposium on kinematic systems in Geodesy, geomatics and navigation, pp 246–254
Teunissen PJG (2003a) Integer aperture GNSS ambiguity resolution. Artif Satell 38(3):79–88
Teunissen PJG (2003b) Theory of integer equivariant estimation with application to GNSS. J Geodes 77(7–8):402–410
Teunissen PJG (2003c) Towards a unified theory of GNSS ambiguity resolution. J Global Position Systems 2(1):1–12
Teunissen PJG, Verhagen S (2009) The GNSS ambiguity ratio-test revisited: a better way of using it. Surv Rev 41(312):138–151
Teunissen PJG, Joosten P, Tiberius CCJM (1999) Geometry-free ambiguity success rates in case of partial fixing. In: Proceedings of the 1999 National Technical Meeting of the Institute of Navigation, pp 201–207
Tiberius CCJM, De Jonge PJ (1995) Fast positioning using the LAMBDA method. In: Proceedings of DSNS-95, Bergen, Norway
Verhagen S (2004a) The GNSS integer ambiguities: estimation and validation. Dissertation, Technische Universiteit Delft
Verhagen S (2004b) Integer ambiguity validation: an open problem? GPS Solut 8(1):36–43
Verhagen S, Teunissen PJG (2006) New global navigation satellite system ambiguity resolution method compared to existing approaches. J Guidance Control Dynamics 29(4):981–991
Verhagen S, Teunissen PJG (2013) The ratio test for future GNSS ambiguity resolution. GPS Solut 17(4):535–548
Verhagen S, Teunissen PJG, van der Marel H, Li B (2011) GNSS ambiguity resolution: which subset to fix. In: Proceedings of IGNSS Symposium, Sydney, Australia
Viterbo E, Boutros J (1999) A universal lattice code decoder for fading channels. IEEE Trans Inf Theory 45(5):1639–1642
Wang J, Stewart M, Tsakiri M (1998) A discrimination test procedure for ambiguity resolution on-the-fly. J Geodes 72(11):644–653
Wang J, Stewart M, Tsakiri M (2000) A comparative study of the integer ambiguity validation procedures. Earth Planets Space 52(10):813–818
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brack, A., Günther, C. Generalized integer aperture estimation for partial GNSS ambiguity fixing. J Geod 88, 479–490 (2014). https://doi.org/10.1007/s00190-014-0699-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-014-0699-7