Abstract
The strength of the GNSS precise positioning model degrades in cases of a lack of visible satellites, poor satellite geometry or uneliminated atmospheric delays. The least-squares solution to a weak GNSS model may be unreliable due to a large mean squared error (MSE). Recent studies have reported that Tikhonov’s regularization can decrease the solution’s MSE and improve the success rate of integer ambiguity resolution (IAR), as long as the regularization matrix (or parameter) is properly selected. However, there are two aspects that remain unclear: (i) the optimal regularization matrix to minimize the MSE and (ii) the IAR performance of the regularization method. This contribution focuses on these two issues. First, the “optimal” Tikhonov’s regularization matrix is derived conditioned on an assumption of prior information of the ambiguity. Second, the regularized integer least-squares (regularized ILS) method is compared with the integer least-squares (ILS) method in view of lattice theory. Theoretical analysis shows that regularized ILS can increase the upper and lower bounds of the success rate and reduce the upper bound of the LLL reduction complexity and the upper bound of the search complexity. Experimental assessment based on real observed GPS data further demonstrates that regularized ILS (i) alleviates the LLL reduction complexity, (ii) reduces the computational complexity of determinate-region ambiguity search, and (iii) improves the ambiguity fixing success rate.
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Data availability statement
The datasets in this contribution are available from the Hong Kong Geodetic Survey Service, http://www.geodetic.gov.hk/tc/satref/satref.htm.
References
Agrell E, Eriksson T, Vardy A, Zeger K (2002) Closest point search in lattices. IEEE Trans Inf Theory 48(8):2201–2214
Akaike H (1980) Likelihood and the Bayes procedure. Trabajos De Estadistica Y De Investigacion Operativa 31:143–166
Akhavi A (2003) The optimal LLL algorithm is still polynomial in fixed dimension. Theor Comput Sci 297(1–3):3–23
Chang XW, Yang X, Zhou T (2005) MLAMBDA: a modified LAMBDA algorithm for integer least-squares estimation. J Geodesy 79(9):552–565
Chang XW, Wen J, Xie X (2013) Effects of the LLL reduction on the success probability of the Babai point and on the complexity of sphere decoding. IEEE Trans Inf Theory 59(8):4915–4926
Counselman C, Gourevitch S (1981) Miniature interferometer terminals for earth surveying: ambiguity and multipath with global positioning system. IEEE Trans Geosci Remote Sens 19(4):244–252
Dermanis A, Rummel R (2008) Data analysis methods in geodesy book: geomatic method for the analysis of data in the earth sciences. Springer, Heidelberg, pp 17–92
Dermanis A, Sansò F, Grün A (2000) An overview of data analysis methods in geomatics book: geomatic methods for the analysis of data in earth sciences. Springer, Heidelberg, pp 1–16
Euler H, Landau H (1992) Fast GPS ambiguity resolution on-the-fly for real-time application. In: Proceedings of the 6th international geodesy symposium on satellite positioning, Columbus, Ohio, 17–20 March, pp 650–659
Frei E, Beutler G (1990) Rapid static positioning based on the fast ambiguity resolution approach ‘FARA’: theory and first results. Manuscr Geodesy 15(6):326–356
Golden G, Foschini C, Valenzuela R, Wolniansky PW (1999) Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture. Electron Lett 35(1):14–16
Grafarend EW (2000) Mixed integer-real valued adjustment (IRA) problems: GPS initial cycle ambiguity resolution by means of the LLL algorithm. GPS Solut 4(2):31–44
Gruber PM, Wills JM (1993) Handbook of convex geometry, vol B. Elsevier, North Holland
Gui QM, Han SH (2007) New algorithm of GPS rapid positioning based on double-k-type ridge estimation. J Surv Eng 133:67–72
Hansen P (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34:561–580
Hassibi A, Boyed S (1998) Integer parameter estimation in linear models with applications to GPS. IEEE Trans Signal Proc 46:2938–2952
Hermite C (1850) Extraits de lettres de m. ch. hermite à m. jacobi sur différents objects de la théorie des nombres. (continuation). Journal Für Die Reine Und Angewandte Mathematik 1850(40):279–315
Hervé D, Brigitte V (1994) An upper bound on the average number of iterations of the lll algorithm. Theor Comput Sci 123(1):95–115
Horn RA, Johnson CR (1993) Matrix analysis. Cambridge University Press, Cambridge
