Geoid (geology)

The shape of the sea-level surface, over the oceans and under the continents, is given by the geoid. This shape differs from the best-fitting ellipsoid by amounts ranging up to approximately one hundred meters, and these variations provide valuable information concerning models of the convection and tectonics of the planet.

Approximating Earth's Shape

The geoid is an imaginary surface that is at sea level everywhere on the earth. Over the oceans, it is generally at mean sea level; under the continents, it is the elevation the sea would have if all the continents were cut by narrow sea-level canals. It is usually represented as the difference in elevation between sea level and some ellipsoid representing the average shape of the earth, and its relief is on the order of one hundred meters. It is important in surveying and geodesy because elevations are measured above or below this surface, and it is important in geology and geophysics because its departures from a perfect ellipsoid reveal information about the earth's interior.

The shape of the earth can be approximated, with varying degrees of complexity and with different levels of success, by different mathematically defined shapes. If represented as a sphere with a radius of 6,371 kilometers, the shape is very simple, but it will have a radius that will be 7 kilometers too small at the equator and 15 kilometers too large at the poles. Nonetheless, this is adequate enough to be used for many scientific purposes.

For some purposes, however, a spherical shape is entirely inadequate. Gravity varies with the radius of the earth. Surveys seeking to detect density variations beneath the surface using sensitive measurements of gravity need a way to account for the gradual increase in radius from the poles to the equator. This change in radius, usually called the earth's “flattening,” is obtained by dividing the difference between the equatorial radius and the polar radius by the equatorial radius. A modification to the spherical shape is obtained by letting the radius vary slightly with latitude, using a straightforward function that includes the value for flattening. This shape is known as the spheroid and sometimes called the “niveau spheroid”; and for many years it was used in gravity surveys. It permitted data reduction at a time when computers filled rooms, if not buildings. This simple formula is actually an approximation of a slightly more complex shape, the ellipsoid.

Equipotential Surfaces

An ellipsoid of revolution is the shape of a solid produced by rotating an ellipse about one of its axes. If an ellipse with a minor axis equal to the polar diameter and a major axis equal to the equatorial diameter is rotated about the poles, the resulting shape is the earth's ellipsoid. This ellipsoid is used in studies when the sphere or spheroid is inadequate, and it forms the basis for the geoid.

The geoid is an example of an equipotential surface. If there was some way of sliding a mass around on its surface without any friction, no work would be done in moving that mass from place to place, because the mass would stay at the same potential. It is not difficult to calculate the shape of this surface for various idealized situations. It is also possible to measure the shape of the geoid. Much can be learned by comparing the observed shape with the shapes generated by the models.

If the earth were a stationary sphere of uniform density, the geoid would also be a sphere. If the earth were a rotating sphere of uniform density, the geoid would become an ellipsoid. This is because the rotation produces centrifugal force. Rotation involves an acceleration, which, when multiplied by a mass, must be balanced by a centripetal force—in this case, one supplied by gravity. For these purposes, using the non-Newtonian centrifugal force will prove simpler. This centrifugal force acts in a direction perpendicular to the rotation axis. At the equator, it would be directly opposed to the gravitational pull of the earth. An equipotential surface would need to be higher there to make up for this force. The equator would be farther from the center of the earth, and the poles would be closer to the center of the earth, but because the geoid is an equipotential surface, traveling from the equator to one of the poles would not involve going downhill. If the earth formed a rigid sphere, oceans would be much deeper at the equator than at the poles. However, the scale of the earth and the fact that it has existed for billions of years allowed it to deform much as if it were a fluid.

Evolving Knowledge of the Geoid

Suppose a model is allowed to assume the equilibrium shape of a fluid with the earth's mean density, rotating in space once a day. In 1686, Sir Isaac Newton determined that such a model would form an ellipsoid with a flattening of 1 part in 230. His solution piqued considerable interest. This much flattening would result in differences in the length of a degree of latitude between the equator and the poles, which should have been measurable using the techniques available in the early part of the eighteenth century. Expeditions were made to Lapland and Peru to do just this. The results showed a flattening, but of only about 1 part in 300. The current value is 1 part in 298.257, and many geoids are presented in terms of elevation above or below this reference ellipsoid.

