Geodesy

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An old geodetic pillar (1855) at Ostend, Belgium Geodetisch station1855.jpg
An old geodetic pillar (1855) at Ostend, Belgium

Geodesy ( /ˈɒdɪsi/ ) [1] is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space and gravitational field. [2] The field also incorporates studies of how these properties change over time and equivalent measurements for other planets (known as planetary geodesy). Geodynamical phenomena include crustal motion, tides and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques and relying on datums and coordinate systems.

Contents

Definition

The word "geodesy" comes from the Ancient Greek word γεωδαισίαgeodaisia (literally, "division of Earth").

It is primarily concerned with positioning within the temporally varying gravity field. Geodesy in the German-speaking world is divided into "higher geodesy" ("Erdmessung" or "höhere Geodäsie"), which is concerned with measuring Earth on the global scale, and "practical geodesy" or "engineering geodesy" ("Ingenieurgeodäsie"), which is concerned with measuring specific parts or regions of Earth, and which includes surveying. Such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding Earth's geometric shape, orientation in space, and gravity field.

To a large extent, the shape of Earth is the result of rotation, which causes its equatorial bulge, and the competition of geological processes such as the collision of plates and of volcanism, resisted by Earth's gravity field. This applies to the solid surface, the liquid surface (dynamic sea surface topography) and Earth's atmosphere. For this reason, the study of Earth's gravity field is called physical geodesy.

History

Geoid and reference ellipsoid

The geoid is essentially the figure of Earth abstracted from its topographical features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents and air pressure variations, and continued under the continental masses. The geoid, unlike the reference ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the reference ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS 80 ellipsoid.

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 = ab/a, where b is the semi-minor axis (polar radius), is a purely geometrical one. The mechanical ellipticity of Earth (dynamical flattening, symbol J2) can be determined to high precision by observation of satellite orbit perturbations. Its relationship with the geometrical flattening is indirect. The relationship depends on the internal density distribution, or, in simplest terms, the degree of central concentration of mass.

The 1980 Geodetic Reference System (GRS 80) 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 (GPS) and is thus also in widespread use outside the geodetic community. The numerous systems that countries have used to create maps and charts are becoming obsolete as countries increasingly move to global, geocentric reference systems using the GRS 80 reference ellipsoid.

The geoid is "realizable", 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 real surface. The reference ellipsoid, however, has many possible instantiations and is not readily realizable, therefore it is an abstract surface. The third primary surface of geodetic interest—the topographic surface of Earth—is a realizable surface.

Coordinate 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 systems are typically geocentric: the Z-axis is aligned with Earth's (conventional or instantaneous) rotation axis.

Prior to the era of satellite geodesy, the coordinate systems associated with a geodetic datum attempted to be geocentric, but their origins differed from the geocenter by hundreds of meters, due to regional deviations in the direction of the plumbline (vertical). These regional geodetic data, such as ED 50 (European Datum 1950) or NAD 27 (North American Datum 1927) 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 are themselves computed in such a system.

Geocentric coordinate systems used in geodesy can be divided naturally into two classes:

  1. 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. Co-rotating, also ECEF ("Earth Centred, Earth Fixed"), where 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 is described to good approximation by (apparent) sidereal time, which takes into account variations in Earth's axial rotation (length-of-day variations). A more accurate description also takes polar motion into account, a phenomenon closely monitored by geodesists.

Coordinate systems in the plane

A Munich archive with lithography plates of maps of Bavaria Litography archive of the Bayerisches Vermessungsamt.jpg
A Munich archive with lithography plates of maps of Bavaria

In surveying and mapping, important fields of application of geodesy, two general types of coordinate 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 coordinates 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 coordinates can be obtained from three-dimensional coordinates using the artifice of a map projection. It is not possible 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 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 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", an irregular, physically defined surface. Heights come in the following variants:

  1. Orthometric heights
  2. Normal heights
  3. Geopotential heights

Each has its advantages and disadvantages. Both orthometric and normal heights are heights in metres above sea level, whereas geopotential numbers are measures of potential energy (unit: m2 s−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 is 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 data

Because geodetic point coordinates (and heights) are always obtained in a system that has been constructed itself using real observations, geodesists introduce the concept of a "geodetic datum": a physical realization of a coordinate system used for describing point locations. The realization is the result of choosing conventional 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 data like the NAP (Normaal Amsterdams Peil), the North American Vertical Datum 1988 (NAVD 88), the Kronstadt datum, the Trieste datum, and so on.

