geoid The geoid is defined as a gravitational equipotential surface that coincides with mean sea level. Any massive body in or around the Earth has a gravitational potential (a measure of the work that has to be done to move it against the gravitational force) that is due to the Earth's gravitational field. This potential varies in inverse proportion to the distance from the centre of mass of the Earth. We know that the potential energy of a body is greater on top of a hill than at its foot (that is why bodies fall under gravity); so gravitational potential energy increases away from the Earth.

It follows that if the Earth were spherically symmetrical and were not spinning, any given equipotential surface would be a sphere, and one could imagine successive equipotential surfaces of steadily increasing potential energy extending away from the Earth like the skins of an onion. Since sea water is free to flow in the gravitational field, it will flow (all else being equal) until every point in its surface is at the same minimum potential energy. This is the mean sea-level surface, and is also clearly a gravitational equipotential. The geoid is simply this one among the infinite number of ‘skins’ that coincides with mean sea level. Although the geoid is defined in terms of sea level, since it is a continuous surface one can determine mathematically its height over land, and can think of this as the level that water would reach in a network of canals connected directly to the sea.

In fact the Earth is a slightly flattened sphere (technically, an oblate spheroid) with its equatorial radius some 21 km longer than the polar one. This flattening, combined with the spin of the Earth, modifies the gravitational potential somewhat, so that the geoid is also slightly flattened. To a good approximation (about 1 part in 60 000), the geoid is an oblate spheroid whose major axis is about 0.3 per cent longer than the minor one. Cartographers, surveyors, and geodecists normally use a perfect oblate spheroid, which is a best fit to the actual geoid, as the datum for height measurements on the Earth. Nevertheless, there are important departures of the actual geoid from this simple shape.

Geoid anomalies

These departures, or geoid ‘anomalies’, are the result of the non-uniform distribution of mass within the Earth. A simple way to picture this is to imagine a uniformly deep ocean in which there is an isolated underwater mountain (a sea mount; Fig. 1). In the absence of the sea mount, the sea surface would have a uniform height everywhere. However, the sea mount, being denser than water, exerts a gravitational attraction on the surrounding water, which therefore tends to pile up above the sea mount, raising the geoid at that point. Another way of looking at this is to realize that the Earth's gravitational field is slightly enhanced over the sea mount because of its greater density, so the same gravitational attraction is found at a slightly greater distance from the Earth's centre.

The increase in geoid height over a large sea mount can be several tens of centimetres, but large, deep-seated mass anomalies in the Earth can distort it by much more. Figure 2 shows a map of geoid anomalies (measured relative to the reference spheroid), which range up to about 100 m (just south of India). Such large anomalies must reflect planet-scale variations in mass distribution, perhaps arising from thermally induced density variations due to mantle convection, or from substantial topography on the core–mantle boundary.

Determining the geoid

The foregoing description shows that there is an intimate relationship between the height of the geoid, the Earth's gravitational attraction, and the distribution of mass within the Earth. There is, in fact, a unique mathematical relationship between the geoid and the gravity field, although the relationship between either and the mass distribution is not unique (because a variety of mass distributions can give rise to a given gravitational field). Nevertheless, the shape of the geoid does place important constraints on the mass distribution. Moreover, since the local vertical (as shown by a plumb-line or spirit-level) is defined as the direction of the gravitational attraction, which is always perpendicular to the geoid, geoid anomalies cause ‘deflections of the vertical’ (relative to an astronomical framework), which are of vital practical importance for surveyors. Determining the geoid is therefore an important objective in geodesy and geophysics.

In the pre-satellite era the geoid was determined from the gravity field, which entailed measuring gravitational attraction or deflections of the vertical (or both) at a large number of locations around the Earth. Accuracy and resolution were limited by the number and distribution of observations that were feasible. Since the advent of artificial satellites, geoid determinations have become easier, more accurate, and of higher resolution. There are basically two methods. In the first (and earliest), satellites are tracked precisely from ground stations, and the geoid inferred from small fluctuations in their measured orbits.

More recently, sea level can be measured directly by a radar-altimeter mounted on a satellite. It is important to remove the effects of waves, storms, ocean currents, and other similar effects, but ultimately this method gives a precision of a few centimetres in geoid height. Although they cannot be used over land, such measurements have yielded unprecedented high-resolution coverage of the geoid over the sea. Along-track resolution can be as good as a few kilometres; cross-track resolution is limited by track spacing, which can be of the order of 10 km in the best cases. This method has been used in recent years to provide very high-resolution maps of the gravity field over the oceans. Since the main source of short-wavelength gravity anomalies there is the varying topography of the sea bed, the method has revealed this topography in unprecedented detail.

Roger Searle

Bibliography

Heiskanen, W. and and Moritz, H. (1967) Physical geodesy. W. H. Freeman, New York.
Sandwell, D. T. and and Smith, W. H. F. (1997) Marine gravity anomaly from Geosat and ERS 1 satellite altimetry. Journal of Geophysical Research, 102, (B5), 10 039–54.

geoid The figure the Earth would have if it were entirely covered by water at mean sea level. The geoid is approximately an ellipsoid, but there are departures from it caused by the gravitational attraction of mountains, as well as differences in density within the Earth. The concept of the geoid is now being applied to other planetary bodies as well. In such cases, an arbitrary datum is used in place of sea level, such as the planet's mean radius or, for gaseous worlds, the altitude at which a given atmospheric pressure is found.

geoid The gravitational equipotential surface corresponding to mean sea level, including the level at which the sea would stand in a continental area if it were able to do so.

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