International Geomagnetic Reference Field

Spherical Harmonics

The IGRF models the geomagnetic field B → ( r , ϕ , θ , t ) as a gradient of a magnetic scalar potential V ( r , ϕ , θ , t )

B → ( r , ϕ , θ , t ) = − ∇ V ( r , ϕ , θ , t )

The magnetic scalar potential model consists of the Gauss coefficients which define a spherical harmonic expansion of V

V ( r , ϕ , θ , t ) = a ∑ ℓ = 1 L ∑ m = 0 ℓ ( a r ) ℓ + 1 ( g ℓ m ( t ) cos ⁡ m ϕ + h ℓ m ( t ) sin ⁡ m ϕ ) P ℓ m ( cos ⁡ θ )

where r is radial distance from the Earth's center, L is the maximum degree of the expansion, ϕ is East longitude, θ is colatitude (the polar angle), a is the Earth's radius, g ℓ m and h ℓ m are Gauss coefficients, and P ℓ m ( cos ⁡ θ ) are the Schmidt normalized associated Legendre functions of degree l and order 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.