# Geoid

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C71_geoid_smooth_Earth-Gravity-ESA.jpg
The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earth's gravity field. (ESA image)

A geoid is a close representation or physical model of the figure of the Earth. According to C.F. Gauss, it is the "mathematical figure of the Earth", in fact, of her gravity field. It is that equipotential surface (surface of fixed potential value) which coincides on average with mean sea level.

The geoid surface is more irregular than the ellipsoid of revolution 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 geoid varies by only about ±100 m about the reference ellipsoid of revolution.

Because the force of gravity is everywhere perpendicular to the geoid (being an equipotential surface), sea water, if left to itself, would assume a surface equal to it—even through the continental land masses if sea water were allowed to freely penetrate them, e.g., by tunnels. In reality it can not, of course; still, geodesists are able to derive the heights of continental points above this imaginary, yet physically defined, surface by a technique called spirit levelling.

When travelling by ship, one does not notice the undulations of the geoid; the local vertical is always perpendicular to it, and the local horizon tangential to it. A GPS receiver on board may show the height variations relative to the (mathematically defined) reference ellipsoid, the centre of which coincides with the Earth's centre of mass, the centre of orbital motion of GPS satellites.

## 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. It contains a full set of coefficients to degree and order 360, describing details in the global geoid as small as 55 km.

The mathematical description of this model is

[itex]

V=\frac{GM}{r}\left(1+{\sum_{n=2}^{360}}\left(\frac{a}{r}\right){\sum_{m=0}^n} \overline{P}_{nm}(\sin\phi)\left[\overline{C}_{nm}\cos m\lambda+\overline{S}_{nm}\sin m\lambda\right]\right), [itex]

where [itex]\phi\ [itex] and [itex]\lambda\ [itex] are geocentric latitude and longitude respectively, [itex]\overline{P}_{nm}[itex] are the fully normalized Legendre functions of degree [itex]n\ [itex] and order [itex]m\ [itex], and [itex]\overline{C}_{nm}[itex] and [itex]\overline{S}_{nm}[itex] are the coefficients of the model. One easily counts that there are approx. [itex]\begin{matrix} \frac{1}{2} \end{matrix} n(n+1)\approx [itex] 65,000 different coefficients. The above formula produces the Earth's gravitational potential [itex]V\ [itex] at location [itex]\phi,\;\lambda,\;r,\ [itex] the co-ordinate [itex]r\ [itex] being the geocentric radius, i.e, distance from the Earth's centre.

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