geoid

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Map of the undulations of the geoid, in meters (based on the EGM96 gravity model and the WGS84 reference ellipsoid).[1]


The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were to be extended through the continents (such as with very narrow canals). According to C.F. Gauss, who first described it, it is the "mathematical figure of the Earth," a smooth but highly irregular surface that corresponds not to the actual surface of the Earth's crust, but to a surface which can only be known through extensive gravitational measurements and calculations. Despite being an important concept for almost two hundred years in the history of geodesy and geophysics, it has only been defined to high precision in recent decades, for instance by works of P. Vaníček and others. It is often described as the true physical figure of the Earth, in contrast to the idealized geometrical figure of a reference ellipsoid.

Description

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1. Ocean
2. Ellipsoid
3. Local plumb
4. Continent
5. Geoid
The geoid surface is irregular, unlike the reference ellipsoids often used to approximate the shape of the physical Earth, but considerably smoother than Earth's physical surface. While the latter has excursions of +8,000 m (Mount Everest) and −11,000 m (Mariana Trench), the total variation in the geoid is less than 200 m (-106 to +85 m[2](compared to a perfect mathematical ellipsoid).

Sea level, if undisturbed by tides and weather, would assume a surface equal to the geoid. If the continental land masses were criss-crossed by a series of tunnels or narrow canals, the sea level in these canals would also coincide with the geoid. In reality the geoid does not have a physical meaning under the continents, but geodesists are able to derive the heights of continental points above this imaginary, yet physically defined, surface by a technique called spirit leveling.

Being an equipotential surface, the geoid is by definition a surface to which the force of gravity is everywhere perpendicular. This means that when travelling by ship, one does not notice the undulations of the geoid; the local vertical is always perpendicular to the geoid and the local horizon tangential component to it. Likewise, spirit levels will always be parallel to the geoid.

Note that 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. This is because GPS satellites, orbiting about the center of gravity of the Earth, can only measure heights relative to a geocentric reference ellipsoid. To obtain one's geoidal height, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a tidal measurement station, as in traditional land surveying, will always be geoidal height.

Spherical harmonics representation

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Three-dimensional visualization of geoid undulations, using units of gravity.


Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM96 (Earth Gravity Model 1996)[3], determined in an international collaborative project led by NIMA. The mathematical description of the non-rotating part of the potential function in this model is



where and are geocentric (spherical) latitude and longitude respectively, are the fully normalized Legendre functions of degree and order , and and are the coefficients of the model. Note that the above equation describes the Earth's gravitational potential , not the geoid itself, at location the co-ordinate being the geocentric radius, i.e, distance from the Earth's centre. The geoid is a particular[4] equipotential surface, and is somewhat involved to compute. The gradient of this potential also provides a model of the gravitational acceleration. EGM96 contains a full set of coefficients to degree and order 360, describing details in the global geoid as small as 55 km (or 110 km, depending on your definition of resolution). One can show there are



different coefficients (counting both and , and using the EGM96 value of ). For many applications the complete series is unnecessarily complex and is truncated after a few (perhaps several dozen) terms.

New even higher resolution models are currently under development. For example, many of the authors of EGM96 are working on an updated model[5] that should incorporate much of the new satellite gravity data (see, e.g., GRACE), and should support up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients).

Precise geoid

The 1990-ies had seen important discoveries in theory of geoid computation. The Precise Geoid Solution [6] by Vaníček and co-workers improved on the Stokesian approach to geoid computation. Their solution enables millimetre-to-centimetre accuracy in geoid computation, an-order-of-magnitude improvement from previous classical solutions [7] [8] [9].

References

1. ^ data from [1]
2. ^ [2] visited 2007-10-11
3. ^ NIMA Technical Report TR8350.2, Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems, Third Edition, 4 July 1997. [Note that confusingly, despite the title, versions after 1991 actually define EGM96, rather than the older WGS84 standard, and also that, despite the date on the cover page, this report was actually updated last in June 23 2004. Available electronically at: [3]]
4. ^ There is no such thing as "The" EGM96 geoid
5. ^ Pavlis, N.K., S.A. Holmes. S. Kenyon, D. Schmit, R. Trimmer, "Gravitational potential expansion to degree 2160", IAG International Symposium, gravity, geoid and Space Mission GGSM2004, Porto, Portugal, 2004.
6. ^ UNB Precise Geoid Determination Package, page accessed 02 October 2007
7. ^ Vaníček, P., Kleusberg, A. The Canadian geoid-Stokesian approach, Pages 86-98, Manuscripta Geodaetica, Volume 12, Number 2 (1987)
8. ^ Vaníček P., Martinec Z. Compilation of a precise regional geoid (pdf), Pages 119-128, Manuscripta Geodaetica, Volume 19 (1994)
9. ^ Vaníček et al. Compilation of a precise regional geoid (pdf), pp.45, Report for Geodetic Survey Division - DSS Contract: #23244-1-4405/01-SS, Ottawa (1995)

