View source for Geoid

{{Short description|Ocean shape without winds and tides}}
{{Distinguish|text=[[geode]], a crystal-filled rock}}
{{Distinguish|text=[[GEOID]], a geocoding scheme}}
[[File:Earth_Gravitational_Model_1996.png|thumb|upright=1.5|Map of the undulation of the geoid in meters (based on the [[EGM96]] gravity model and the [[WGS84]] reference ellipsoid).<ref>{{cite web|url=http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html|title=WGS 84, N=M=180 Earth Gravitational Model|work=NGA: Office of Geomatics|publisher=National Geospatial-Intelligence Agency|access-date=17 December 2016|archive-date=8 August 2020|archive-url=https://web.archive.org/web/20200808025531/https://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html|url-status=dead}}</ref>]]
{{Geodesy}}
{{Geophysics|all}}
{{Use dmy dates|date=May 2022}}

The '''geoid''' ({{IPAc-en|ˈ|dʒ|iː|.|oɪ|d}} {{respell|JEE|oyd}}) is the shape that the [[ocean]] surface would take under the influence of the [[gravity of Earth]], including [[gravitational attraction]] and [[Earth's rotation]], if other influences such as winds and [[tide]]s were absent. This surface is extended through the [[continent]]s (such as might be approximated with very narrow hypothetical [[canal]]s). According to [[Carl Friedrich Gauss]], who first described it, it is the "mathematical [[figure of the Earth]]", a smooth but irregular [[surface]] whose shape results from the uneven distribution of mass within and on the surface of Earth.<ref name="Gauß1828">{{cite book | last=Gauß | first=C.F. | title=Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector | publisher=Vandenhoeck und Ruprecht | year=1828 | url=https://books.google.com/books?id=tIg_AAAAcAAJ&pg=PA73 | language=de | access-date=2021-07-06 | page=73}}</ref> It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of [[geodesy]] and [[geophysics]], it has been defined to high precision only since advances in [[satellite geodesy]] in the late 20th century.

The geoid is often expressed as a '''geoid undulation''' or '''geoidal height''' above a given [[reference ellipsoid]], which is a slightly flattened sphere whose [[equatorial bulge]] is caused by the planet's rotation. Generally the geoidal height rises where the Earth's material is locally more dense and exerts greater gravitational force than the surrounding areas. The geoid in turn serves as a reference [[coordinate surface]] for various [[vertical coordinate]]s, such as ''[[orthometric height]]s'', ''[[geopotential height]]s'', and ''[[dynamic height]]s'' (see [[Geodesy#Heights]]).

All points on a geoid surface have the same [[geopotential]] (the sum of [[gravitational energy|gravitational potential energy]] and [[centrifugal force|centrifugal]] potential energy). At this surface, apart from temporary tidal fluctuations, the [[force of gravity]] acts everywhere perpendicular to the geoid, meaning that [[plumb bob|plumb lines]] point perpendicular and [[bubble level]]s are parallel to the geoid. 
Being an [[equigeopotential]] means the geoid corresponds to the [[free surface]] of water at rest (if only the Earth's gravity and rotational acceleration were at work); this is also a sufficient condition for a ball to remain at rest instead of rolling over the geoid.
Earth's gravity acceleration (the [[vertical derivative]] of geopotential) is thus non-uniform over the geoid.<ref name="BookGeodesyConcepts">''Geodesy: The Concepts.''  Petr Vanicek and E.J. Krakiwsky.  Amsterdam: Elsevier.  1982 (first ed.): {{ISBN|0-444-86149-1}}, {{ISBN|978-0-444-86149-8}}.  1986 (third ed.): {{ISBN|0-444-87777-0}}, {{ISBN|978-0-444-87777-2}}.  {{ASIN|0444877770}}.</ref>

[[File:Geoid undulation 10k scale.jpg|thumb|Geoid undulation in [[pseudocolor]], [[shaded relief]] and [[vertical exaggeration]] (10000 vertical scaling factor).]]
[[File:Geoid undulation to scale.jpg|thumb|Geoid undulation in pseudocolor, without vertical exaggeration.]]

