Geoid Undulation To Scale
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The geoid ( ) is the shape that the
ocean The ocean is the body of salt water that covers approximately 70.8% of Earth. The ocean is conventionally divided into large bodies of water, which are also referred to as ''oceans'' (the Pacific, Atlantic, Indian Ocean, Indian, Southern Ocean ...
surface would take under the influence of the
gravity of Earth The gravity of Earth, denoted by , is the net force, net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a Eucl ...
, including
gravitational attraction In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force be ...
and
Earth's rotation Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in progra ...
, if other influences such as winds and
tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...
s were absent. This surface is extended through the
continent A continent is any of several large geographical regions. Continents are generally identified by convention (norm), convention rather than any strict criteria. A continent could be a single large landmass, a part of a very large landmass, as ...
s (such as might be approximated with very narrow hypothetical
canal Canals or artificial waterways are waterways or engineered channels built for drainage management (e.g. flood control and irrigation) or for conveyancing water transport vehicles (e.g. water taxi). They carry free, calm surface ...
s). According to
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
, who first described it, it is the "mathematical
figure of the Earth In geodesy, the figure of the Earth is the size and shape used to model planet Earth. The kind of figure depends on application, including the precision needed for the model. A spherical Earth is a well-known historical approximation that is ...
", a smooth but irregular
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
whose shape results from the uneven distribution of mass within and on the surface of Earth. 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 Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
and
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
, it has been defined to high precision only since advances in
satellite geodesy Satellite geodesy is geodesy by means of artificial satellites—the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques ...
in the late 20th century. The geoid is often expressed as a geoid undulation or geoidal height above a given
reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximation ...
, which is a slightly flattened sphere whose
equatorial bulge An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On ...
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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
for various vertical coordinates, such as ''
orthometric height The orthometric height (symbol ''H'') is the vertical distance along the plumb line from a point of interest to a reference surface known as the ''geoid'', the vertical datum that approximates mean sea level. Orthometric height is one of the sci ...
s'', ''
geopotential height Geopotential height, also known as geopotential altitude or geopotential elevation, is a vertical coordinate (with dimension of length) representing the work involved in lifting one unit of mass over one unit of length through a hypothetical spac ...
s'', and ''
dynamic height Dynamic height (symbol H^\text or H^\text) is a way of specifying the vertical position of a point above a vertical datum; it is an alternative for orthometric height or normal height. It can be computed (in SI units of metre) by dividing the loca ...
s'' (see Geodesy#Heights). All points on a geoid surface have the same
geopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vect ...
(the sum of
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy an object with mass has due to the gravitational potential of its position in a gravitational field. Mathematically, it is the minimum Work (physics), mechanical work t ...
and
centrifugal Centrifugal (a key concept in rotating systems) may refer to: *Centrifugal casting (industrial), Centrifugal casting (silversmithing), and Spin casting (centrifugal rubber mold casting), forms of centrifigual casting *Centrifugal clutch *Centrifug ...
potential energy). At this surface, apart from temporary tidal fluctuations, the
force of gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
acts everywhere perpendicular to the geoid, meaning that plumb lines point perpendicular and
bubble level A spirit level, bubble level, or simply a level, is an instrument designed to indicate whether a surface is horizontal (level) or vertical ( plumb). Two basic designs exist: ''tubular'' (or ''linear'') and '' bull's eye'' (or ''circular'' ...
s are parallel to the geoid. Being an
equigeopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vector ...
means the geoid corresponds to the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
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.''Geodesy: The Concepts.'' Petr Vanicek and E.J. Krakiwsky. Amsterdam: Elsevier. 1982 (first ed.): , . 1986 (third ed.): , . .


