Geodesy is the science of the Earth’s shape and size. Altimetry makes it possible to compute Mean Sea Surface; such a surface includes the geoid, i.e. the shape of the sea surface, assuming a complete absence of any perturbing forces (tides, winds, currents, etc.). The geoid reflects the Earth’s gravitational field. It varies in height by as much as 100 metres over distances of several thousand kilometres due to uneven mass distribution within the planet’s crust, mantle and core. Other less pronounced irregularities are also visible over smaller distances. These mostly reflect the ocean bottom topography.

Mean Sea Surface, representing the sea level resulting from constant phenomena, computed from 20 years of altimetry data. This Mean Sea Surface is shaped by permanent ocean currents and, above all, by the gravity field. Differences below the surface of the Earth (for example, variations in magma temperature) can generate sea level variations of over 100 metres between two ocean regions thousands of kilometres apart. At smaller scales (a few kilometres), we can also observe on this surface (highlighted here so as to be visible) the influence of ocean floor topography (see bathymetry) which causes variations of several metres at the ocean surface.
(Credits CNES/CLS)

According to the laws of physics, if we set aside any perturbing forces (tides, winds, currents, etc.), the surface of the ocean becomes an equipotential surface of the earth’s gravity field. Basically this means that if we could place balls all over the surface of the ocean, none of the balls would roll down the hills of this surface because they would all be on the same “level” (i.e. at the same gravity) and subsequently, that the waters of the currents would not flow due to geoid height variations. This equipotential surface deviates by up to 100 metres from the reference ellipsoid, the ideal shape which fits the rotating Earth most closely. These hills and valleys in the ocean’s surface are caused by minute variations in the earth’s gravitational field. For example the extra gravitational attraction of a massive mountain on the ocean floor attracts water towards it causing a local bump in the ocean surface; a 2,000-m-tall undersea volcano causes a bump about 2 m high, with a radius of about 20 km. This bump cannot be seen with the naked eye because the slope of the ocean surface is very low. In practice, altimetry data, collected by different satellites over many years, are combined to achieve high data density and to average out sea surface disturbing factors such as waves, winds, tides, and ocean variability. The only other component of mean sea surface that is not the geoid is then the static currents (mean dynamic topography), which have to be explained using different methods and then subtracted (see Altimetry Basic Principles, Large-scale ocean circulation, Applications).

## Long wavelengths geoid undulations

The greatest geoid heights and those most visible on a map, reflect deeply-buried density variations.

EIGEN-CG01C Geoid. (Credits GFZ Potsdam)

## Gravity anomalies

In order to enhance small-scale features, the high-precision geoid can be converted into a gravity anomaly. Gravity anomaly computations are quite complex, based on laws of physics, geometry and statistics (e. g. see [Sandwell and Smith, 1997]).

EIGEN-CG01C Free Air Gravity Anomalies. (Credits GFZ Potsdam)

### References:

Green, C. M., Fairhead, J. D., and Maus, S., Satellite-derived gravity: Where we are and what’s next: The Leading Edge, 17, 77-79, 1998.
Haxby, W. F., Karner, G. D., LaBrecque, J. L., and Weissel, J. K., Digital images of combined oceanic and continental data sets and their use in tectonic studies: EOS, 64, 995-1004, 1983.
Li, Xiong, and Hans-Jürgen Götzez, Tutorial: Ellipsoid, geoid, gravity, geodesy, and geophysics, Geophysics, vol. 66, no. 6 (november-december 2001); p. 1660-1668, 2001.
Sandwell,D.T., and Smith,W.H.F., Marine gravity anomaly from Geosat and ERS-1 satellites: J. Geophys. Res., 102, 10039-10054, 1997.
Yale, M. M., Sandwell, D. T., and Herring, A. T., What are the limitations of satellite altimetry?: The Leading Edge, 17, 73-76, 1998.

Further information:
Tapley, B.D. and M.C. Kim, Applications to geodesy, Satellite altimetry and Earth sciences, L.L. Fu and A. Cazenave Ed., Academic Press, 2001