==Description==
The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as an [[ellipsoid]]), but is considerably smoother than Earth's physical surface. Although the "ground" of the Earth has excursions on the order of +8,800 m ([[Mount Everest]]) and −11,000 m ([[Marianas Trench]]), the geoid's deviation from an ellipsoid ranges from +85&nbsp;m (Iceland) to −106&nbsp;m (southern India), less than 200 m total.<ref name="TexasGeoid">{{cite web | url=http://www2.csr.utexas.edu/grace/gravity/gravity_definition.html | title=Earth's Gravity Definition | publisher=Center for Space Research ([[University of Texas at Austin]]) / Texas Space Grant Consortium | work=GRACE – Gravity Recovery and Climate Experiment | date=11 February 2004 | access-date=22 January 2018}}</ref>

If the ocean were of constant density and undisturbed by tides, currents or weather, its surface would resemble the geoid. The permanent deviation between the geoid and [[mean sea level]] is called [[ocean surface topography]]. If the continental land masses were crisscrossed by a series of tunnels or canals, the sea level in those canals would also very nearly coincide with the geoid. [[Geodesist]]s are able to derive the heights of continental points above the geoid by [[spirit leveling]].

Being an [[equipotential surface]], the geoid is, by definition, a surface upon which the force of gravity is perpendicular everywhere, apart from temporary tidal fluctuations. This means that when traveling by ship, one does not notice the [[#Undulation|undulation of the geoid]]; neglecting tides, the local vertical (plumb line) is always perpendicular to the geoid and the local horizon [[tangential component|tangential]] to it. Likewise, spirit levels will always be parallel to the geoid.

===Simplified example===
[[File:Geoida.svg|thumb|{{olist
    |Ocean
    |Ellipsoid
    |Local plumb line
    |Continent
    |Geoid
}}]]
Earth's gravitational field is not uniform. An [[oblate spheroid]] is typically used as the idealized Earth, but even if the Earth were spherical and did not rotate, the strength of gravity would not be the same everywhere because density varies throughout the planet. This is due to magma distributions, the density and weight of different [[geological]] compositions in the [[Earth's crust]], mountain ranges, deep sea trenches, crust compaction due to glaciers, and so on.

If that sphere were then covered in water, the water would not be the same height everywhere. Instead, the water level would be higher or lower with respect to Earth's center, depending on the integral of the strength of gravity from the center of the Earth to that location. The geoid level coincides with where the water would be. Generally the geoid rises where the Earth's material is locally more dense, exerts greater gravitational force, and pulls more water from the surrounding area.

==Formulation{{anchor|Shape|Undulation}}==
[[File:Geoundnsrp.png|thumb|Meridional profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also: [[Earth's pear shape]].]]

The '''geoid undulation''' (also known as '''geoid height''' or '''geoid anomaly'''), ''N'', is the height of the geoid relative to a given [[ellipsoid of reference]]. 
<math display="block">N=h-H</math>
The undulation is not standardized, as different countries use different mean sea levels as reference, but most commonly refers to the [[EGM96]] geoid.

In maps and common use, the height over the mean sea level (such as [[orthometric height]], ''H'') is used to indicate the height of elevations while the [[ellipsoidal height]], ''h'', results from the [[GPS]] system and similar [[GNSS]]:
<math display="block">H=h-N</math>
(An analogous relationship exists between [[normal height]]s and the ''[[quasigeoid]]'', which disregards local density variations.)
In practice, many handheld GPS receivers [[spatial interpolation|interpolate]] ''N'' in a pre-computed ''geoid map'' (a [[lookup table]]).<ref>{{cite web|last1=Wormley|first1=Sam|title=GPS Orthometric Height|url=http://www.edu-observatory.org/gps/height.html|website=edu-observatory.org|access-date=15 June 2016|url-status=dead|archive-url=https://web.archive.org/web/20160620102914/http://www.edu-observatory.org/gps/height.html|archive-date=20 June 2016}}</ref>

So a [[GPS receiver]] on a ship may, during the course of a long voyage, indicate height variations, even though the ship will always be at sea level (neglecting the effects of tides). That is because GPS [[satellite]]s, orbiting about the center of gravity of the Earth, can measure heights only relative to a geocentric reference ellipsoid. To obtain one's [[orthometric height]], a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a [[tide gauge]], as in traditional land surveying, is closer to orthometric height. Modern GPS receivers have a grid implemented in their software by which they obtain, from the current position, the height of the geoid (e.g., the EGM96 geoid) over the [[World Geodetic System]] (WGS) ellipsoid. They are then able to correct the height above the WGS ellipsoid to the height above the EGM96 geoid. When height is not zero on a ship, the discrepancy is due to other factors such as ocean tides, [[atmospheric pressure]] (meteorological effects), local [[sea surface topography]], and measurement uncertainties.

