Local solutions to inverse problems in geodesy: The impact of the noise covariance structure upon the accuracy of estimation

Journal of Geodesy

Volumn 81, Issue 1, 2007, Pages 39-51

  Bauer, Frank ^a   Mathé, Peter ^b   Pereverzev, Sergei ^c  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^c JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

Gaussian random noise; Ill posed inverse problems; Regularization by local data; Satellite gravity gradiometry

Indexed keywords

ACCURACY ASSESSMENT; COVARIANCE ANALYSIS; ESTIMATION METHOD; GEODESY; GRAVITY FIELD; HARMONIC ANALYSIS; INVERSE PROBLEM; NOISE; SATELLITE DATA;

EID: 33845882269     PISSN: 09497714     EISSN: 14321394     Source Type: Journal    
DOI: 10.1007/s00190-006-0049-5     Document Type: Article

References (35)

1. Anderssen R (1986) The linear functional strategy for improperly posed problems. In: Cannon J, Hornung U (eds) Inverse problems, proc. conference, Oberwolfach/Ger. 1986, international series of numerical mathematics, vol 77. Birkhäuser, Basel, Boston, Stuttgart, pp 11-30
2. Bauer F (2004) An alternate approach to the oblique derivative problem in potential theory. PhD Thesis, Universtiy of Kaiserslautern, Department of Mathematics, Geomathematics Group
3. Bauer F, Hohage T (2005) A Lepskij-type stopping rule for regularized Newton methods. Inverse Probl 21:1975-1991
4. Bauer F, Pereverzev S (2005a) Regularization without preliminary knowledge of smoothness and error behavior. Eur J Appl Math 16(3):303-317
5. Bauer F, Pereverzev S (2005b) An utilization of a rough approximation of a noise covariance within the framework of multi-parameter regularization (submitted). Int J Tomogr Stat, preprint: Ftp:// ftpnummathuni-goettingende/pub/preprints/bauer/mainpdf
6. Böttcher A, Hofmann B, Tautenhahn U, Yamamoto M (2006) Convergence rates for Tikhonov regularization from different kind of smoothness conditions. Appl Anal 85, publ. electronically
7. Donoho D (1994) Statistical estimation and optimal recovery. Ann Stat 22(1):238-270
8. Driscoll J, Healy D (1994) Computing fourier transforms and convolutions on the 2-sphere. Adv Appl Math 15(2):202-250
9. Engl H, Neubauer A (1994) A parameter choice strategy for (iterated) Tikhonov regularization of ill-posed problems leading to superconvergence with optimal rates. Appl Anal 27(1-3):5-18
10. Engl H, Hanke M, Neubauer A (1996) Regularization of inverse problems. In: Mathematics and its applications. Kluwer, Dordrecht, Boston, London
11. Freeden W (1999) Multiscale modelling of spaceborne geodata. In: European consortium for mathematics in industry. B.G. Teubner, Stuttgart Leipzig
12. Freeden W, Mayer T (2002) On multiscale denoising of spherical functions: Basic theory and numerical aspects. ETNA 14:56-78
13. Freeden W, Pereverzev S (2001) Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise. J Geod 74:730-736
14. Goldenshluger A, Pereverzev S (2000) Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probab Theory Relat Fields 118(2):169-186
15. Holmes S, Featherstone W (2002) A unified approach to the clenshaw summation and the recursive computation of very-high degree and order normalised associated legendre functions. J Geod 76(5): 279-299
16. Klees R, Ditmar P, Broersen P (2003) How to handle colored noise in large least-squares problems. J Geod 76: 629-640
17. Kusche J, Klees R (2002) Regularization of gravity field estimation from satellite gravity gradients. J Geod 76:359-368
18. Lemoine F, Kenyon S, Factor J, Trimmer R, Pavlis N, Chinn D, Cox C, Klosko S, Luthcke S, Torrence M, Wang Y, Williamson R, Rapp R, Olson T (1998) The development of the joint nasa/gsfc and the national imagery and mapping agency (NIMA) geopotential models, EGM96. NASA, Greenbelt TP-1998-206861, pp 575
19. Lepskii O (1990) On a problem of adaptive estimation in Gaussian white noise. Theory Probab Appl 35(3):454-466
20. Lonkhuyzen M, Klees R, Bouman J (2001) Regularization for the gravity field recovery from goce observations. In: Proceedings of the IAG international Symposium of Gravity, geoid geodynamics 2000, Banff. Springer, Berlin Heidelberg New York
21. Mathé P (2004) Saturation of regularization methods for linear ill-posed problems in Hilbert spaces. SIAM J Numer Anal 42(3):968-973
22. Mathé P, Pereverzev S (2002) Direct estimation of linear functionals from indirect noisy observations. J Complex 18(2):501-516
23. Mathé P, Pereverzev S (2003) Geometry of linear ill-posed problems in variable hilbert spaces. Inverse Probl 19(3):789-803
24. Mathé P, Pereverzev S (2005) Regularization of some linear inverse problems with discretized random noise data (to appear Math. of Comput.)
25. Pail R, Plank G (2002) Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform. J Geod 76:462-474
26. Pereverzev S, Schock E (1999) Error estimates for band-limited spherical regularization wavelets in an inverse problem of satellite geodesy. Inverse Probl 15(4):881-890
27. Rebhan H, Aguirre M, Johannessen J (2000) The gravity field and steady-state ocean circulation explorer mission - GOCE. ESA Earth Obs Q 66:6-11
28. Rummel R, Reigber C, Ilk K (1978) The use of satellite-to-satellite tracking for gravity parameter recovery. In: ESA workshop on space oceanography, navigation and geodynamics (SONG), ESA-SP-137, pp 151-161
29. Rummel R, Balmino G, Johannessen J, Visser P, Woodworth P (2002) Dedicated gravity field missions - principles and aims. J Geodyn 33:3-20
30. Sansò F, Sona G (1995) The theory of optimal linear estimation for continuous fields of measurements. Manuscri Geodaet 20:204-230
31. Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02 - an improved earth gravity field model from GRACE. J Geod 79(8):467-478
32. Tscherning C (1986) Mathematical and numerical techniques in physical geodesy. In: Functional methods for gravity field approximation. Springer, Berlin Heidelberg New York, pp 3-47
33. Tsybakov A (2000) On the best rate of adaptive estimation in some inverse problems. C R Acad Sci Paris Sér I Math 330(9):835-840
34. Xu P (1998) Truncated SVD methods for linear discrete ill-posed problems. Geophys J Int 135:505-514
35. Xu P, Rummel R (1994) A generalized ridge regression method with applications in determination of potential fields. Manuscr Geodaet 20(1):8-20