Physical geodesy


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.

Measurement procedure

Traditional geodetic instruments such as theodolites rely on the gravity field for orienting their vertical axis along the local plumb line or local vertical direction with the aid of a spirit level. After that, vertical angles (zenith angles or, alternatively, elevation angles) are obtained with respect to this local vertical, and horizontal angles in the plane of the local horizon, perpendicular to the vertical.

Levelling instruments again are used to obtain geopotential differences between points on the Earth's surface. These can then be expressed as "height" differences by conversion to metric units.

The geopotential

The Earth's gravity field can be described by a potential as follows:

:mathbf{g} = abla W = mathrm{grad} W = frac{partial W}{partial X}mathbf{i}+frac{partial W}{partial Y}mathbf{j}+frac{partial W}{partial Z}mathbf{k}

which expresses the gravitational acceleration vector as the gradient of W, the potential of gravity. The vector triad {mathbf{i},mathbf{j},mathbf{k}} is the orthonormal set of base vectors in space, pointing along the X,Y,Z coordinate axes.

Note that both gravity and its potential contain a contribution from the centrifugal pseudo-force due to the Earth's rotation. We can write

:W = V + Phi,

where V is the potential of the "gravitational" field, W that of the "gravity" field, and Phi that of the centrifugal force field.

The centrifugal force is given by

:mathbf{g}_c = omega^2 mathbf{p},


:mathbf{p} = Xmathbf{i}+Ymathbf{j}+0cdotmathbf{k}

is the vector pointing to the point considered straight from the Earth's rotational axis. It can be shown that this pseudo-force field, in a reference frame co-rotating with the Earth, has a potential associated with it that looks like this:

:Phi = frac{1}{2} omega^2 (X^2+Y^2).

This can be verified by taking the gradient ( abla) operator of this expression.

Here, X, Y and Z are geocentric coordinates.

Units of gravity and geopotential

Gravity is commonly measured in units of m s-2, (metres per second squared). This also can be expressed as newtons per kilogram of attracted mass.

Potential is expressed as gravity times distance, m2 s-2. Travelling one metre in the direction of a gravity vector of strength 1 m s-2 will increase your potential by 1 m2 s-2.

A more convenient unit is the GPU, or geopotential unit: it equals 10 m2 s-2. This means that travelling one metre in the vertical direction, i.e., the direction of the 9.8 m s-2 ambient gravity, will "approximately" change your potential by 1 GPU. Which again means that the difference in geopotential, in GPU, of a point with that of sea level can be used as a rough measure of height "above sea level" in metres.

The normal potential

To a rough approximation, the Earth is a sphere, or to a much better approximation, an ellipsoid. We can similarly approximate the gravity field of the Earth by a spherically symmetric field:

:W approx frac{GM}{R}

of which the "equipotential surfaces" -- the surfaces of constant potential value -- are concentric spheres.

It is more accurate to approximate the geopotential by a field that has "the Earth reference ellipsoid" as one of its equipotential surfaces, however. The most recent Earth reference ellipsoid is GRS80, or Geodetic Reference System 1980, which the Global Positioning system uses as its reference. Its geometric parameters are: semi-major axis a=6378137.0 m, and flattening f = 1/298.257222101.

A geopotential field U is constructed, being the sum of a gravitational potential Psi and the known centrifugal potential Phi, that "has the GRS80 reference ellipsoid as one of its equipotential surfaces". If we also require that the enclosed mass is equal to the known mass of the Earth (including atmosphere) "GM" = 3986005 × 108 m3s-2, we obtain for the "potential at the reference ellipsoid:"

:U_0=62636860.850 {, extrm m^2 s^{-2

Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (R ightarrowinfty), as is common in physics. For practical purposes it makes more sense to choose the zero point of normal gravity to be that of the reference ellipsoid, and refer the potentials of other points to this.

Disturbing potential and geoid

Once a clean, smooth geopotential field U has been constructed matching the known GRS80 reference ellipsoid with an equipotential surface -- we call such a field a "normal potential" -- we can subtract it from the true potential W of the real Earth. The result is the "disturbing potential":


which is numerically a whole lot smaller, and captures the detailed, complex variations of the gravity field of the really existing Earth, as distinguished from the overall global trend captured well by the normal potential.

