Abstract
Discontinuity detection is of great relevance at different stages of the processing and analysis of geodetic time-series of data. This paper is essentially a review of two possible methods. The first method follows a stochastic approach and exploits the Bayesian theory to compute the posterior distributions of the discontinuity parameters. The epoch and the amplitude of the discontinuity are then selected as maximum a posteriori (MAP). The second method follows a variational approach based on the Mumford and Shah functional to segment the time-series and to detect the discontinuities. Whereas the original formulation was developed in a continuous form, discrete approaches are also available presenting some interesting connections with robust regressions. Both the methods have been applied to identify the occurrence of cycle-slips in GNSS phase measurements. Simulated and real data have been processed to compare the performance and to evaluate pros and cons of the two approaches. Results clearly show that both the methods can successfully identify cycle-slips.
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References
Albertella A, Cazzaniga N, Crespi M, Luzietti L, Sacerdote F, Sansò F (2007) Deformations detection by a Bayesian approach: prior information representation and testing criteria definition. In: International Association of Geodesy Symposia, vol 131, Springer, Berlin, pp 30–37.
Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: A new release of the International Terrestrial Reference Frame based on time series of stations positions and Earth orientation parameters. J Geophys Res 112, B09401
Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variations and free discontinuity problems. Oxford University Press, Oxford
Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. Bollettino dell’Unione Matematica Italiana 6-B (7):105–123
Blake A, Zisserman A (1987) Visual Reconstruction. The MIT Press, Cambridge
Box GEP, Tiao GC (1992) Bayesian inference in statistical analysis. Wiley, New York
Braides A (1998) Approximation of Free-Discontinuity Problems. Lecture Notes in Mathematics 1694. Springer, Berlin
Chambolle A (1995) Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J Appl Math 55(3):827–863
Dermanis A, Kotsakis C (2007) Estimating crustal deformation parameters from geodetic data: review of existing methodologies, open problems and new challenges. In: International Association of Geodesy Symposia, vol 131, Springer, Berlin, pp 7–18
Gao Y, Li Z (1999) Cycle slip detection and ambiguity resolution algorithm for dual-frequency GPS data processing. Mar Geodes 22(3):169–181
De Giorgi E, Carriero M, Leaci A (1989) Existence theorem for a minimum problem with free discontinuity set. Arch Rational Mech Anal 108:195–218
De Giorgi E, Franzoni T (1975) Su un tipo di convergenza variazionale. Atti Accademia Nazionale dei Lincei - Rend Cl Sci 58:842–850
Koch KR (2007) Introduction to Bayesian Statistics, 2nd edn. Springer, Berlin
de Lacy MC, Reguzzoni M, Sansò F, Venuti G (2008) The Bayesian detection of discontinuities in a polynomial regression and its application to the cycle-slip problem. J Geodes 82(9):527–542
Lichtenegger H, B Hofmann-Wellenhof (1989) GPS-data preprocessing for cycle-slip detection. In: International Association of Geodesy Symposia, vol 102, Springer, Berlin, pp 57–68
Modica L, Mortola S (1977) Un esempio di Gamma-convergenza. Bollettino dell’Unione Matematica Italiana 14-B (5):285–299
Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Comm Pure Appl Math 42:577–685
Perfetti N (2006) Detection of coordinate discontinuities within the Italian Fiducial Network. J Geodes 80(7):381–396
Serpelloni E, Anzidei M, Baldi P, Casula G, Galvani A (2005) Crustal velocity and strain-rate fields in Italy and surrounding regions: new results from the analysis of permanent and non-permanent GPS networks. Geophys J Int 161(3):861–880
Teunissen PJ, Kleusberg A (1998) GPS for Geodesy, 2nd edn. Springer, Berlin
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Benciolini, B., Reguzzoni, M., Venuti, G., Vitti, A. (2012). Bayesian and Variational Methods for Discontinuity Detection: Theory Overview and Performance Comparison. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds) VII Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 137. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22078-4_22
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DOI: https://doi.org/10.1007/978-3-642-22078-4_22
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