As additional geodetic surveys, gravity surveys, and satellite orbit determinations were done, knowledge of the geoid evolved. It is now known that the Indian Ocean just off the southern tip of India is about 100 meters beneath the ellipsoid. Other ocean lows exist in the western North Atlantic Ocean (⁻50 meters), the eastern North Pacific Ocean (⁻50 meters), and the Ross Sea near Antarctica (⁻60 meters). On the continents, lows are present in central Asia (⁻60 meters) and northern Canada (⁻50 meters). High areas of the geoid occur over New Guinea (⁺75 meters), southeast of Africa halfway to Antarctica (⁺50 meters), in the North Atlantic Ocean (⁺60 meters), in western South America (⁺40 meters), and in southern Alaska (⁺20 meters). These highs and lows dominate the geoid. Their existence and locations have been known since the 1970s.

Undulations and Geoid Anomalies

The huge areas over which individual highs and lows extend require that they be produced by large, deep-seated density variations. Their existence suggests the earth is not in hydrostatic equilibrium, which in turn suggests that they result from density variations that do not persist for more than a few hundred million years. Therefore, mantle convection seems to be the most likely cause of these undulations, and many geophysical studies of the long-wavelength undulations of the geoid have concentrated on determining what they tell us about this convection. In general, lows on the geoid are above areas of rapid spreading, and highs are above subducted slabs. Although still evolving and hence subject to change, most of these investigations seem to suggest that mantle convection is driven by descending, not ascending, plumes; that convection involves the whole mantle, not just the seismically active (less than 670 kilometers deep) mantle; and that viscosity increases by a factor of about ten at some depth, probably 670 kilometers.

Other research involves the smaller-wavelength geoid anomalies, particularly over ocean areas. The data are filtered to remove the larger effects, and what remains is usually an excellent indicator of sea-bottom topography. The additional mass produced by a mountain on the sea floor attracts extra water, which piles up and causes a high on the geoid. The geoid is depressed above trenches because trench areas have less mass than the normal sea floor, so they attract less water. This small-wavelength low, with an amplitude of ten meters or so, is usually superimposed on a much larger wavelength high produced by the huge mass excess of the cold slab descending into the mantle nearby.

Undulations of the geoid give some of the best evidence for lateral density variations in the mantle. Seismic data also reveal lateral variations within the mantle, but these are variations in seismic wave velocities, which may or may not correspond directly to density variations. Eventually, the two lines of investigation promise to reveal the inner workings of the planet.

Laplace's Equation

Because gravity depends on the shape of the geoid, if there were enough accurate determinations of gravity at sea level, mathematical manipulations could be performed to find the shape of the geoid. However, gravity measurements taken above sea level can be adjusted in ways that convert them to equivalent values for sea-level readings. By the middle of the twentieth century, great progress had been made in mapping the gravity field over much of North America and Europe. This gave some indication of how the geoid undulated locally, but accurate determinations of geoid heights actually require considerable knowledge of gravity from around the entire earth. When the first satellite was launched in 1957, a new technique suddenly became available that permitted global gravity—and geoid height—to be calculated.

A satellite's orbit is influenced by the distribution of mass beneath it. The motion of the satellite in its orbit and the precession (gyration) and nutation (wobble) of the orbit are all influenced by the gravitational field it experiences. If the orbit is carefully tracked, it reveals much about the earth's shape and gravity. This requires a considerable mathematical effort, seeking solutions to a partial differential equation called Laplace's equation. These solutions take the form of coefficients of Legendre polynomials and associated polynomials. To see how they can describe the geoid, consider a surface suspended in space above a chessboard.

To describe the topography of this surface, each square could be designated by its row and column number and its elevation listed. However, there is another way to do this. The description can begin with the average elevation for the whole surface. Once that is determined, the surface is divided into quarters labeled “1” through “4”; the difference between the average elevation for each quarter and the average for the whole surface is then listed. Each quarter is then divided into four parts labeled “A” through “D,” and the difference between the average elevation for each part and the average for the whole quarter is again listed. If this is done once more, labeling the divisions “a” through “d,” elevation data will exist for all sixty-four squares of the chessboard. If the quarters are numbered clockwise from the upper left, then the square in row 3, column 7 would be designated 2Ca, and its elevation would equal the total surface average plus the difference for quarter 2, plus the difference for sixteenth 2C, plus the difference for sixty-fourth 2Ca.

One advantage this system has is that the earlier results do not change as the description gets more and more detailed. The geoid uses a similar system; however, it divides a spherical grid rather than a square one, so instead of repeatedly quartering, it uses Legendre polynomials. Instead of dividing the board horizontally, the geoid goes to the next degree; instead of dividing the board vertically, the geoid goes to the next order. A geoid determined to degree and order 12 will have much more detail than one determined to degree and order 6, but the coefficients for the first six degrees and orders should be the same.