Description

The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as an
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
), 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 Mount Everest (), known locally as Sagarmatha in Nepal and Qomolangma in Tibet, is Earth's highest mountain above sea level. It lies in the Mahalangur Himal sub-range of the Himalayas and marks part of the China–Nepal border at it ...
) and −11,000 m (
Marianas Trench The Mariana Trench is an oceanic trench located in the western Pacific Ocean, about east of the Mariana Islands; it is the deepest oceanic trench on Earth. It is crescent-shaped and measures about in length and in width. The maximum known ...
), the geoid's deviation from an ellipsoid ranges from +85 m (Iceland) to −106 m (southern India), less than 200 m total. 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 A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
is called
ocean surface topography Ocean surface topography or sea surface topography, also called ocean dynamic topography, are highs and lows on the ocean surface, similar to the hills and valleys of Earth's land surface depicted on a topographic map. These variations are exp ...
. 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 by spirit leveling. Being an
equipotential surface In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can als ...
, 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 of the geoid The geoid ( ) 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 tides were absent. This surface is exten ...
; neglecting tides, the local vertical (plumb line) is always perpendicular to the geoid and the local horizon
tangential In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on ...
to it. Likewise, spirit levels will always be parallel to the geoid.


Simplified example

Earth's gravitational field is not uniform. An
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
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 Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth s ...
compositions in the
Earth's crust Earth's crust is its thick outer shell of rock, referring to less than one percent of the planet's radius and volume. It is the top component of the lithosphere, a solidified division of Earth's layers that includes the crust and the upper ...
, 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

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. N=h-H The undulation is not standardized, as different countries use different mean sea levels as reference, but most commonly refers to the
EGM96 The Earth Gravitational Models (EGM) are a series of geopotential models of the Earth published by the National Geospatial-Intelligence Agency (NGA). They are used as the geoid reference in the World Geodetic System. The NGA provides the mo ...
geoid. In maps and common use, the height over the mean sea level (such as
orthometric height The orthometric height (symbol ''H'') is the vertical distance along the plumb line from a point of interest to a reference surface known as the ''geoid'', the vertical datum that approximates mean sea level. Orthometric height is one of the sci ...
, ''H'') is used to indicate the height of elevations while the
ellipsoidal height Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as g ...
, ''h'', results from the
GPS The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31. It is one of the global navigation satellite systems (GNSS) that provide geol ...
system and similar
GNSS A satellite navigation or satnav system is a system that uses satellites to provide autonomous geopositioning. A satellite navigation system with global coverage is termed global navigation satellite system (GNSS). , four global systems are op ...
: H=h-N (An analogous relationship exists between
normal height Normal heights (symbol H^* or H^N; SI unit metre, m) is a type of height above sea level introduced by the Soviet scientist Mikhail Molodenskii. The normal height of a point is defined as the quotient of a point's geopotential number ''C'' (i.e. it ...
s and the '' quasigeoid'', which disregards local density variations.) In practice, many handheld GPS receivers
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a ...
''N'' in a pre-computed ''geoid map'' (a
lookup table In computer science, a lookup table (LUT) is an array data structure, array that replaces runtime (program lifecycle phase), runtime computation of a mathematical function (mathematics), function with a simpler array indexing operation, in a proc ...
). So a
GPS receiver A satellite navigation (satnav) device or GPS device is a device that uses satellites of the Global Positioning System (GPS) or similar global navigation satellite systems (GNSS). A satnav device can determine the user's geographic coordinat ...
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 A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...
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 The orthometric height (symbol ''H'') is the vertical distance along the plumb line from a point of interest to a reference surface known as the ''geoid'', the vertical datum that approximates mean sea level. Orthometric height is one of the sci ...
, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a
tide gauge A tide gauge is a device for measuring the change in sea level relative to a vertical datum. It is also known as a mareograph, marigraph, and sea-level recorder. When applied to freshwater continental water body, water bodies, the instrument may ...
, 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 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also describ ...
(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 Atmospheric pressure, also known as air pressure or barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1,013. ...
(meteorological effects), local
sea surface topography Ocean surface topography or sea surface topography, also called ocean dynamic topography, are highs and lows on the ocean surface, similar to the hills and valleys of Earth's land surface depicted on a topographic map. These variations are exp ...
, and measurement uncertainties.


