Reconstruction and Decomposition of Scalar and Vectorial Potential Fields on the Sphere

A Brief Overview
  • Christian GerhardsEmail author
  • Roger Telschow
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)


Wir geben einen kurzen Überblick über Approximationsmethoden auf der Sphäre, welche Anwendung in verschiedenen geophysikalischen Fragestellungen finden. Im Speziellen geht es um Methoden mit Bezug zu Potentialfeldpro- blemen und Lokalisierung auf der Sphäre (z. B. Splines, Multiskalenmethoden und Slepian Funktionen). Des Weiteren führen wir zwei bekannte Vektorfeldzerlegungen (Helmholtz und Hardy-Hodge) ein und stellen die Verbindung zu einigen neueren Resultaten her. Abschliessend illustrieren wir unsere Ansätze an zwei geophysikalischen Beispielen: der Bestimmung des Störpotentials aus Lotabweichungen und der Approximation des Magnetfelds, welches durch Ozeangezeiten erzeugt wird.


Approximation on the sphere Spatial localization on the sphere Spherical multiscale expansions Spherical function systems Spherical vector field decompositions Potential theory on the sphere 


We give a brief overview on approximation methods on the sphere that can be used in a variety of geophysical setups. A particular focus is on methods related to potential field problems and spatial localization, such as spherical splines, multiscale methods, and Slepian functions. Furthermore, we introduce the common Helmholtz and Hardy-Hodge decompositions of spherical vector fields together with some related recent results. The methods are illustrate for two different examples: determination of the disturbing potential from deflections of the vertical and approximation of magnetic fields induced by oceanic tides.



This work was partly supported by DFG grant GE 2781/1-1.


