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 gnm and hnm 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 RE is the mean radius of the Earth (6371.2 km) and Pnm (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 Pnm(x):
The associated Legendre function becomes:
The 'Gauss' normalized coefficients gnm and hnm 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).
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