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Geodesy

This article should include material from Erdmessung.

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.

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The figure of the Earth

Primitive ideas about the figure of the Earth, still found in young children, hold the Earth to be flat, and the heavens a physical dome spanning over it. Already the ancient Greeks were aware of the spherical shape of the Earth. Lunar eclipses, e.g., always have a circular edge of appox. three times the radius of the lunar disc; as these always happen when the Earth is between Sun and Moon, it suggests that the object casting the shadow is the Earth and must be spherical (and four times the size of the Moon, the lunar and solar discs being the same size).

Also an astronomical event like a lunar eclipse which happened high in the sky in one end of the Mediterranean world, was close to the horizon in the other end, also suggesting curvature of the Earth's surface. Finally, Eratosthenes determined a remarkably accurate value for the radius of the Earth at around 200 BC.

The Renaissance brought the invention of the telescope and the theodolite, making possible triangulation and grade measurement . Of the latter especially should be mentioned the expedition by the French Academy of Sciences to determine the flattening of the Earth. One expedition was sent to Lapland as far North as possible under Pierre Louis Maupertuis (1736-37), the other under Pierre Bouguer was sent to Peru, near the equator (1735-44).

At the time there were two competing theories on the precise figure of the Earth: Isaac Newton had calculated that, based on his theory of gravitation, the Earth should be flattened at the poles to a ratio of 1:230. On the other hand the astronomer Jean Dominique Cassini held the view that the Earth was elongated at the poles. Measuring the length, in linear units, of a degree of change in north-south direction of the astronomical vertical, at two widely differing latitudes would settle the issue: on a flattened Earth the length of a degree grows toward the poles.

The flattening found by comparing the results of the two grade measurement expeditions confirmed that the Earth was flattened, the ratio found being 1:210. Thus the next approximation to the true figure of the Earth after the sphere became the flattened, biaxial ellipsoid of revolution .

In South America Bouguer noticed, as did George Everest in India, that the astronomical vertical tended to be "pulled" in the direction of large mountain ranges, obviously due to the gravitational attraction of these huge piles of rock. As this vertical is everywhere perpendicular to the idealized surface of mean sea level, or the geoid, this means that the figure of the Earth is even more irregular than an ellipsoid of revolution. Thus the study of the "undulations of the geoid" became the next great undertaking in the science of studying the figure of the Earth.

Geoid and reference ellipsoid

The geoid is essentially the figure of the Earth abstracted from its topographic features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. and continued under the continental masses. The geoid, unlike the ellipsoid, is irrgular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between it and the reference ellipsoid is called the geoidal undulation. It varies globally between $\pm$ 110 m.

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 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). It is essentially the basis for geodetic positioning by the Global Positioning System and is thus in extremely widespread use also outside the geodetic community.

The numerous other systems which have been used by diverse countries for their maps and charts are gradually dropping out of use as more and more countries move to global, geocentric reference systems using the GRS80 reference ellipsoid.

Co-ordinate systems in space

The locations of points in three-dimensional space are most conveniently described by three cartesian or rectangular coordinates, X,Y and Z. Since the advent of satellite positioning, such coordinate sytems are typically geocentric: the Z axis is aligned with the Earth's (conventional or instantaneous) rotation axis.

Before the satellite geodesy era, the coordinate systems associated with geodetic datums attempted to 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 coordinate system defined by satellite geodetic means, as the satellite positions in space are themselves computed in such a system.

Geocentric co-ordinate systems used in geodesy can be divided naturally into two classes:

1. Inertial reference systems, where the co-ordinate 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. Co-rotating, also ECEF ("Earth Centred, Earth Fixed"), where the axes are attached to the solid body of the Earth. The X axis lies within the Greenwich observatory's meridian plane.

The co-ordinate transformation between these two systems is described to good approximation by (apparent) sidereal time. A more accurate description takes also length-of-day variations and polar motion into account, phenomena currently closely monitored by geodesists.

