SIZE AND SHAPE OF EARTH FROM SATELLITES

Modern evidence abounds that clearly supports the idea that the Earth is a spherical body. The memorable photographs of the Earth taken by the Apollo astronauts from the lunar surface and from cislunar space must motivate serious consideration of the sphericity notion even among the hardest skeptic.
But the stirring evidence presented in a photograph from space was unavailable to the ancients who had to deduce the ideas of sphericity from subtle effects (such as the sails of a ship slowly sinking at the horizon), from obvious comparative forms (the Moon and the Sun appear to be spheres), and from philosophical ideas (the sphere is a perfect geometric object, hence the Earth must be a sphere). Although these early approaches spurred the development of the science of geodesy, the revolutionary work of Isaac Newton in the seventeenth century brought the first renaissance in geodesy. The second renaissance coincides with the space age, which brought new tools to the pursuit of ancient questions. The new discipline of space geodesy appeared in the early 1960s, but the areas ofgeodetic interest go far beyond the classical study that focused on the size and shape of the Earth. Modern geodesy includes scholarly pursuits into the mass distribution within the Earth, as manifested by the external gravity field, the rotation of the Earth, and temporal changes in the positions of points on the Earth’s surface. The modern tools and techniques of geodesy are used by other scientific disciplines, including tectonophysics, oceanography, and glaciology, to name a few. And the methodologies and concepts are applied to other planets in the discipline known as planetary geodesy.


Historical View

The notion of a spherical the Earth is often attributed to Pythagoras in the sixth century B.C., but Eratosthenes made the first serious attempt to measure the geometric characteristics of such a model in the third century B.C. As the
librarian at Alexandria, Egypt, it is thought that he was one of the most learned men of antiquity (1). He made the subtle observation that the Sun cast a shadow of different lengths from an upright rod at two different locations, even though observed at the same time of year.
Eratosthenes must have observed the changes in the length of the shadow during the course of a year near his Alexandria home. But the shadow cast under similar conditions near today’s Aswan Dam was noticeably different. Assuming that the Sun’s rays arrived parallel at both Alexandria and Aswan, coupled with knowledge of the distance between the two sites (about 800 km), Eratosthenes computed an approximate radius of the Earth. This was a remarkable application of known geometric properties of the day, but assessing the accuracy of the radius by comparison with a modern estimate of the Earth’s radius depends on the conversion between distance units that Eratosthenes used and those used in modern geodesy. Although the precise conversion is debated by scholars of science history, the worst case suggests that Eratosthenes was in error by 15%, a rather remarkable achievement in the third century B.C. Eratosthenes is sometimes referred to as the father of geodesy, a well-justified title.
In the second century B.C., the astronomer Hipparchus determined a subtle motion of the Earth that remained unexplained for 1500 years (1). He noted that star positions had shifted systematically since the determinations by others 150 years earlier. This shift, now known as the precession of the equinoxes, amounts to only 1.47century. The explanation of this motion of the Earth had to wait for the brilliant work of Isaac Newton, who calculated the rate produced by lunisolar forces in response to an interaction with the Earth’s mass distribution and, hence, its shape. Other more subtle motions of the Earth, analogous to those of a gyroscope, which include nutations, also result from the interaction of the Sun and Moon with the mass distribution of the Earth (2).
The First Renaissance. There is little documentation of efforts to measure the radius of the Earth in the 1500 years after Eratosthenes. Nevertheless, the voyages of exploration in the fifteenth and sixteenth centuries were clear demonstrations of the spherical nature of the Earth.
In the seventeenth century, geodesy was invigorated by several technological and scientific developments. These developments included technologies for measuring the length of a specified latitudinal displacement, such as 1°. Using the technology to measure the difference in latitude between two points, a second technology was needed for geodetic applications to measure the distance between these points. The latitude measurements were made with astronomical instruments, known as transit circles, and early versions ofangle measuring surveying equipment were developed to support the distance measurement.
