Geodesy

Geodesy, also called geodetics, is the scientific discipline that deals with the measurement and representation of the earth, its gravitational field and geodynamic phenomena (polar motion, earth tides, and crustal motion) in three-dimensional time varying space. Geodesy is primarily concerned with positioning and the gravity field and geometrical aspects of their temporal variations, although it can also include the study of the Earth's magnetic field.

Wolfgang Torge quotes in his 2001 textbook Geodesy (3rd edition) Friedrich Robert Helmert as defining geodesy as "the science of the measurement and mapping of the earth's surface."

As Torge also remarks, the shape of the earth is to a large extent the result of its gravity field. This applies to the solid surface (orogeny; few mountains are higher than 10 km, few deep sea trenches deeper than that). It affects similarly the liquid surface (dynamic sea surface topography) and the earth's atmosphere. For this reason, the study of the Earth's gravity field is seen as a part of geodesy, called physical geodesy.

Geoid and reference ellipsoid

The geoid is essentially the shape of the earth abstracted from its topographic features. It is an idealized equilibrium surface. The geoid, unlike the ellipsoid, is too complicated to serve as the computational surface on which to solve geometrical problems like point positioning.

A reference ellipsoid, customarily chosen to be the same size (volume) as the geoid, is described by its semi-major axis (equatorial radius) $a$ and flattening $f$. The quantity $f=(a-b)/a$, where $b$ is the semi-minor axis (polar radius) is a purely geometrical one. The mechanical ellipticity of the earth (dynamical flattening) is determined by observation and differs from the geometrical because the earth is not of uniform density.

The geoid is an irregular surface. The geometrical separation between it and the reference ellipsoid is called the geoidal undulation. It varies globally between $\pm$ 110 m.

The 1967 Geodetic Reference System posited a 6,378,160 m semi-major axis and a 1:298.247 flattening. The 1980 Geodetic Reference System (GRS80) posited a 6,378,137 m semi-major axis and a 1:298.257 flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics (IUGG).

Numerous other systems have been used by diverse countries for their maps and charts. The 1979 International Astronomic Union (IAU) values are 6,378,140 m and 1:298.257.

Co-ordinate systems in space

The locations of points in three-dimensional space are most conveniently described by three cartesian or rectangular co-ordinates, $X,Y$ and $Z$. Since the advent of satellite positioning, such co-ordinate sytems are typically geocentric: the $Z$ axis is aligned with the Earth's (conventional or instantaneous) rotation axis, while the $X$ axis lies within the Greenwich observatory's meridian plane.

Before the satellite geodesy era, the co-ordinate systems associated with geodetic datums attempted to be be geocentric, but their origins differed from the geocentre by hundreds of metres, due to regional deviations in the direction of the plumbline (vertical). These regional geodetic datums, such as ED50 (European Datum 1950) or NAD83 (North American Datum 1983) have ellipsoids associated with them that are regional 'best fits' to the geoids within their areas of validity, minimising the deflections of the vertical over these areas.

It is only because GPS satellites orbit about the geocentre, that this point becomes naturally the origin of a co-ordinate system defined by satellite geodetic means, as the satellite positions in space are themselves computed in such a system.

Co-ordinate systems in the plane

In surveying and mapping, important aspects of geodesy, two general types of co-ordinate systems are used in the plane:

1. Plano-polar, in which points in a plane are defined by distance from a specified point along a ray having a specified direction with respect to a base line or axis;
2. Rectangular, points are defined by distances from two perpendicular axes called x and y. It is geodetic practice -- contrary to the mathematical convention -- to let the $x$ axis point to the North and the $y$ axis to the East.

Rectangular co-ordinates in the plane can be used intuitively with respect to one's current location, in which case the x axis will point to the local North. More formally, such co-ordinates can be obtained from three-dimensional co-ordinates using the artifice of a map projection. It is not possible to map the curved surface of the Earth onto a flat map surface without deformation. The compromise chosen -- called a conformal projection -- is most often to preserve angles and length ratios, so small spheres are mapped as small spheres and small squares as squares.

An example of such a projection is UTM (Universal Transverse Mercator). Within the map plane, we have rectangular co-ordinates $x$ and $y$. In this case the North direction used for reference is the map North, not the local North. The difference between the two is called meridian convergence.

It is easy enough to "translate" between polar and rectangular co-ordinates in the plane: let direction and distance be $\alpha$ and $s$ respectively, then we have

${\begin{matrix}x&=&s\cos \alpha ,\\y&=&s\sin \alpha .\end{matrix}}$

The reverse translation is slightly more tricky.

Geodetic heights are "above sea level", an irregular, physically defined surface. Therefore a geodetic height should not be referred to as a co-ordinate. It is more like a physical quantity, and though it can be tempting to treat height as the vertical co-ordinate z, in addition to the horizontal co-ordinates x and y, and though this actually is a good approximation of physical reality in small areas, it becomes quickly invalid in larger areas.

Geodetic datums

Because geodetic point co-ordinates (and heights) are always obtained in a system that has been constructed itself using real observations, we have to introduce the concept of a geodetic datum: a physical realization of a co-ordinate system used for describing point locations. The realization is the result of choosing conventional co-ordinate values for one or more datum points.

In the case of height datums, it suffices to choose one datum point: the reference bench mark, typically a tide gauge at the shore. Thus we have vertical datums like the NAP (Normaal Amsterdams Peil), the North American Vertical Datum 1988 (NAVD88), the Kronstadt datum, the Trieste datum, etc.