  1. 1.
    Alfeld, P., Neamtu, M., Shumaker, L.L.: Fitting data on sphere-like surfaces using splines. J. Comput. Appl. Math. 73, 5–43 (1995)CrossRefGoogle Scholar
  2. 2.
    Atkinson, K.: Numerical integration on the sphere. J. Austr. Math. Soc. 23, 332–347 (1982)CrossRefGoogle Scholar
  3. 3.
    Backus, G., Parker, R., Constable, C.: Foundations of Geomagnetism. Cambridge University Press, Cambridge (1996)Google Scholar
  4. 4.
    Baratchart, L., Gerhards, C.: On the recovery of crustal and core contributions in geomagnetic potential fields. SIAM J. Appl. Math. 77, 1756–1780 (2017)CrossRefGoogle Scholar
  5. 5.
    Baratchart, L., Hardin, D.P., Lima, E.A., Saff, E.B., Weiss, B.P.: Characterizing kernels of operators related to thin plate magnetizations via generalizations of Hodge decompositions. Inverse Prob. 29, 015004 (2013)CrossRefGoogle Scholar
  6. 6.
    Bauer, F., Gutting, M., Lukas, M.A.: Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  7. 7.
    Bauer, F., Reiß, M.: Regularization independent of the noise level: an analysis of quasi-optimality. Inverse Prob. 24, 055009 (2008)CrossRefGoogle Scholar
  8. 8.
    Berkel, P., Fischer, D., Michel, V.: Spline multiresolution and numerical results for joint gravitation and normal-mode inversion with an outlook on sparse regularisation. GEM Int. J. Geomath. 1, 167–204 (2011)CrossRefGoogle Scholar
  9. 9.
    Chambodut, A., Panet, I., Mandea, M., Diamet, M., Holschneider, M., Jamet, O.: Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163, 875–899 (2005)CrossRefGoogle Scholar
  10. 10.
    Dahlke, S., Dahmen, W., Schmitt, E., Weinreich, I.: Multiresolution analysis and wavelets on S 2 and S 3. Num. Func. Anal. Appl. 16, 19–41 (1995)CrossRefGoogle Scholar
  11. 11.
    Driscoll, J.R., Healy, M.H. Jr.: Computing fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)CrossRefGoogle Scholar
  12. 12.
    Fehlinger, T., Freeden, W., Mayer, C., Michel, D., Schreiner, M.: Local modelling of sea surface topography from (geostrophic) ocean flow. ZAMM 87, 775–791 (2007)CrossRefGoogle Scholar
  13. 13.
    Fehlinger, T., Freeden, W., Mayer, C., Schreiner, M.: On the local multiscale determination of the earths disturbing potential from discrete deflections of the vertical. Comput. Geosci. 12, 473–490 (2009)CrossRefGoogle Scholar
  14. 14.
    Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry – case study: spatial and temporal mass variations in South America. Inverse Prob. 28, 065012 (2012)CrossRefGoogle Scholar
  15. 15.
    Fischer, D., Michel, V.: Automatic best-basis selection for geophysical tomographic inverse problems. Geophys. J. Int. 193, 1291–1299 (2013)CrossRefGoogle Scholar
  16. 16.
    Freeden, W.: On integral formulas of the (unit) sphere and their application to numerical computation of integrals. Computing 25, 131–146 (1980)CrossRefGoogle Scholar
  17. 17.
    Freeden, W.: On approximation by harmonic splines. Manuscr. Geod. 6, 193–244 (1981)Google Scholar
  18. 18.
    Freeden, W.: Multiscale Modelling of Spaceborne Geodata. Teubner, Leipzig (1999)Google Scholar
  19. 19.
    Freeden, W., Gerhards, C.: Poloidal and toroidal field modeling in terms of locally supported vector wavelets. Math. Geosc. 42, 818–838 (2010)CrossRefGoogle Scholar
  20. 20.
    Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2012)CrossRefGoogle Scholar
  21. 21.
    Freeden, W., Gerhards, C.: Romberg extrapolation for Euler summation-based cubature on regular regions. GEM Int. J. Geomath. 8, 169–182 (2017)CrossRefGoogle Scholar
  22. 22.
    Freeden, W., Gerhards, C., Schreiner, M.: Disturbing potential from deflections of the vertical: from globally reflected surface gradient equation to locally oriented multiscale modeling. In: Grafarend, E. (ed.) Encyclopedia of Geodesy. Springer International Publishing (2015)Google Scholar
  23. 23.
    Freeden, W., Gervens, T.: Vector spherical spline interpolation – basic theory and computational aspects. Math. Methods Appl. Sci. 16, 151–183 (1993)CrossRefGoogle Scholar
  24. 24.
    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere (with Applications to Geomathematics). Oxford Science Publications. Clarendon Press, Oxford (1998)Google Scholar
  25. 25.
    Freeden, W., Gutting, M.:Special Functions of Mathematical (Geo-)Physics. Applied and Numerical Harmonic Analysis. Springer, Basel (2013)CrossRefGoogle Scholar
  26. 26.
    Freeden, W., Hesse, K.: On the multiscale solution of satellite problems by use of locally supported kernel functions corresponding to equidistributed data on spherical orbits. Stud. Sci. Math. Hungar. 39, 37–74 (2002)Google Scholar
  27. 27.
    Freeden, W., Michel, V.: Constructive approximation and numerical methods- in geodetic research today – an attempt at a categorization based on an uncertainty principle. J. Geod. 73, 452–465 (1999)CrossRefGoogle Scholar
  28. 28.
