Advertisement

Regularization and Adjustment

  • Yunzhong Shen
  • Guochang Xu
Chapter

Abstract

The linear observation equation is usually expressed as
$$ {\user2{l}} = {\user2{Ax}} + {\user2{e}} $$
(6.1)
where the non-random design matrix \( {\user2{A}} \in R^{m \times n} , \) the vector of unknown parameters \( {\user2{x}} \in R^{n \times 1} , \) the vector of measurements \( {\user2{l}} \in R^{m \times 1} \)and contaminated by random error vector \( {\user2{e}} \) with zero mean and variance–covariance matrix \( \sigma_{0}^{2} {\user2{P}}^{ - 1} , \) where P is the weight matrix and \( \sigma_{0}^{2} \) is the variance of unit weight. If the coefficient matrix A of the observation equation possesses very large condition number, the observation equation is ill-conditioned, which is defined as ill-posed problems by Hadamard (1932). In geodesy ill-posed problems are frequently encountered in satellite gravimetry due to downward continuation, or in geodetic date procession due to the colinearity among parameters that are to be estimated. Most useful and necessary adjustment algorithms for data processing are outlined in the second part of this chapter. The adjustment algorithms discussed here include least squares adjustment, sequential application of least squares adjustment via accumulation, sequential least squares adjustment, conditional least squares adjustment, a sequential application of conditional least squares adjustment, block-wise least squares adjustment, a sequential application of block-wise least squares adjustment, a special application of block-wise least squares adjustment for code-phase combination, an equivalent algorithm to form the eliminated observation equation system and the algorithm to diagonalize the normal equation and equivalent observation equation. A priori constrained adjustment and filtering are discussed for solving the rank deficient problems. After a general discussion on the a priori parameter constraints, a special case of the so-called a priori datum method is given. A quasi-stable datum method is also discussed. A summary is given at the end of this part of the chapter.