Determination

The undulation of the geoid ''N'' is closely related to the disturbing potential ''T'' according to Bruns' formula (named after Heinrich Bruns): : N=T/\gamma\,, where \gamma is the force of normal gravity, computed from the normal field potential U. Another way of determining ''N'' is using values of ''
gravity anomaly The gravity anomaly at a location on the Earth's surface is the difference between the observed value of gravity and the value predicted by a theoretical model. If the Earth were an ideal oblate spheroid of uniform density, then the gravity meas ...
'' \Delta g, differences between true and normal reference gravity, as per (or Stokes' integral), published in 1849 by
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish mathematician and physicist. Born in County Sligo, Ireland, Stokes spent his entire career at the University of Cambridge, where he served as the Lucasi ...
: : N=\frac\iint_\sigma \Delta g \,S(\psi)\, d\sigma. The
integral kernel In mathematics, an integral transform is a type of transform (mathematics), transform that maps a function (mathematics), function from its original function space into another function space via integral, integration, where some of the propert ...
''S'', called ''Stokes function'', was derived by Stokes in closed analytical form. Note that determining N anywhere on Earth by this formula requires \Delta g 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 The International Association of Geodesy (IAG) is a constituent association of the International Union of Geodesy and Geophysics focusing on the science which measures and describes the Figure of the Earth, Earth's shape, its rotation and gravity ...
(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 The Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) was the first of European Space Agency, ESA's Living Planet Programme satellite, heavy satellites intended to map in unprecedented detail the Earth's gravity field. The spacecr ...
and
GRACE Grace may refer to: Places United States * Grace, Idaho, a city * Grace (CTA station), Chicago Transit Authority's Howard Line, Illinois * Little Goose Creek (Kentucky), location of Grace post office * Grace, Carroll County, Missouri, an uni ...
. 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. The precise geoid solution by Petr Vaníček and co-workers improved on the Stokesian approach to geoid computation. Their solution enables millimetre-to-centimetre
accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value''. ''Precision'' is how close the measurements are to each other. The ...
in geoid
computation A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, hist ...
, an order-of-magnitude improvement from previous classical solutions. Geoid undulations display uncertainties which can be estimated by using several methods, e.g.,
least-squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
collocation (LSC),
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
,
artificial neural networks In machine learning, a neural network (also artificial neural network or neural net, abbreviated ANN or NN) is a computational model inspired by the structure and functions of biological neural networks. A neural network consists of connected ...
,
radial basis function In mathematics a radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), o ...
s (RBF), and
geostatistical Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petro ...
techniques. Geostatistical approach has been defined as the most-improved technique in prediction of geoid undulation.


Relationship to mass density

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
orogenic belts An orogenic belt, orogen, or mobile belt, is a zone of Earth's crust affected by orogeny. An orogenic belt develops when a continental plate crumples and is uplifted to form one or more mountain ranges; this involves a series of geological proc ...
produced by
continental collision In geology, continental collision is a phenomenon of plate tectonics that occurs at Convergent boundary, convergent boundaries. Continental collision is a variation on the fundamental process of subduction, whereby the subduction zone is destroy ...
) is positive, opposite to what should be expected if the thickening affects the entire
lithosphere A lithosphere () is the rigid, outermost rocky shell of a terrestrial planet or natural satellite. On Earth, it is composed of the crust and the lithospheric mantle, the topmost portion of the upper mantle that behaves elastically on time ...
. Mantle convection also changes the shape of the geoid over time. The surface of the geoid is higher than the
reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximation ...
wherever there is a positive
gravity anomaly The gravity anomaly at a location on the Earth's surface is the difference between the observed value of gravity and the value predicted by a theoretical model. If the Earth were an ideal oblate spheroid of uniform density, then the gravity meas ...
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). 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. The largest absolute deviation can be found in the Indian Ocean Geoid Low, 106 meters below the average sea level. 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 The Late Cenozoic Ice Age,National Academy of Sciences - The National Academies Press - Continental Glaciation through Geologic Times https://www.nap.edu/read/11798/chapter/8#80 or Antarctic Glaciation, began 34 million years ago at the Eocene� ...
.