In the early 1600s, Willebrord Snellius (or Snell) published a new method that would significantly contribute to determining the shape of the Earth. This method allowed accurate inference of the distance between two points at different latitudes along the same meridian of longitude. This innovative method diminished the effect of local topographic obstacles that were inherent in the direct measurement of distance between two points separated by 100 km (1° of latitude) or more. In essence, Snell’s method made an accurate measure of the distance between two points on flat terrain, usually separated by a kilometer or two. This baseline measurement was made by placing a series of calibrated rods end to end. Using a series of triangles and the measurement of all angles within the triangles, the single baseline distance could be used to determine the lengths of all triangle sides, regardless of the intervening terrain. Another baseline measurement was usually made with the rods elsewhere in the network to verify the result from the triangles. This method of triangulation was the basic methodology for determining the lengths of various meridian arcs that were made during two centuries.
Using the new methods, an important set of arcs was measured in France during the late 1600s and early 1700s. The measurements made by the Cassini family within France tended to support the view that the Earth was not a perfect sphere; instead, the results suggested that the Earth was somewhat elongated along the polar axis and squeezed inward at the equator. A geometrical figure that matched the Cassini results was a prolate spheroid, which can be created by an ellipse rotated about its major axis. In this model, the Earth’s polar axis would coincide with the major axis of the ellipse.
At the same time, Newton published his monumental Principia in 1687 (3). In this work, Newton showed that the centrifugal force resulting from rotation of a sphere would tend to distort the sphere around the equator, so that the body would be characterized as an oblate spheroid. This body is generated by rotating an ellipse about its minor axis; and the minor axis coincides with the polar axis.
Proponents and detractors were attracted to the Cassini view that the Earth was a prolate spheroid, whereas others supported the Newtonian view of an oblate spheroid. The scientific debate occasionally degraded into nationalistic fervor to support one or the other view. Because of heated interest in the topic, the Royal Academy of France in 1735 suggested that expeditions be undertaken to measure the length of arc meridians at different latitudes. The first expedition was directed to take place in South America near the equator (modern Ecuador), creating what became known as the Peruvian arc. A second expedition was dispatched to Lapland, in polar Scandinavia. If the Earth was an oblate spheroid, then 1° of latitude will be slightly longer in the polar region than in the equatorial region. Information about the expeditions has been summarized by Tod-hunter (4) and Butterfield (5).
Both expeditions experienced great hardships. The Lapland expedition left a year after the Peruvian group, but it was completed within a year, thereby giving it the honor of publishing the first results in support of the Newtonian view. The Peruvian expedition endured many difficulties, which were compounded by rugged terrain and infighting among the participants. It took 10 years for the expedition to return to Paris and make known their results. But the results again reinforced the Newtonian oblate spheroidal model of the Earth. The philosopher Voltaire made the observation about the expeditions: ”You have found by prolonged toil, what Newton had found without even leaving his home” (6).
The oblate spheroid model of the Earth is characterized by (1) the equatorial radius (the ellipse semimajor axis, but it is a circle when rotated about the polar axis); and (2) the polar axis (the ellipse semiminor axis). Instead of using the specific polar dimension, an alternate parameter known as flattening is used, which is related to the eccentricity of the ellipse. If a represents the semimajor axis and b is the semiminor axis, then flattening f is (a — b)/a.
Although the Peruvian and Lapland expeditions showed that the Earth was more appropriately represented by an oblate rather than prolate geometric figure, the equatorial radius and flattening parameters that resulted from the expeditions were significantly different, ranging from 1/178 to 1/266 for f.In the following 200 years, geodesists whose names became associated with them made various determinations of the ellipsoid parameters: Airy, Everest, Clarke, Helmert, and Hayford to name a few (6).
Perhaps the most innovative presatellite determination was made by Sir Harold Jeffreys (7). He used the measurements of arc lengths, and he also included measurements based on the changes in pendulum motions as a function of latitude and the precession of the equinoxes. His combination solution gave 297.1for 1/f.

The Shape and Gravity Field of the Earth

Newton’s law of gravity states that the magnitude of the gravitational force F between two particles separated by a distance d is GM1M2/d2, where G is the constant of gravitation and M1 and M2 are the masses of the particles. The mathematical model for the particle is a point mass, that is, it is assumed that all of the mass of the body is concentrated at a point. Gravity is an attractive force, so that each mass experiences a force of  the same magnitude, but the force on M2 is directed toward M1, for example. The acceleration of gravity g experienced by M2 as a falling object is given by GM 1/d2.If M1 represents the Earth and d is the distance between the center of the Earth (assumed as the location of the point mass model), a falling body M2 will experience an acceleration of 9.8m/s2 near the surface of Earth.
