Editing Geoid (section)

==Spherical harmonics representation==
{{Further|Geopotential spherical harmonic model}}

[[Spherical harmonic]]s are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is [[EGM2020]] (Earth Gravitational Model 2020), determined in an international collaborative project led by the National Imagery and Mapping Agency (now the [[National Geospatial-Intelligence Agency]], or NGA). The mathematical description of the non-rotating part of the potential function in this model is:<ref name="noaa.gov">{{cite book|first=Dru A.|last=Smith|chapter-url=http://www.ngs.noaa.gov/PUBS_LIB/EGM96_GEOID_PAPER/egm96_geoid_paper.html|chapter=There is no such thing as 'The' EGM96 geoid: Subtle points on the use of a global geopotential model|title=IGeS Bulletin No. 8|publisher=International Geoid Service |location=Milan, Italy |pages=17–28|year=1998|access-date=16 December 2016}}</ref>

<math display="block">
V=\frac{GM}{r}\left(1+{\sum_{n=2}^{n_\text{max}}}\left(\frac{a}{r}\right)^n{\sum_{m=0}^n}
\overline{P}_{nm}(\sin\phi)\left[\overline{C}_{nm}\cos m\lambda+\overline{S}_{nm}\sin m\lambda\right]\right),
</math>

where <math>\phi\ </math> and <math>\lambda\ </math> are ''geocentric'' (spherical) latitude and longitude respectively, <math>\overline{P}_{nm}</math> are the fully normalized [[associated Legendre polynomials]] of degree <math>n\ </math> and order <math>m\ </math>, and <math>\overline{C}_{nm}</math> and <math>\overline{S}_{nm}</math> are the numerical coefficients of the model based on measured data. The above equation describes the Earth's gravitational [[potential]] <math>V</math>, not the geoid itself, at location <math>\phi,\;\lambda,\;r,\ </math> the co-ordinate <math>r\ </math> being the ''geocentric radius'', i.e., distance from the Earth's centre. The geoid is a particular [[equipotential]] surface,<ref name="noaa.gov"/> and is somewhat involved to compute. The gradient of this potential also provides a model of the gravitational acceleration. The most commonly used EGM96 contains a full set of coefficients to degree and order 360 (i.e., <math>n_\text{max} = 360</math>), describing details in the global geoid as small as 55&nbsp;km (or 110&nbsp;km, depending on the definition of resolution). The number of coefficients, <math>\overline{C}_{nm}</math> and <math>\overline{S}_{nm}</math>, can be determined by first observing in the equation for <math>V</math> that for a specific value of <math>n</math> there are two coefficients for every value of <math>m</math> except for <math>m=0</math>. There is only one coefficient when <math>m=0</math> since <math> \sin (0\lambda) = 0</math>. There are thus <math>(2n+1)</math> coefficients for every value of <math>n</math>. Using these facts and the formula, <math display="inline">\sum_{I=1}^{L}I = \frac{1}{2}L(L + 1)</math>, it follows that the total number of coefficients is given by

<math display="block">\sum_{n=2}^{n_\text{max}}(2n+1) = n_\text{max}(n_\text{max}+1) + n_\text{max} - 3 = 130317</math> using the EGM96 value of <math>n_\text{max} = 360</math>.

For many applications, the complete series is unnecessarily complex and is truncated after a few (perhaps several dozen) terms.

Still, even higher resolution models have been developed. Many of the authors of EGM96 have published EGM2008. It incorporates much of the new satellite gravity data (e.g., the [[Gravity Recovery and Climate Experiment]]), and supports up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients),<ref>Pavlis, N. K.; Holmes, S. A.; Kenyon, S.; Schmit, D.; Trimmer, R. "Gravitational potential expansion to degree 2160". ''IAG International Symposium, gravity, geoid and Space Mission GGSM2004''. Porto, Portugal, 2004.</ref> with additional coefficients extending to degree 2190 and order 2159.<ref name="nga.mil">{{cite web|url=http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html|title=Earth Gravitational Model 2008 (EGM2008)|publisher=[[National Geospatial-Intelligence Agency]]|access-date=9 September 2008|archive-date=8 May 2010|archive-url=https://web.archive.org/web/20100508002312/http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html|url-status=dead}}</ref> EGM2020 is the international follow-up that was originally scheduled for 2020 (still unreleased in 2024), containing the same number of harmonics generated with better data.<ref>{{Cite journal|url=https://ui.adsabs.harvard.edu/abs/2015AGUFM.G34A..03B/abstract|bibcode = 2015AGUFM.G34A..03B|title = Earth Gravitational Model 2020|last1 = Barnes|first1 = D.|last2 = Factor|first2 = J. K.|last3 = Holmes|first3 = S. A.|last4 = Ingalls|first4 = S.|last5 = Presicci|first5 = M. R.|last6 = Beale|first6 = J.|last7 = Fecher|first7 = T.|journal = AGU Fall Meeting Abstracts|year = 2015|volume = 2015|pages = G34A–03}}</ref>