Geodesy

Geodesy is the science concerned with the exact size and shape of the surface of the Earth. It also involves the study of variations of the Earth's gravity.
These precise knowledge and measurements were unimportant to early navigators, because of the relative inaccuracy of their methods. The precision of today's navigation systems and the global nature of satellite and other long-range positioning methods demand a more complete understanding of geodesy.

The Shape of the Earth

The Greek philosophers were the first to theorize that the Earth was round. However, in their speculation and theorizing, the shape of the Earth ranged from the flat disc advocated by Homer to Pythagoras' spherical figure - an idea supported one hundred years later by Aristotles. Pythagoras was a mathematician and to him the most perfect figure was a sphere. He reasoned that the gods would create a perfect figure and therefore the Earth was created to be spherical in shape. Anaximenes, an early Greek scientist, believed strongly that the Earth was rectangular in shape.

Since the spherical shape was the most widely supported during the Greek Era, efforts to determine its size followed. Plato determined the circumference of the Earth to be 400 000 stadia while Archimedes estimated 300 000 stadia. Plato's figure was a guess and Archimedes' a more conservative approximation. Meanwhile, in Alexandria (Egypt), a Greek scholar and philosopher, Eratosthenes, set out to make more explicit measurements.
Eratosthenes born 275 BC in Cyrene - then Greece now in Lybia - studied at Alexandria and Athens and became director of the Great Library at Alexandria in 236 BC. Inspired by his readings of the works of Posidonius, Eratosthenes was the first who tried to experimentally estimate the dimensions of the Earth.

From his readings he had learnt that once a year - on the day of the summer solstice - the bottom of a well situated in Syene (now Aswan on the Nile in Upper Egypt) was illuminated by the Sun. However, in Alexandria, this never happened. Eratosthenes then set up the following experiment: he assumed that Alexandria and Syene were on the same Meridian (the difference in longitude is actually around 3°) and he postulated that the Sun is far enough away from the Earth such that the sunlight reaches the Earth as parallel beams (an idea that was already commonly held by ancient Greek mathematicians).

Further he knew from the trading caravans that the distance from Syene to Alexandria was 5000 stadia. 100 stadia was the distance an average caravan of camels would travel in one day. Although our idea of the exact value of the stadium - which was not the same at Athens, Alexandria or Rome - is fairly vague, it is believed to be around 180 metres.

On the summer solstice day (around the 21th of June) at local noon, Eratosthenes measured the length of a gnomon (probably he used an obelisk with a known height) at Alexandria. The measurement showed that the length of the shadow was 1/8th of the height of the gnomon, yielding an incident angle of 7.12°. From this he concluded that the circumference of the Earth must be 360°/7.12° = 50.6 times the distance Syene-Alexandria. Eratosthenes had assumed that this distance was 5000 stadia, fixing the terrestrial circumference to 252800 stadia. Using the consensus value of 180 metres for one stadium, this corresponds to 45500 km. Today we know, the distance Aswan-Alexandria is about 840 km, with a resulting circumference of 42400 km. So with this simple experiment - and some luck, because some errors advantageously cancelled out - Eratosthenes obtained a reasonable good value for the size of the Earth.

Ancient Greek philosophers concluded that the Earth could only be a sphere because that, in their opinion, was the "most perfect" shape. Today we know that the shape of the Earth is much more complex and consists of a very complex and irregular topographic surface.
The topographic surface is generally the concern of topographers and hydrographers. The irregular shape of the topographic surface is simplified in a first step by defining a geoid. The geoid is a surface along which gravity is constant and to which the direction of gravity is perpendicular. The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular. The later is particularly significant because optical instruments containing levelling devices are commonly used to make geodetic measurements.

The geoid is that surface to which the oceans would conform over the Earth if free to adjust to the combined effect of the Earth's mass attraction and the centrifugal force of the Earth's rotation. Uneven distribution of the Earth's mass makes the geoidal surface irregular. The surface of the geoid, with some exceptions, tends to rise under mountains and to dip above ocean basins.