Jaldén J, Seethaler D, Matz G (2008) Worst- and average-case complexity of LLL lattice reduction in MIMO wireless systems. In: Proceedings of IEEE international conference on acoustics, Las Vegas, March 30—April 4 2008, pp 2685–2688
Jazaeri S, Amiri-Simkooei A, Sharifi MA (2012) Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory. J Geodesy 86(2):123–136
Jazaeri S, Amiri-Simkooei A, Sharifi MA (2014) On lattice reduction algorithms for solving weighted integer least squares problems: comparative study. GPS Solut 18(1):105–114
Jonge PD, Tiberius C (1996) Integer ambiguity estimation with the lambda method. Springer, Berlin
Kannan R (1983) Improved algorithms for integer programming and related problems. In: 15th ACM symposium on theory of computing, pp 193–206
Korkine A, Zolotarev G (1873) Surles Formes Quadratiques. Math Ann 6(3):366–389
Lacy MCD, Sansò F, Rodriguez-Caderot G, Gil AJ (2002) The Bayesian approach applied to GPS ambiguity resolution. A mixture model for the discrete–real ambiguities alternative. J Geodesy 76(2):82–94
Lenstra AK, Lenstra HW, Lovász L (1982) Factoring polynomials with rational coefficients. Math Ann 261(4):515–534
Li BF, Shen YZ, Feng YM (2010) Fast GNSS ambiguity resolution as an ill-posed problem J. Geod 84:683–698
Ling C, Mow WH (2009) A unified view of sorting in lattice reduction: from V-BLAST to LLL and beyond. In: Proceedings of IEEE information theory workshop, pp 529–533
Minkowski H (1896) Geometrie der Zahlen. Teubner-Verlag, Stuttgart
Schnorr CP (2006) Fast LLL-type lattice reduction. Inf Comput 204(1):1–25
Schnorr CP (2009) Progress on LLL and lattice reduction. In: Nguyen P, Vallée B (eds) The LLL algorithm. Information security and cryptography. Springer, Berlin
Schnorr CP, Euchner M (1994) Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math Prog 66:181–199
Seethaler D, Jalden J, Studer C, Boelcskei H (2011) On the complexity distribution of sphere decoding. IEEE Trans Inf Theory 57(9):5754–5768
Shannon CE (1959) Probability of error for optimal codes in a Gaussian channel. Bell Syst Technol J 38:611–656
Shen YZ, Li BF (2007) Regularized solution to fast GPS ambiguity resolution. J Surv Eng 133:168–172
Shores TS (2007) Applied linear algebra and matrix analysis. Springer, New York
Talagrand O (2003) Bayesian estimation. Optimal interpolation statistical linear estimation. In: Swinbank R, Shutyaev V, Lahoz WA (eds) Data assimilation for the earth system. NATO science series (series IV: earth and environmental sciences), vol 26. Springer, Dordrecht
Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70(1–2):65–82
Teunissen PJG (1999) An optimality property of the integer least-squares estimator. J Geod 73(11):587–593
Teunissen PJG (2001) Integer estimation in the presence of biases. J Geodesy 75(7–8):399–407
Teunissen PJG (2010) Integer least-squares theory for the GNSS compass. J Geodesy 84:433–447
Teunissen PJG (1993) Least squares estimation of integer GPS ambiguities. Sect IV theory and methodology, IAG general meeting, Beijing
Tikhonov AN (1963) Regularization of ill-posed problems. Dokl Akad Nauk SSSR 151(1):49–52
Verhagen S, Li BF, Teunissen PJG (2013) Ps-LAMBDA: ambiguity success rate evaluation software for interferometric applications. Comput Geosci 54:361–376
Wang J, Feng YM (2013) Orthogonality defect and reduced search-space size for solving integer least-squares problems. GPS Solut 17(2):261–274
Wu ZM, Bian SF (2015) GNSS integer ambiguity validation based on posterior probability. J Geodesy 89(10):961–977
Wu ZM, Bian SF, Ji B, Xiang CB, Jiang DF (2015) Short baseline GPS multi-frequency single-epoch precise positioning: utilizing a new carrier phase combination method. GPS Solut 20(3):373–384
Wu ZM, Bian SF, Xiang CB, Tong YD (2013) A new method for TSVD regularization truncated parameter selection. Math Probl Eng 2013:161834. https://doi.org/10.1155/2013/161834
Wu ZM, Li HP, Bian SF (2017) Cycled efficient V-Blast GNSS ambiguity decorrelation and search complexity estimation. GPS Solut 21:1829–1840
Xu PL (1998) Truncated SVD methods for discrete linear ill-posed problems. Geophys J Int 135:505–514
Xu PL (2001) Random simulation and GPS decorrelation. J Geod 75(7–8):408–423
Xu PL (2006) Voronoi cells, probabilistic bounds and hypothesis testing in mixed integer linear models. IEEE Trans Inf Theory 52(7):3122–3138
Xu PL (2012) Parallel Cholesky-based reduction for the weighted integer least squares problem. J Geodesy 86(1):35–52