The point mass concept applied to the Earth is a serious stretch of the imagination when one views the Earth as a resident on the surface. Nevertheless, it can be shown that a spherical body that has a uniform mass density is gravitationally equivalent to the point mass concept. But if the body is an oblate spheroid, even with uniform density, the force experienced by a point mass external to the spheroid will not correspond exactly to the simple inverse square of the distance law. This statement does not mean that Newton’s law of gravity is invalid.
Consider that the oblate spheroid figure is subdivided into a very large number of point masses or a set of finite mass elements. Each point mass interacts with an external mass, M, in accordance with Newton’s law of gravity. But the total force experienced by M is the result of the summation of the individual finite element contributions. This net force will be different from the force of gravity computed from the total mass of the spheroid, the distance from the spheroid’s center, and the mass M. The source of the difference is caused by the fact that a point mass mathematical model is not a satisfactory representation of an oblate spheroid or a general geometric body that has an arbitrary distribution of interior mass, but the model does apply to the interaction between the external mass and the finite mass elements. It can be shown that a scalar potential function U can be used to describe the gravitational field of a body that results from any mass distribution within the body. The usual expression for U is (8)
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where
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When m = 0, an alternate set of coefficients is often used such that Jl = — Ci>0. These coefficients are referred to as zonal harmonics because the representation divides the gravitational effect into zones of latitude. When l = m, the coefficients are known as sectoral harmonics because the gravity field is divided into sectors of longitude. Finally, all other coefficients are referred to as tesseral harmonics. It is important to note that zonal harmonics are associated with terms that have no longitudinal dependence. For an oblate spheroid of uniform density, the gravitational potential requires only terms of even degree l. Using this scalar function, the force of gravity that a point mass M will experience is
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A fundamental question now arises: What is meant by the shape of the Earth? On the one hand, it is immediately evident that if detailed account is taken of all of the topographic surface variations (mountains, valleys, ravines, human land-forms, and structures), even at the scale of 1-kilometer, there is no simple geometric figure analogous to the oblate spheroid. An alternative description is based on a shape that reflects the mass distribution within the Earth, that is, the gravity field. As it turns out, a simple geometric figure cannot be assigned to either description unless one is not particularly concerned how accurately well the figure represents the actual Earth.
The shape of the Earth defined by the gravity field uses the potential function U. If a surface of constant potential is defined such that U=const everywhere on the surface, the shape of this equipotential surface will serve as a surrogate shape of the physical Earth. The selection of the constant value remains, but if the constant is chosen so that the surface coincides with mean sea level, the surface has a readily understood relevance to everyday experiences. This particular surface is known as the geoid, but the gravitational potential given before must be augmented by a term that will account for the centrifugal force due to the Earth’s rotation. The definition of the geoid is based on an augmented potential W given by U + ^ m2r2 cos2 j, such that W = const = W0, and o is the rate of the Earth’s rotation based on the sidereal day (2p/86164rad/s). The gradient of W, multiplied by the mass of a body at rest on the rotating Earth,yields the weight of the body, which includes the force of gravity as well as the centrifugal force caused by the Earth’s rotation.
The gravitational field of an oblate spheroid, an ellipse of revolution that has uniform density, is represented by the potential function
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where l is an even number. To a reasonable level of approximation, the flattening of the ellipsoid representing the geoid is related to the second-degree zonal harmonic J2 by the relationship
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where K is related to the ratio of centrifugal to gravitational acceleration at the equator, 0.0034498 (9). Although the representation has little practical use in modern geodesy, it illustrates the conceptual relationship between the parameters.

The Space Age

The First Decade. The second renaissance in geodesy began with the launches of the first artificial satellites in the late 1950s. Whereas the determination of the Earth’s shape in the eighteenth century was based on measurements of meridian arcs, the opportunity presented by artificial satellites required understanding the influence of the Earth’s gravity on the satellite orbits. A fundamental question is: How will the orbit of a satellite respond to the mass distribution of an oblate spheroid?
