# Geoid

three-dimensional plot of Geoid undulations

A geoid is an equipotential surface which (approximately) coincides with the mean ocean surface. It is often referred to as 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 the gravity field. It is that equipotential surface (surface of fixed potential value) which coincides on average with mean sea level.

## 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 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 leveling.

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)[1], 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

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

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 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[2] 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

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

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[3] that will incorporate much of the new satellite gravity data (see, e.g., GRACE), and will support up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients).

## References

1. ^ 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, was actually updated last in June 23 2004. Available electronically at: http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.html]
2. ^ There is no such thing as "The" EGM96 geoid
3. ^ 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.