# International Geomagnetic Reference Field

The International Geomagnetic Reference Field (IGRF) is a standard mathematical description of the large-scale structure of the Earth's main magnetic field and its secular variation. It was created by fitting parameters of a mathematical model of the magnetic field to measured magnetic field data from surveys, observatories and satellites across the globe. The IGRF has been produced and updated under the direction of the International Association of Geomagnetism and Aeronomy (IAGA) since 1965.[1]

## Spherical Harmonics

The IGRF models the geomagnetic field ${\displaystyle {\vec {B}}(r,\phi ,\theta ,t)}$ as a gradient of a magnetic scalar potential ${\displaystyle V(r,\phi ,\theta ,t)}$

${\displaystyle {\vec {B}}(r,\phi ,\theta ,t)=-\nabla V(r,\phi ,\theta ,t)}$

The magnetic scalar potential model consists of the Gauss coefficients which define a spherical harmonic expansion of ${\displaystyle V}$[1]

${\displaystyle V(r,\phi ,\theta ,t)=a\sum _{\ell =1}^{L}\sum _{m=0}^{\ell }\left({\frac {a}{r}}\right)^{\ell +1}\left(g_{\ell }^{m}(t)\cos m\phi +h_{\ell }^{m}(t)\sin m\phi \right)P_{\ell }^{m}\left(\cos \theta \right)}$

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

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. The 12th edition of the IGRF model covers the 2015 epoch.[1]