    Freeden, W., Schneider, F.: Regularization wavelets and multiresolution. Inverse Prob. 14, 225–243 (1998)CrossRefGoogle Scholar
  29. 29.
    Freeden, W., Schreiner, M.: Local multiscale modeling of geoidal undulations from deflections of the vertical. J. Geod. 78, 641–651 (2006)CrossRefGoogle Scholar
  30. 30.
    Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences. Springer, Berlin/Heidelberg (2009)Google Scholar
  31. 31.
    Freeden, W., Windheuser, U.: Combined spherical harmonics and wavelet expansion – a future concept in Earth’s gravitational potential determination. Appl. Comput. Harm. Anal. 4, 1–37 (1997)CrossRefGoogle Scholar
  32. 32.
    Gemmrich, S., Nigam, N., Steinbach, O.: Boundary integral equations for the Laplace-Beltrami operator. In: Munthe-Kaas, H., Owren, B. (eds.) Mathematics and Computation, a Contemporary View. Proceedings of the Abel Symposium 2006. Springer, Berlin (2008)Google Scholar
  33. 33.
    Gerhards, C.: Spherical decompositions in a global and local framework: theory and an application to geomagnetic modeling. GEM Int. J. Geomath. 1, 205–256 (2011)CrossRefGoogle Scholar
  34. 34.
    Gerhards, C.: Locally supported wavelets for the separation of spherical vector fields with respect to their sources. Int. J. Wavel. Multires. Inf. Process. 10, 1250034 (2012)CrossRefGoogle Scholar
  35. 35.
    Gerhards, C.: A combination of downward continuation and local approximation for harmonic potentials. Inverse Prob. 30, 085004 (2014)CrossRefGoogle Scholar
  36. 36.
    Gerhards, C.: Multiscale modeling of the geomagnetic field and ionospheric currents. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  37. 37.
    Gerhards, C.: On the unique reconstruction of induced spherical magnetizations. Inverse Prob. 32, 015002 (2016)CrossRefGoogle Scholar
  38. 38.
    Gerhards, C.: On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere. Inv. Probl. Sci. Engin., to appear.CrossRefGoogle Scholar
  39. 39.
    Gerhards, C.: Spherical potential theory: tools and applications. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of mathematical geodesy – functional analytic and potential theoretic methods, Birkhäuser, Basel (2018)Google Scholar
  40. 40.
    Gerhards, C., Pereverzyev, S. Jr., Tkachenko, P.: A parameter choice strategy for the inversion of multiple observations. Adv. Comp. Math. 43, 101–112 (2017)CrossRefGoogle Scholar
  41. 41.
    Gerhards, C., Pereverzyev, S. Jr., Tkachenko, P.: Joint inversion of multiple observations. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of mathematical geodesy – functional analytic and potential theoretic methods, Birkhäuser, Basel (2018)Google Scholar
  42. 42.
    Gubbins, D., Ivers, D., Masterton, S.M., Winch, D.E.: Analysis of lithospheric magnetization in vector spherical harmonics. Geophys. J. Int. 187, 99–117 (2011)CrossRefGoogle Scholar
  43. 43.
    Gutkin, E., Newton, K.P.: The method of images and green’s function for spherical domains. J. Phys. A: Math. Gen. 37, 11989–12003 (2004)CrossRefGoogle Scholar
  44. 44.
    Gutting, M.: Fast spherical/harmonic spline modeling. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  45. 45.
    Gutting, M., Kretz, B., Michel, V., Telschow, R.: Study on parameter choice methods for the RFMP with respect to downward continuation. Front. Appl. Math. Stat. 3, 10 (2017)CrossRefGoogle Scholar
  46. 46.
    Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69, 331–371 (1910)CrossRefGoogle Scholar
  47. 47.
    Hesse, K., Sloan, I., Womersley, R.: Numerical integration on the sphere. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  48. 48.
    Hofmann-Wellenhof, B., Moritz, H.: Physical Geodesy, 2nd edn. Springer, Vienna (2006)Google Scholar
  49. 49.
    Holschneider, M.: Continuous wavelet transforms on the sphere. J. Math. Phys. 37, 4156–4165 (1996)CrossRefGoogle Scholar
  50. 50.
    Hubbert, S., LeGia, Q.T., Morton, T.: Spherical Radial Basis Functions, Theory and Applications. Springer International Publishing (2015)Google Scholar
  51. 51.
    Kamman, P., Michel, V.: Time-dependent Cauchy-Navier splines and their application to seismic wave front propagation. ZAMM J. Appl. Math. Mech. 88, 155–178 (2008)CrossRefGoogle Scholar
  52. 52.
    Kidambi, R., Newton, K.P.: Motion of three point vortices on a sphere. Phys. D 116, 143–175 (1998)CrossRefGoogle Scholar
  53. 53.
    Kidambi, R., Newton, K.P.: Point vortex motion on a sphere with solid boundaries. Phys. Fluids 12, 581–588 (2000)CrossRefGoogle Scholar
  54. 54.
    Kuvshinov, A.V.: 3-D global induction in the ocean and solid earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric, and ococean origin. Surv. Geophys. 29, 139–186 (2008)CrossRefGoogle Scholar
  55. 55.
    LeGia, Q.T., Sloan, I., Wendland, H.: Multiscale analysis on sobolev spaces on the sphere. SIAM J. Num. Anal. 48, 2065–2090 (2010)CrossRefGoogle Scholar
  56. 56.