The geoid refers to the actual size and shape of the Earth, but such an irregular surface has serious limitations as a mathematical Earth model. For mapping and charting purposes, it is necessary to use a regular geometric shape which closely approximates the shape of the geoid either on a local or global scale, and which has a specific mathematical expression. This "mathematical" model is called the ellipsoid.

Geodetic Systems

Since the Earth is in fact flattened slightly at the poles and bulges somewhat at the equator, the geometrical figure used in geodesy to most nearly approximate the shape of the Earth is an ellipsoid of revolution. The ellipsoid of revolution is the figure which would be obtained by rotating an ellipse about its shorter axis.

An ellipsoid of revolution is uniquely defined by two parameters. Geodesists, by convention use the parameters "semi-major axis" and "flattening". The size of the ellipsoid is determined by the semi-major axis, which will be the Earth radius at the Equator. The shape is given by the flattening, which indicates how closely an ellipsoid approaches a perfect spherical shape. The flattening for the Earth is about 1/300 and the ratio of the two axis of the ellipsoid is about 299/300.

Since the ellipsoid is used to approximate the irregular surface of the geoid, an ellipsoid can provide only a good approximation for a part of the geoidal surface. The ellipsoid that fits best in North America is different from the ellipsoid that fits best for Europe. Therefore a number of different reference ellipsoids are used in geodesy and mapping.

The ellipsoids 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. The international ellipsoid was developed by Hayford in 1910 and adopted by the International Union of Geospatial Sciences Division (IUGG) which recommended it for international use.

At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 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 Geodetic Reference System 1967 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 South America for the South American Datum 1969.

The ellipsoid called GRS-80 (Geodetic Reference System 1980) was approved and adopted at the 1979 meeting of the IUGG held in Canberra, Australia. The ellipsoids used to define WGS 66 and WGS 72 are discussed in ...

Notice: Referencing geodetic coordinates to the wrong datum can result in position errors of hundreds of meters!

Global Reference Systems and Reference Frames

An important underlying concept is that definitions of reference systems are purely definitions and must be "realised" through some defined process. At the most fundamental level, two types of reference systems are of interest. The first is the Celestial Reference System (CRS) which is a space fixed system to which the positions of celestial objects are referred.

The second reference system of relevance is the Conventional Terrestrial Reference System (CTRS). The International Terrestrial Reference System (ITRS) is a particular realisation of the CTRS. The ITRS has the following characteristics:

• The origin is at the centre of mass of the whole Earth including the oceans and atmosphere. The unit of length is the metre.

• The orientation of its axes is consistent with that of the Bureau International de l'Heure (noe IERS) at the beginning of 1984.

• Changes in orientation over time are such that there is no residual rotation with respect to the horizontal movement of the Earth's crust.

The International Earth Rotation Service (IERS) has been established since 1988 jointly by the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). The IERS mission is to provide to the worldwide scientific and technical community reference values for Earth orientation parameters and reference realizations of internationally accepted celestial and terrestrial reference systems.

The IERS is in charge to realize, use and promote the International Terrestrial Reference System (ITRS) as defined by the IUGG resolution No 2 adopted in Vienna,1991.

In the geodetic terminology, a reference frame is a set of points with their coordinates which realize an ideal reference system. The frames produced by IERS as realizations of ITRS are named International Terrestrial Reference Frames (ITRF).
Three particularly relevant realisations of the ITRS are the

• International Terrestrial Reference Frame (ITRF),

• WGS84 as used for GPS and

• PZ90 as used for GLONASS.

WGS-84

WGS-84 is an Earth fixed global reference frame, including an Earth model, which was established by the US Defense Mapping Agency (now the National Imaging and Mapping Agency, NIMA). It is defined by a set of primary and secondary parameters:

• the primary parameters define the shape of an Earth ellipsoid, its angular velocity, and the Earth mass which is included in the ellipsoid reference

• the secondary parameters define a detailed gravity model of the Earth.