In case of plane or spatial coordinates, we typically need several datum points. A regional, ellipsoidal datum like ED 50 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, so to make them refer to another datum, is called a datum transformation. In the case of vertical data, 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 coordinate system as used in mathematics and geodesy is called a "coordinate system" in ISO terminology, whereas the International Earth Rotation and Reference Systems Service (IERS) uses the term "reference system". When these coordinates are realized by choosing datum points and fixing a geodetic datum, ISO says "coordinate reference system", while IERS says "reference frame". The ISO term for a datum transformation again is a "coordinate transformation". [3]

Point positioning

Geodetic Control Mark (example of a deep benchmark) Geodetic Control Mark.jpg
Geodetic Control Mark (example of a deep benchmark)

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 computation 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 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 traverses (polygons), into which local mapping surveying measurements, usually with measuring tape, corner prism, and the familiar[ where? ] 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 "zero 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 and are normally documented by national mapping agencies. Surveyors involved in real estate and insurance will use these to tie their local measurements.

Geodetic problems

In geometric geodesy, two standard problems exist—the first (direct or forward) and the second (inverse or reverse).

First (direct or forward) 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 coordinates of) that second point.
Second (inverse or reverse) geodetic problem
Given two points, determine the azimuth and length of the line (straight line, arc or geodesic) that connects them.

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

On the ellipsoid of revolution, geodesics may be written in terms of elliptic integrals, which are usually evaluated in terms of a series expansion—see, for example, Vincenty's formulae. In the general case, the solution is called the geodesic for the surface considered. The differential equations for the geodesic can be solved numerically.

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.

Measurements

A NASA project manager talks about his work for the Space Geodesy Project, including an overview of its four fundamental techniques: GPS, VLBI, SLR, and DORIS.

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 the 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 to even robotic 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. Real-time kinematic (RTK) GPS techniques are used as well. Data collected are tagged and recorded digitally for entry into a Geographic Information System (GIS) database.

Geodetic GPS receivers produce directly three-dimensional coordinates in a geocentric coordinate frame. Such a frame is, e.g., WGS84, or the frames that are regularly produced and published by the International Earth Rotation and Reference Systems 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 ranging (SLR) and lunar laser ranging (LLR) and very-long-baseline interferometry (VLBI) techniques. All these techniques also serve to monitor irregularities in Earth's rotation as well as plate tectonic motions.

Gravity is measured using gravimeters, of which there are two kinds. First, "absolute gravimeters" are based on measuring the acceleration of free fall (e.g., of a reflecting prism in a vacuum tube). They are used to establish the vertical geospatial control and can be used in the field. Second, "relative gravimeters" are spring-based and are more common. They are used in gravity surveys over large areas for establishing the figure of the geoid over these areas. The most accurate relative gravimeters are called "superconducting" gravimeters, which are sensitive to one-thousandth of one-billionth of Earth-surface gravity. Twenty-some superconducting gravimeters are used worldwide for studying Earth's tides, rotation, interior, and ocean and atmospheric loading, as well as for verifying the Newtonian constant of gravitation.

In the future, gravity and altitude, will be measured by relativistic time dilation measured by strontium optical clocks.

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 Earth is used.

One geographical mile, defined as one minute of arc on the equator, equals 1,855.32571922 m. One 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 the shortest at the equator as is the nautical mile.

A metre was originally defined as the 10-millionth part of the length from equator to North Pole along the meridian through Paris (the target was not quite reached in actual implementation, so that is off by 200 ppm in the current definitions). This means that one kilometre is roughly equal to (1/40,000) * 360 * 60 meridional minutes of arc, which equals 0.54 nautical mile, though this is not exact because the two units are defined on different bases (the international nautical mile is defined as exactly 1,852 m, corresponding to a rounding of 1,000/0.54 m to four digits).

Temporal change

In geodesy, temporal change can be studied by a variety of techniques. Points on Earth's surface change their location due to a variety of mechanisms:

The science of studying deformations and motions of Earth's crust and its solidity as a whole is called geodynamics. Often, study of Earth's irregular rotation is also included in its definition.

Techniques for studying geodynamic phenomena on the global scale include:

Notable geodesists

Mathematical geodesists before 1900

20th century geodesists

Unlisted

See also

Fundamentals
Governmental agencies
International organizations
Other

Related Research Articles

Latitude The angle between zenith at a point and the plane of the equator

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Geographic coordinate system Coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

Geoid irregular surface approximating the mean sea level

The geoid is the shape that the ocean surface would take under the influence of the gravity and rotation of Earth alone, if other influences such as winds and tides were absent. This surface is extended through the continents. According to Gauss, who first described it, it is the "mathematical figure of the Earth", a smooth but irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of geodesy and geophysics, it has been defined to high precision only since advances in satellite geodesy in the late 20th century.

Physical geodesy The study of the physical properties of the Earths gravity field

Physical geodesy is the study of the physical properties of the gravity field of the Earth, the geopotential, with a view to their application in geodesy.