Xu PL (2013) Experimental quality evaluation of lattice basis reduction methods for decorrelating low-dimensional integer least squares problems. EURASIP J Adv Signal Process 2013(1):137
Xu PL, Rummel R (1994) A simulation study of smoothness methods in recovery of regional gravity fields. Geophys J Int 117:472–486
Xu PL, Cannon E, Lachapelle G (1995) Mixed integer programming for the resolution of GPS carrier phase ambiguities. Presented at IUGG95 assembly, 2–14 July, Boulder, CO, USA
Xu PL, Chi C, Liu J (2012) Integer estimation methods for GPS ambiguity resolution: an application-oriented review and improvement. Surv Rev 44(324):59–71
Xu PL, Du F, Shu YM, Zhang HP, Shi Y (2021) Regularized reconstruction of peak ground velocity and acceleration from very high-rate GNSS precise point positioning with applications to the 2013 Lushan Mw6.6 earthquake. J Geodesy 95:17
Zhu J, Ding X, Chen Y (2001) Maximum-likelihood ambiguity resolution based on Bayesian principle. J Geodesy 75(4):175–187
Acknowledgements
This work was supported by the National Natural Science Foundation of China (nos. 41504029 and 41631072) and the Natural Science Foundation for Distinguished Young Scholars of Hubei Province of China (no. 2019CFA086). I would like to express my appreciation to Prof. Athanasios Dermanis and two other reviewers, Editor-in-Chief Jürgen Kusche, and Associate Editor Mattia Crespi for their helpful suggestions.
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Conceptualization, software, and writing were performed by ZW; methodology and experiment assessment were performed by ZW and SB. All the authors have read and approved the final manuscript.
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Appendix
Appendix
The original GNSS mathematical model can be expressed as a nonlinear vector function as follows:
where \({\varvec{s}}\in {\mathbb{R}}^{l}\) is the original double-differenced carrier phase and code observables; \({\varvec{x}}={\left[{\mathfrak{a}}^{T}\boldsymbol{ }\boldsymbol{ }{\mathfrak{b}}^{T}\right]}^{T}\) is a vector of unknown parameters, with integer-valued \(\mathfrak{a}\in {\mathbb{Z}}^{n}\) and real-valued \(\mathfrak{b}\in {\mathbb{R}}^{m}\); and \({\varvec{\varepsilon}}\) is a vector of observation errors. \(\mathrm{f}({\varvec{x}})\) is a nonlinear measurement vector function of parameter vector \({\varvec{x}}\). The VC matrix of observations in \({\varvec{s}}\) is \({{\varvec{Q}}}_{{\varvec{s}}{\varvec{s}}}\), which is regarded as known a priori. The equation can be extended using a Taylor series around an initial parameter vector \({{\varvec{x}}}_{(0)}\) as
where \({\varvec{F}}\) is a partial derivatives matrix of \(\mathrm{f}({\varvec{x}})\) with respect to \({\varvec{x}}\) at \({\varvec{x}}={{\varvec{x}}}_{(0)}\)
If the initial parameters are adequately near the true values, the second and further terms of the Taylor series can be neglected. We can approximate the following linear equation as
Then, we can obtain the estimated unknown parameter vector increment \(\delta {\widehat{{\varvec{x}}}}_{(0)}\) by
If the initial parameters \({{\varvec{x}}}_{(0)}\) are not sufficiently near the true values, we can iteratively improve the estimated parameters as
with
until all the elements in the observed-minus-computed vector \({\varvec{s}}-\mathrm{f}({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\boldsymbol{^{\prime}}})\) are within a user-defined post fit threshold
where means rounding to the nearest integer, \(\alpha \) is a user-defined post fit threshold, and \({{\varvec{e}}}_{k}\) is the unit vector with its \(k\)th entry a 1 and other entries 0. Using \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}\) instead of \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\mathrm{^{\prime}}}\) in the iterations guarantees the integer nature of the ambiguity vector in each step. However, we also need to use \({\widehat{{\varvec{x}}}}_{\left(i+1\right)}^{\mathrm{^{\prime}}}\) to judge when the iterations could terminate. This process is a modification of the well-known Gauss–Newton iteration. If (64) is satisfied after \(j\) iterations, the linearized GNSS model is obtained as
with
and the float solution of the parameter vector is
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Wu, Z., Bian, S. Regularized integer least-squares estimation: Tikhonov’s regularization in a weak GNSS model. J Geod 96, 22 (2022). https://doi.org/10.1007/s00190-021-01585-7
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DOI: https://doi.org/10.1007/s00190-021-01585-7