These additional parameters are needed because WGS-84 is used not only for defining coordinates in surveying, but, for example, also for determining the orbits of GPS navigation satellites.

The significance of WGS-84 comes about because GPS receivers rely on WGS-84. The satellites send their positions in WGS-84 as part of the broadcast signal recorded by the receivers (the so-called Broadcast Ephemeris) and all calculations internal to receivers are performed in WGS-84.

From a technical point of view, WGS-84 is a particular realization of the CTRS (conventional terrestrial reference system). It is established by the National Imagery and Mapping Agency (NIMA) of the US Department of Defence. The initial realization of WGS-84 relied on Transit System observations and was only accurate at the meter level. Since 1994 (start of GPS Week 730) the use of a revised value of the gravitation constant along with improved coordinates for the Air Force and NIMA GPS tracking stations led to the WGS-84 G730 geodetic system. That realization was shown to be consistent with the ITRF (International terrestrial reference frame) at the 10 centimetre level.

Further improvements to the tracking station coordinates in 1996 led to WGS-84 G873. The G873 coordinates were implemented in the GPS Operational Control Segment on 29 January 1997. Tests have shown WGS-84 G873 to be coincident with the ITRF94 at a level better than 2cm.

It should also be noted that the ellipsoid used for WGS-84 agrees with that of the Geodetic Reference System of 1980 (GRS-80) except for a very small difference in the flattening term. GRS80 is the reference ellipsoid associated with ITRF.

It should be noted that there are only two ways to directly produce WGS-84 coordinates. The first is by GPS surveying measurements relative to the US DoDs GPS tracking stations. However, the GPS data from those DoD stations is not typically available to civilians. The second way is by point positioning using a GPS receiver. However, the accuracy of point positions performed by civilians is limited by the policy of Selective Availability to +/- 100m at 95% confidence. Only US DoD or allied military agencies can perform point positioning with centimetre to decimetre accuracy.

Civilian surveyors often require WGS-84 coordinates to an accuracy better than that available from point positioning. For example, a common requirement for accurate WGS-84 coordinates is to seed the processing of GPS surveying baselines. However as outlined above, civilians cannot access WGS-84 directly with high accuracy and must therefore resort to indirect means to produce WGS-84 compatible coordinates.

One way to obtain more accurate WGS-84 compatible coordinates is to use local datum coordinates and a published transformation process. In practice, a transformation process is derived between data sets on both datum and any errors in those data sets affect the transformation process. The quasi WGS84 coordinates that result from a transformation process can be in error in an absolute sense by as much as several metres but are usually more accurate in a relative sense. Transformation processes in common use include the three parameter Molodensky method (or block shift), seven parameter (or similarity) transformation, multiple regression equations and surface fitting approaches.

The most rigorous way for civilian surveyors to produce WGS84 compatible coordinates is to perform GPS surveying measurements relative to control stations with published ITRF coordinates. That will produce ITRF coordinates for any new stations. As outlined above, ITRF94 (or later) coordinates can then be claimed to be

An important mechanism allowing the ITRF to be accessible for geodetic networks anywhere in the world is the ability to access precise ephemeris for the GPS satellites and precise station coordinates from the International GPS for Geodynamics Service (IGS). The IGS has a global network of stations with high quality receivers observing GPS continuously (Zumberger et al 1995).

Given widespread use of GPS, there is a trend for the working geodetic datum to be consistent with recent ITRF and therefore with WGS84. This trend was set with the North American Datum of 1983 as a geocentric datum using the GRS80 ellipsoid. Recent implementations have taken advantage of the continued development of the various ITRF (e.g. for European developments see Seeger, 1994). Australia is also progressing toward adoption of an ITRF based geocentric datum by the year 2000 (Manning and Harvey, 1994). In such cases where the modern geodetic datum is based on a recent ITRF it will be compatible with WGS84 at the few centimetre level.