The orbit of a satellite is characterized by six orbital elements: three that describe the orbit geometry and a reference time, plus three that describe the angular orientation of the orbit in space. The former three are semimajor axis, eccentricity, and a reference time at perigee passage. The spatial orientation of the orbit is defined by the location of the ascending node, the inclination of the orbit plane with respect to the equator, and the angular location of perigee (10). The ascending node O is an angle measured eastward at the equator from a fixed direction, usually the vernal equinox, to the point where the satellite crosses the equator from the Southern Hemisphere into the Northern Hemisphere. This angle is usually referred to as the right ascension of the ascending node. If the Earth were a perfect sphere of constant density (gravitational equivalent of a point mass) and the satellite were influenced only by the Earth’s gravity, then the six orbital elements for the satellite orbit would be constant, and the specific elements depend on the position and velocity of the satellite at some time. This is the major result of the classical problem of two bodies.
The mass distribution of an oblate spheroid, for example, perturbs the orbit from the ideal two-body motion. The consequence of this perturbation is that the six orbital elements are functions of time. Time variations in the orbital elements are usually characterized by (1) linear variation in the average value over time; and (2) periodic variations, which are usually dominated by a cyclic change that goes through two cycles in one orbital revolution (a twice per revolution effect). It can be shown that the linear variation, known as secular change, depends on all even-degree zonal harmonics, but in the case of the Earth, the second degree zonal harmonic is of order 10 ~ 3, which is 1000 times larger than any of the other zonal harmonics. Consequently, this degree-two term dominates the description of secular motion. Only the location of the ascending node and perigee exhibit the secular changes that are of interest here. It can be shown that the secular temporal rate of change O in the right ascension of the ascending node of a satellite orbit is
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where the satellite orbit parameters are the mean motion n, semimajor axis a, eccentricity e, and inclination i. The value of J2 will be positive for an oblate spheroid and negative for a prolate spheroid. Consideration of the node rate equation shows that the ascending node will regress for an oblate spheroid and posigrade inclination. In other words, the ascending node will move in a westward direction with respect to the stars.
If the node rate of a satellite is observed and the orbital elements (a, e, i) are known, then the gravity coefficient J2 can be determined from Equation 5. Using J2, the flattening f can be found from Equation 4 that relates f and J2. This is an oversimplified illustration of how the Earth’s shape can be determined from observations of a satellite’s motion.
The launch of Sputnik I by the Soviet Union on 4 October 1957 opened the space age. The satellite orbit decayed because of atmospheric drag and a low perigee and the satellite was destroyed in atmospheric reentry on 4 January 1958. The 83-kg satellite operated on batteries and transmitted temperature data for 23 days in orbit. Even though the orbit of the satellite was not well determined, the westward motion of O was evident, and the satellite provided an early assessment of the prelaunch parameters that described the shape and mass distribution of the Earth.
One month after Sputnik I, the much larger Sputnik II (507 kg) was launched in November 1957, carrying a dog. The solar-illuminated satellite was photographed against the stars, thereby enabling a reasonably accurate determination of the orbit. Early comparisons by Merson and King-Hele (11) of the predicted node rate (based on the best available flattening) with the observed rate suggested a discrepancy in flattening of about 1%, compared to the presat-ellite value.
Although the United States was stunned by the Soviet achievements, work had already been underway to launch a satellite for U.S. participation in the International Geophysical Year. As early as 1954, Major John O’Keefe of the Army Map Service suggested that a satellite would allow studying the size and shape of the Earth and the intensity of its gravitational field (12).
In January 1958, the United States launched Explorer I, followed by Vanguard I on 17 March of the same year. The Vanguard Project was under intense scrutiny because of early launch failures, but the project had been well planned. A network of tracking stations, known as Minitrack, had been deployed that was ready for postlaunch operations, and the use of a state-of-the-art IBM 704 digital computer had been arranged to determine the orbit. The instrumentation carried on the 2-kg satellite included batteries, solar cells, radio transmitters, and temperature sensors. The satellite represented a significant achievement in miniaturization, and the Minitrack system proved capable of providing accurate measurements of angles for determining the orbit by using a radio interfero-metric technique. Nine Minitrack stations were distributed across North and South America, plus a station in Australia and one in South Africa. The objectives of Vanguard I were to ”determine atmospheric density and the shape of the Earth, to evaluate satellite thermal design parameters and to check the life of solar cells in orbit” (13).