External links

See also

Equipotential surfaces are surfaces of constant scalar potential. They are used to visualize an (n)-dimensional scalar potential function in (n-1) dimensional space. The gradient of the potential, denoting the direction of greatest increase, is perpendicular to the surface.
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Johann Carl Friedrich Gauss

Carl Friedrich Gauss, painted by Christian Albrecht Jensen
Born 30 March 1777(1777--)
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Gravitation is a natural phenomenon by which all objects with mass attract each other. In everyday life, gravitation is most familiar as the agency that endows objects with weight.
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Geodesy (IPA North American English /dʒiˈɑdɪsi/; British, Australian English etc. /dʒɪˈɒdəsi/), also called geodetics
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Geophysics, a branch of Earth sciences, is the study of the Earth by quantitative physical methods, especially by seismic, electromagnetic, and radioactivity methods. The theories and techniques of geophysics are employed extensively in the planetary sciences in general.
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The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas.
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In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network
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In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network
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Mariana Trench (or Marianas Trench) is the deepest known submarine trench, with a maximum depth of about 11 km (6.8 mi), and the deepest location on the surface of the Earth's crust.
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Geodesy (IPA North American English /dʒiˈɑdɪsi/; British, Australian English etc. /dʒɪˈɒdəsi/), also called geodetics
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Spirit leveling is a technique for determining differences in height between points on the Earth's surface. It works by using a spirit level, an instrument consisting of a telescope and a tube level like that used by carpenters, rigidly connected.
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Equipotential surfaces are surfaces of constant scalar potential. They are used to visualize an (n)-dimensional scalar potential function in (n-1) dimensional space. The gradient of the potential, denoting the direction of greatest increase, is perpendicular to the surface.
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tangential component of the vector, and another one perpendicular to the surface, called the normal component of the vector.

More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of
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Global Positioning System (GPS) is the only fully functional Global Navigation Satellite System (GNSS). Utilizing a constellation of at least 24 medium Earth orbit satellites that transmit precise microwave signals, the system enables a GPS receiver to determine its
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satellite is an object which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon.
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Spherical Harmonic is a fantasy novel from the Saga of the Skolian empire series of books by Catherine Asaro which tells the story of Pharaoh Dyhianna Selei (Dehya), ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord.
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EGM96 (or Earth Geopotential Model 1996) is a geopotential model of the Earth consisting of spherical harmonic coefficients complete to degree and order 360. It is a composite solution, consisting of: (1) a combination solution to degree and order 70; (2) a block diagonal solution
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The National Geospatial-Intelligence Agency (NGA) is an agency of the United States Government with the primary mission of collection, analysis, and distribution of geospatial intelligence (GEOINT) in support of national security.
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associated Legendre functions are the canonical solutions of the general Legendre equation



or



where the indices and m
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potential may refer to the scalar potential or to the vector potential. In either case, it is a field defined in space, from which many important physical properties may be derived.
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Equipotential or isopotential in mathematics and physics (especially electronics) refers to a region in space where every point in it is at the same potential. This usually refers to a scalar potential, although it can also be applied to vector potentials.
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The goal of the Gravity Recovery And Climate Experiment (GRACE) space mission is to obtain accurate global and high-resolution determination of both the static and the time-variable components of the Earth's gravity field.
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accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value. Accuracy is closely related to precision, also called reproducibility or repeatability, the degree to which further measurements or calculations show the same or similar
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Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking.
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An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. The ratio most commonly used is 10.
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Physical geodesy is the study of the physical properties of the gravity field of the Earth, the geopotential, with a view to their application in geodesy.

Measurement procedure


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Geodesy (IPA North American English /dʒiˈɑdɪsi/; British, Australian English etc. /dʒɪˈɒdəsi/), also called geodetics
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