Abstract
Integer ambiguity validation is an essential quality control step for high-precision positioning and navigation with global navigation satellite systems (GNSS). In order to validate the resolved integer ambiguities, statistical tests, such as the R-ratio test, F-ratio test, W-ratio test, difference test, and projector test, have been favored. In practice, the critical values for these statistical tests are determined either empirically or from the assumed distributions. However, previous research has revealed that some of these statistics have upper bounds, which can be obtained from simulations. In this contribution, we find that under the framework of the integer aperture estimation, the upper bounds for these ambiguity validation statistical tests can be derived without actual measurements or simulation. As a result, the assumed distributions for these statistical tests are inappropriate. According to the derivation, it has been concluded that the upper bounds of these ambiguity validation tests depend only on the ambiguity geometry (e.g., the float ambiguity variance–covariance matrix) and can be obtained at the design stage of GNSS positioning. Thus, the critical value for these ambiguity validation statistics has a rigorous range, and it should be chosen to be smaller than a priori derived upper bound. Otherwise, no integer ambiguities can be obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Tiberius C, de Jonge, P (1995) Fast positioning using the LAMBDA method. In: Proceedings of 4th international conference differential satellite systems, paper 30, Bergen, Norway, 24–28 April
Euler H, Goad C (1992) On optimal filtering of GPS dual-frequency observations without orbit information. Bull Geod 65(2):130–143
Euler H, Schaffrin B (1991) On a measure for the discernibility between different ambiguity solutions in the static-kinematic GPS mode. IAG Symposia no 107, kinematic systems in geodesy, surveying, and remote sensing. Springer, Berlin, pp 285–295
Feng S, Ochieng W, Samson J, Tossaint M, Hernandez M, Juan J, Sanz J, Aragon A, Ramos P, Jofre M (2012) Integrity monitoring for carrier phase ambiguities. J Navig 65(1):41–58
Frei E, Beutler G (1990) Rapid static positioning based on the fast ambiguity resolution approach FARA: theory and first results. Manuscr Geod 15(4):325–356
Han S (1997) Quality control issues relating to instantaneous ambiguity resolution for real-time GPS kinematic positioning. J Geod 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
Li T, Wang J (2012a) Some remarks on GNSS integer ambiguity validation methods. Surv Rev 44(326):230–238
Li T, Wang J (2012b) Theoretical upper bound and lower bound for integer aperture estimation fail-rate and practical implications. J Navig. doi:10.1017/S0373463312000513
Teunissen P (1993) Least-squares estimation of the integer GPS ambiguities. Delft Geodetic Computer Centre LGR series, No. 6, p 16
Teunissen P (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70(1–2):65–82
Teunissen P (1998) Success probability of integer GPS ambiguity rounding and bootstrapping. J Geod 73(11):587–593
Teunissen P (1999) An optimality property of the integer least-squares estimator. J Geod 73(11):587–593
Teunissen P (2002) The parameter distributions of the integer GPS model. J Geod 76:41–48
Teunissen P (2003a) Integer aperture GNSS ambiguity resolution. Artif Satell 38(3):79–88
Teunissen P (2003b) A carrier phase ambiguity estimator with easy-to-evaluate fail-rate. Artif Satell 38(3):89–96
Teunissen P, Verhagen S (2009) The GNSS ambiguity ratio-test revisited: a better way of using it. Surv Rev 41(312):138–151
Verhagen S (2004) Integer ambiguity validation: an open problem? GPS Solut 8(1):36–43
Verhagen S (2005) The GNSS integer ambiguities: estimation and validation. PhD thesis, Publications on Geodesy 58 Netherland Geodetic Commission, Delft
Verhagen S, Teunissen P (2006a) New global navigation satellite system ambiguity resolution method compared to existing approaches. J Guid Control Dyn 29(4):981–991
Verhagen S, Teunissen P (2006b) On the probability density function of the GNSS ambiguity residuals. GPS Solut 10:21–28
Wang J, Stewart M, Tsakiri M (1998) A discrimination test procedure for ambiguity resolution on-the-fly. J Geod 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):813–817
Xu P (2006) Voronoi cells, probabilistic bounds and hypothesis testing in mixed integer linear models. IEEE Trans Info Theory 52(2):3122–3138
Acknowledgments
The first author would like to acknowledge the Chinese Scholarship Council (CSC) for supporting his studies at The University of New South Wales, Sydney, Australia. The comments from two anonymous reviewers are greatly appreciated. Thanks to Joseph Gauthier for proofreading.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, T., Wang, J. Analysis of the upper bounds for the integer ambiguity validation statistics. GPS Solut 18, 85–94 (2014). https://doi.org/10.1007/s10291-013-0312-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10291-013-0312-1