Satellite Data

As satellites have been tracked through more orbits by better technology, the geoid has become known to increasingly better precision. It was known to degree and order 8 by 1961, 16 by 1971, and 20 by 1985. In the twenty-first century, supercomputers work through 10,000 parameters, yielding some results to degree and order 360. There are differences between them; however, typically they vary on the order of one-half meter at degree and order 50 or so. Much of the difficulty lies in correctly merging data from more than thirty satellites and tens of thousands of individual gravity measurements made on land.

Satellite data are influenced by factors such as atmospheric drag and refraction effects, and considerable thought and engineering have gone into finding ways around such problems. Geodetic satellites must be in near-Earth orbits to be influenced significantly by the undulations in the geoid, and in such orbits atmospheric drag is significant. One approach has been to design satellites out of uranium, which is denser than lead, giving them a great mass for a small cross-sectional area. Another solution has been to suspend a massive core inside a spherical shell. Atmospheric drag will act on the shell, pushing it closer to the core on the side experiencing the drag. Appropriate use of thrusters built into the shell can then ensure that these effects are perfectly canceled. Refraction effects also result from the atmosphere and can be minimized by determining the satellite's position relative to other satellites in higher orbit. Lasers are used for this, reflecting off corner cubes embedded in the outer shell of the satellite.

Time variations in the geoid were measured by the GRACE (Gravity Recovery and Climate Experiment) satellites, which operated from 2002 until 2017, and the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite launched in 2009. The GOCE satellite uses a highly sensitive gradiometer to measure variations in density, and can measure the geoid to an accuracy of 1-2 cm. Time variations of the geoid are used in studies of hydrological cycles and of glacial melting.

Ground-based gravity data are influenced by instrument design, operator expertise, and the care with which topographic effects have been removed. Accurate elevations are essential, and much of the earth's land surface has not been mapped with sufficient topographic precision to provide this control. This, combined with political and economic realities, has resulted in a data set that includes very little information from Asia and not enough from South America. The global positioning system (GPS) constellation of satellites promises great improvement in the data set by providing better satellite tracking as well as better elevation control in poorly mapped areas.

Predictive Value

Better knowledge of the geoid will lead to a better understanding of plate motions and convection in the mantle, permit more accurate placement of satellites in orbit, and provide an enhanced base for surveys on Earth. One of the outstanding questions in geophysics concerns the scale of convection in the mantle. No one knows if the convection cells extend throughout the entire mantle or if there are two convection regimes—one above a depth of 670 kilometers, the other below it. Unequivocal answers have yet to be found, but the study of the geoid offers great promise. Models for most of the highs of the geoid use the mass of the descending slabs to generate them. It appears that a slab reaching only to depths of 670 kilometers may not have sufficient mass. This somewhat tentative result favors deep mantle convection. As knowledge of the geoid improves, particularly in continental areas for which only sparse gravity data exist, the validity of this result may be established. An improved grasp of convection in the mantle will increase understanding of the plate-tectonic theory. As that develops, it may well produce a capability to predict earthquakes and volcanoes and to mitigate the death and destruction they cause.

As communications technology develops, the need for many more satellites to relay the ever-increasing traffic will grow. Just as minor perturbations in satellite orbits help us to define the geoid, a better understanding of the geoid will permit much better predictions of just how a satellite will behave in orbit. This is true even for the very high communications satellites. As the population of satellites grows, the need for such refinements in orbit calculations will become necessary to avoid collisions.

Ocean currents such as the Gulf Stream are powered by differences in elevation of the sea surface. These differences, on the order of a meter or so, occur because huge, warm lenses of water float on denser water below. Water at the surface, which in this case is not an equipotential surface, tries to flow down the slope, is affected by the earth's rotation, and ends up going around the lens of warmer water instead. Measuring such tiny variations in the level of the sea is difficult. As our knowledge of the geoid improves, however, we should be able to observe these discrepancies. This should provide important data from which changes in ocean currents can be predicted. Finally, because all surveys on land measure elevations with respect to sea level, improvements in charting sea level will make the elevation measures on land much better.

Principal Terms

ellipsoid of revolution: a three-dimensional shape produced by rotating an ellipse around one of its axes

equipotential surface: a surface on which every point is at the same potential, used here to include gravitational and rotational effects; no work is done when moving along an equipotential surface

global positioning system (GPS): a group of satellites that go around Earth every twenty-four hours and that send out signals that can be used to locate places on Earth and in near-Earth orbits

Legendre polynomials: mathematical functions used to describe equipotential surfaces on spheres

mantle convection: thermally driven flow in the earth's mantle thought to be the driving force of plate tectonics

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