Keywords

Mean Square Error Regularization Parameter Normal Equation Tikhonov Regularization Observation Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Publications Inc., New YorkGoogle Scholar
  2. Albertella A, Sacerdote F (1995) Spectral analysis of block averaged data in geopotential global model determination. J Geodesy 70(3):166–175CrossRefGoogle Scholar
  3. Arikan F, Erol CB, Arikan O (2003) Regularized estimation of vertical total electron content from global positioning system data. J Geophys Res 108(A12): SIA20/1–12Google Scholar
  4. Axelsson O (1994) Iterative solution methods. Cambridge University Press, CambridgeGoogle Scholar
  5. Ayres F (1975) Differential- und Integralrechnung, schaum’s outline. McGraw-Hill Book, New YorkGoogle Scholar
  6. Blewitt G (1998) GPS data processing methodology. In: Teunissen PJG, Kleusberg A (eds) GPS for geodesy. Springer, Berlin, pp 231–270CrossRefGoogle Scholar
  7. Bronstein IN, Semendjajew KA (1987) Taschenbuch der Mathematik.B. G. Teubner Verlagsgesellschaft, Leipzig, ISBN 3-322-00259-4Google Scholar
  8. Cross PA, Ramjattan AN (1995) A Kalman filter model for an integrated land vehicle navigation sys-tem. In: Proceedings of the 3rd international workshop on high precision navigation: High precision navigation 95. University of Stuttgart, April 1995, Bonn, pp 423–434Google Scholar
  9. Cui X, Yu Z, Tao B, Liu D (1982) Adjustment in surveying. Surveying Press, Peking, (in Chinese)Google Scholar
  10. Davis P, Rabinowitz P (1984) Methods of numerical integration, 2nd edn. Academic, New YorkGoogle Scholar
  11. Davis PJ (1963) Interpolation and approximation. Dover Publications Inc., New YorkGoogle Scholar
  12. Ding X, Coleman R (1996) Multiple outlier detection by evaluating redundancy contributions of observations. J Geodesy 708:489–498Google Scholar
  13. Faruqi FA, Turner KJ (2000) Extended Kalman filter synthesis for integrated global positioning/iner-tial navigation systems. Appl Math Comput 115(2–3):213–227CrossRefGoogle Scholar
  14. Gleason DM (1996) Avoiding numerical stability problems of long duration DGPS/INS Kalman filters. J Geodesy 70(5):263–275CrossRefGoogle Scholar
  15. Gotthardt E (1978) Einführung in die Ausgleichungsrechnung. Herbert Wichmann Verlag, KarlsruheGoogle Scholar
  16. Hadamard J (1932) Lecture on Cauchy’s problem in linear partial differential equations, Yale University Press, reprinted by Dover, New York, 1952Google Scholar
  17. Hansen P (1996) Rank-deficient and ill-posed problems, PHD thesis of the technical university of DenmarkGoogle Scholar
  18. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–67CrossRefGoogle Scholar
  19. Hostetter GH (1987) Handbook of digital signal processing, engineering applications. Academic, New YorkGoogle Scholar
  20. Hotine M (1991) Differential geodesy. Springer, BerlinCrossRefGoogle Scholar
  21. Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101CrossRefGoogle Scholar
  22. Knickmeyer ET, Knickmeyer EH, Nitschke M (1996) Zur Auswertung kinematischer Messungen mit dem Kalman-Filter. Schriftenreihe des Deutschen Vereins für Vermessungswesen, Bd. 22, Stuttgart, pp 141–166Google Scholar
  23. Koch KR (1980) Parameterschätzung und Hypothesentests in linearen Modellen. Dümmler-Verlag, BonnGoogle Scholar
  24. Koch KR (1988) Parameter estimation and hypothesis testing in linear models. Springer, BerlinGoogle Scholar
  25. Koch KR (1996) Robuste Parameterschätzung. Allgemeine Vermessungsnachrichten 103(1):1–18Google Scholar
  26. Koch KR, Yang Y (1998a) Konfidenzbereiche und Hypothesentests für robuste Parameterschätzungen. ZfV 123(1):20–26Google Scholar
  27. Koch KR, Yang Y (1998b) Robust Kalman filter for rank deficient observation model. J Geodesy 72:436–441CrossRefGoogle Scholar
  28. Lemmens R (2004) Book review: GPS—theory, algorithms and applications, Xu G 2003. Int J Appl Earth Obs Geoinf 5: 165–166Google Scholar
  29. Li B, Shen Y, Feng Y (2010) Fast GNSS ambiguity resolution as an ill-posed problem. J Geodesy 84:683–698CrossRefGoogle Scholar
  30. Ludwig R (1969) Methoden der Fehler- und Ausgleichsrechnung. Vieweg and Sohn, BraunschweigGoogle Scholar
  31. Masreliez CJ, Martin RD (1977) Robust Bayesian estimation for the linear model and robustifying the Kalman filter. IEEE T Automat Contr AC-22:361–371Google Scholar
  32. Miller K (1970) Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math Anal 1:52–74Google Scholar
  33. Mohamed AH, Schwarz KP (1999) Adaptive Kalman filtering for INS/GPS. J Geodesy 73:193–203CrossRefGoogle Scholar
  34. Morozov VA (1984) Methods for solving incorrectly posed problems. Springer, BerlinCrossRefGoogle Scholar
  35. Ou JK, Wang ZJ (2004) An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers. Chinese Sci Bull 49(2):196–200CrossRefGoogle Scholar
  36. Reigber C, Schmidt R, Flechtner F, König R, Meyer U, Neumayer KH, Schwintzer P, Zhu SY (2005) An earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. J Geodynamics 39:1–10CrossRefGoogle Scholar
  37. Rothacher M, Schaer S (1995) GPS-Auswertetechniken. Schriftenreihe des Deutschen Vereins für Vermessungswesen, Bd. 18, pp 107–121Google Scholar
  38. Schaffrin B (1980) Tikhonov regularization in geodesy, an example. Boll Geod Sci Aff 39:207–216Google Scholar
  39. Schaffrin B (1991) Generating robustified Kalman filters for the integration of GPS and INS. Techni-cal Report, No. 15, Institute of Geodesy, University of StuttgartGoogle Scholar
  40. Schaffrin B (1995) On some alternative to Kalman filtering. In: Sanso F (ed) Geodetic theory today. Springer, Berlin, pp 235–245Google Scholar
  41. Schaffrin B (2008) Minimum mean squared error (MSE) adjustment and the optimal Tikhonov-Phillips regularization parameter via reproducing best invariant quadratic uniformly unbiased estimates (repro-BIQUE). J. Geod 82:113–121CrossRefGoogle Scholar