In February 1959, using Vanguard I data, O’Keefe (by then associated with NASA), published the surprising result that suggested the existence of J3, which would relate to a pear shape of the Earth. In September 1959, he published values for the zonal harmonics up to degree four, the first such determinations for the third- and fourth- degree zonal coefficients (14,15). The key to determining J3 was the fact that odd-degree zonal harmonics produce asymmetry in a geoid with respect to the equator. Whereas the oblateness of the Earth is characterized by symmetry with respect to the equator, which produces a secular, or linear, change in the location of the ascending node and perigee location of the orbit, the effect of asymmetry is quite different. Examination of the eccentricity variation in the Vanguard I orbit revealed a long-period change of 82 days, the interval required for perigee to make one revolution in response to the oblateness, or J2. O’Keefe estimated that the pear shape corresponded to a 15-meter undulation of the geoid. This undulation compares to the dominant characteristic of J2, which is associated with the fact that the difference between the equatorial and polar axes is about 21 km.
The objectives of Vanguard I were achieved. The pear shape contribution to the shape of the Earth was hailed as one of the major, and perhaps unexpected, results of Vanguard. The refinement of a model for the figure of the Earth continued through the early 1960s, but the studies were limited by a paucity of data. It was fortunate that Vanguard I had been designed with solar cells because the battery failed after a few months in orbit. But Vanguard continued transmitting for several years, thereby enabling observations on the nature of its long-term orbit evolution. It has been predicted that Vanguard I, now quiet, will remain in orbit until well into the twenty-third century or longer.
Thus began the discipline known as Satellite Geodesy. Project Vanguard foretold the requirements for continued improvements in satellite geodesy: satellite instrumentation interacting with a global network of tracking stations to provide accurate observations related to satellite position and digital computers to analyze the observations. In April 1962, the first international symposium on the use of artificial satellites for geodesy was convened in Washington, D.C. (16). The rapidly expanding discipline attracted presentations by almost 50 individuals and gave strong evidence of the synergy with celestial mechanics, geophysics, satellite tracking systems, reference frames, and numerical analysis. The publication by Kaula (17) of a gravity field for the Earth to degree and order eight based on Sputnik II and Vanguard I was an early indicator of the contributions that satellites would make in determining the shape of the Earth as reflected by the geoid. The Earth’s parameters determined by Kaula were equatorial radius = 6378163 m and 1/f = 298.24.
The oblate shape of the Earth and the pear shape produce distinctly different effects on the motion of an artificial satellite. But other unusual effects exist. Even though the first geosynchronous satellite (an orbital period equal to the Earth’s rotational period) did not begin operation until July 1963 (Syncom 2), the special interaction between the ellipticity of the Earth’s equator and satellite motion was highlighted at the 1962 symposium. If the equator was elliptical, there would be only four equilibrium locations where a geosynchronous satellite would remain stationary (the points are on opposite sides along the major and minor axes), although the problem of two bodies predicted an infinite number. As it turned out, the two points along the minor axes are dynamically stable, which implies that the geosynchronous satellite will librate around these equilibrium points. The ellipticity of the equator is represented by the degree two and order two sectoral coefficients (C2j2 and <S2j2), which is further characterized by the equatorial moments and products of inertia of the Earth. Blitzer et al. (18) showed the existence of the equilibrium points and noted that geosynchronous satellites could be especially useful in determining C2j2 and S2j2. The ellipticity of the equator is a 100-meter undulation of the geoid at the equator. It is known now that the major axis of the elliptical equatorial cross-section is oriented at 15° west of the Greenwich meridian. From the dynamic point of view, the orientation of the principal axis determines where the Earth’s equatorial products of inertia are zero.
