Editing Earth's magnetic field (section)

=== Statistical models ===
Each measurement of the magnetic field is at a particular place and time. If an accurate estimate of the field at some other place and time is needed, the measurements must be converted to a model and the model used to make predictions.

==== Spherical harmonics ====
{{See also|Multipole expansion}}
[[File:Spherical harmonics positive negative.svg|thumb|Schematic representation of spherical harmonics on a sphere and their nodal lines. {{math|<var>P</var><sub>ℓ <var>m</var></sub>}} is equal to 0 along {{math|<var>m</var>}} [[great circle]]s passing through the poles, and along {{math|ℓ-<var>m</var>}} circles of equal latitude. The function changes sign each ℓtime it crosses one of these lines.]]
[[File:VFPt four charges.svg|thumb|Example of a quadrupole field. This can also be constructed by moving two dipoles together.]]

The most common way of analyzing the global variations in the Earth's magnetic field is to fit the measurements to a set of [[spherical harmonics]]. This was first done by Carl Friedrich Gauss.<ref name=Wallace2003>{{harvnb|Campbell|2003}}, p.&nbsp;1.</ref> Spherical harmonics are functions that oscillate over the surface of a sphere. They are the product of two functions, one that depends on latitude and one on longitude. The function of longitude is zero along zero or more great circles passing through the North and South Poles; the number of such ''[[nodal line]]s'' is the absolute value of the ''order'' {{math|<var>m</var>}}. The function of latitude is zero along zero or more latitude circles; this plus the order is equal to the ''degree'' ℓ. Each harmonic is equivalent to a particular arrangement of magnetic charges at the center of the Earth. A ''[[Magnetic monopole|monopole]]'' is an isolated magnetic charge, which has never been observed. A ''[[Magnetic dipole|dipole]]'' is equivalent to two opposing charges brought close together and a ''[[quadrupole]]'' to two dipoles brought together. A quadrupole field is shown in the lower figure on the right.<ref name=MMMch2>{{harvnb |Merrill|McElhinny|McFadden|1996 |loc=Chapter 2}}</ref>

Spherical harmonics can represent any [[scalar field]] (function of position) that satisfies certain properties. A magnetic field is a [[vector field]], but if it is expressed in Cartesian components {{math|<var>X, Y, Z</var>}}, each component is the derivative of the same scalar function called the ''[[magnetic scalar potential|magnetic potential]]''. Analyses of the Earth's magnetic field use a modified version of the usual spherical harmonics that differ by a multiplicative factor. A least-squares fit to the magnetic field measurements gives the Earth's field as the sum of spherical harmonics, each multiplied by the best-fitting  ''Gauss coefficient'' {{math|<var>g<sub>m</sub></var><sup>ℓ</sup>}} or {{math|<var>h<sub>m</sub></var><sup>ℓ</sup>}}.<ref name="MMMch2" />

The lowest-degree Gauss coefficient, {{math|<var>g</var><sub>0</sub><sup>0</sup>}}, gives the contribution of an isolated magnetic charge, so it is zero. The next three coefficients – {{math|<var>g</var><sub>1</sub><sup>0</sup>}}, {{math|<var>g</var><sub>1</sub><sup>1</sup>}}, and {{math|<var>h</var><sub>1</sub><sup>1</sup>}} – determine the direction and magnitude of the dipole contribution. The best fitting dipole is tilted at an angle of about 10° with respect to the rotational axis, as described earlier.<ref name="MMMch2" />

==== Radial dependence ====
Spherical harmonic analysis can be used to distinguish internal from external sources if measurements are available at more than one height (for example, ground observatories and satellites). In that case, each term with coefficient {{math|<var>g<sub>m</sub></var><sup>ℓ</sup>}} or {{math|<var>h<sub>m</sub></var><sup>ℓ</sup>}} can be split into two terms: one that decreases with radius as {{math|1/<var>r</var><sup>ℓ+1</sup>}} and one that ''increases'' with radius as {{math|<var>r</var><sup>ℓ</sup>}}. The increasing terms fit the external sources (currents in the ionosphere and magnetosphere). However, averaged over a few years the external contributions average to zero.<ref name="MMMch2" />

The remaining terms predict that the potential of a dipole source ({{math|ℓ{{=}}1}}) drops off as {{math|1/<var>r</var><sup>2</sup>}}. The magnetic field, being a derivative of the potential, drops off as {{math|1/<var>r</var><sup>3</sup>}}. Quadrupole terms drop off as {{math|1/<var>r</var><sup>4</sup>}}, and higher order terms drop off increasingly rapidly with the radius. The radius of the outer core is about half of the radius of the Earth. If the field at the core-mantle boundary is fit to spherical harmonics, the dipole part is smaller by a factor of about 8 at the surface, the quadrupole part by a factor of 16, and so on. Thus, only the components with large wavelengths can be noticeable at the surface. From a variety of arguments, it is usually assumed that only terms up to degree {{math|14}} or less have their origin in the core. These have wavelengths of about {{cvt|2000|km}} or less. Smaller features are attributed to crustal anomalies.<ref name="MMMch2" />

