Earth ellipsoid

Ellipsoid parameters

In geophysics, geodesy, and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used. For bodies that cannot be well approximated by an ellipsoid of revolution a triaxial (or scalene) ellipsoid is used.

The shape of an ellipsoid of revolution is determined by the shape parameters of that ellipse. The semi-major axis of the ellipse, a, becomes the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b, becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid.

In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius) a and the flattening f, defined as:

${\displaystyle f=\frac{a-b}{a}.}$

That is, f is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/m; m = 1/f then being the "inverse flattening". A great many other ellipse parameters are used in geodesy but they can all be related to one or two of the set a, b and f.

A great many ellipsoids have been used to model the Earth in the past, with different assumed values of a and b as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the Earth's center of mass and of its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids.

The ellipsoid WGS-84, widely used for mapping and satellite navigation has f close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to a difference of the major and minor semi-axes of approximately 21 km (13 miles) (more precisely, 21.3846857548205 km). For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825, while Jupiter is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto, is highly flattened, with f between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of the equatorial.

Determination

Arc measurement is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a reference ellipsoid. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the radii of curvature so obtained would be related to the equatorial radius and the polar radius, respectively a and b (see: Earth polar and equatorial radius of curvature). Then, the flattening would readily follow from its definition:

${\displaystyle f=(a-b)/a}$.

For two arc measurements each at arbitrary average latitudes ${\displaystyle {\phi }_i}$, ${\displaystyle i=1,2}$, the solution starts from an initial approximation for the equatorial radius ${\displaystyle a_0}$ and for the flattening ${\displaystyle f_0}$. The theoretical Earth's meridional radius of curvature ${\displaystyle M_0({\phi }_i)}$ can be calculated at the latitude of each arc measurement as:

${\displaystyle M_0({\phi }_i)=\frac{a(1-e^2)}{(1-e_0^2{\sin }^2{\phi }_i{)}^{\frac{3}{2}}}}$

where ${\displaystyle e_0^2=2f_0-f_0^2}$. Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as ${\displaystyle \delta M_i=M_i-M_0({\phi }_i)}$. Finally, corrections for the initial equatorial radius ${\displaystyle \delta a}$ and the flattening ${\displaystyle \delta f}$ can be solved by means of a system of linear equations formulated via linearization of ${\displaystyle M}$:

${\displaystyle \delta M_i\approx \delta a(\mathrm{\partial }M/\mathrm{\partial }a)+\delta f(\mathrm{\partial }M/\mathrm{\partial }f)}$

where the partial derivatives are:

${\displaystyle \mathrm{\partial }M/\mathrm{\partial }a\approx 1}$

${\displaystyle \mathrm{\partial }M/\mathrm{\partial }f\approx -2a_0(1-1.5{\sin }^2{\phi }_i)}$

Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares adjustment. The parameters determined are usually the semi-major axis, ${\displaystyle a}$, and any of the semi-minor axis, ${\displaystyle b}$, flattening, or eccentricity.

Regional-scale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical, as explored in astrogeodetic leveling.

Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem.

Modern geodesy no longer uses simple meridian arcs or ground triangulation networks, but the methods of satellite geodesy, especially satellite gravimetry.

Geodetic coordinates

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal height h (also known as geodetic height). The triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates).

History

In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate") ellipsoid of revolution, generated by an ellipse rotated around its minor diameter; a shape which he termed an oblate spheroid.

In 1669, Jean Picard found the first accurate and reliable value for the radius of Earth as 6,365.6 kilometres. Picard's geodetic observations had been confined to the determination of the magnitude of the Earth considered as a sphere, but the discovery made by Jean Richer turned the attention of mathematicians to the Earth's deviation from a spherical form. Christiaan Huygens found out the centrifugal force which explained variations of gravitational acceleration depending on latitude.

In 1743, Alexis Clairaut proposed a theorem which suggested that the study of variations in gravitational acceleration was a way to determine the figure of the Earth, whose crucial parameter was the flattening of the Earth ellipsoid. Towards the end of the 18th century, the geodesists sought to reconcile the values of flattening drawn from the measurements of meridian arcs with that given by Clairaut's theorem drawn from the measurement of gravity. The Weights and Measures Commission would, in 1799, adopt a flattening of ⁠1/334⁠ based on analysis by Pierre-Simon Laplace who combined the arc of Peru and the data of the meridian arc of Delambre and Méchain.

Historical Earth ellipsoids

The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for the individual who derived them and the year of development is given. In 1887 the English surveyor Colonel Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.

At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 (Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of the IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the South American Datum 1969.

The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid) ${\displaystyle a}$, total mass ${\displaystyle GM}$, dynamic form factor ${\displaystyle J_2}$ and angular velocity of rotation ${\displaystyle \omega }$, making the inverse flattening ${\displaystyle 1/f}$ a derived quantity. The minute difference in ${\displaystyle 1/f}$ seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to differ slightly from the GRS-80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for ${\displaystyle J_2}$, was truncated to eight significant digits in the normalization process.

An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing geodetic datum. For example, the older ED-50 (European Datum 1950) is based on the Hayford or International Ellipsoid. WGS-84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct.

Note that the same ellipsoid may be known by different names. It is best to mention the defining constants for unambiguous identification.