Through the 1960s, numerous developments occurred in both theory and analysis. In 1966, Kaula published a landmark topic on satellite geodesy. He developed a linear theory that described the influence on orbital motion that results from any term in the gravity field of the Earth. The theory enabled characterizing the perturbations in frequency space, that is, the frequency and period for changes in the satellite orbital elements produced by any degree and order set of coefficients could be readily determined. With this relationship, short-period effects could be removed from the orbital model to elucidate the long-term variations caused by, for example, odd-degree zonal harmonics. And even though the theory did not predict the existence of equilibrium solutions for a geosynchronous satellite, it identified the conditions for both shallow resonance (Kaula’s linear theory is valid) and deep resonance (such as the geosynchronous problem).
As more satellites were launched in the 1960s, a revolution was also taking place in the accuracy of tracking systems. The U.S. Navy launched a system of satellites, initially known as the Navy Navigation Satellite System (NNSS) and later known as Transit, to support the navigation of its fleet of submarines. The tracking system was based on the apparent shift in satellite transmitter frequency associated with the relative motion between the satellite and a radio receiver. During the decade, a new application of lasers was also made, which produced an accurate distance measurement to satellites. Satellite laser ranging (SLR) is based on measuring the time required for a short laser pulse to travel from a ground-based transmitter to a satellite where it is reflected back to the source. The initial experiments with SLR showed that the system could measure the distance from the station to the satellite as accurately as a few meters, but within two decades, the systems were measuring with better than centimeter accuracy.
In the 1960s, several satellites were launched that carried SLR reflector arrays (LRA, laser reflector array), that direct the photons from an illuminating laser back to the source. But the series of Explorer satellites known as GEOS were the first satellites after Vanguard I to focus on geodetic problems. These satellites carried a Doppler tracking system similar to the NNSS, but operating at different frequencies, and a LRA.
By the early 1970s, the gravity field of the Earth had been determined to degree and order 16 (19). The determination of gravity and the corresponding improvements in the shape of the Earth, as reflected by the geoid, were enabled by improvements in the applied technologies foretold by Project Vanguard. The undulations in the geoid introduced by the various degree and order coefficients were found at the few meter level, except for the terms noted previously. Satellite Geodesy from 1970-2000. Still other geodetic satellites were launched in the 1970s whose primary objectives were the gravity field and shape of Earth. To minimize the influence of nongravitational forces (atmospheric drag and solar radiation pressure), these satellites were spheres of small cross-sectional area, but high mass, which gave them a low ratio of area to mass, an important parameter in the magnitude of a nongravitational force. The first such satellite, known as Starlette, was launched by the French space agency in 1975 into a 50° inclination orbit and an altitude of about 1000 km.The surface of the 24-cm diameter passive sphere was covered with laser corner cubes to support ground-based SLR tracking, and the core (constructed of uranium 238) gave the small satellite a mass of about 48 kg. In 1976, NASA launched a 60-cm diameter spherical satellite known as LAGEOS-I (LAser GEOdynamics Satellite) into an orbit that had an altitude of about 5800 km, but whose area to mass ratio was comparable to that of Starlette. This satellite was also covered with laser corner cubes.
In the 1990s, both Starlette and LAGEOS-I were followed by launches of twin satellites into different orbit planes. The Starlette twin, known as Stella, was launched into an inclination of about 98°, and LAGEOS-II was launched into an inclination of 50°. In 1989, the Soviet Union launched spherical geodetic satellites, known as Etalon-I and Etalon-II, into high-altitude orbits similar to those used by the Soviet navigation satellite system, GLONASS. One additional spherical satellite that had a geodetic purpose was launched by Japan, it was known as Ajisai. Still another spherical geodetic satellite was placed into a low-altitude orbit from the Mir space station in April 1995. This German satellite from Geo Forschungs Zentrum Potsdam, known as GFZ-1, was destroyed on reentry in 1999.
A new geodetic tool, a radar altimeter, was tested in Earth orbit on the space station precursor, known as Skylab, in 1974. For the first time, this altimeter directly measured the distance between a satellite-borne instrument and the surface of the Earth by emitting microwave radiation and measuring the time between transmission of the pulse and the arrival of the echo from the Earth’s surface, also known as a ”time of flight” measurement. The radar altimeter carried on Skylab showed the technical feasibility of such instrumentation, and it hinted at the ability to measure directly the shape of the Earth in the ocean regions. Because space-borne radar altimeters are nadir pointed, it is apparent that the altimeter can be used to map the surface after removing the effects of orbital motion and other corrections, such as atmospheric delays.
The Geodynamics Experimental Ocean Satellite GEOS-3 was launched in 1975 to extend the Skylab experiment into a regular operation with radar al-timetry. Tracking data from the GEOS-3 Doppler system and SLR tracking instrumentation, as well as the altimetry data, played a prominent role in determining parameters in updated versions of the gravity field, particularly the Goddard Earth Model series, known as GEM (20).
The GEOS-3 launch was followed by a sequence of satellites that carried radar altimeters: Seasat in 1978, Geosat in 1985, ERS-1 in 1991, TOPEX/POSEIDON in 1992, and ERS-2 in 1995. These altimeters were all designed for optimal performance over oceans, large inland seas, and lakes. The parameters that describe the oblate spheroid character of the Earth could be determined by a least-squares fit of an ellipsoid to the ever increasing volumes of altimeter data in the ocean areas that accounted for 70% of the Earth’s surface. Whereas Newton’s prediction about the oblate nature of the Earth was originally confirmed by arc measurements in Lapland and Ecuador, direct confirmation from most of the ocean areas was made by satellite altimetry. Beyond this aspect, satellite al-timetry has revolutionized oceanography, as described by Fu and Cazenave (21). Images of the ocean surface created from altimeter data revealed bathymetric features, including seamounts and ocean trenches, gravitational reflections of the ocean bottom on the ocean surface.
By the start of the twenty-first century, based primarily on the body of geodetic data collected for 40 years, the International Association of Geo-
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shape during the period from 1960 to 2000. There is no evidence to support this suggestion. What has taken place in this interval is a significant improvement in the observational accuracy, so that the changes in the Earth model parameters reflect a convergence toward more accurate values rather than a change in the physical Earth.
In the reflection of the shape of the Earth through the gravity field and the corresponding geoid, dramatic improvements have been achieved in both the accuracy and resolution of gravity (and geoid) models derived from altimeter data. The most common gravity fields in use in 2001 were JGM-3 (23), EGM-96 (24), and GRIM-4 (25). As one representative model of the shape of the Earth, Fig. 1 shows the geoid computed from the gravity model of JGM-3, based on coefficients to degree and order 70, which corresponds to a horizontal resolution of about 600km.This figure shows a ”dip” in the geoid in the Indian Ocean of about 115 meters with respect to the reference ellipsoid. Likewise, a rise in the surface can be seen in the North Atlantic region. The lack of strong correlation between the geoid and surface topography is evident, although some correlation exists, such as in the Andes, because the satellite data have been augmented by surface gravity measurements. The difference between the geoid and surface topography in the Indian Ocean, for example, can be explained by the reminder that the geoid represents the shape of the gravity field, which, in turn, depends on subsurface mass distribution that does not always correlate well with surface features on the Earth.
The shape of the Earth is illustrated by the geoid, a surface determined from the JGM-3 gravity field of the Earth. The geoid, which is a reflection of the mass distribution within the Earth, coincides with mean sea level in the ocean regions. The figure shows departures of the geoid from an oblate spheroid, or ellipse of revolution. Though the geoid is a suitable surrogate for the physical shape of the Earth in the ocean areas, local land topography (such as Mt. Everest) must be added to represent fully the topographic characteristics in landmass areas. This figure is available in full color at http://www.mrw. interscience wiley com/esst
Figure 1. The shape of the Earth is illustrated by the geoid, a surface determined from the JGM-3 gravity field of the Earth. The geoid, which is a reflection of the mass distribution within the Earth, coincides with mean sea level in the ocean regions. The figure shows departures of the geoid from an oblate spheroid, or ellipse of revolution. Though the geoid is a suitable surrogate for the physical shape of the Earth in the ocean areas, local land topography (such as Mt. Everest) must be added to represent fully the topographic characteristics in landmass areas.
