International Geomagnetic Reference Field

The International Geomagnetic Reference Field (IGRF) is a standard mathematical description of the large-scale structure of the Earth's main magnetic field and its secular variation. It was created by fitting parameters of a mathematical model of the magnetic field to measured magnetic field data from surveys, observatories and satellites across the globe. The IGRF has been produced and updated under the direction of the International Association of Geomagnetism and Aeronomy (IAGA) since 1965.
The IGRF model covers a significant time span, and so is useful for interpreting historical data. (This is unlike the World Magnetic Model, which is intended for navigation in the next few years.) It is updated at 5-year intervals, reflecting the most accurate measurements available at that time. The current 13th edition of the IGRF model was released in December 2019 and is valid from 1900 until 2025. For the interval from 1945 to 2015, it is "definitive", meaning that future updates are unlikely to improve the model in any significant way.
Spherical Harmonics.
The IGRF models the geomagnetic field formula_1 as a gradient of a magnetic scalar potential formula_2
The magnetic scalar potential model consists of the Gauss coefficients which define a spherical harmonic expansion of formula_4
where formula_6 is radial distance from the Earth's center, formula_7 is the maximum degree of the expansion, formula_8 is East longitude,
formula_9 is colatitude (the polar angle), formula_10 is the Earth's radius,
formula_11 and formula_12 are Gauss coefficients, and formula_13
are the Schmidt normalized associated Legendre functions of degree formula_14 and order formula_15. The Gauss coefficients are assumed to vary linearly over the time interval specified by the model.