World Geodetic System geodetic reference system

The World Geodetic System (WGS) is a standard for use in cartography, geodesy, and satellite navigation including GPS. This standard includes the definition of the coordinate system's fundamental and derived constants, the ellipsoidal (normal) Earth Gravitational Model (EGM), a description of the associated World Magnetic Model (WMM), and a current list of local datum transformations.

Figure of the Earth Size and shape used to model the Earth for geodesy

Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. The sphere is an approximation of the figure of the Earth that is satisfactory for many purposes. Several models with greater accuracy have been developed so that coordinate systems can serve the precise needs of navigation, surveying, cadastre, land use, and various other concerns.

Reference ellipsoid An ellipsoid that approximates the figure of the Earth

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

Vertical deflection

The vertical deflection at a point on the Earth is a measure of how far the gravity direction has been shifted by local anomalies such as nearby mountains. They are widely used in geodesy, for surveying networks and for geophysical purposes.

Geodetic datum reference frame used in geodesy, surveying, chartography and navigation

A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used for locating places on the Earth. An approximate definition of sea level is the datum WGS 84, an ellipsoid, whereas a more accurate definition is Earth Gravitational Model 2008 (EGM2008), using at least 2,159 spherical harmonics. Other datums are defined for other areas or at other times; ED50 was defined in 1950 over Europe and differs from WGS 84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there.

Satellite geodesy geodesy by means of artificial satellites

Satellite geodesy is geodesy by means of artificial satellites — the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques. It belongs to the broader field of space geodesy. Traditional astronomical geodesy is not commonly considered a part of satellite geodesy, although there is considerable overlap between the techniques.

The orthometric height of a point is the distance H along a plumb line from the point to a reference height. When the reference height is a geoid model, orthometric height is for practical purposes "height above sea level".

North American Datum series of geodesic datums in 1901, 1923, and 1988 versions

The North American Datum (NAD) is the horizontal datum now used to define the geodetic network in North America. A datum is a formal description of the shape of the Earth along with an "anchor" point for the coordinate system. In surveying, cartography, and land-use planning, two North American Datums are in use for making lateral or "horizontal" measurements: the North American Datum of 1927 (NAD 27) and the North American Datum of 1983 (NAD 83). Both are geodetic reference systems based on slightly different assumptions and measurements.

ECEF Earth-centered, Earth-fixed reference frame

ECEF, also known as ECR, is a geographic and Cartesian coordinate system and is sometimes known as a "conventional terrestrial" system. It represents positions as X, Y, and Z coordinates. The point is defined as the center of mass of Earth, hence the term geocentric coordinates. The distance from a given point of interest to the center of Earth is called the geocentric radius or geocentric distance.

North American Vertical Datum of 1988

The North American Vertical Datum of 1988 is the vertical datum for orthometric heights established for vertical control surveying in the United States of America based upon the General Adjustment of the North American Datum of 1988.

Helmert transformation transformation method within a three-dimensional space

The Helmert transformation is a transformation method within a three-dimensional space. It is frequently used in geodesy to produce distortion-free transformations from one datum to another. The Helmert transformation is also called a seven-parameter transformation and is a similarity transformation.

The Bessel ellipsoid is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geodesy.

Earth ellipsoid ellipsoid of rotation that approximates the figure of the Earth

An Earth ellipsoid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.

The term fundamental station is used for special observatories which combine several space positioning techniques like VLBI, satellite laser ranging, GPS, Glonass, etc. They are the basis of plate tectonic analysis, allowing the monitoring of continental drift rates with milimetre accuracies. A fundamental point is the geometric origin of a geodetic network and defines the geodetic datum of a national survey.

Vertical datum reference surface for vertical positions

A vertical datum or height datum is a reference surface for vertical positions, such as the elevations of Earth features including terrain, bathymetry, water level, and man-made structures; in any particular case one must be assigned even if arbitrarily, and commonly adopted criteria for a vertical datum include the following approaches:

<i>Normalhöhennull</i> standard reference level, the equivalent of sea level, used in Germany to measure height

Normalhöhennull or NHN is a vertical datum used in Germany.

References

  1. "geodesy | Definition of geodesy in English by Lexico Dictionaries". Lexico Dictionaries | English. Retrieved 2019-08-15.
  2. "What Is Geodesy". National Ocean Service . Retrieved 8 February 2018.
  3. (ISO 19111: Spatial referencing by coordinates).
  4. "DEFENSE MAPPING AGENCY TECHNICAL REPORT 80-003". Ngs.noaa.gov. Retrieved 8 December 2018.
  5. "Guy Bomford tribute". Bomford.net. Retrieved 8 December 2018.

Further reading

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