Map of the undulations of the geoid, in meters (based on the EGM96 gravity model and the WGS84 reference ellipsoid). EGM96 ( Earth Gravitational Model 1996) is a Geopotential model of the Earth consisting of Spherical harmonic coefficients complete to degree and order 360 The World Geodetic System defines a reference frame for the earth for use in Geodesy and Navigation. [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). Equipotential surfaces are Surfaces of constant Scalar potential. 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. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Gravitation is a natural Phenomenon by which objects with Mass attract one another 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. Geodesy (dʒiːˈɒdɪsi also called geodetics, a branch of Earth sciences, is the scientific discipline that deals Geophysics, a major discipline of Earth sciences, is the study of the Earth by quantitative physical methods especially by seismic, electromagnetic Petr Vaníček (born 1935 in Sušice, Czechoslovakia, today in Czech Republic) is a Czech It is often described as the true physical figure of the Earth, in contrast to the idealized geometrical figure of a reference ellipsoid. 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 In Geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the Geoid, the truer Figure of the Earth, or other planetary body

## Description

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. In Geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the Geoid, the truer Figure of the Earth, or other planetary body 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). Mount Everest, also called Sagarmatha (सगरमाथा meaning Head of the Sky) or Chomolungma, Qomolangma or Zhumulangma (in The Mariana Trench (or Mariana's Trench) is the deepest part of the world's Oceans and the deepest location on the surface of the Earth 's

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. Geodesy (dʒiːˈɒdɪsi also called geodetics, a branch of Earth sciences, is the scientific discipline that deals Spirit leveling is a technique for determining differences in height between points on the Earth's surface

Being an equipotential surface, the geoid is by definition a surface to which the force of gravity is everywhere perpendicular. Equipotential surfaces are Surfaces of constant Scalar potential. 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. In Mathematics, given a vector at a point on a Surface, that vector can be decomposed uniquely as a sum of two vectors one Tangent to the surface called Likewise, spirit levels will always be parallel to the geoid.

## Spherical harmonics representation

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

Spherical harmonics are often used to approximate the shape of the geoid. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of 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. EGM96 ( Earth Gravitational Model 1996) is a Geopotential model of the Earth consisting of Spherical harmonic coefficients complete to degree and order 360 The National Geospatial-Intelligence Agency ( NGA) is an agency of the United States Government with the primary mission of collection analysis and The mathematical description of the non-rotating part of the potential function in this model is

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

where $\phi\$ and $\lambda\$ are geocentric (spherical) latitude and longitude respectively, $\overline{P}_{nm}$ are the fully normalized Legendre functions of degree $n\$ and order $m\$, and $\overline{C}_{nm}$ and $\overline{S}_{nm}$ are the coefficients of the model. Note This article describes a very general class of functions Note that the above equation describes the Earth's gravitational potential $V\$, not the geoid itself, at location $\phi,\;\lambda,\;r,\$ the co-ordinate $r\$ being the geocentric radius, i. The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical e, distance from the Earth's centre. The geoid is a particular[4] equipotential surface, and is somewhat involved to compute. Equipotential or isopotential in Mathematics and Physics (especially Electronics) refers to a region in space where every point in it is at the 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

$\sum_{k=2}^n 2k+1 = n(n+1) + n - 3 = 130,317$

different coefficients (counting both $\overline{C}_{nm}$ and $\overline{S}_{nm}$, and using the EGM96 value of n = nmax = 360). 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). 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

## Precise geoid

The 1990s saw 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. Petr Vaníček (born 1935 in Sušice, Czechoslovakia, today in Czech Republic) is a Czech Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist Their solution enables millimetre-to-centimetre accuracy in geoid computation, an order-of-magnitude improvement from previous classical solutions [7] [8] [9]. Computation is a general term for any type of Information processing. 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

## References

1. ^ data from http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html
2. ^ http://www.csr.utexas.edu/grace/gravity/gravity_definition.html 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: http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.html]
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)