Skip to main content
SpringerLink
Log in
Book cover

Handbook of Geomathematics pp 1–34Cite as

Correlation Modeling of the Gravity Field in Classical Geodesy

  • 445 Accesses

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.

This is a preview of subscription content, log in via an institution.

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

    Article  MATH  MathSciNet  Google Scholar 

  • Baranov V (1957) A new method for interpretation of aeromagnetic maps: pseudo-gravimetric anomalies. Geophysics 22:359–383

    Article  MathSciNet  Google Scholar 

  • Brown RG (1983) Introduction to random signal analysis and Kalman filtering. Wiley, New York

    Google Scholar 

  • de Coulon F (1986) Signal theory and processing. Artech House, Dedham

    MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Géod 59(4):342–360

    Article  Google Scholar 

  • Forsberg R (1987) A new covariance model, for inertial gravimetry and gradiometry. J Geophys Res 92(B2):1305–1310

    Article  Google Scholar 

  • Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere, with applications in geomathematics. Clarendon, Oxford

    Google Scholar 

  • Heller WG, Jordan SK (1979) Attenuated white noise statistical gravity model. J Geophys Res 84(B9):4680–4688

    Article  Google Scholar 

  • Helmert FR (1884) Die Mathematischen und Physikalischen Theorien der Höheren Geodäsie, vol 2. BD Teubner, Leipzig

    Google Scholar 

  • Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, Berlin

    Google Scholar 

  • Jeffreys H (1955) Two properties of spherical harmonics. Q J Mech Appl Math 8(4):448–451

    Article  MATH  MathSciNet  Google Scholar 

  • Jekeli C (1991) The statistics of the Earth’s gravity field, revisited. Manuscr Geod 16(5):313–325

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Jordan SK, Moonan PJ, Weiss JD (1981) State-space models of gravity disturbance gradients. IEEE Trans Aerosp Electron Syst AES 17(5):610–619

    Article  Google Scholar 

  • Kaula WM (1966) Theory of satellite geodesy. Blaisdell, Waltham

    Google Scholar 

  • Lauritzen SL (1973) The probabilistic background of some statistical methods in physical geodesy. Report no. 48, Geodaestik Institute, Copenhagen

    Google Scholar 

  • Lyche T, Schumaker LL (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J Sci Comput 22(2):724–746

    Article  MATH  MathSciNet  Google Scholar 

  • Mandelbrot B (1983) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  • Marple SL (1987) Digital spectral analysis with applications. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Springer, Berlin

    Google Scholar 

  • Maybeck PS (1979) Stochastic models, estimation, and control, vols I and II. Academic, New York

    Google Scholar 

  • Milbert DG (1991) A family of covariance functions based on degree variance models and expressible by elliptic integrals. Manuscr Geod 16:155–167

    Google Scholar 

  • Moritz H (1976) Covariance functions in least-squares collocation. Report no. 240, Department of Geodetic Science, Ohio State University, Columbus

    Google Scholar 

  • Moritz H (1978) Statistical foundations of collocation. Report no. 272, Department of Geodetic Science, Ohio State University, Columbus

    Google Scholar 

  • Moritz H (1980) Advanced physical geodesy. Abacus Press, Tunbridge Wells

    Google Scholar 

  • Olea RA (1999) Geostatistics for engineers and earth scientists. Kluwer Academic, Boston

    Book  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Priestley MB (1981) Spectral analysis and time series analysis. Academic, London

    MATH  Google Scholar 

  • Rummel R, Yi W, Stummer C (2011) GOCE gravitational gradiometry. J Geod 85:777–790

    Article  Google Scholar 

  • Schreiner M (1997) Locally supported kernels for spherical spline interpolation. J Approx Theory 89:172–194

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Geodetic Science, School of Earth Sciences, Ohio State University, Columbus, OH, USA

    Christopher Jekeli

Authors
  1. Christopher Jekeli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher Jekeli .

Editor information

Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

Rights and permissions

Reprints 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

Policies and ethics

Chapter history

  1. 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

  2. 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

Search

Navigation