==== Global models ====
The [[International Association of Geomagnetism and Aeronomy]] maintains a standard global field model called the [[International Geomagnetic Reference Field]] (IGRF). It is updated every five years. The 11th-generation model, IGRF11, was developed using data from satellites ([[Ørsted (satellite)|Ørsted]], [[CHAMP (satellite)|CHAMP]] and SAC-C) and a world network of geomagnetic observatories.<ref>{{cite journal|last1=Finlay|first1=CC|last2=Maus|first2=S|last3=Beggan|first3=CD|last4=Hamoudi|first4=M.|last5=Lowes|first5=FJ|last6=Olsen|first6=N|last7=Thébault|first7=E.|date=2010|title=Evaluation of candidate geomagnetic field models for IGRF-11|journal=Earth, Planets and Space|volume=62|issue=10|pages=787–804|bibcode=2010EP&S...62..787F|doi=10.5047/eps.2010.11.005|s2cid=530534|doi-access=free}}</ref> The spherical harmonic expansion was truncated at degree 10, with 120 coefficients, until 2000. Subsequent models are truncated at degree 13 (195 coefficients).<ref name=irgf_health>{{cite web |url=https://www.ncei.noaa.gov/products/international-geomagnetic-reference-field/health-warning |title=The International Geomagnetic Reference Field: A "Health" Warning |date=January 2010 |publisher=National Geophysical Data Center |access-date=13 October 2011}}</ref>

Another global field model, called the [[World Magnetic Model]], is produced jointly by the United States [[National Centers for Environmental Information]] (formerly the National Geophysical Data Center) and the [[British Geological Survey]]. This model truncates at degree 12 (168 coefficients) with an approximate spatial resolution of 3,000&nbsp;kilometers. It is the model used by the [[United States Department of Defense]], the [[Ministry of Defence (United Kingdom)]], the United States [[Federal Aviation Administration]] (FAA), the [[North Atlantic Treaty Organization]] (NATO), and the [[International Hydrographic Organization]] as well as in many civilian navigation systems.<ref>{{cite web |title=The World Magnetic Model |url=https://www.ncei.noaa.gov/products/world-magnetic-model |publisher=National Geophysical Data Center |access-date=14 October 2011}}</ref>

The above models only take into account the "main field" at the core-mantle boundary. Although generally good enough for navigation, higher-accuracy use cases require smaller-scale [[magnetic anomalies]] and other variations to be considered. Some examples are (see geomag.us ref for more):<ref>{{cite web |title=Geomagnetic and Electric Field Models |url=https://geomag.us/models/index.html |website=geomag.us}}</ref>
* The "comprehensive modeling" (CM) approach by the [[Goddard Space Flight Center]] ([[NASA]] and [[GSFC]]) and the [[Danish Space Research Institute]]. CM attempts to reconcile data with greatly varying temporal and spatial resolution from ground and satellite sources. The latest version as of 2022 is CM5 of 2016. It provides separate components for main field plus [[lithosphere]] (crustal), [[Tide#Principal lunar semi-diurnal constituent|M2 tidal]], and primary/induced magnetosphere/ionosphere variations.<ref>{{cite web |title=Model information |url=https://ccmc.gsfc.nasa.gov/models/modelinfo.php?model=CM5 |website=ccmc.gsfc.nasa.gov |access-date=2022-01-12 |archive-date=2021-12-09 |archive-url=https://web.archive.org/web/20211209100224/https://ccmc.gsfc.nasa.gov/models/modelinfo.php?model=CM5 |url-status=dead }}</ref>
* The US [[National Centers for Environmental Information]] developed the [[Enhanced Magnetic Model]] (EMM), which extends to degree and order 790 and resolves [[magnetic anomalies]] down to a wavelength of 56&nbsp;kilometers. It was compiled from satellite, marine, aeromagnetic and ground magnetic surveys. {{As of|2018}}, the latest version, EMM2017, includes data from The European Space Agency's Swarm satellite mission.<ref>{{cite web |title=The Enhanced Magnetic Model|url=https://www.ncei.noaa.gov/products/enhanced-magnetic-model |publisher=United States [[National Centers for Environmental Information]] |access-date=29 June 2018}}</ref>

For historical data about the main field, the IGRF may be used back to year 1900.<ref name="irgf_health" /> A specialized GUFM1 model estimates back to year 1590 using ship's logs.<ref>{{cite journal |last1=Jackson |first1=Andrew |last2=Jonkers |first2=Art R. T. |last3=Walker |first3=Matthew R. |title=Four centuries of geomagnetic secular variation from historical records |journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences |date=15 March 2000 |volume=358 |issue=1768 |pages=957–990 |doi=10.1098/rsta.2000.0569 |bibcode=2000RSPTA.358..957J |s2cid=40510741}}</ref> [[Paleomagnetic]] research has produced models dating back to 10,000&nbsp;BCE.<ref>{{cite web |title=The GEOMAGIA database |url=https://geomagia.gfz-potsdam.de/models.php |website=geomagia.gfz-potsdam.de}}</ref>