Mathematical Properties Relevant to Geomagnetic Field Modeling

  • Terence J. Sabaka
  • Gauthier Hulot
  • Nils Olsen


Geomagnetic field modeling consists in converting large numbers of magnetic observations into a linear combination of elementary mathematical functions that best describes those observations. The set of numerical coefficients defining this linear combination is then what one refers to as a geomagnetic field model. Such models can be used to produce maps. More importantly, they form the basis for the geophysical interpretation of the geomagnetic field, by providing the possibility of separating fields produced by various sources and extrapolating those fields to places where they cannot be directly measured. In this chapter, the mathematical foundation of global (as opposed to regional) geomagnetic field modeling is reviewed, and the spatial modeling of the field in spherical coordinates is focussed. Time can be dealt with as an independent variable and is not explicitly considered. The relevant elementary mathematical functions are introduced, their properties are reviewed, and how they can be used to describe the magnetic field in a source-free (such as the Earth’s neutral atmosphere) or source-dense (such as the ionosphere) environment is explained. Completeness and uniqueness properties of those spatial mathematical representations are also discussed, especially in view of providing a formal justification for the fact that geomagnetic field models can indeed be constructed from ground-based and satellite-born observations, provided those reasonably approximate the ideal situation where relevant components of the field can be assumed perfectly known on spherical surfaces or shells at the time for which the model is to be recovered.


Spherical Shell Toroidal Field Internal Origin Gauss Coefficient Geomagnetic Field Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This is IPGP contribution 2596.


