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

${\displaystyle {\vec {B}}(r,\phi ,\theta ,t)=-\nabla V(r,\phi ,\theta ,t)=-\left({\dfrac {\partial V}{\partial r}},{\dfrac {1}{r}}{\dfrac {\partial V}{\partial \theta }},{\dfrac {1}{r\sin \theta }}{\dfrac {\partial V}{\partial \phi }}\right)}$ 

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

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

where ${\displaystyle r}$  is radial distance from the Earth's center, ${\displaystyle N}$  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, ${\displaystyle g_{n}^{m}}$  and ${\displaystyle h_{n}^{m}}$  are Gauss coefficients, and ${\displaystyle P_{n}^{m}\left(\cos \theta \right)}$  are the Schmidt quasi-normalized associated Legendre functions of degree ${\displaystyle n}$  and order ${\displaystyle m}$ 

${\displaystyle P_{n}^{m}(\cos \theta )=K_{n}\sin ^{m}\theta \ {\frac {d^{m+n}}{d(\cos \theta )^{m+n}}}(\cos ^{2}\theta -1)^{n}}$ 

where

${\displaystyle K_{n}={\begin{cases}{\frac {1}{2^{n}}}&{\text{; }}m=0\\{\frac {1}{2^{n}}}{\sqrt {\frac {2(n-m)!}{(n+m)!}}}&{\text{; }}m>0\end{cases}}}$ 

is the normalization coefficient for the Schmidt quasi-normalized formulation.

The ${\displaystyle (-1)^{m}}$  term usually present in the associated Legendre functions is omitted in this formulation and is instead accounted for in the Gauss coefficients.

The Gauss coefficients are modelled as a piecewise-linear function of time with a 5-year step size.^[2]

• Geomagnetic latitude
• Magsat
• Ørsted (satellite)
• CHAMP (satellite)

• IGRF Model Description by IAGA
• Notes concerning correct use and limitations of IGRF