  42. Schaffrin B, Grafarend E (1986) Generating classes of equivalent linear models by nuisance parameter elimination. Manuscr Geodaet 11:262–271Google Scholar
  43. Shen Y, Li B (2007) Regularized solution to fast GPS ambiguity resolution. J Surveying Eng 133(4):168–172CrossRefGoogle Scholar
  44. Strang G, Borre K (1997) Linear algebra, geodesy, and GPS. Cambridge Press, WellesleyGoogle Scholar
  45. Tarantola A (2005) Inverse problem theory. SIAM, PhildelphiaGoogle Scholar
  46. Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70(1–2):65–82CrossRefGoogle Scholar
  47. Tikhonov AN (1963a) Regularization of ill-posed problems, English translation of Dokl. Akad Nauk SSSR 151(1):49–52Google Scholar
  48. Tikhonov AN (1963b) Solution of incorrectly formulated problems and the regularization method, English translation of Dokl. Akad Nauk SSSR 151(3):501–504Google Scholar
  49. Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. Wiley, New YorkGoogle Scholar
  50. Tikhonov AN, Goncharsky AV, Steppanov VV, Yagola AG (1995) Numerical methods for the solution of ill-posed problems. Kluwer Academic Publishers, NetherlandsGoogle Scholar
  51. Tsai C, Kurz L (1983) An adaptive robustifing approach to Kalman filtering. Automatica 19:279–288CrossRefGoogle Scholar
  52. Wahba G (1983) Bayesian “confidence intervals” for the cross-validated smoothing spline. J R Stat Soc B45:133–150Google Scholar
  53. Wang LX, Fang ZD, Zhang MY, Lin GB, Gu LK, Zhong TD, Yang XA, She DP, Luo ZH, Xiao BQ, Chai H, Lin DX (1979) Mathematic handbook. Educational Press, Peking. ISBN 13012-0165Google Scholar
  54. Wang G, Chen Z, Chen W, Xu G (1988) The principle of GPS precise positioning system. Surveying Press, Peking, p 345. ISBN 7-5030-0141-0/P.58 (in Chinese)Google Scholar
  55. Xu G (2002a) GPS data processing with equivalent observation equations. GPS Solutions, vol 6, No. 1–2, 6:28–33Google Scholar
  56. Xu G (2002b) A general criterion of integer ambiguity search. J GPS 1(2):122–131CrossRefGoogle Scholar
  57. Xu G (2003) A diagonalization algorithm and its application in ambiguity search. J. GPS 2(1):35–41CrossRefGoogle Scholar
  58. Xu G (2007) GPS–theory, algorithms and applications. Springer, Berlin, pp xix + 340Google Scholar
  59. Xu G, Qian Z (1986) The application of block elimination adjustment method for processing of the VLBI Data. Crustal Deformation and Earthquake, Vol. 6, No. 4, (in Chinese)Google Scholar
  60. Xu P (1992) Determination of surface gravity anomalies using gradiometric observables. Geophys J Int 110:321–332CrossRefGoogle Scholar
  61. Xu P (1998) Truncated SVD methods for discrete linear ill-posed problems. Geophys J Int 135:505–514CrossRefGoogle Scholar
  62. Xu P, Rummel R (1994) Generalized ridge regression with applications in determination of potential fields. Manuscr Geod 20:8–20Google Scholar
  63. Xu P, Shen Y, Fukuda Y, Liu Y (2006) Variance components estimation in linear inverse ill-posed models. J Geod 80:69–81CrossRefGoogle Scholar
  64. Yang Y (1991) Robust Bayesian estimation. B Geod 65:145–150Google Scholar
  65. Yang Y (1993) Robust estimation and its applications. Bayi Publishing House, PekingGoogle Scholar
  66. Yang Y (1994) Robust estimation for dependent observations. Manuscr Geodaet 19:10–17Google Scholar
  67. Yang Y (1997a) Estimators of covariance matrix at robust estimation based on influence functions. ZfV 122(4):166–174Google Scholar
  68. Yang Y (1997b) Robust Kalman filter for dynamic systems. J Zhengzhou Inst Surveying Mapping 14:79–84Google Scholar
  69. Yang Y (1999) Robust estimation of geodetic datum transformation. J Geodesy 73:268–274CrossRefGoogle Scholar
  70. Yang Y, Cui X (2008) Adaptively robust filter with multi adaptive factors. Surv Rev 40(309):260–270CrossRefGoogle Scholar
  71. Yang Y, Gao W (2005) Comparison of adaptive factors on navigation results. J Navigation 58:471–478CrossRefGoogle Scholar
  72. Yang Y, Gao W (2006) An optimal adaptive kalman filter with applications in navigation. J Geodesy 80:177–183CrossRefGoogle Scholar
  73. Yang Y, Gao W (2006) A new learning statistic for adaptive filter based on predicted residuals. Prog Nat Sci 16(8):833–837CrossRefGoogle Scholar
  74. Yang Y, He H, Xu G (2001) Adaptively robust filtering for kinematic geodetic positioning. J Geodesy 75:109–116CrossRefGoogle Scholar
  75. Yang Y, Tang Y, Li Q, Zou Y (2006) Experiments of adaptive filters for kinemetic GPS positioning applied in road information updating in GIS. J Surv Eng (in press)Google Scholar
  76. Yang YX et al (2005a) Combined adjustment project of national astronomical geodetic networks and 2000’ national GPS control network. Prog Nat Sci 15(5):435–441CrossRefGoogle Scholar
  77. Yang YX, Xu TH, Song LJ (2005) Robust estimation of variance components with application in global positioning system network adjustment. J Surv Eng ASCE 131(4): 107–112Google Scholar
  78. Zhou J (1985) On the Jie factor. Acta Geodaetica et Geophysica 5 (in Chinese)Google Scholar
  79. Zhou J (1989) Classical theory of errors and robust estimation. Acta Geod Cartogr Sinica 18:115–120Google Scholar
  80. Zhou J, Huang Y, Yang Y, Ou J (1997) Robust least squares method. Publishing House of Huazhong University of Science and Technology, WuhanGoogle Scholar
  81. Zhu J (1996) Robustness and the robust estimate. J Geodesy 70(9):586–590CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Surveying and Geo-Informatics EngineeringTongJi UniversityShanghai PR. China
  2. 2.Deptartment 1 Geodesy and Remote SensingGFZ German Research Centre for GeosciencesPotsdamGermany
  3. 3.Chinese Academy of Space TechnologyBeijingP.R. China

Personalised recommendations