# Canadian Geomagnetic Reference Field

The Canadian Geomagnetic Reference Field (CGRF) is a regional magnetic reference field model that covers Canada and adjacent areas. Because it is produced for a region that is small compared to the earth the CGRF is capable of depicting spatial variations in the magnetic field that are small compared to those represented by most global models - 833 km compared to 4000 for the IGRF The CGRF also uses data not normally used in global models which rely primarily on satellite data. Although satellite data are used in the production of the CGRF extensive use is made of many other types of data gathered on or near the surface of land and sea.

Between 1985 and 2010, spherical cap harmonic analysis (Haines, 1985) was used to produce the CGRF. Spherical cap harmonic analysis resembles traditional spherical harmonic analysis. However, whereas spherical harmonics apply to the whole sphere, spherical cap harmonics apply to a portion of a sphere. In both cases the basis functions comprise associated Legendre functions in colatitude, trigonometric functions in longitude and powers of radial distance, but for spherical cap harmonics, the degree of the associated Legendre function is allowed to be non-integer.

Thus, over a spherical cap, the magnetic potential can be expressed as:

$V=a\sum _{k=0}^{K}\sum _{m=0}^{k}{\left(\frac{a}{r}\right)}^{{n}_{k}\left(m\right)+1}{P}_{{n}_{k}\left(m\right)}^{m}\left(\mathrm{cos}\theta \right)\left[{g}_{k}^{m}\mathrm{cos}\left(m\lambda \right)+{h}_{k}^{m}\mathrm{sin}\left(m\lambda \right)\right]$

For the CGRF the expansion is taken to a maximum K of 16, which allows spatial structure of 830 km to be depicted when the spherical cap has a radius of 30°.

The coefficients, ${g}_{k}^{m}$ and ${h}_{k}^{m}$ are functions of time so the CGRF is valid over the time interval 1960 to 2010.

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