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Abstrakty
In adjusted geodetic networks, cases of local configuration defects (defects in the geometric structure of the network due to missing data or errors in point numbering) can be encountered, which lead to the singularity of the normal equation system in the least-squares procedure. Numbering errors in observation sets cause the computer program to define the network geometry incorrectly. Another cause of a defect may be accidental omission of certain data records, causing local indeterminacy or lowering of local reliability rates in a network. Obviously, the problem of a configuration defect may be easily detectable in networks with a small number of points. However, it becomes a real problem in large networks, where manual checking of all data becomes a very expensive task. The paper presents a new strategy for the detection of configuration defects with the use of the Tikhonov regularization method. The method was implemented in 1992 in the GEONET system (www.geonet.net.pl).
Czasopismo
Rocznik
Tom
Strony
19--25
Opis fizyczny
Bibliogr. 30 poz., rys., tab.
Twórcy
autor
- Department of Geodesy and Geotechnics, Rzeszów University of Technology, 12 PowstańcówWarszawy St. 35-959 Rzeszów, Poland
Bibliografia
- [1] Baarda, W. (1968). A testing procedure for use in geodetic networks. Publication on Geodesy, New Series, 2.
- [2] Bakushinskii, A. B. (1992). The problem of the convergence of the iteratively regularized Gauss-Newton method. Computational Mathematics and Mathematical Physics, 32(9):1353–1359.
- [3] Bell, I. F. and Roberts, D. (1973). Some notes on the application of orthogonal matrix to the least squares problem. In Symposium on Computational Methods in geometric Geodesy. Oxford.
- [4] Bjerhammar, A. (1973). Theory of errors and generalized matrix inverses. Elsevier Scientific Publishing Co., Amsterdam-London-New York.
- [5] Bossler, J. D. (1972). Bayesian inference in geodesy. The Ohio State University.
- [6] Deutsch, R. (1965). Estimation theory. Prentice-Hall, Inc. Englewood Cliffs, N.J.
- [7] George, S. (2010). On convergence of regularized modified Newton's method for nonlinear ill-posed problems. Journal of Inverse & Ill-Posed Problems, 18(2).
- [8] Hampel, F. R. (1971). A general qualitative definition of robustness. The Annals of Mathematical Statistics, 42(6):1887–1896, doi:10.1214/aoms/1177693054.
- [9] Hanke, M. and Groetsch, C. W. (1998). Nonstationary iterated Tikhonov regularization. Journal of Optimization Theory and Applications, 98(1):37–53, doi:10.1023/A:1022680629327.
- [10] Hausbrandt, S. (1954). Adjustment of trigonometric networks with the rejection of the assumption of faultlessness of the tie points (in Polish). Geodezja i Kartografia, 1.
- [11] Holland, P. W. and Welsch, R. E. (1977). Robust regression using iteratively reweighted least-squares. Communications in Statistics – Theory and Methods, 6(9):813–827, doi:10.1080/03610927708827533.
- [12] Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1):73-101, doi:10.1214/aoms/1177703732.
- [13] Janusz, W. (1958). The weighting problem when aligning geodetic networks with the rejection of the assumption that the tie points are faultless (in Polish). Przegląd Geodezyjny, 12(14):462–464.
- [14] Kadaj, R. (1978). Adjustment with outliers (in Polish). Przegląd Geodezyjny, 8:252–253.
- [15] Kadaj, R. (1979). Two-stage method of adjustment of horizontal geodetic networks with the division of the system into sub-sets (in Polish). Zeszyty Naukowe AGH, 59.
- [16] Kadaj, R. (1988). Eine Klasse von Schaetzverfahren mit praktischen Anwendungen. ZfV, H4.
- [17] Levenberg, K. (1944). A method for the solution of certain nonlinear problems in least squares. Quarterly of applied mathematics, 2(2):164–168.
- [18] Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the society for Industrial and Applied Mathematics, 11(2):431–441, doi:10.1137/0111030.
- [19] Marquardt, D. W. (1970). Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics, 12(3):591–612, doi:10.1080/00401706.1970.10488699.
- [20] Mehsner, A. (2013). Comparison of Different Optimization and Regularization Methods for the Solution of Inverse Problems. PhD thesis, Institut für Grundlagen und Theorie der Elektrotechnik Technische Universität Graz, 8010 Graz.
- [21] Mittermayer, E. (1972). Zur Ausgleichung freier Netze. ZfV, 11.
- [22] Moore, E. H. (1920). On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc., 26:394–395.
- [23] Penrose, R. (1955). A generalized inverse for matrices. In Mathematical proceedings of the Cambridge philosophical society, volume 51, pages 406–413. Cambridge University Press, doi:10.1017/S0305004100030401.
- [24] Phillips, D. L. (1962). A technique for the numerical solution of certain integral equations of the first kind. Journal of the ACM (JACM), 9(1):84–97.
- [25] Prószyński, W. and Kwaśniak, M. (2019). The effect of observation correlations upon the basic characteristics of reliability matrix as oblique projection operator. Journal of Geodesy, 93(8):1197–1206, doi:10.1007/s00190-019-01236-y.
- [26] Scherzer, O. (1993). Convergence rates of iterated Tikhonov regularized solutions of nonlinear ill-posed problems. Numerische Mathematik, 66(1):259–279, doi:10.1007/BF01385697.
- [27] Tikhonov, A. (1965). Application of the regularization method in nonlinear problems. J. Comp. Math. Math. Phys, 5(3):363–373.
- [28] Tikhonov, A. N. (1963). On the solution of ill-posed problems and the method of regularization. In Doklady Akademii Nauk, volume 151, pages 501–504. Russian Academy of Sciences.
- [29] Wilkinson, J. H. (1994). Rounding errors in algebraic processes. Courier Corporation, Prentice Hall.
- [30] Wiśniewski, Z. (2013). Advanced Methods for Developing Geodetic Observations with Examples. UWM in Olsztyn.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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