[[File:Geoundaequrp.png|thumb|Equatorial profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also: [[triaxial Earth]].]]

==Determination==
{{See also|Earth ellipsoid#Determination}}

The undulation of the geoid ''N'' is closely related to the [[disturbing potential]] ''T'' according to '''Bruns' formula''' (named after [[Heinrich Bruns]]):{{anchor|Bruns formula}}
:<math>
N=T/\gamma\,,
</math>
where <math>\gamma</math> is the force of [[normal gravity]], computed from the normal field potential <math>U</math>.

Another way of determining ''N'' is using values of ''[[gravity anomaly]]'' <math>\Delta g</math>, differences between true and normal reference gravity, as per '''{{vanchor|Stokes formula}}''' (or '''Stokes' integral'''), published in 1849 by [[George Gabriel Stokes]]:

:<math>
N=\frac{R}{4\pi \gamma_0}\iint_\sigma \Delta g \,S(\psi)\, d\sigma.
</math>

The [[integral kernel]] ''S'', called ''Stokes function'', was derived by Stokes in closed analytical form.<ref name="Wang 2016 pp. 1–8">{{cite book | last=Wang | first=Yan Ming | title=Encyclopedia of Geodesy | chapter=Geodetic Boundary Value Problems | publisher=Springer International Publishing | publication-place=Cham | year=2016 | isbn=978-3-319-02370-0 | doi=10.1007/978-3-319-02370-0_42-1 | pages=1–8}}</ref>
Note that determining <math>N</math> anywhere on Earth by this formula requires <math>\Delta g</math> to be known ''everywhere on Earth'', including oceans, polar areas, and deserts. For terrestrial gravimetric measurements this is a near-impossibility, in spite of close international co-operation within the [[International Association of Geodesy]] (IAG), e.g., through the International Gravity Bureau (BGI, Bureau Gravimétrique International).

Another approach for geoid determination is to ''combine'' multiple information sources: not just terrestrial gravimetry, but also satellite geodetic data on the figure of the Earth, from analysis of satellite orbital perturbations, and lately from satellite gravity missions such as [[GOCE]] and [[Gravity Recovery and Climate Experiment|GRACE]]. In such combination solutions, the low-resolution part of the geoid solution is provided by the satellite data, while a 'tuned' version of the above Stokes equation is used to calculate the high-resolution part, from terrestrial gravimetric data from a neighbourhood of the evaluation point only.

