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# Magnetic field reference models

Magnetic reference field models provide an easy way to calculate magnetic declination and other components of the magnetic field. A reference field model is a mathematical algorithm whose parameters are based on an analysis of magnetic observations either over the entire world or a part of the world. Spherical harmonic analysis is the most common method used for producing global models. The International Geomagnetic Reference Field (IGRF) and the World Magnetic Model (WMM) are the most commonly used models for navigational purposes. Models are traditionally updated every five years. The Canadian Geomagnetic Reference Field (CGRF) is a model of the magnetic field over the Canadian region for the time period 1985-2010. It was produced using denser data over Canada than were used for the IGRF, and because the analysis was carried out over a smaller region, the CGRF can reproduce smaller spatial variations in the magnetic field than can the IGRF.

Since magnetic field models such as the IGRF and CGRF are approximations to observed data, a value of declination computed using either of them is likely to differ somewhat from the "true" value at that location. It is generally agreed that the IGRF achieves an overall accuracy of better than 1° in declination; the accuracy is better than this in densely surveyed areas such as Europe and North America, and worse in oceanic areas such as the south Pacific. The accuracy of the CGRF, in southern Canada, is about 0.5°. The accuracy of all models is worse in the Arctic near the North Magnetic Pole.

Magnetic field models are used to calculate magnetic declination by means of computer programs such as the Magnetic Information Retrieval Program (MIRP), a software package developed by the Geomagnetism Program of the Geological Survey of Canada. The user inputs the year, latitude and longitude and MIRP calculates the declination. MIRP is able to compute values for any location on the Earth in the time period 1900 to the present. For all locations, MIRP computes values using the IGRF. Two versions of MIRP are available on-line: one to calculate declination; the other to calculate all components of the magnetic field.

## Spherical harmonics

In 1838 the German mathematician and magnetician Frederick Gauss developed a method of representing the magnetic field in terms of a converging series whose terms were functions of latitude, longitude and radial distance from the centre of the earth. In modern notation, the representation is:

$V=a\sum _{n=1}^{\mathrm{N max}}{\left(\frac{a}{r}\right)}^{n+1}\left[{g}_{n}^{m}\mathrm{cos}\left(m\phi \right)+{h}_{n}^{m}\mathrm{sin}\left(m\phi \right)\right]{P}_{n}^{m}\left(\theta \right)$
• φ refers to longitude
• θ refers to latitude
• r is the radial distance
• n is the degree of the term
• m is the order of the term
• V is called the scalar potential

The ${P}_{n}^{m}$ are called associated Legendre polynomials which look very much like distorted sine waves. The ${g}_{n}^{m}$ and ${h}_{n}^{m}$ are called Gauss coefficients which are determined through a least-squares analysis of a world-wide distribution of magnetic observations.

In theory the series goes to infinity; in practice some maximum degree, Nmax is chosen so that the series is able to reproduce the observed field to the desired resolution and accuracy. For example, for the IGRF, Nmax = 10. To reproduce the field originating within the core of the Earth requires Nmax = 15. To reproduce crustal anomalies visible in magnetic data at satellite altitudes requires Nmax = 80.

The magnetic field components (X, Y and Z) can be calculated from the scalar potential through the following derivatives:

$X=\frac{1}{r}\frac{\partial V}{\partial \theta }$
$Y=\frac{1}{rsin\theta }\frac{\partial V}{\partial \phi }$
$Z=\frac{\partial V}{\partial r}$