    Lima, E.A., Weiss, B.P., Baratchart, L., Hardin, D.P., Saff, E.B.: Fast inversion of magnetic field maps of unidirectional planar geological magnetization. J. Geophys. Res. Solid Earth 118, 1–30 (2013)CrossRefGoogle Scholar
  57. 57.
    Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)CrossRefGoogle Scholar
  58. 58.
    Masterton, S., Gubbins, D., Müller, R.D., Singh, K.H.: Forward modelling of oceanic lithospheric magnetization. Geophys. J. Int. 192, 951–962 (2013)CrossRefGoogle Scholar
  59. 59.
    Mayer, C., Maier, T.: Separating inner and outer Earth’s magnetic field from CHAMP satellite measurements by means of vector scaling functions and wavelets. Geophys. J. Int. 167, 1188–1203 (2006)CrossRefGoogle Scholar
  60. 60.
    Michel, V.: Lectures on Constructive Approximation – Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball. Birkhäuser, Boston (2013)Google Scholar
  61. 61.
    Michel, V., Simons, F.: A general approach to regularizing inverse problems with regional data using Slepian wavelets. Inverser Prob. 33, 125016 (2018)CrossRefGoogle Scholar
  62. 62.
    Michel, V., Telschow, R.: A non-linear approximation method on the sphere. GEM Int. J. Geomath. 5, 195–224 (2014)CrossRefGoogle Scholar
  63. 63.
    Michel, V., Telschow, R.: The regularized orthogonal functional matching pursuit for ill-posed inverse problems. SIAM J. Num. Anal. 54, 262–287 (2016)CrossRefGoogle Scholar
  64. 64.
    Michel, V., Wolf, K.: Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophys. J. Int. 173, 1–16 (2008)CrossRefGoogle Scholar
  65. 65.
    Müller, C.: Spherical Harmonics. Springer, New York (1966)CrossRefGoogle Scholar
  66. 66.
    Olsen, N., Glassmeier, K-H., Jia, X.: Separation of the magnetic field into external and internal parts. Space Sci. Rev. 152, 135–157 (2010)CrossRefGoogle Scholar
  67. 67.
    Olsen, N., Lühr, H., Finlay, C.C., Sabaka, T.J., Michaelis, I., Rauberg, J., Tøffner-Clausen, L.: The CHAOS-4 geomagnetic field model. Geophys. J. Int. 197, 815–827 (2014)CrossRefGoogle Scholar
  68. 68.
    Plattner, A., Simons, F.J.: slepian_golf version 1.0.0.
  69. 69.
    Plattner, A., Simons, F.J.: Spatiospectral concentration of vector fields on a sphere. Appl. Comp. Harm. Anal. 36, 1–22 (2014)CrossRefGoogle Scholar
  70. 70.
    Plattner, A., Simons, F.J.: Potential-field estimation from satellite data using scalar and vector Slepian functions. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  71. 71.
    Plattner, A., Simons, F.J.: Internal and external potential-field estimation from regional vector data at varying satellite altitude. Geophys. J. Int. 211, 207–238 (2017)Google Scholar
  72. 72.
    Sabaka, T., Tyler, R., Olsen, N.: Extracting ocean-generated tidalmagnetic signals from Swarm data through satellite gradiometry. Geophys. Res. Lett. 43, 3237–3245 (2016)CrossRefGoogle Scholar
  73. 73.
    Sabaka, T.J., Olsen, N., Tyler, R.H., Kuvshinov, A.: CM5, a pre-Swarm comprehensive geomagnetic field model derived from over 12 years of CHAMP, ørsted, SAC-C and observatory data. Geophys. J. Int. 200, 1596–1626 (2015)Google Scholar
  74. 74.
    Schreiner, M. Locally supported kernels for spherical spline interpolation. J. Approx. Theory 89, 172–194 (1997)CrossRefGoogle Scholar
  75. 75.
    Shure, L., Parker, R.L., Backus, G.E.: Harmonic splines for geomagnetic modeling. Phys. Earth Planet. Inter. 28, 215–229 (1982)CrossRefGoogle Scholar
  76. 76.
    Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral localization on a sphere. SIAM Rev. 48, 505–536 (2006)CrossRefGoogle Scholar
  77. 77.
    Simons, F.J., Plattner, A.: Scalar and vector slepian functions, spherical signal estimation and spectral analysis. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)Google Scholar
  78. 78.
    Sloan, I., Womersley, R.: Filtered hyperinterpolation: a constructive polynomial approximation on the sphere. GEM Int. J. Geomath. 3, 95–117 (2012)CrossRefGoogle Scholar
  79. 79.
    Telschow, R.: An Orthogonal Matching Pursuit for the Regularization of Spherical Inverse Problems. PhD thesis, University of Siegen (2014)Google Scholar
  80. 80.
    Tyler, R., Maus, S., Lühr, H.: Satellite observations of magnetic fields due to ocean tidal flow. Science 299, 239–240 (2003)CrossRefGoogle Scholar
  81. 81.
    Vervelidou, F, Lesur, V., Morschhauser, A.,Grott, M., Thomas, P.: On the accuracy of paleopole estimations from magnetic field measurements. Geophys. J. Int. 211, 1669–1678 (2017)CrossRefGoogle Scholar
  82. 82.
    Wahba, G.: Spline inteprolation and smoothing on the sphere. SIAM J. Sci. Stat. Comput. 2, 5–16 (1981)CrossRefGoogle Scholar
  83. 83.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)CrossRefGoogle Scholar
  84. 84.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Computational Science CenterUniversity of ViennaViennaAustria

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations