Abstract
The mathematical framework for a spline method combining interpolation and smoothing of heterogeneous data is presented. The method is demonstrated for a spherical earth model. A spline approximation for the gravitational field is obtained by using a Hilbert space with topology induced by the (Laplace-) Beltrami operator of the sphere.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.M. ANSELONE, P.J. LAURENT: A general method for the construction of interpolating and smoothing spline functions, Num. Math.,12, 66–82 (1968).
A. BJERHAMMAR: Discrete approaches to the solution of the boundary value problem in physical geodesy. Boll. Geod. Sci. Aff.,34, 185–240 (1974).
K. BÖHMER: Spline-Funktionen, Stuttgart: B.G. Teubner (1974).
C. DE BOOR, R.E. LYNCH: On splines and their minimum properties. J. Math. Mech.,15, 953–969 (1966).
J. EEG, T. KRARUP: Integrated geodesy, Methoden und Verfahren der math. Physik, (editors: B. Brosowski, E. Martensen). B. I. Wissenschaftsverlag, Bd. 13, 77–123 (1975).
W. FREEDEN: Über eine Klasse von Integralformeln der mathematischen Geodäsie. Veröff. Geod. Inst. RWTH Aachen, Report No. 27 (1979).
W. FREEDEN: On integral formulas of the unit sphere and their application to numerical computation of integrals. Computing.25, 131–146 (1980).
W. FREEDEN: On spherical spline interpolation and approximation. Math. Meth. in the Appl. Sci.,3, 551–575, (1981a).
W. FREEDEN: On approximation by harmonic splines. Manuscripta geodaetica, 193–244, (1981b).
T.N.E. GREVILLE: Introduction to spline functions in: Theory and applications of spline functions. Academic Press, (1969).
T. KRARUP: A contribution to the mathematical foundation of physical geodesy. Geod. Inst. Kobenhavn, Meddelelse No. 44, (1969).
P. MEISSL: Elements of functional analysis, in: Methoden und Verfahren der math. Physik (editors: B. Brosowski, E. Martensen) B.I. Wissenschaftsverlag, Bd. 12, 19–78 (1975).
P. MEISSL: Hilbert spaces and their application to geodetic least squares problems. Boll. Geod. Sci. Aff., Vol. XXXV, No. 1, 181–201 (1976).
H. MORITZ: Advanced least-squares methods. Dept. of Geod. Science, OSU, Report No. 175 (1972).
H. MORITZ: Stepwise and sequential collocation. Dept. of Geod. Science, OSU, Report No. 203, (1973).
H. MORITZ: Interpolation and approximation, in: Approximation methods in geodesy (editors: H. Moritz, H. Sünkel), Sammlung Wichmann, Bd. 10, 1–45, (1978).
H. MORITZ: Advanced physical geodesy. Sammlung Wichmann, Bd. 13, (1980).
I.J. SCHOENBERG: On trigonometric spline interpolation. J. Math. Mech.,13, 785–825 (1964).
L. SJÖBERG: On the discrete boundary value problem of physical geodesy with harmonic reductions to an internal sphere. The Royal Institute of Technology, Division of Geodesy, Stockholm (1975).
C.C. TSCHERNING: A FORTRAN IV program for the determination of the anomalous potential using stepwise least squares collocation. Dept. of Geodetic Science, OSU, Report No. 212 (1974).
C.C. TSCHERNING: A note on the choice of norm when using collocation for the computation of approximations to the anomalous potential. Bull. Geod.,51, 137–147 (1977).
C.C. TSCHERNING, R.H. RAPP: Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical implied by anomaly degree-variance models. Dept. of Geodetic Science, OSU, Report No. 208, (1974).
E.T. WHITTAKER: On a new method of graduation. Proc. Edinburgh Math. Soc.,41, 63–75 (1923).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Freeden, W., Witte, B. A combined (spline-) interpolation and smoothing method for the determination of the external gravitational potential from heterogeneous data. Bull. Geodesique 56, 53–62 (1982). https://doi.org/10.1007/BF02525607
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02525607