Thegeoid (/ˈdʒiː.ɔɪd/JEE-oyd) is the shape that theocean surface would take under the influence of thegravity of Earth, includinggravitational attraction andEarth's rotation, if other influences such as winds andtides were absent. This surface is extended through thecontinents (such as might be approximated with very narrow hypotheticalcanals). According toCarl Friedrich Gauss, who first described it, it is the "mathematicalfigure of the Earth", a smooth but irregularsurface whose shape results from the uneven distribution of mass within and on the surface of Earth.[2] It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history ofgeodesy andgeophysics, it has been defined to high precision only since advances insatellite geodesy in the late 20th century.
All points on a geoid surface have the samegeopotential (the sum ofgravitational potential energy andcentrifugal potential energy). At this surface, apart from temporary tidal fluctuations, theforce of gravity acts everywhere perpendicular to the geoid, meaning thatplumb lines point perpendicular andbubble levels are parallel to the geoid. Being anequigeopotential means the geoid corresponds to thefree 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 (thevertical derivative of geopotential) is thus non-uniform over the geoid.[3]
The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as anellipsoid), 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 m (Iceland) to −106 m (southern India), less than 200 m total.[4]
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 andmean sea level is calledocean 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.Geodesists are able to derive the heights of continental points above the geoid byspirit leveling.
Being anequipotential 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 theundulation of the geoid; neglecting tides, the local vertical (plumb line) is always perpendicular to the geoid and the local horizontangential to it. Likewise, spirit levels will always be parallel to the geoid.
Earth's gravitational field is not uniform. Anoblate 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 differentgeological compositions in theEarth'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.
Meridional profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also:Earth's pear shape.
Thegeoid undulation (also known asgeoid height orgeoid anomaly),N, is the height of the geoid relative to a givenellipsoid of reference.The undulation is not standardized, as different countries use different mean sea levels as reference, but most commonly refers to theEGM96 geoid.
In maps and common use, the height over the mean sea level (such asorthometric height,H) is used to indicate the height of elevations while theellipsoidal height,h, results from theGPS system and similarGNSS:(An analogous relationship exists betweennormal heights and thequasigeoid, which disregards local density variations.)In practice, many handheld GPS receiversinterpolateN in a pre-computedgeoid map (alookup table).[5]
So aGPS 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 GPSsatellites, orbiting about the center of gravity of the Earth, can measure heights only relative to a geocentric reference ellipsoid. To obtain one'sorthometric height, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from atide 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 theWorld 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), localsea surface topography, and measurement uncertainties.
Equatorial profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also:triaxial Earth.
where is the force ofnormal gravity, computed from the normal field potential.
Another way of determiningN is using values ofgravity anomaly, differences between true and normal reference gravity, as perStokes formula (orStokes' integral), published in 1849 byGeorge Gabriel Stokes:
Theintegral kernelS, calledStokes function, was derived by Stokes in closed analytical form.[6]Note that determining anywhere on Earth by this formula requires to be knowneverywhere 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 theInternational Association of Geodesy (IAG), e.g., through the International Gravity Bureau (BGI, Bureau Gravimétrique International).
Another approach for geoid determination is tocombine 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 asGOCE andGRACE. 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.[7][8]The precise geoid solution byPetr Vaníček and co-workers improved on theStokesian approach to geoid computation.[9] Their solution enables millimetre-to-centimetreaccuracy in geoidcomputation, anorder-of-magnitude improvement from previous classical solutions.[10][11][12][13]
Gravity and Geoid anomalies caused by various crustal and lithospheric thickness changes relative to a reference configuration. All settings are under localisostatic 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, inorogenic belts produced bycontinental collision) is positive, opposite to what should be expected if the thickening affects the entirelithosphere. Mantle convection also changes the shape of the geoid over time.[15]
The surface of the geoid is higher than thereference ellipsoid wherever there is a positivegravity anomaly or negativedisturbing potential (mass excess) and lower than the reference ellipsoid wherever there is a negative gravity anomaly or positive disturbing potential (mass deficit).[16]
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.[17]
The largest absolute deviation can be found in theIndian Ocean Geoid Low, 106 meters below the average sea level.[18]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 theLate Cenozoic Ice Age.[19]
Recent satellite missions, such as theGravity Field and Steady-State Ocean Circulation Explorer (GOCE) andGRACE, 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.[20][21] 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 theTechnical University of Munich, Germany.[22] Studies using the time-variable geoid computed from GRACE data have provided information on global hydrologic cycles,[23] mass balances ofice sheets,[24] andpostglacial rebound.[25] From postglacial rebound measurements, time-variable GRACE data can be used to deduce theviscosity ofEarth's mantle.[26]
Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients isEGM2020 (Earth Gravitational Model 2020), determined in an international collaborative project led by the National Imagery and Mapping Agency (now theNational Geospatial-Intelligence Agency, or NGA). The mathematical description of the non-rotating part of the potential function in this model is:[27]
where and aregeocentric (spherical) latitude and longitude respectively, are the fully normalizedassociated Legendre polynomials of degree and order, and and are the numerical coefficients of the model based on measured data. The above equation describes the Earth's gravitationalpotential, not the geoid itself, at location the co-ordinate being thegeocentric radius, i.e., distance from the Earth's centre. The geoid is a particularequipotential surface,[27] 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.,), describing details in the global geoid as small as 55 km (or 110 km, depending on the definition of resolution). The number of coefficients, and, can be determined by first observing in the equation for that for a specific value of there are two coefficients for every value of except for. There is only one coefficient when since. There are thus coefficients for every value of. Using these facts and the formula,, it follows that the total number of coefficients is given by
using the EGM96 value of.
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., theGravity Recovery and Climate Experiment), and supports up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients),[28] with additional coefficients extending to degree 2190 and order 2159.[29] EGM2020 is the international follow-up that was originally scheduled for 2020 (still unreleased in 2025), containing the same number of harmonics generated with better data.[30]
^Vaníček, P.; Kleusberg, A.; Martinec, Z.; Sun, W.; Ong, P.; Najafi, M.; Vajda, P.; Harrie, L.; Tomasek, P.; ter Horst, B.Compilation of a Precise Regional Geoid(PDF) (Report). Department of Geodesy and Geomatics Engineering, University of New Brunswick. 184. Retrieved22 December 2016.
^Schmidt, R.; Schwintzer, P.; Flechtner, F.; Reigber, C.; Guntner, A.; Doll, P.; Ramillien, G.;Cazenave, A.; et al. (2006). "GRACE observations of changes in continental water storage".Global and Planetary Change.50 (1–2):112–126.Bibcode:2006GPC....50..112S.doi:10.1016/j.gloplacha.2004.11.018.
^Ramillien, G.; Lombard, A.;Cazenave, A.; Ivins, E.; Llubes, M.; Remy, F.; Biancale, R. (2006). "Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE".Global and Planetary Change.53 (3): 198.Bibcode:2006GPC....53..198R.doi:10.1016/j.gloplacha.2006.06.003.
^Vanderwal, W.; Wu, P.; Sideris, M.; Shum, C. (2008). "Use of GRACE determined secular gravity rates for glacial isostatic adjustment studies in North-America".Journal of Geodynamics.46 (3–5): 144.Bibcode:2008JGeo...46..144V.doi:10.1016/j.jog.2008.03.007.
^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.
