Earth's Geoid

Geoid: The geoid is a representation of the surface of the earth that it would assume if the sea covered the earth, also known as surface of equal gravitational attraction and mean sea level. Remember, sea level isn't level! The vertical coordinate, Z (elevation), is referenced to the geoid.

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Can be defined as "the shape a fluid Earth would have if it had exactly the gravity field of the Earth." It is:

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an equipotential surface

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roughly the sea-level surface - dynamic effects such as waves, and tides, must be excluded

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geoid on continents lies below continents - corresponds to level of nearly massless fluid if narrow channels were cut through continents

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geoid highs are gravity highs

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g (vector gravity), or vertical, is perpendicular to the geoid: "What's up?"  "Perpendicular to the geoid."  This because:

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potential at a point is "work done to take a unit mass from that point to infinity." Over a density high, this would require starting at a higher point

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gravity will be deflected toward a dense body

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fluid will be attracted toward a dense body, raising its surface

From scienceworld.wolfram.com:

The shape of an object's gravitational equipotential surface. For the Earth, the reference geoid is

where is the colatitude. The most complete model for the earths gravitational field, based on an expansion in a Laplace series, is given by the GEM-T2 model. It contains 600 coefficients above degree 36.

Geoid Anomaly:  A change in the height of a portion of the geoid compared to its height for the majority of the body. On Earth, substantial geoid anomalies are found at subduction zones and hotspots. In continental regions, they do not correlate with topography because of isostatic compensation . On both Venus and Mars, however, geoid anomalies are correlated with topography.  © Eric W. Weisstein

References

Marsh, J. G.; Lerch, F. J.; Putney, B. H.; Felsentreger, T. L.; Sanchez, B. V.; et al. "The GEM-T2 Gravitational Model." J. Geophys. Res. 95, B13, 22043-22071, 1990.

Spherical Harmonic Coefficients

Zonal harmonics have axial symmetry and are represented by the m=0 (order 0) terms; Sectorial harmonics have mirror symmetry with respect to equator, and are represented by l=m terms. Tesseral harmonics are the general case, m <> 0 and l <> m. From the Greek "tesserae" or tiles.

Before looking at the geoid, which is dominated by the J2 term, that term is removed, which amounts to removing an ellipsoid of flattening of (as currently determined) 1/298.5:

Stokes's Theorem

It is possible to estimate the geoid at a point from gravity values by means of the Theorem of Stokes (from Heiskanen and Vening Meinesz: The Earth and Its Gravity Field, p. 65):

where r and g are mean values over the geoid.

We thus have obtained the theorem of Stokes, which is of fundamental importance for geodesy because it allows us to determine the geoid from gravity. Thus we can find the geoid even at sea, which is not possible by any other method.

Attention should be drawn to several features of the theorem. In the first place, because of the manner in which it has been derived, the theorem is valid only if no masses are present outside the geoid. Therefore reductions must be applied to gravity anomalies and to the geoid. The simplest kind of reduction is that suggested by Helmert, who suggests considering that the outside topographic masses are condensed in an infinitely thin layer just inside the geoid. This mass displacement is so small that its effect on the geoid can be neglected. In Sec. 3-12 we return to these problems in a further discussion of Stokes' theorem.

In the second place, Stokes' theorem provides us with the distance N0 from the geoid to an earth spheroid which has the same volume as the geoid and of which the center coincides with the center of gravity of the earth, and so N0 contains no zero- or 1st-order spherical harmonics. This fact implies that the theorem cannot be used to find a more correct figure for the equatorial radius a of the earth spheroid; i.e., we cannot determine the scale of our model of the geoid, nor can we get a figure for the equatorial value of normal gravity ge. For the flattening a, on the other hand, we can get a new figure.

In the third place, although the applicability of Stokes' theorem seems to be impeded by the fact that the integral must be extended over the whole earth and that gravity is still unknown over great parts of it, actually, however, for great distances from the station where N0 is to be determined the function F0 is small, and so distant anomalies have only a slight effect.

Geoid of Other Bodies

The geoid, and gravity, can be determined for other planets from satellite data.

Moon

Venus

Geoid model, derived from Magellan orbit data, spherical harmonic fit (1° resolution).

[From the Jules Verne Voyager:  http://jules.unavco.org/Voyager/Venus?grd=6]

Ellipsoid

An ellipsoid is a smooth elliptical model of the earth's surface. X,Y (horizontal coordinates) are referenced to an ellipsoid. GRS80 is currently the most commonly used elliptical model used for the globe,though a new ellipsoid has recently been developed by the National Geodetic Survey and will likely replace GRS80 for future projects. The Clarke 1866 ellipsoid is a predecessor to the GRS80 ellipsoid.

Reference Ellipsoids used in Geodesy
 

Name of ellipsoid

semimajor
axis a[m]

flattening
f = (a-b)/a

applied for

Geodetic Reference System 1980 (GRS80) 6 378 137. 1 : 298.25722 World Geodetic System 1984
World Geodetic System 1972 (WGS72) 6 378 135. 1 : 298.26 World Geodetic System 1972
Geodetic Reference System 1967 6 378 160. 1 : 298.25 Australian Datum 1966
South American Datum 1969
Krassovski (1942) 6 378 245. 1 : 298.3 Pulkovo Datum 1942
International (Hayford 1924) 6 378 388. 1 : 297.0 European Datum 1950
Clark (1866) 6 378 206. 1 : 294.98 North American Datum 1927
Bessel (1841) 6 377 397. 1 : 299.15 German DHDN