Calculating the undulation is mathematically challenging.<ref>{{Cite book | doi=10.1007/978-90-481-8702-7_154| chapter=Geoid Determination, Theory and Principles| title=Encyclopedia of Solid Earth Geophysics| pages=356–362| series=Encyclopedia of Earth Sciences Series| year=2011| last1=Sideris| first1=Michael G.| isbn=978-90-481-8701-0| s2cid=241396148}}</ref><ref>{{Cite book |doi = 10.1007/978-90-481-8702-7_225|chapter = Geoid, Computational Method|title = Encyclopedia of Solid Earth Geophysics|pages = 366–371|series = Encyclopedia of Earth Sciences Series|year = 2011|last1 = Sideris|first1 = Michael G.|isbn = 978-90-481-8701-0}}</ref>
The precise geoid solution by [[Petr Vaníček]] and co-workers improved on the [[George Gabriel Stokes|Stokesian]] approach to geoid computation.<ref>{{cite web|url=http://www2.unb.ca/gge/Research/GRL/GeodesyGroup/SHGeo.html |title=UNB Precise Geoid Determination Package |access-date=2 October 2007 }}</ref> Their solution enables millimetre-to-centimetre [[accuracy]] in geoid [[computation]], an [[order of magnitude|order-of-magnitude]] improvement from previous classical solutions.<ref>{{cite journal |last=Vaníček |first=P. |author2=Kleusberg, A. |date=1987 |title=The Canadian geoid-Stokesian approach |journal=Manuscripta Geodaetica |volume=12 |issue=2 |pages=86–98 |doi=10.1007/BF03655117 }}</ref><ref>{{cite journal |last=Vaníček |first=P. |author2=Martinec, Z. |date=1994 |title=Compilation of a precise regional geoid |journal=Manuscripta Geodaetica |volume=19 |pages=119–128 |doi=10.1007/BF03655333 |url=http://gge.unb.ca/Personnel/Vanicek/StokesHelmert.pdf }}</ref><ref>{{cite report|first1=P.|last1=Vaníček|first2=A.|last2=Kleusberg|first3=Z.|last3=Martinec|first4=W.|last4=Sun|first5=P.|last5=Ong|first6=M.|last6=Najafi|first7=P.|last7=Vajda|first8=L.|last8=Harrie|first9=P.|last9=Tomasek|first10=B.|last10=ter Horst|title=Compilation of a Precise Regional Geoid|publisher=Department of Geodesy and Geomatics Engineering, University of New Brunswick |docket=184|url=http://gge.unb.ca/Personnel/Vanicek/GeoidReport950327.pdf|access-date=22 December 2016}}</ref><ref>{{cite book|last1=Kopeikin|first1=Sergei|last2=Efroimsky|first2=Michael|last3=Kaplan|first3=George|title=Relativistic celestial mechanics of the solar system|url=https://archive.org/details/relativisticcele00kope|url-access=limited|date=2009|publisher=[[Wiley-VCH]]|location=Weinheim|isbn=9783527408566|page=[https://archive.org/details/relativisticcele00kope/page/n735 704]}}</ref>

Geoid undulations display uncertainties which can be estimated by using several methods, e.g., [[Least squares|least-squares]] collocation (LSC), [[fuzzy logic]], [[Artificial neural network|artificial neural networks]], [[radial basis function]]s (RBF), and [[Geostatistics|geostatistical]] techniques. Geostatistical approach has been defined as the most-improved technique in prediction of geoid undulation.<ref>{{Cite journal|last1=Chicaiza|first1=E.G.|last2=Leiva|first2=C.A.|last3=Arranz|first3=J.J.|last4=Buenańo|first4=X.E.|date=2017-06-14|title=Spatial uncertainty of a geoid undulation model in Guayaquil, Ecuador|journal=Open Geosciences|volume=9|issue=1|pages=255–265|doi=10.1515/geo-2017-0021|issn=2391-5447|bibcode=2017OGeo....9...21C|doi-access=free}}</ref>

==Relationship to mass density==
{{further|Gravity anomaly}}
[[File:Gravity, geoid anomaly synthetic cases with local isostasy 2.gif|thumb|[[Gravity anomaly|Gravity]] and Geoid anomalies caused by various crustal and lithospheric thickness changes relative to a reference configuration. All settings are under local [[isostasy|isostatic]] compensation.]]

Variations in the height of the geoidal surface are related to anomalous density distributions within the Earth. Geoid measures thus help understanding the internal structure of the planet. Synthetic calculations show that the geoidal signature of a thickened crust (for example, in [[orogen|orogenic belts]] produced by [[continental collision]]) is positive, opposite to what should be expected if the thickening affects the entire [[lithosphere]]. Mantle convection also changes the shape of the geoid over time.<ref>{{cite journal |last1=Richards |first1=M. A. |first2=B. H. |last2=Hager |year=1984 |title=Geoid anomalies in a dynamic Earth |journal=Journal of Geophysical Research |volume=89 |issue=B7 |pages=5987–6002 |doi=10.1029/JB089iB07p05987|bibcode=1984JGR....89.5987R }}</ref>

[[File:Gravity anomalies on Earth.jpg|thumb|Three-dimensional visualization of [[Gravity anomaly|gravity anomalies]] in [[Gal (unit)|units of Gal.]], using [[pseudo color]] and [[shaded relief]].]]

