Abstract
The Least-squares collocation (LSC) method is commonly used in geodesy, but generally associated with globally supported covariance functions, i.e. with dense covariance matrices. We consider locally supported radial covariance functions, which yield sparse covariance matrices. Having many zero entries in the covariance matrice can both greatly reduce computer storage requirements and the number of floating point operations needed in computation. This paper reviews some of the most well-known compactly supported radial covariance functions (CSRCFs) that can be easily substituted to the usually used covariance functions. Numerical experiments reveals that these finite covariance functions can give good approximations of the Gaussian, second- and third-order Markov models. Then, interpolation of KMS02 free-air gravity anomalies in Azores Islands shows that dense covariance matrices associated with Gaussian model can be replaced by sparse matrices from CSRCFs resulting in memory savings of one-fortieth and with 90% of the solution error less than 0.5 mGal.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Achlioptas D, McSherry F and Schölopf B (2002). Sampling techniques for kernel methods. Adv Neur Inf Proc Syst 14: 335–342
Andersen OB, Knudsen P, Trimmer R (2004) Improving high resolution altimetric gravity field mapping (KMS2002 Global marine gravity field). In: Sansò F (ed.) A window on the future of geodesy. Springer, Berlin. IAG Symposia 128:326–331
Arabelos D and Tscherning CC (1996). Collocation with finite covariance functions. Int Geoid Serv Bull 5: 117–136
Arabelos D and Tscherning CC (1999). Gravity field recovery from airborne gradiometer data using collocation and taking into account correlated errors. Phys Chem Earth(A) 24(1): 19–25
Askey R (1973). Radial characteristic functions. MRC Report 1262, University of Wisconsin, Madison
Buhmann MD (2000) A new class of radial basis functions with compact support. Math Comput 70:307–318. doi:10.1090/S0025-5718-00-01251-5
Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 24th National Conference on ACM, New York, pp 157–172
Fieguth PW, Karl WC, Willsky AS and Wunsch C (1995). Milterosolution optimal interpolation and statistical analysis of TOPEX/ POSEIDON satellite altimetry. IEEE Tans Geosci Remote Sens 33(2): 280–292
Floater MS and Iske A (1996). Multistep scattered data interpolation using compactly supported radial basis functions. J Comp Appl Math 73: 65–78
Fornefett M, Rohr K, Stiehl HS (1999) Elastic registration of medical images using radial basis functions with compact support. In: Proceedings of the Computer Vision and Pattern Recognition, Fort Collins, USA, June 23–25, pp 402–407
Furrer R, Genton MG, Nychka D (2005) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15(3):502–523. doi:10.1198/106186006X132178
Gaspari G and Cohn SE (1999). Construction of correlation functions in two and three dimensions. Quart J R Meteorol Soc 125: 723–757
Gaspari G, Cohn SE, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Quart J R Meteorol Soc 132:1815–1838. doi:10.1256/qj.05.08
George A and Liu JWH (1981). Computer solution of large sparse positive definite matrices. Prentice Hall, Englewood Cliffs
George A and Liu JWH (1989). The evolution of the minimum degree algorithm. SIAM Rev 31: 1–19
Gneiting T (1999). Correlation functions for atmospheric data analysis. Q J R Meteorol Soc 125: 2449–2464
Hon YC and Zhou X (2000). A comparison on using various radial basis functions for options pricing. Int J Appl Sci Comput 7: 29–47
Jonge de PJ (1992). A comparative study of algorithms for reducing the fill-in during Cholesky factorization. Bull Geod 66: 296–305
Jordan SK (1972). Self-consistent statistical models for the gravity anomaly, vertical deflections and undulation of the geoid. J Geophys Res 77: 3660–3670
Kasper JF (1971). A second-order Markov gravity anomaly model. J Geophys Res 76(32): 7844–7849
Kusche J (2001). Implementation of multigrid solvers for satellite gravity anomaly recovery. J Geod 74: 773–782
Moreaux G, Tscherning CC and F Sansò (1999). Approximation of harmonic covariance functions on the sphere by non-harmonic locally supported ones. J Geod 73: 555–567
Moreaux G (2001). Some preconditioners of harmonic spherical spline problems. Inverse Problems 17: 157–177
Moritz H (1989). Advanced physical geodesy 2nd edn. Wichmann, Karlsruhe
Morse BS, Yoo TS, Rheingans P, Chen DT, Subramanian KR (2001) Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. In: Proceedings of the International conference on shape modeling and applications, Genova, Italy, pp 89–98
Rygaard-Hjalsted C, Constable CG and Parker RL (1997). The influence of correlated crustal signals in modelling the main geomagnetic field. Geophys J Int 130: 717–726
Saad Y (1996) Iterative methods for sparse linear system 1st edn. The PWS Series in Computer SCience, PWS Publishing, Boston, http://www-users.cs.umm.edu/~saad/books.html
Sansò F and Schuh WD (1987). Finite covariance functions. Bull Geod 61: 331–347
Schaback R, Wendland H (1993) Special cases of compactly supported radial basis functions. Manuscript, Göttingen
Schreiner M (1997). Locally supported kernels for spherical spline interpolation. J Approx Theory 89: 172–194
Tscherning CC, Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical implied by anomaly degree variance models. Department of Geodetic Science, Report 208, The Ohio State University, Columbus
Tóth G, Völgyesi L (2007) Local gravity field modeling using surface gravity gradient measurements. Tregoning P, Rizos C (eds.) Dynamic planet monitoring and understanding a dynamic planet with geodetic and oceanographic tools. Springer, Berlin. IAG Symposia 130:424–429
Wachowiak MP, Wang X, Fenster A, Peters TM (2004) Compact support radial basis functions for soft tissue deformation. In: Proceedings of IEEE International Symposium on Biomedical Imaging, Arlington, Virginia, pp 1259–1262
Weber RO and Talkner P (1993). Some remarks on spatial correlation function models. Mon Weather Rev 121: 2611–2617
Wendland H (1995). Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Adv Comput Math 4: 389–396
Wendland H (1998). Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J Approx Theory 93: 258–272
Wendland H (2002) Compactly supported correlation functions. J Multivariate Anal 83:493–508. doi:10.1006/jmva.2001.2056
Wong SM, Hon YC and Li TS (1999). Radial basis functions with compact support and multizone decomposition: applications to environmental modelling. Bound Elem Technol 13: 355–364
Wu Z (1995). Compactly supported positive definite radial functions. Adv Comput Math 4: 283–292
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is dedicated to Cerbère.
Rights and permissions
About this article
Cite this article
Moreaux, G. Compactly supported radial covariance functions. J Geod 82, 431–443 (2008). https://doi.org/10.1007/s00190-007-0195-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-007-0195-4