Statistische Schätzung und Hypothesentest für das gemischte ganzzahlige lineare Modell mittels GPS-Trägerphasenmessung
Final Report Abstract
High accurate GPS (Global Positioning System) geodesy and navigafion are based on the very precise carrier phase observables, where the central problem of the determination ofthe phase ambiguities must be first solved. The GPS phase observations are measured on the unit circle which are actually von Mises normally distributed. According to this fact a proper method of estimation and hypothesis test for the mixed model (a part unknown parameters in a physical GPS model, a second part of bias terms, partially "integer", in a mixed integral and real valued model) should be developed. The existed validation procedures within Least Squares Ambiguity Search (LSAS) approaches are performed under the assumption that the measured phases or phase differences are approximately Gauss-Laplace normally distributed, and must therefore be improved accordingly. Through the research work in the two phase DFG project we have achieved the following main five results: (1) The fundamental recognition and related statistical validation of the stafistical properties of the von Mises (circular) normal distribution of GPS carrier phase observables and (2) the fi-actional parts of the float solution of carrier phase ambiguities with respect to their fixed solution; (3) the successful construcfton of the related hypothesis tests with bootstrapping residuals and bootstrapping pairs for linear model; (4) the new total optimal search criterion in solving the GPS mixed linear model; together with (5) the application of the optimal regularizafion method in GNSS rapid static positioning. These contributions provide a complete solution to the estimation and hypothesis tests on the parameters of the GPS mixed integer linear models in the directional context, which will not only substantially improve the theoretical background and application reality of GPS geodesy and navigation, but also be significant to the forthcoming Galileo, Europe's new civilian-managed Global Navigation Satellite System (GNSS). The increasing scientific and commercial applications will certainly benefit from the highest-possible GPS positioning accuracies and integrity standards. As we have introduced in the section 2.1, up to now, most of published or applied hypothesis tests and validation of the estimations with GPS phase observation are performed under the assumption that the measured phases or phase differences have a Gauss-Laplace normal distribution. Even in the more recently published papers from Verhagen and Teunissen (2006) and Teunissen and Verhagen (2008) they have applied the same assumption. Although Gabor and Nerem (2004) and Keshin (2004) discussed the directional characteristics of satellite-satellite single-difference widelane phase biases and Yun and Kim (2007) investigated the non-Gaussian measurements, there is no other research group which has studied the statistical property of the GPS geodetic relative positioning with carrier phase observables, while we have systematically studied them with GPS real phase observations in the first phase's researching work. The fundamental recognition and related statistical validation of the statistical properties of the von Mises (circular) normal distribution of GPS carrier phase observables and the fractional parts of the float solution of carrier phase ambiguities with respect to their fixed solution, together with the successful constmction of the related hypothesis tests with bootstrapping residuals and bootstrapping pairs for linear model provide a complete solufion to the estimation and hypothesis tests ofthe GPS mixed integer linear models in the direcfional context. After we have submitted the project applicafion in 11. 2004, in one technical report G. Xu (12. 2004) emphasized once his equivalent ambiguity searching criterion with the same argument that the generally applied LSAS searching criterion is only sub-optimal. Morujao and Mendesand (2007) applied it in one investigation of instantaneous ambiguity resolution with the GPS/Galileo combination for precise kinematic positioning. These show that there is remaining the necessity to study the priority of both criteria in the sense of optimality and unbiasness. Our contribution with development of total optimal search criterion in solving the mixed integer linear model provides an explicit answer to the ongoing debate about the priority of both criteria in the sense of optimality and unbiasness. In addition to the initial objectives we have obtained another important result via the application of our newly developed A-optimal regularization methods in GNSS rapid static positioning.
Publications
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(2006): A spherical problem of algebraic representation - inconsistent system of directional observational equations - overdetermined system of nonlinear equations on curved manifolds. Chapter 7 in the book "Linear and Nonlinear Models - Fixed Effects, Random Effects and Mixed Models", pp.327-346, Walter de Gruyter, Berlin, New York
Grafarend, E.
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(2007): Directional statistics and its application in the hypothesis testing of GPS integer ambiguity resolution, Geophysical Research Abstracts, Vol. 9, 08768, European Geosciences Union 2007
Cai J. and Grafarend E.
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(2007): The Optimal Regularization Method and its Application in GNSS Rapid Static Positioning, ION GNSS 2007 Meeting Proceedings "ION GNSS 20th International Technical Meeting of the Satellite Division, 25-28, Sep. 2007, Fort Worth, TX, pp. 299-305
Cai J., Hu C. and Grafarend E.
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(2007): The Statistical Property of the GNSS Carrier Phase Observations and its Effects on the Hypothesis Testing ofthe Related Estimators, ION GNSS 2007 Meeting Proceedings "ION GNSS 20th Intemational Technical Meeting of the Satellite Division, 25-28, Sep. 2007, Fort Worth, TX, pp. 331-338
Cai J., E. Grafarend and C. Hu
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(2008): The uniform Tykhonov-Phillips regularization (aweighted S-homBLE) and its application in GPS rapid static positioning, lAG Symposia 132 "VI Hotine-Mamssi Symposium on Theoretical and Computational Geodesy", Wuhan, China, 29 May-2 June, 2006, eds. P. Xu, J. Liu and A. Dermanis, pp.221-229, Springer, Berlin, Heidelberg
Cai J., C. Hu, E. Grafarend and J. Wang
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(2009): Bootstrap method and its application to the hypothesis testing in GPS mixed integer linear model, Geophysical Research Abstracts, Vol. 11, EGU2009-9135, European Geosciences Union 2009
Cai J. and Grafarend E.
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(2009): The total optimal search criterion in solving the mixed integer linear model with GNSS carrier phase observations, Journal of GPS Solutions, Vol.13, 221-230,
Cai J., E. Grafarend and C. Hu
