# International Geomagnetic Reference Field

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