In summary, the shape of the Earth cannot be described with high accuracy using a simple geometric figure. At least in the ocean areas, the mean sea surface closely coincides with the geoid (within 7 125 meters everywhere). The instantaneous ocean surface departs from the geoid because of temporal effects (e.g., tides and meteorology) at the few meter level. For landmasses, the physical topography of the surface exhibits large variations in elevation from sea level to Mt. Everest (9 km) and other striking changes in elevation over short horizontal distances (e.g., the Grand Canyon). Nevertheless, the geoid surface is contained within 7 125 meters of the oblate spheroid given by Groten (22), even in land areas. For applications that require a highly accurate representation of the surface topography instead of the geoid surface, accounting for the local land topography can be applied, such as a 1-kilometer resolution Digital Elevation Model (26).
The Future. The confluence of several technologies in the late 1950s enabled dramatic improvements in describing the Earth’s shape: artificial satellites,networks of ground-based satellite tracking systems, space-borne radar altimeters, and high-speed computers. By the end of the twentieth century, new technologies had been developed to support the ever increasing accuracies required for scientific investigations.
Applications of satellites to the determination of the size and shape of the Earth were enabled by the Minitrack system at the beginning of the space age, but by the beginning of the twenty first century the Global Positioning System (GPS) was playing an increasingly prominent role. GPS receivers carried on satellites in low Earth orbit (LEO) provide global and continuous observations of the orbital perturbations introduced by the Earth’s gravity field and other forces. The GPS receiver carried on TOPEX/POSEIDON contributed to the JGM-3 gravity model, for example, and such receivers carried on other LEO satellites will contribute to further improvements in knowledge of the size and shape of the Earth. Geodesy itself is undergoing a revolution because of GPS, with a variety of applications in the Earth sciences that require determination of position with an accuracy at the centimeter-level or better.
The most detailed descriptions of the topography of Mars were obtained by using a laser altimeter carried on the Mars Surveyor in the closing years of the twentieth century. The altimeter, known as MOLA (Mars Orbiter Laser Altimeter), provided the most detailed topographic map of Mars to date. Although similar in principle to the ”time of flight” radar altimeter measurement, the narrow beam laser allows surface averaging over much smaller areas within the beam, an important factor over rough terrain. Analysis of MOLA data showed a dramatic difference in surface elevation between the high Southern and low Northern Hemispheres of Mars (27), which suggests a dichotomy in the evolution of each. In terms of a simple geometric figure, Mars can be reasonably described as a triaxial ellipsoid that has 2- to 5-kilometer differences in the axes. The center of figure is offset from the center of mass by 3 km in the polar axis direction. The solar system is ripe for applications to other celestial bodies in the twenty-first century.
Mapping of the Earth’s topography is undergoing a revolution with instrumentation that provides improved horizontal and vertical resolution. In 2000, a Space Shuttle mission carried the Shuttle Radar Topography Mission, a radar interferometric system designed to map most of the land surface between 56° S and 60° N. A digital elevation model that has 30-m horizontal resolution and 15-m vertical accuracy will be produced from the data (28).
Whereas the geodetic emphasis in the first two decades of the space age was on characterizing the Earth’s shape, the emphasis at the end of the twentieth century was moving toward detecting topographic change and scientific interpretation of the changes. The twenty-first century will bear witness to the application of new technologies to detect change at even greater accuracy. For example, the Geoscience Laser Altimeter System will begin operations in 2003 on the Ice, Cloud and Land Elevation Satellite (ICESat) to provide highly accurate elevations of the land surface, especially to detect topographic change in the major ice sheets of Greenland and Antarctica (29). The growth or diminishment of the ice sheets has a direct relation to sea level and climate change.
It took almost 2000 years to refine human knowledge of the shape of the Earth from a sphere to an ellipse of revolution. After the first artificial satellites were launched, the next refinement represented by the pear shape took only two years. In successive years since the first artificial satellites were launched, the shape of the Earth has been refined into high-resolution digital models. Determinations of temporal changes in the topographic characteristics that were begun in the late twentieth century will blossom in the twenty-first century as new and more accurate measuring tools are developed. The tools will provide the fundamental measurements, but scientific interpretation of temporal changes in topography and the relation of topography to other phenomena, such as the evolution of El Nino in the Earth’s oceans and the surface evolution of celestial bodies, will be the areas emphasized in the twenty-first century.

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