  1. Abramowitz M, Stegun I A (1964) Handbook of mathematical functions. Dover, New York, 1964zbMATHGoogle Scholar
  2. Alberto P, Oliveira O, Pais MA (2004) On the non-uniqueness of main geomagnetic field determined by surface intensity measurements: the Backus problem. Geophys J Int 159: 558–554. 10.1111/j.1365-246X.2004.02413.xCrossRefGoogle Scholar
  3. Backus GE (1968) Applications of a non-linear boundary value problem for Laplace’s equation to gravity and geomagnetic intensity surveys. Q J Mech Appl Math 21: 195–221zbMATHCrossRefMathSciNetGoogle Scholar
  4. Backus GE (1970) Non-uniqueness of the external geomagnetic field determined by surface intensity measurements. J Geophys Res 75 (31): 6339–6341CrossRefGoogle Scholar
  5. Backus GE (1974) Determination of the external geomagnetic field from intensity measurements. Geophys Res Lett 1 (1): 21CrossRefGoogle Scholar
  6. Backus G (1986) Poloidal and toroidal fields in geomagnetic field modeling. Rev Geophys 24: 75–109CrossRefMathSciNetGoogle Scholar
  7. Backus G, Parker R, Constable C (1996) Foundations of geomagnetism. Cambridge Univ. Press, New YorkGoogle Scholar
  8. Barraclough DR, Nevitt C (1976) The effect of observational errors on geomagnetic field models based solely on total-intensity measurements. Phys Earth Planet Int 13: 123–131CrossRefGoogle Scholar
  9. Blakely RJ (1999) Potential theory in gravity and magnetic applications. Cambridge Press, CambridgeGoogle Scholar
  10. Bloxham J (1985) Geomagnetic secular variation. PhD thesis, Cambridge UniversityGoogle Scholar
  11. Dahlen F, Tromp J (1998) Theoretical global seismology. Princeton University Press, PrincetonGoogle Scholar
  12. Edmonds A (1996) Angular momentum in quantum mechanics. Princeton University Press, PrincetonzbMATHGoogle Scholar
  13. Elsasser W (1946) Induction effects in terrestrial magnetism. Part I. Theory. Phys Rev 69 (3-4): 106–116Google Scholar
  14. Ferrers NM (1877) An elementary treatise on spherical harmonics and subjects connected with them. Macmillan, LondonzbMATHGoogle Scholar
  15. Friis-Christensen E, Lühr H, Hulot G (2006) Swarm: a constellation to study the Earth’s magnetic field. Earth Planets Space 58: 351–358Google Scholar
  16. Friis-Christensen E, Lühr H, Hulot G, Haagmans R, Purucker M (2009) Geomagnetic research from space. Eos 90: 25CrossRefGoogle Scholar
  17. Gauss CF (1839) Allgemeine Theorie des Erdmagnetismus. Resultate aus den Beobachtungen des Magnetischen Vereins im Jahre 1838. Göttinger Magnetischer Verein, LeipzigGoogle Scholar
  18. Goldie AHR, Joyce JW (1940) In: Proceedings of the 1939 Washington Assembly of the Association of Terrestrial Magnetism and Electricity of the International Union of Geodesy and Geophysics, vol 11 (6). Neill & Co, EdinburghGoogle Scholar
  19. Granzow DK (1983) Spherical harmonic representation of the magnetic field in the presence of a current density. Geophys J R Astron Soc 74: 489–505Google Scholar
  20. Harrison CGA (1987) The crustal field. In: Jacobs JA (ed) Geomagnetism, vol 1. Academic, London, pp 513–610Google Scholar
  21. Holme R, Bloxham J (1995) Alleviation of the backus effect in geomagnetic field modelling. Geophys Res Lett 22: 1641–1644CrossRefGoogle Scholar
  22. Holme R, James MA, Lühr H (2005) Magnetic field modelling from scalar-only data: resolving the Backus effect with the equatorial electrojet. Earth Planets Space 57: 1203–1209Google Scholar
  23. Hulot G, Khokhlov A, Le Mouël JL (1997) Uniqueness of mainly dipolar magnetic fields recovered from directional data. Geophys J Int 129: 347–354CrossRefGoogle Scholar
  24. Hulot G, Sabaka TJ, Olsen N (2007) The present field. In: Kono M (ed) Treatise on geophysics, vol 5. Elsevier, AmsterdamGoogle Scholar
  25. Jackson J (1998) Classical electrodynamics. Wiley, New YorkGoogle Scholar
  26. Jackson A, Finlay CC (2007) Geomagnetic secular variation and its application to the core. In: Kono M, (ed) Treatise on geophysics, vol 5. Elsevier, AmsterdamGoogle Scholar
  27. Jackson A, Jonkers ART, Walker MR (2000) Four centuries of geomagnetic secular variation from historical records. Phil Trans R Soc Lond A 358: 957–990Google Scholar
  28. Kellogg OD (1954) Foundations of potential theory. Dover, New YorkGoogle Scholar
  29. Khokhlov A, Hulot G, Le Mouël JL (1997) On the Backus effect - I. Geophys J Int 130: 701–703CrossRefGoogle Scholar
  30. Khokhlov A, Hulot G, Le Mouël JL (1999) On the Backus effect - II. Geophys J Int 137: 816–820CrossRefGoogle Scholar
  31. Kono M (1976) Uniqueness problems in the spherical analysis of the geomagnetic field direction data. J Geomagn Geoelectr 28: 11–29Google Scholar
  32. Langel RA (1987) The main field. In: Jacobs JA (ed) Geomagnetism, vol 1. Academic, London, pp 249–512Google Scholar
  33. Langel RA, Hinze WJ (1998) The magnetic field of the Earth’s lithosphere: the satellite perspective. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  34. Lesur V, Wardinski I, Rother M, Mandea M (2008) GRIMM: the GFZ reference internal magnetic model based on vector satellite and observatory data. Geophys J Int 173: 382–294CrossRefGoogle Scholar
  35. Lorrain P, Corson D (1970) Electromagnetic fields and waves. WH Freeman, San FranciscoGoogle Scholar
  36. Lowes FJ (1966) Mean-square values on sphere of spherical harmonic vector fields. J Geophys Res 71: 2179Google Scholar
  37. Lowes FJ (1974) Spatial power spectrum of the main geomagnetic field, and extrapolation to the core. Geophys J R Astr Soc 36: 717–730Google Scholar
