# Extensive Definition

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

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(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

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), 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

V=\frac\left(1+\left(\frac\right)^n
\overline_(\sin\phi)\left[\overline_\cos m\lambda+\overline_\sin
m\lambda\right]\right),

where \phi\ and \lambda\ are geocentric
(spherical) latitude and longitude respectively, \overline_ are the
fully normalized Legendre
functions of degree n\ and order m\ , and \overline_ and
\overline_ are the coefficients of the model. 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.e, distance from the Earth's centre. The
geoid is a particular 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

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

different coefficients (counting both \overline_
and \overline_, and using the EGM96 value of n=n_=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 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 1990s saw important discoveries in theory of geoid computation. The Precise Geoid Solution 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 .## References

## External links

## See also

geoid in Asturian: Xeoide

geoid in Bengali: ভূগোলক

geoid in Czech: Geoid

geoid in German: Geoid

geoid in Estonian: Geoid

geoid in Spanish: Geoide

geoid in Esperanto: Geoido

geoid in French: Géoïde

geoid in Korean: 지오이드

geoid in Italian: Geoide

geoid in Luxembourgish: Geoid

geoid in Hungarian: Geoid

geoid in Dutch: Geoïde

geoid in Japanese: ジオイド

geoid in Norwegian: Geoide

geoid in Polish: Geoida

geoid in Portuguese: Geóide

geoid in Russian: Геоид

geoid in Slovak: Geoid

geoid in Slovenian: Geoid

geoid in Finnish: Geoidi

geoid in Swedish: Geoid

geoid in Thai: จีออยด์

geoid in Ukrainian: Геоїд