The surface of the geoid is higher than the [[reference ellipsoid]] wherever there is a positive [[gravity anomaly]] or negative [[disturbing potential]] (mass excess) and lower than the reference ellipsoid wherever there is a negative gravity anomaly or positive disturbing potential (mass deficit).<ref>{{cite book|last=Fowler|first=C.M.R.|author-link=Mary Fowler (geologist)|title=The Solid Earth; An Introduction to Global Geophysics|date=2005|publisher=[[Cambridge University Press]]|location=United Kingdom|isbn=9780521584098|page=214}}</ref>

This relationship can be understood by recalling that gravity potential is defined so that it has negative values and is inversely proportional to distance from the body.
So, while a mass excess will strengthen the gravity acceleration, it will decrease the gravity potential. As a consequence, the geoid's defining equipotential surface will be found displaced away from the mass excess.
Analogously, a mass deficit will weaken the gravity pull but will increase the geopotential at a given distance, causing the geoid to move towards the mass deficit.

The presence of a localized inclusion in the background medium will rotate the gravity acceleration vectors slightly towards or away from a denser or lighter body, respectively, causing a bump or dimple in the equipotential surface.<ref name="Lowrie 1997 p. 50">{{cite book | last=Lowrie | first=W. | title=Fundamentals of Geophysics | publisher=Cambridge University Press | year=1997 | isbn=978-0-521-46728-5 | url=https://books.google.com/books?id=7vR2RJSIGVoC&pg=PA50 | access-date=2022-05-02 | page=50}}</ref>

The largest absolute deviation can be found in the [[Indian Ocean Geoid Low]], 106 meters below the average sea level.<ref name="Raman 2017">{{cite web | last=Raman | first=Spoorthy | title=The missing mass -- what is causing a geoid low in the Indian Ocean? | website=GeoSpace | date=2017-10-16 | url=https://blogs.agu.org/geospace/2017/10/16/missing-mass-causing-geoid-low-indian-ocean/ | access-date=2022-05-02}}</ref>
Another large feature is the North Atlantic Geoid High (or North Atlantic Geoid Swell), caused in part by the weight of ice cover over North America and northern Europe in the [[Late Cenozoic Ice Age]].<ref name="x153">{{cite journal | last1=Carminati | first1=Eugenio | last2=Doglioni | first2=Carlo | title=North Atlantic geoid high, volcanism and glaciations | journal=Geophysical Research Letters | volume=37 | issue=3 | date=2010 | issn=0094-8276 | doi=10.1029/2009GL041663 | page=| bibcode=2010GeoRL..37.3302C }}</ref>