  38. Lowes FJ (1975) Vector errors in spherical harmonic analysis of scalar data. Geophys J R Astron Soc 42: 637–651zbMATHGoogle Scholar
  39. Lowes FJ, De Santis A, Duka B (1999) A discussion of the uniqueness of a Laplacian potential when given only partial information on a sphere. Geophys J Int 121: 579–584CrossRefGoogle Scholar
  40. Lühr H, Maus S, Rother M (2002) First in-situ observation of night-time F region currents with the CHAMP satellite. Geophys Res Lett 29 ((10):127.1–127.4. doi: 10.1029/2001 GL 013845)Google Scholar
  41. Maeda H, Iyemori T, Araki T, Kamei T (1982) New evidence of a meridional current system in the equatorial ionosphere. Geophys Res Lett 9: 337–340CrossRefGoogle Scholar
  42. Malin S (1987) Historical introduction to geomagnetism. In: Jacobs JA (ed) Geomagnetism, vol 1. Academic, London, pp 1–49Google Scholar
  43. Mauersberger P (1956) Das Mittel der Energiedichte des geomagnetischen Hauptfeldes an der Erdoberfläche und seine säkulare Änderung. Gerl Beitr Geophys 65: 207–215Google Scholar
  44. Maus S (2007) CHAMP magnetic mission. In: Gubbins D, Herrero-Bervera E (eds) Encyclopedia of geomagnetism and paleomagnetism. Springer, HeidelbergGoogle Scholar
  45. Maus S, Lühr H (2006) A gravity-driven electric current in the earth’s ionosphere identified in champ satellite magnetic measurements. Geophys Res Lett 33: L02812. doi:10.1029/2005GL024436CrossRefGoogle Scholar
  46. Maus S, Yin F, Lühr H, Manoj C, Rother M, Rauberg J, Michaelis I, Stolle C, Müller R (2008) Resolution of direction of oceanic magnetic lineations by the sixth-generation lithospheric magnetic field model from CHAMP satellite magnetic measurements. Geochem Geophys Geosyst 9 (7): Q07021CrossRefGoogle Scholar
  47. Merrill R, McElhinny M (1983) The Earth’s magnetic field. Academic, LondonGoogle Scholar
  48. Merrill R, McFadden P, McElhinny M (1998) The magnetic field of the Earth: paleomagnetism, the core, and the deep mantle. Academic, LondonGoogle Scholar
  49. Mie G (1908) Considerations on the optic of turbid media, especially colloidal metal sols. Ann Phys(Leipzig) 25: 377–442Google Scholar
  50. Morse P, Feshbach H (1953) Methods of theoretical physics. International series in pure and applied physics. McGraw-Hill, New YorkzbMATHGoogle Scholar
  51. Olsen N (1997) Ionospheric F region currents at middle and low latitudes estimated from Magsat data. J Geophys Res 102 (A3): 4563–4576Google Scholar
  52. Olsen N, Mandea M, Sabaka TJ, Tøffner-Clausen L (2009) CHAOS-2—a geomagnetic field model derived from one decade of continuous satellite data. Geophys J Int 199(3): 1477–1487. doi:10.1111/j.1365-246X.2009.04386.xCrossRefGoogle Scholar
  53. Proctor MRE, Gubbins D (1990) Analysis of geomagnetic directional data. Geophys J Int 100: 69–77CrossRefGoogle Scholar
  54. Purucker M, Whaler K (2007) Crustal magnetism. In: Kono M (ed) Treatise on geophysics, vol 5. Elsevier, Amsterdam, pp 195–235CrossRefGoogle Scholar
  55. Richmond AD (1995) Ionospheric electrodynamics using magnetic Apex coordinates. J Geomagn Geoelectr 47: 191–212Google Scholar
  56. Sabaka TJ, Olsen N, Langel RA (2002) A comprehensive model of the quiet-time near-Earth magnetic field: Phase 3. Geophys J Int 151: 32–68CrossRefGoogle Scholar
  57. Sabaka TJ, Olsen N, Purucker ME (2004) Extending comprehensive models of the Earth’s magnetic field with Ørsted and CHAMP data. Geophys J Int 159: 521–547. doi: 10.1111/j.1365–246X.2004.02421.xCrossRefGoogle Scholar
  58. Schmidt A (1935) Tafeln der Normierten Kugelfunktionen. Engelhard-Reyher Verlag, GothazbMATHGoogle Scholar
  59. Stern DP (1976) Representation of magnetic fields in space. Rev Geophys 14: 199–214CrossRefGoogle Scholar
  60. Stern DP, Bredekamp JH (1975) Error enhancement in geomagnetic models derived from scalar data. J Geophys Res 80: 1776–1782CrossRefGoogle Scholar
  61. Stern DP, Langel RA, Mead GD (1980) Backus effect observed by Magsat. Geophys Res Lett 7: 941–944CrossRefGoogle Scholar
  62. Stolle C, Lühr H, Rother M, Balasis G (2006) Magnetic signatures of equatorial spread F, as observed by the CHAMP satellite. J Geophys Res 111: A02304. doi:10.1029/2005JA011184CrossRefGoogle Scholar
  63. Thomson AWP, Lesur V (2007) An improved geomagnetic data selection algorithm for global geomagnetic field modelling. Geophys J Int 169 (3): 951–963CrossRefGoogle Scholar
  64. Ultré-Guérard P, Hamoudi M, Hulot G (1998) Reducing the Backus effect given some knowledge of the dip-equator. Geophys Res Lett 22 (16): 3201–3204CrossRefGoogle Scholar
  65. Walker AD (1992) Comment on “Non-uniqueness of the external geomagnetic field determined by surface intensity measurements” by Georges E. Backus. J Geophys Res 97 (B10): 13991Google Scholar
  66. Watson GN (1966) A treatise on the theory of Bessel function. Cambridge University Press, LondonGoogle Scholar
  67. Winch D, Ivers D, Turner J, Stening R (2005) Geomagnetism and Schmidt quasi-normalization. Geophys J Int 160 (2): 487–504CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Terence J. Sabaka
    • 1
  • Gauthier Hulot
    • 2
  • Nils Olsen
    • 3
  1. 1.Planetary Geodynamics LaboratoryGreenbelt, MDUSA
  2. 2.Equipe de GéomagnétismeInstitut de Physique du Globe de Paris, Institut de recherche associé au CNRS et a l’Université Paris 7ParisFrance
  3. 3.CopenhagenDenmark

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