Geodesy


Geodesy or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D space. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems. Geodetic job titles include geodesist and geodetic surveyor.
Through highly accurate observations, geodesy provides the scientific basis for mapping, navigation, and positioning, and supports applications such as infrastructure development, natural resource management, mineral exploration, and geophysics. Its measurements underpin modern geospatial reference frames used in transportation, satellite systems, global trade, and timekeeping.
Geodynamic phenomena, including crustal motion, tides, and polar motion, are studied through global and national control networks, space geodesy and terrestrial geodetic techniques, and the use of datums and coordinate systems.

History

Geodesy began in pre-scientific antiquity, so the very word geodesy comes from the Ancient Greek word γεωδαισία or geodaisia.
Early ideas about the figure of the Earth held the Earth to be flat and the heavens a physical dome spanning over it. Two early arguments for a spherical Earth were that lunar eclipses appear to an observer as circular shadows and that Polaris appears lower and lower in the sky to a traveler headed South.

Definition

Geodesy refers to the science of measuring and representing geospatial information, while geomatics encompasses practical applications of geodesy on local and regional scales, including surveying.
Geodesy originated as the science of measuring and understanding Earth's geometric shape, orientation in space, and gravitational field; it is now also applied to other astronomical bodies in the Solar System.
To a large extent, Earth's shape is the result of rotation, which causes its equatorial bulge, and the competition of geological processes such as the collision of plates, as well as of volcanism, resisted by Earth's gravitational field. This applies to the solid surface, the liquid surface, and Earth's atmosphere. For this reason, the study of Earth's gravitational field is called physical geodesy.

Geoid and reference ellipsoid

The geoid essentially is the figure of Earth abstracted from its topographical features. It is an idealized equilibrium surface of seawater, the mean sea level surface in the absence of currents and air pressure variations, and continued under the continental masses. Unlike a reference ellipsoid, the geoid is irregular and too complicated to serve as the computational surface for solving geometrical problems like point positioning. The geometrical separation between the geoid and a reference ellipsoid is called geoidal undulation, and it varies globally between ±110 m based on the GRS 80 ellipsoid.
A reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f. The quantity f =, where b is the semi-minor axis, is purely geometrical. The mechanical ellipticity of Earth can be determined to high precision by observation of satellite orbit perturbations. Its relationship with geometrical flattening is indirect and depends on the internal density distribution or, in simplest terms, the degree of central concentration of mass.
The 1980 Geodetic Reference System, adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics, posited a 6,378,137 m semi-major axis and a 1:298.257 flattening. GRS 80 essentially constitutes the basis for geodetic positioning by the Global Positioning System and is thus also in widespread use outside the geodetic community. Numerous systems used for mapping and charting are becoming obsolete as countries increasingly move to global, geocentric reference systems utilizing the GRS 80 reference ellipsoid.
The geoid is a "realizable" surface, meaning it can be consistently located on Earth by suitable simple measurements from physical objects like a tide gauge. The geoid can, therefore, be considered a physical surface. The reference ellipsoid, however, has many possible instantiations and is not readily realizable, so it is an abstract surface. The third primary surface of geodetic interest — the topographic surface of Earth — is also realizable.

Coordinate systems in space

The locations of points in 3D space most conveniently are described by three cartesian or rectangular coordinates, X, Y, and Z. Since the advent of satellite positioning, such coordinate systems are typically geocentric, with the Z-axis aligned to Earth's rotation axis.
Before the era of satellite geodesy, the coordinate systems associated with a geodetic datum attempted to be geocentric, but with the origin differing from the geocenter by hundreds of meters due to regional deviations in the direction of the plumbline. These regional geodetic datums, such as ED 50 or NAD 27, have ellipsoids associated with them that are regional "best fits" to the geoids within their areas of validity, minimizing the deflections of the vertical over these areas.
It is only because GPS satellites orbit about the geocenter that this point becomes naturally the origin of a coordinate system defined by satellite geodetic means, as the satellite positions in space themselves get computed within such a system.
Geocentric coordinate systems used in geodesy can be divided naturally into two classes:
  1. The inertial reference systems, where the coordinate axes retain their orientation relative to the fixed stars or, equivalently, to the rotation axes of ideal gyroscopes. The X-axis points to the vernal equinox.
  2. The co-rotating reference systems, in which the axes are "attached" to the solid body of Earth. The X-axis lies within the Greenwich observatory's meridian plane.
The coordinate transformation between these two systems to good approximation is described by sidereal time, which accounts for variations in Earth's axial rotation. A more accurate description also accounts for polar motion as a phenomenon closely monitored by geodesists.

Coordinate systems in the plane

In geodetic applications like surveying and mapping, two general types of coordinate systems in the plane are in use:
  1. Plano-polar, with points in the plane defined by their distance, s, from a specified point along a ray having a direction α from a baseline or axis.
  2. Rectangular, with points defined by distances from two mutually perpendicular axes, x and y. Contrary to the mathematical convention, in geodetic practice, the x-axis points North and the y-axis East.
One can intuitively use rectangular coordinates in the plane for one's current location, in which case the x-axis will point to the local north. More formally, such coordinates can be obtained from 3D coordinates using the artifice of a map projection. It is impossible to map the curved surface of Earth onto a flat map surface without deformation. The compromise most often chosen — called a conformal projection — preserves angles and length ratios so that small circles get mapped as small circles and small squares as squares.
An example of such a projection is UTM. Within the map plane, we have rectangular coordinates 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 coordinates in the plane: let, as above, direction and distance be α and s respectively; then we have:
The reverse transformation is given by:

Heights

In geodesy, point or terrain heights are "above sea level" as an irregular, physically defined surface.
Height systems in use are:
  1. Orthometric heights
  2. Dynamic heights
  3. Geopotential heights
  4. Normal heights
Each system has its advantages and disadvantages. Both orthometric and normal heights are expressed in metres above sea level, whereas geopotential numbers are measures of potential energy and not metric. The reference surface is the geoid, an equigeopotential surface approximating the mean sea level as described above. For normal heights, the reference surface is the so-called quasi-geoid, which has a few-metre separation from the geoid due to the density assumption in its continuation under the continental masses.
One can relate these heights through the geoid undulation concept to ellipsoidal heights, representing the height of a point above the reference ellipsoid. Satellite positioning receivers typically provide ellipsoidal heights unless fitted with special conversion software based on a model of the geoid.

Geodetic datums

Because coordinates and heights of geodetic points always get obtained within a system that itself was constructed based on real-world observations, geodesists introduced the concept of a "geodetic datum" : a physical realization of a coordinate system used for describing point locations. This realization follows from choosing coordinate values for one or more datum points. In the case of height data, it suffices to choose one datum point — the reference benchmark, typically a tide gauge at the shore. Thus we have vertical datums, such as the NAVD 88, NAP, the Kronstadt datum, the Trieste datum, and numerous others.
In both mathematics and geodesy, a coordinate system is a "coordinate system" per ISO terminology, whereas the International Earth Rotation and Reference Systems Service uses the term "reference system" for the same. When coordinates are realized by choosing datum points and fixing a geodetic datum, ISO speaks of a "coordinate reference system", whereas IERS uses a "reference frame" for the same. The ISO term for a datum transformation again is a "coordinate transformation".