In case of plane or spatial co-ordinates, we typically need several datum points. A regional, ellipsoidal datum like ED50 can be fixed by prescribing the undulation of the geoid and the deflection of the vertical in one datum point, in this case the Helmert Tower in Potsdam. However, an overdetermined ensemble of datum points can also be used.

Changing the co-ordinates of a point set referring to one datum, to make them refer to another datum, is calleda datum transformation. In the case of vertical datums, this consists of simply adding a constant shift to all height values. In the case of plane or spatial co-ordinates, datum transformation takes the form of a similarity or Helmert transformation, consisting of a rotation and scaling operation in addition to a simple translation. In the plane, a Helmert transformation has four parameters, in space, seven.

Point positioning

Point positioning is the determination of the coordinates of a point on land, at sea, or in space with respect to a coordinate system. Point position is solved by compution from measurements linking the known positions of terrestrial or extraterrestrial points with the unknown terrestrial position. This may involve transformations between or among astronomical and terrestrial coordinate systems.

The known points used for point positioning can be, e.g., triangulation points of a higher order network, or GPS satellites.

Traditionally, a hierarchy of networks has been built to allow point positioning within a country. Highest in the hierarchy were triangulation networks. These were densified into networks of polygons, into which local mapping surveying measurements, usually with measuring tape, corner prism and the familiar red and white poles, are tied.

Nowadays all but special measurements (e.g., underground or high precision engineering measurements) are performed with GPS. The higher order networks are measured with static GPS, using differential measurement to determine vectors between terrestrial points. These vectors are then adjusted in traditional network fashion. A global polyhedron of permanently operating GPS stations under the auspices of the IERS is used to define a single global, geocentric reference frame which serves as the "zeroth order" global reference to which national measurements are attached.

For surveying mappings, frequently Real Time Kinematic GPS is employed, tying in the unknown points with known terrestrial points close by in real time.

One purpose of point positioning is the provision of known points for mapping measurements, also known as (horizontal and vertical) control. In every country, thousands of such known points exist in the terrain and are documented by the national mapping agencies. Constructors and surveyors involved in real estate will use these to tie their local measurements to.

Geodetic problems

In geometric geodesy we formulate two standard problems: the geodetic principal problem and the geodetic inverse problem.

Geodetic principal problem

Given a point (in terms of its co-ordinates) and the direction (azimut) and distance from that point to a second point, determine (the co-ordinates of) that second point.

Geodetic inverse problem

Given two points, determine the azimut and length of the line (straight line, great circle or geodesic) that connects them.

In the case of plane geometry (valid for small areas on the Earth's surface) the solutions to both problems reduce to simple trigonometry. On the sphere, the solution is significantly more complex, e.g., in the inverse problem the azimuths will differ between the two end points of the connecting great circle arc.

On the ellipsoid of revolution, closed solutions do not exist; series expansions have been traditionally used that converge rapidly. Alternatively, the differential equations for the geodesic can be solved numerically, e.g., in MatLab(TM).

Geodetic observational concepts

Here we define some basic observational concepts, like angles and co-ordinates, defined in geodesy (and astronomy as well) from the viewpoint of the local observer.

• The plumbline or vertical is the direction of local gravity. It is slightly curved.

• The zenith is the point on the celestial sphere where the direction of the gravity vector in a point, extended upwards, intersects it. More correct is to call it a <direction> rather than a point.

• The nadir is the opposite point (or rather, direction), where the direction of gravity extended downward intersects the (invisible) celestial sphere.

• The celestial horizon is a plane perpendicular to a point's gravity vector.

• Azimut is the direction angle within the plane of the horizon, typically counted clockwise from the North (in geodesy) or South (in astronomy and France).

• Elevation is the angular height of an object above the horizon, Alternatively zenith distance, being equal to 90 degrees minus elevation.

• Local topocentric co-ordinates are azimut (direction angle within the plane of the horizon) and elevation angle (or zenith angle) as well as distance if known.

• The North celestial pole is the extension of the Earth's (precessing and nutating) instantaneous spin axis extended Northward to intersect the celestial sphere. (Similarly for the South celestial pole.)

• The celestial equator is the intersection of the (instantaneous) Earth equatorial plane with the celestial sphere.

• An astronomical meridian is any plane perpendicular to the celestial equator and containing the celestial poles.

• The local meridian is the plane containing the direction to the zenith and the direction to the celestial pole.

• Celestial co-ordinates are rightascension (longitude along the celestial equator) and declination (latitude from the celestial equator).

Zero right ascension is the position of the Sun at the instant of vernal equinox -- the beginning of spring, when the Sun crosses the equatorial plane from South to North.

Units and measures on the ellipsoid

Geographical latitude and longitude are stated in the units degree, minute of arc, and second of arc. They are angles, not metric measures, and describe the direction of the local normal to the reference ellipsoid of revolution. This is approximately the same as the direction of the plumbline, i.e., local gravity, which is also the normal to the geoid surface. For this reason, astronomical position determination, measuring the direction of the plumbline by astronomical means, works fairly well provided an ellipsoidal model of the figure of the Earth is used.

A geographic mile, defined as one minute of arc on the equator, equals 1,855.32571922 m. A nautical mile is one minute of astronomical latitude. The radius of curvature of the ellipsoid varies with latitude, being the longest at the pole and shortest at the equator as is the nautical mile.

A metre was originally defined as the 40 millionth part of the length of a meridian. This means that a kilometre is equal to (1/40,000) * 360 * 60 meridional minutes of arc, which equals 0.54 nautical miles. Similarly a nautical mile is on average 1/0.54 = 1.85185... km.

See also

• History of Geodesy
• Geodesy for the layman
• Important publications in geodesy

External Links

• The Geodesy Page.