Temporal change

Recent satellite missions, such as the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and
GRACE Grace may refer to: Places United States * Grace, Idaho, a city * Grace (CTA station), Chicago Transit Authority's Howard Line, Illinois * Little Goose Creek (Kentucky), location of Grace post office * Grace, Carroll County, Missouri, an uni ...
, 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. 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 The Technical University of Munich (TUM or TU Munich; ) is a public research university in Munich, Bavaria, Germany. It specializes in engineering, technology, medicine, and applied and natural sciences. Established in 1868 by King Ludwig II ...
, Germany. Studies using the time-variable geoid computed from GRACE data have provided information on global hydrologic cycles, mass balances of
ice sheet In glaciology, an ice sheet, also known as a continental glacier, is a mass of glacier, glacial ice that covers surrounding terrain and is greater than . The only current ice sheets are the Antarctic ice sheet and the Greenland ice sheet. Ice s ...
s, and
postglacial rebound Post-glacial rebound (also called isostatic rebound or crustal rebound) is the rise of land masses after the removal of the huge weight of ice sheets during the last glacial period, which had caused isostatic depression. Post-glacial rebound ...
. From postglacial rebound measurements, time-variable GRACE data can be used to deduce the
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
of
Earth's mantle Earth's mantle is a layer of silicate mineral, silicate rock between the Earth's crust, crust and the Earth's outer core, outer core. It has a mass of and makes up 67% of the mass of Earth. It has a thickness of making up about 46% of Earth's ...
.


Spherical harmonics representation

Spherical harmonic In mathematics and Outline of physical science, physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The tabl ...
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 The National Geospatial-Intelligence Agency (NGA) is a combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) to support national se ...
, or NGA). The mathematical description of the non-rotating part of the potential function in this model is: V=\frac\left(1+\left(\frac\right)^n \overline_(\sin\phi)\left overline_\cos m\lambda+\overline_\sin m\lambda\rightright), where \phi\ and \lambda\ are ''geocentric'' (spherical) latitude and longitude respectively, \overline_ are the fully normalized
associated Legendre polynomials In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently ...
of degree n\ and order m\ , and \overline_ and \overline_ are the numerical coefficients of the model based on measured data. The above equation describes the Earth's gravitational
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
V, not the geoid itself, at location \phi,\;\lambda,\;r,\ the co-ordinate r\ being the ''geocentric radius'', i.e., distance from the Earth's centre. The geoid is a particular
equipotential In mathematics and physics, an equipotential or isopotential refers to a region (mathematics), region in space where every point is at the same Electric potential, potential. This usually refers to a scalar potential (in that case it is a level ...
surface, 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., n_\text = 360), describing details in the global geoid as small as 55 km (or 110 km, depending on the definition of resolution). The number of coefficients, \overline_ and \overline_, can be determined by first observing in the equation for V that for a specific value of n there are two coefficients for every value of m except for m=0. There is only one coefficient when m=0 since \sin (0\lambda) = 0. There are thus (2n+1) coefficients for every value of n. Using these facts and the formula, \sum_^I = \fracL(L + 1), it follows that the total number of coefficients is given by \sum_^(2n+1) = n_\text(n_\text+1) + n_\text - 3 = 130317 using the EGM96 value of n_\text = 360. 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 The Gravity Recovery and Climate Experiment (GRACE) was a joint mission of NASA and the German Aerospace Center (DLR). Twin satellites took detailed measurements of Earth's gravity field anomalies from its launch in March 2002 to the end of it ...
), and supports up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients), with additional coefficients extending to degree 2190 and order 2159. 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.


See also

* Deflection of the vertical *
Geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame, or terrestrial reference frame) is a global datum reference or reference frame for unambiguously representing the positi ...
*
Geopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vect ...
*
International Terrestrial Reference Frame The International Terrestrial Reference System (ITRS) describes procedures for creating reference frames suitable for use with measurements on or near the Earth's surface. This is done in much the same way that a physical standard might be descr ...
*
Physical geodesy Physical geodesy is the study of the physical properties of Earth's gravity and its potential field (the geopotential), with a view to their application in geodesy. Measurement procedure Traditional geodetic instruments such as theodolites rely ...
* Planetary geoid ** Areoid (Mars's geoid) **
Selenoid Selenography is the study of the surface and physical features of the Moon (also known as geography of the Moon, or selenodesy). Like geography and areography, selenography is a subdiscipline within the field of planetary science. Historically, ...
(Moon's geoid)


References


Further reading

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External links


NGA webpage on Earth Gravitational ModelsNOAA webpage on Geoid ModelsInternational Centre for Global Earth Models (ICGEM)International Service for the Geoid (ISG)
{{Authority control Gravimetry Geodesy Vertical datums Vertical position