Geoid



The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest, and 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 Petr Vaníĝek 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 ellipsoid which is a mathematical idealized representation of the physical Earth, but considerably smoother than Earth's physical surface. Although the former has excursions of +8,000 m (Mount Everest) and −11,000 m (Mariana Trench), the geoid's total variation is less than 200 m (-106 to +85 m) compared to a perfect mathematical ellipsoid.

Sea level, if undisturbed by currents 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 (tides not considered). 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. Modern GPS receivers have a grid implemented inside where they obtain the WGS84 geoid height over the WGS ellipsoid from the current position. Then they are able to correct the height above WGS ellipsoid to the height above WGS84 geoid. In that case when the height is not zero on a ship it is because of the tides.

Simplified Example
The gravity field of the earth is neither perfect nor uniform. A flattened ellipsoid is typically used as the idealized earth, but even if the earth were perfectly spherical, the strength of gravity would not be the same everywhere, because density (and therefore mass) varies throughout the planet. This is due to magma distributions, mountain ranges, deep sea trenches, and so on.

If that perfect sphere were then covered in water, the water would not be the same height everywhere. Instead, the water level would be higher or lower depending on the particular strength of gravity in that location.

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{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 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 $$n=n_{max}=360$$). 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).  NGA has announced the availability of EGM2008, complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Software and data is on the Earth Gravitational Model 2008 (EGM2008) - WGS 84 Version page.

Precise geoid
The 1990s saw important discoveries in theory of geoid computation. The Precise Geoid Solution by Vaníĝek 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.