Abstract
The spatial correlation of the Earth’s gravity field is well known and widely utilized in applications of geophysics and physical geodesy. This paper develops the mathematical theory of correlation functions, as well as covariance functions under a statistical interpretation of the field, for functions and processes on the sphere and plane, with formulation of the corresponding power spectral densities in the respective frequency domains and with extensions into the third dimension for harmonic functions. The theory is applied, in particular, to the disturbing gravity potential with consistent relationships of the covariance and power spectral density to any of its spatial derivatives. An analytic model for the covariance function of the disturbing potential is developed for both spherical and planar application, which has analytic forms also for all derivatives in both the spatial and the frequency domains (including the along-track frequency domain). Finally, a method is demonstrated to determine the parameters of this model from empirical regional power spectral densities of the gravity anomaly.
Similar content being viewed by others
References
Alfeld P, Neamtu M, Schumaker LL (1996) Fitting scattered data on sphere-like surfaces using spherical splines. J Comput Appl Math 73:5–43
Baranov V (1957) A new method for interpretation of aeromagnetic maps: pseudo-gravimetric anomalies. Geophysics 22:359–383
Brown RG (1983) Introduction to random signal analysis and Kalman filtering. Wiley, New York
de Coulon F (1986) Signal theory and processing. Artech House, Dedham
Fengler MJ, Freeden W, Michel V (2004) The Kaiserslautern multiscale geopotential model SWITCH-03 from orbit perturbations of the satellite CHAMP and its comparison to models EGM96, UCPH2002_02_05, EIGEN-1S and EIGEN-2. Geophys J Int 157:499–514
Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Géod 59(4):342–360
Forsberg R (1987) A new covariance model, for inertial gravimetry and gradiometry. J Geophys Res 92(B2):1305–1310
Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere, with applications in geomathematics. Clarendon, Oxford
Heller WG, Jordan SK (1979) Attenuated white noise statistical gravity model. J Geophys Res 84(B9):4680–4688
Helmert FR (1884) Die Mathematischen und Physikalischen Theorien der Höheren Geodäsie, vol 2. BD Teubner, Leipzig
Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, Berlin
Jeffreys H (1955) Two properties of spherical harmonics. Q J Mech Appl Math 8(4):448–451
Jekeli C (1991) The statistics of the Earth’s gravity field, revisited. Manuscr Geod 16(5):313–325
Jekeli C (2005) Spline representations of functions on a sphere for geopotential modeling. Report no. 475, Geodetic Science, Ohio State University, Columbus. http://www.geology.osu.edu/~jekeli.1/OSUReports/reports/report_475.pdf
Jordan SK (1972) Self-consistent statistical models for the gravity anomaly, vertical deflections, and the undulation of the geoid. J Geophys Res 77(20):3660–3669
Jordan SK, Moonan PJ, Weiss JD (1981) State-space models of gravity disturbance gradients. IEEE Trans Aerosp Electron Syst AES 17(5):610–619
Kaula WM (1966) Theory of satellite geodesy. Blaisdell, Waltham
Lauritzen SL (1973) The probabilistic background of some statistical methods in physical geodesy. Report no. 48, Geodaestik Institute, Copenhagen
Lyche T, Schumaker LL (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J Sci Comput 22(2):724–746
Mandelbrot B (1983) The fractal geometry of nature. Freeman, San Francisco
Marple SL (1987) Digital spectral analysis with applications. Prentice-Hall, Englewood Cliffs
Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Springer, Berlin
Maybeck PS (1979) Stochastic models, estimation, and control, vols I and II. Academic, New York
Milbert DG (1991) A family of covariance functions based on degree variance models and expressible by elliptic integrals. Manuscr Geod 16:155–167
Moritz H (1976) Covariance functions in least-squares collocation. Report no. 240, Department of Geodetic Science, Ohio State University, Columbus
Moritz H (1978) Statistical foundations of collocation. Report no. 272, Department of Geodetic Science, Ohio State University, Columbus
Moritz H (1980) Advanced physical geodesy. Abacus Press, Tunbridge Wells
Olea RA (1999) Geostatistics for engineers and earth scientists. Kluwer Academic, Boston
Pavlis NK, Holmes SA, Kenyon SC, Factor JF (2012a) The development and evaluation of earth gravitational model (EGM2008). J Geophys Res 117:B04406. doi:10.1029/2011JB008916
Pavlis NK, Holmes SA, Kenyon SC, Factor JF (2012b) Correction to “The development and evaluation of Earth Gravitational Model (EGM2008)”. J Geophys Res, 118, 2633, doi:10.1002/jgrb.50167
Priestley MB (1981) Spectral analysis and time series analysis. Academic, London
Rummel R, Yi W, Stummer C (2011) GOCE gravitational gradiometry. J Geod 85:777–790
Schreiner M (1997) Locally supported kernels for spherical spline interpolation. J Approx Theory 89:172–194
Schumaker LL, Traas C (1991) Fitting scattered data on sphere-like surfaces using tensor products of trigonometric and polynomial splines. Numer Math 60:133–144
Tscherning CC (1976) Covariance expressions for second and lower order derivatives of the anomalous potential. Report no. 225, Department of Geodetic Science, Ohio State University, Columbus. http://geodeticscience.osu.edu/OSUReports.htm
Tscherning CC, Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical implied by anomaly degree variance models. Report no. 208, Department of Geodetic Science, Ohio State University, Columbus. http://geodeticscience.osu.edu/OSUReports.htm
Turcotte DL (1987) A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars. J Geophys Res 92(B4):E597–E601
Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Jekeli, C. (2013). Correlation Modeling of the Gravity Field in Classical Geodesy. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_28-2
Download citation
DOI: https://doi.org/10.1007/978-3-642-27793-1_28-2
Received:
Accepted:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Online ISBN: 978-3-642-27793-1
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering
Publish with us
Chapter history
-
Latest
Correlation Modeling of the Gravity Field in Classical Geodesy- Published:
- 15 September 2014
DOI: https://doi.org/10.1007/978-3-642-27793-1_28-3
-
Original
Correlation Modeling of the Gravity Field in Classical Geodesy- Published:
- 20 August 2014
DOI: https://doi.org/10.1007/978-3-642-27793-1_28-2