Earth's magnetic field can be approximated by a series expansion of orthogonal spherical harmonics. This expansion is an infinite series in terms of coefficients g_{n}^{m} and h_{n}^{m} where m varies from 0 to n and n varies from 1 to infinity. Typically, the series is truncated at n = 10 or more recently n = 12 (i.e. 12 terms). In the spherical coordinate system, the expansion of Earth's magnetic field in terms of the magnetic potential, V in spherical harmonics can be written as:

where R_{E} is the mean radius of the Earth (6371.2 km) and P_{n}^{m} (cos *θ*) are the Schmidt quasi-normalized associated Legendre functions where *θ* is the colatitude (90-latitude) and *φ* is the longitude. Schmidt (1934) introduced the following normalization constant for the associated Legendre function **P**_{n}^{m}(x):

The associated Legendre function becomes:

The 'Gauss' normalized coefficients g_{n}^{m} and h_{n}^{m} are determined experimentally by combining earth-based and satellite measurements of Earth's magnetic field every 5 years. The most recent set of coefficients has been released for 2015 by the V-MOD Working Group of the *International Association for Geomagnetism and Aeronomy* (IAGA) and is know as the International Geomagnetic Reference Field (IGRF).

#### References

Cain, J. C., S. J. Hendricks, R. A. Langel, and W. V. Hudson, A proposed model for the international geomagnetic reference field-1965, J. Geomag. Geoelectr., 19, 335-355, 1967.

Campbell, W., Introduction to geomagnetic fields, Cambridge University Press, Cambridge, 1997.

Chapman, S. and J. Bartels, Geomagnetism, 2 vols., pp 1049, Oxford University Press, London, 1940.

Langel, R. A., Main Field, Chapter Four in Geomagnetism, ed. J. A. Jacobs, Academic Press, London, 1987.

Maus, S. et al., The 10th generation International Geomagnetic Reference Field, Geophys J. Int., 161, 561-565, 2005.

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Schmidt, A., Der magnetische mittlepunkt de erde und seine Bedeutung, Gerlands Beitr. Geophys., 41, 346-358, 1934.