# International Geomagnetic Reference Field

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The International Geomagnetic Reference Field (IGRF) is a standard mathematical description of the Earth's main magnetic field and its secular variation. It is the product of a collaborative effort between magnetic field modellers and the institutes involved in collecting and disseminating magnetic field data from satellites and from observatories and surveys around the world.

## Spherical Harmonics

Mathematically, the IGRF model consists of the Gauss coefficients which define a spherical harmonic expansion of the magnetic scalar potential.

$V(r,\phi ,\theta )=a\sum _{{\ell =1}}^{L}\sum _{{m=-\ell }}^{\ell }\left({\frac {a}{r}}\right)^{{\ell +1}}\left(g_{\ell }^{m}\cos m\phi +h_{\ell }^{m}\sin m\phi \right)P_{\ell }^{m}\left(\cos \theta \right)$

where $r$ is radial distance from the Earth's center, $L$ is the maximum degree of the expansion, $\phi$ is East longitude, $\theta$ is colatitude (the polar angle), $a$ is the Earth's radius, $g_{\ell }^{m}$ and $h_{\ell }^{m}$ are Gauss coefficients, and $P_{\ell }^{m}\left(\cos \theta \right)$ are the Schmidt normalized associated Legendre functions of degree $l$ and order $m$.

An online calculator is available from NOAA which allows easy evaluation of the most recent (11th generation) IGRF model at any location and time between 1900 and 2015. IGRF models are standardized for a particular year, reflecting the most accurate measurements available at that time, and indicating a small-scale, slow time variation of the Earth's overall magnetic field.