## The Earth's Geoid

Earth's 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 potential, and is essentially mean sea level. Remember, sea level isn't flat! The vertical coordinate, Z (elevation), is referenced to the geoid.

Can be defined as:

• the shape a fluid Earth would have if it had exactly the gravity field of the Earth
• an equipotential surface
• roughly the sea-level surface - dynamic effects such as waves, and tides, must be excluded
• geoid on continents lies below continents - corresponds to level of nearly massless fluid if narrow channels were cut through continents
• geoid highs are gravity highs
• g (vector gravity), or vertical, is perpendicular to the geoid: "What's up?" "Perpendicular to the geoid." This because:
• 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
• gravity will be deflected toward a dense body
• 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.

An equipotential map of the Earth is dominated by the variation in gravity (and hence geoid height, or basically the shape of the Earth) caused by the Earth's rotation and subsequent flattening. It would look something like this:

The wiggles you see on the contour lines are actually just gridding/contour artifacts.

Ellipsoid

• An ellipsoid is a smooth elliptical model of the earth's surface. X,Y (horizontal coordinates) are referenced to an ellipsoid.
• In the past, different regions of the world had adopted "local" versions of the elllipsoid
• The Clarke 1866 ellipsoid is a predecessor to the GRS80 ellipsoid that was used in North America and is still the reference geoid on many maps
• In the the "space age," however, a "universal" system was required
• 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.
 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

## Why does a GPS measure elevation relative to WGS84, or some other reference ellipsoid, whereas a surveyor's estimate is relative to the geoid?

“My dog isn’t a piglet”. Vladimir Putin’s dog Koni prepares to test Russia’s new GPS system Photo: AP

Geoid Anomaly:  A change in the height of a portion of the geoid compared to its height for a flattened ellipsoid. 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

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) 1/298.25:

Below is

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.

Ocean Geoid

Geoid Over Subduction Zones

The most prominent features on most geoid maps (depending on filtering used) are subduction zones:

Cross-sections across subduction-zone geoid anomalies show an asymmetric anomaly low (trench) and an anomaly high (presence of cold, dense slab in lighter asthenospere):

Fig. 1. (A) Free-air gravity anomaly from satellite altimetry for the Tonga-Kermadec region. (B) Free-air gravity anomaly for 3D dynamic model including a low viscosity region in the wedge. (C) Comparison of topography along east west profiles across the subduction zone at 20, 25 and 30°S (thick/blue) to observed topography (thin/black). Model topography has an arbitrary reference height (here set to zero) therefore, observed topography is adjusted to equal zero at the model boundary. (D) Comparison of model geoid anomalies (thick/blue) with observed along east west profiles. An east west linear ramp is removed from each of the observed and model geoid profiles so that the geoid equals zero at the model boundaries.

# Geoid of the United States

 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

The GRACE Experiment

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

 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
 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.

The GOCE (Gravity field and steady-state Ocean Circulation Explorer) mission will measure high-accuracy gravity gradients and provide a global model of the Earth's gravity field and of the geoid. The geoid (the surface of equal gravitational potential of a hypothetical ocean at rest) serves as the classical reference for all topographical features. The accuracy of its determination is important for surveying and geodesy, and in studies of Earth interior processes, ocean circulation, ice motion and sea-level change.
Credits: ESA

 WAPGEO_anom_20_270_360.   Free air anomaly map of the Weddel sea.

## Geoid of Other Bodies

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

### Venus

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

## 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.

 Would gravity still be less due to the equatorial bulge (neglecting rotation effect) for a homogeneous planet? The bulge has 2 effects: it reduces g because one is farther from the center of the planet (free-air effect) but increases g due to the mass of the bulge (Bouguer effect). The "real" Earth has a significant central condensation, i.e., it gets denser toward the center and the bulge alone produces a decrease in gravity approximately equal to the decrease caused by rotation alone. But what about a homogeneous planet? Assume that, to first order, the planet is spherical. The gravity from the surface outward would be given by where R is the planet radius and ρ is density. The decrease of g(r) with r (elevation), would be Evaluated at the planet's surface, r=R, we get The attraction due to the mass of the bulge can be approximated by computing the gravity of an infinite slab (Bouguer effect) which is given by Thus the elevation effect is bigger by