 

egm96_geoid: The Geoid is that equipotential surface of the Earth gravity field that most closely approximates the mean sea surface. At every point the geoid surface is perpendicular to the local plumb line. It is therefore a natural reference for heights - measured along the plumb line. At the same time, the geoid is the most graphical representation of the Earth gravity field.
    The geoid surface is described by geoid heights that refer to a suitable Earth reference ellipsoid. Geoid heights are relative small.The minimum of some -106 meter is located at the Indian Ocean. The maximum geoid height is about 85 meter. The figure below shows a global map with geoid heights of the EGM96 gravity field model, computed relative to the GRS80 ellipsoid

 

GEOID99 is a refined model of the geoid in the United States, which supersedes the previous models GEOID90, GEOID93, and GEOID96. For the conterminous United States (CONUS), GEOID99 heights range from a low of -50.97 meters (magenta) in the Atlantic Ocean to a high of 3.23 meters (red) in the Labrador Strait. However, these geoid heights are only reliable within CONUS due to the limited extents of the data used to compute it. GEOID99 models are also available for Alaska, Hawaii, and Puerto Rico & the U.S. Virgin Islands.

"More than any other data set of the Earth the Geoid shows us the dynamic structure of the Earth's deep interior. The most dramatic feature in the Geoid of North American is the Yellowstone Hot Spot,  believed to be a plume structure rising through the mantle and the main contributor to the Geoid high over Montana. Details of the topographic anomalies of the Western Rockies can be seen superimposed upon this anomaly, although with much less magnitude. The Great San Joaquin Valley of California, formed through the tectonics of the earlier subduction of the Pacific plate by North America is outlined in detail in the Geoid.

Comparison with this feature can be made with those smaller yet similar Geoid lows to the north in Oregon and Washington state. In the midcontinent an ancient rift or suture zone can be seen in sharp outline running from the tip of Lake Superior through Minnesota and continuing to Texas. The Eastern offshore shows some of the oldest portions of the Atlantic Ocean formed some 120 million years ago with its now characteristic Geoid low centered off the Carolinas. Seen also is a deep suture structure running the length of the Hudson River Valley to the opening of the Gulf of Saint Laurence. At the very top of the figure on the right can be seen the outline of the most recently formed feature of Geoid of North America. This is the postglacial Geoid low caused by the depression of the continent under the ice load from the last Ice Age some 20,000 years ago. Because of the viscous nature of the Earth's Mantle this feature will slowly disappear until the end of the next Ice Age when the process will repeat itself again."

By: Allen Joel Anderson
Department of Physics
University of California

From www.gfz-potsdam.de/ news/foto/champ/welcome.html

Die Abweichungen der physikalischen Oberfläche der Erde (Geoid oder 'Normal Null') von einem regelmässigen Ellipsoid, vom Computer mit 15000facher Überhoehung gezeichnet, sind Ausdruck der unregelmässigen Dichte- und Massenverteilung im Erdinnern. Die sich unter dem Einfluss des Erdschwerefeldes ausbildenden Verformungen reichen von -110m im Indischen Ozean bis +90m ueber Südostasien. Die Grossstrukturen dieser Figur der Erde konnten mit dem Mitte 2000 gestarteten deutschen Satelliten CHAMP mit bisher unerreichter Genauigkeit aus Beobachtungen seiner Bahnstörungen ausgemessen werden. Über den Kontinenten ist das
Geoid zur besseren Unterscheidung in Graustufen dargestellt.

 

 

Image Name : ww15mgh;      Boundaries : Lat -90N to 90N;     Lon 0E to 360E;
Color Scale, Upper (Red) : 85.4 meters and higher;       Color Scale, Lower (Magenta) :-107.0 meters and lower
Data Max value : 85.4 meters       Data Min value :-107.0 meters        Illuminated from the : East

This is an image generated from 15'x15' geoid undulations covering the planet Earth. These undulations represent the NIMA/GSFC WGS-84 EGM96 15' Geoid Height File. This file is a global grid of undulations generated from: (a) the EGM96 spherical harmonic coefficients and (b) correction terms that convert pseudo-height anomalies on the ellipsoid to geoid undulations.

This file may be found at: http://164.214.2.59/geospatial/products/GandG/wgs-84/geos.html.  The undulations in this file refer to the WGS-84(G873) reference ellipsoid.  Some interesting features to note about this image are: Even at 15' resolution, some beautiful features of the global geoid are obvious.  The major trench systems have obvious impacts on the geoid, as well as the topography/ ocean boundaries (whose geoid signals closely coincide with the shoreline).

The Hawaiian Island chain may be followed up through its transition into the Emperor Seamounts and toward the western end of the Aleutian Islands.  The structure of seamounts with the Marshall Islands, east of the Mariana Trench, can be seen in the geoid signal in that area.  Finally, the well-known geoid low near the tip of India, and the geoid high over New Guinea stand out, with a great deal of finely detailed structures mixed in with these broad features.    Map and description from the National Geodetic Survey.

 

WAPGEO_anom_20_270_360.   Free air anomaly map of the Weddel sea.

Jules Verne Voyager: http://jules.unavco.org

Welcome to the Jules Map Server

To better understand the inter-relationships of geophysical and geological processes, structures, and measurements with high-precision GPS monument data and solutions, the Data Management and Archiving Group has developed an interactive map tool for virtual exploration of Earth and other worlds:

Global gravity maps and the structure of the Earth, 1985, Carl Bowin, The Utility of Regional and Magnetic Anomaly Maps, William J. Hinze, Ed., S.E.G.