Co-ordinate systems in the plane

In surveying and mapping, important fields of application 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 a distance s from a specified point along a ray having a specified direction α 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 most often chosen -- called a conformal projection -- preserves angles and length ratios, so that small circles are mapped as small circles 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, as above, direction and distance be α 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.

Heights

In geodesy, point or terrain heights are "above sea level", an irregular, physically defined surface. Therefore a height should ideally 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 coordinate 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.

Heights come in the following variants:

1. Orthometric height s
2. Normal height s
3. Geopotential number s

Each have their advantages and disadvantages. Both orthometric and normal heights are heights in metres above sea level, which geopotential numbers are measures of potential energy (unit: m2s - 2) and not metric. Orthometric and normal heights differ in the precise way in which mean sea level is conceptually continued under the continental masses. The reference surface for orthometric heights is the geoid, an equipotential surface approximating mean sea level.

None of these heights are in any way related to geodetic or ellipsoidial heights, which express the height of a point above the reference ellipsoid. Satellite positioning receivers typically provide ellipsoidal heights, unless they are fitted with special conversion software based on a model of the geoid.

Geodetic datums

Because geodetic point c-oordinates (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 coordinates, 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 coordinates of a point set referring to one datum, to make them refer to another datum, is called a 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 coordinates, 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.

A note on terminology

In the abstract, a co-ordinate system as used in mathematics and geodesy is, e.g., in ISO terminology, referred to as a coordinate system. International geodetic organizations like the IERS (International Earth Rotation and Reference Systems Service) speak of a reference system.

When these co-ordinates are realized by choosing datum points and fixing a geodetic datum, ISO uses the terminology coordinate reference system, while IERS speaks of a reference frame. A datum transformation again is referred to by ISO as a coordinate transformation. (ISO 19111: Spatial referencing by coordinates).

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 traverse s (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 (also: first geodetic problem)
Given a point (in terms of its coordinates) and the direction (azimuth) and distance from that point to a second point, determine (the co-ordinates of) that second point.
Geodetic inverse problem (also: second geodetic problem)
Given two points, determine the azimuth 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.

In the general case, the solution is called the geodesic for the surface considered. It may be nonexistent or non-unique. 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 coordinates, defined in geodesy (and astronomy as well), mostly from the viewpoint of the local observer.

• The plumbline or vertical is the direction of local gravity, or the line that results by following it. 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.
• Azimuth 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.
• A meridian plane 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.

Geodetic observing instruments

The level is used for determining height differences and height reference systems, commonly referred to mean sea level. The traditional spirit level produces these practically most useful heights above sea level directly; the more economical use of GPS instruments for height determination requires precise knowledge of the figure of the geoid, as GPS only gives heights above the GRS80 reference ellipsoid. As geoid knowledge accumulates, one may expect use of GPS heighting to spread.

The theodolite is used to measure horizontal and vertical angles to target points. These angles are referred to the local vertical. The tacheometer additionally determines, electronically or electro-optically, the distance to target, and is highly automated in its operations. The method of free station position is widely used.

For local detail surveys, tacheometers are commonly employed although the old-fashioned rectangular technique using angle prism and steel tape is still an inexpensive alternative. More and more, also real time kinematic (RTK) GPS techniques are used. Data collected is tagged and recorded digitally for entry into a Geographic Information System (GIS) data base.

Geodetic GPS receivers produce directly three-dimensional co-ordinates in a geocentric co-ordinate frame. Such a frame is, e.g., WGS84, or the frames that are regularly produced and published by the International Earth Rotation Service (IERS).

GPS receivers have almost completely replaced terrestrial instruments for large-scale base network surveys. For planet-wide geodetic surveys, previously impossible, we can still mention satellite laser and Very Long Baseline Interferometer (VLBI) techniques. All these techniques also serve to monitor Earth rotation irregularities as well as plate tectonic motions.

Gravity is measured using gravimeters . Common field gravimeters are spring based and referred to a relative. Absolute gravimeters, which nowadays can also be used in the field, are based directly on measuring the acceleration of free fall of a reflecting prism in a vacuum tube. Gravity surveys over large areas can serve to establish the figure of the geoid over these areas.

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.