==Temporal change==
{{See also|Geodesy#Temporal change}}
Recent satellite missions, such as the [[Gravity Field and Steady-State Ocean Circulation Explorer]] (GOCE) and 
[[GRACE (satellite)|GRACE]], have enabled the study of time-variable geoid signals. The first products based on GOCE satellite data became available online in June 2010, through the European Space Agency.<ref>{{cite web|url=http://www.esa.int/SPECIALS/GOCE/SEMB1EPK2AG_1.html|title=ESA makes first GOCE dataset available|date=9 June 2010|work=GOCE|publisher=[[European Space Agency]]|access-date=22 December 2016}}</ref><ref>{{cite web|url=http://www.esa.int/SPECIALS/GOCE/SEMY0FOZVAG_0.html|title=GOCE giving new insights into Earth's gravity|date=29 June 2010|work=GOCE|publisher=European Space Agency|access-date=22 December 2016|archive-date=2 July 2010|archive-url=https://web.archive.org/web/20100702013747/http://www.esa.int/SPECIALS/GOCE/SEMY0FOZVAG_0.html|url-status=dead}}</ref> ESA launched the satellite in March 2009 on a mission to map Earth's gravity with unprecedented accuracy and spatial resolution. On 31 March 2011, a new geoid model was unveiled at the Fourth International GOCE User Workshop hosted at the [[Technical University of Munich]], Germany.<ref>{{cite web|url=http://www.esa.int/esaCP/SEM1AK6UPLG_index_0.html|title=Earth's gravity revealed in unprecedented detail|date=31 March 2011|work=GOCE|publisher=European Space Agency|access-date=22 December 2016}}</ref> Studies using the time-variable geoid computed from GRACE data have provided information on global hydrologic cycles,<ref>{{cite journal|last1=Schmidt|first1=R.|last2=Schwintzer|first2=P.|last3=Flechtner|first3=F.|last4=Reigber|first4=C.|last5=Guntner|first5=A.|last6=Doll|first6=P.|last7=Ramillien|first7=G.|last8=Cazenave|first8=A.|author8-link=Anny Cazenave|last9=Petrovic|first9=S.| display-authors = 8|title=GRACE observations of changes in continental water storage|journal=Global and Planetary Change|volume=50|issue=1–2|pages=112–126|date=2006|doi=10.1016/j.gloplacha.2004.11.018|bibcode = 2006GPC....50..112S }}</ref> mass balances of [[ice sheet]]s,<ref>{{cite journal|last1=Ramillien|first1=G.|last2=Lombard|first2=A.|last3=Cazenave|first3=A.|author3-link=Anny Cazenave|last4=Ivins|first4=E.|last5=Llubes|first5=M.|last6=Remy|first6=F.|last7=Biancale|first7=R.|title=Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE|journal=Global and Planetary Change|volume=53|pages=198|date=2006|doi=10.1016/j.gloplacha.2006.06.003|bibcode = 2006GPC....53..198R|issue=3 }}</ref> and [[postglacial rebound]].<ref>{{cite journal|last1=Vanderwal|first1=W.|last2=Wu|first2=P.|last3=Sideris|first3=M.|last4=Shum|first4=C.|title=Use of GRACE determined secular gravity rates for glacial isostatic adjustment studies in North-America|journal=Journal of Geodynamics|volume=46|pages=144|date=2008|doi=10.1016/j.jog.2008.03.007|bibcode = 2008JGeo...46..144V|issue=3–5 }}</ref> From postglacial rebound measurements, time-variable GRACE data can be used to deduce the [[viscosity]] of [[Earth's mantle]].<ref>{{cite journal|last1=Paulson|first1=Archie|last2=Zhong|first2=Shijie|last3=Wahr|first3=John|author-link3=John M. Wahr|title=Inference of mantle viscosity from GRACE and relative sea level data|journal=[[Geophysical Journal International]]|volume=171|pages=497|date=2007|doi=10.1111/j.1365-246X.2007.03556.x|bibcode = 2007GeoJI.171..497P|issue=2 |doi-access=free}}</ref>

==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>

==See also==
*[[Deflection of the vertical]]
*[[Geodetic datum]]
*[[Geopotential]]
*[[International Terrestrial Reference Frame]]
*[[Physical geodesy]]
*[[Planetary geoid]]
**[[Areoid]] (Mars's geoid)
**[[Selenoid]] (Moon's geoid)

==References==
{{Reflist}}

==Further reading==
{{Refbegin}}
* {{cite journal |first1=Xiong |last1=Li |first2=Hans-Jürgen |last2=Götze |url=https://geored2.sgc.gov.co/Articulos%20y%20documentacion/Li_G_Tut.pdf |doi=10.1190/1.1487109 |journal=Geophysics |date=November 2001 |volume=66 |number=6 |pages=1660–1668 |title=Ellipsoid, geoid, gravity, geodesy, and geophysics|bibcode=2001Geop...66.1660L }}
* {{cite journal |first=H. |last=Moritz |date=March 2011 |title=A contemporary perspective of geoid structure |journal=Journal of Geodetic Science |volume=1 |number=1 |pages=82–87 |doi=10.2478/v10156-010-0010-7 |bibcode = 2011JGeoS...1...82M |doi-access=free }}
* {{cite book |title=Geodesy for the Layman |chapter=Physical Geodesy |chapter-url=https://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003C.HTM |publisher=NOAA |date=1984}}
{{Refend}}

==External links==
{{Wiktionary}}
*[https://earth-info.nga.mil/ NGA webpage on Earth Gravitational Models]
*[https://cddis.nasa.gov/926/egm96/egm96.html NASA webpage on EGM96]
*[https://www.ngs.noaa.gov/GEOID/ NOAA webpage on Geoid Models]
*[https://icgem.gfz-potsdam.de/home International Centre for Global Earth Models (ICGEM)]
*[https://www.isgeoid.polimi.it/ International Service for the Geoid (ISG)]

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[[Category:Gravimetry]]
[[Category:Geodesy]]
[[Category:Vertical datums]]
[[Category:Vertical position]]