Due to the irregularity of the Earth's true gravity field, the equilibrium figure of sea water, or the geoid, will also be of irregular form. In some places, like west of Ireland, the geoid -- mathematical mean sea level -- sticks out as much as 100 m above the regular, rotationally symmetric reference ellipsoid of GRS80; in other places, like close to Ceylon, it dives under the ellipsoid by nearly the same amount. The separation between these two surfaces is called the undulation of the geoid, symbol N, and is closely related to the disturbing potential.

According to the famous Bruns formula, we have


where gamma is the force of gravity computed from the normal field potential U.

In 1849, the mathematician George Gabriel Stokes published the following formula named after him:

:N=frac{R}{4pi G}iint_sigma Delta g ,S(psi), dsigma.

In this formula, Delta g stands for "gravity anomalies", differences between true and normal (reference) gravity, and "S" is the "Stokes function", a kernel function derived by Stokes in closed analytical form. (Note that determining N anywhere on Earth by this formula requires Delta g to be known "everywhere on Earth". Welcome to the role of international co-operation in physical geodesy.)

The geoid, or mathematical mean sea surface, is defined not only on the seas, but also under land; it is the equilibrium water surface that would result, would sea water be allowed to move freely (e.g., through tunnels) under the land. Technically, an "equipotential surface" of the true geopotential, chosen to coincide (on average) with mean sea level.

As mean sea level is physically realized by tide gauge bench marks on the coasts of different countries and continents, a number of slightly incompatible "near-geoids" will result, with differences of several decimetres to over one metre between them, due to the dynamic sea surface topography. These are referred to as "vertical" or "height datums".

For every point on Earth, the local direction of gravity or vertical direction, materialized with the plumb line, is "perpendicular" to the geoid. On this is based a method, "astrogeodetic levelling", for deriving the local figure of the geoid by measuring "deflections of the vertical" by astronomical means over an area.

Gravity anomalies

Above we already made use of "gravity anomalies" Delta g. These are computed as the differences between true (observed) gravity g=|vec{g}|, and calculated (normal) gravity gamma=|vec{gamma}|=| abla U|. (This is an oversimplification; in practice the location in space at which γ is evaluated will differ slightly from that where "g" has been measured.) We thus get

:Delta g = g - gamma.,

These anomalies are called free-air anomalies, and are the ones to be used in the above Stokes equation.

In geophysics, these anomalies are often further reduced by removing from them the "attraction of the topography", which for a flat, horizontal plate (Bouguer plate) of thickness "H" is given by

:a_B=2pi G ho H,,

The Bouguer reduction to be applied as follows:

:Delta g_B = Delta g_{FA} - a_B,,

so-called Bouguer anomalies. Here, Delta g_{FA} is our earlier Delta g, the free-air anomaly.

In case the terrain is not a flat plate (the usual case!) we use for "H" the local terrain height value but apply a further correction called the terrain correction ("TC").

See also

* Friedrich Robert Helmert
* gravimetry
* satellite geodesy


* B. Hofmann-Wellenhof and H. Moritz, Physical Geodesy, Springer-Verlag Wien, 2005. (This text is an updated edition of the 1967 classic by W.A. Heiskanen and H. Moritz).

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Geodesy — (pronEng|dʒiːˈɒdɪsi [OED] ), also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three dimensional time varying… …   Wikipedia

  • Physical science — is an encompassing term for the branches of natural science and science that study non living systems, in contrast to the biological sciences. However, the term physical creates an unintended, somewhat arbitrary distinction, since many branches… …   Wikipedia

  • Physical geography — For other uses, see Physiographic regions of the world. True color image of the Earth s surface and atmosphere. NASA Goddard Space Flight Center image. Physical geography (also known as geosystems or physiography) is one of the two major… …   Wikipedia

  • geodesy — The science that deals mathematically with the size and shape of the earth and with the precise location of points on its surface. It includes geometrical methods of ordinary surveying together with observations of physical phenomena, such as the …   Aviation dictionary

  • History of geodesy — See also the main article on Geodesy. Humanity has always been interested in the Earth. During very early times this interest was limited, naturally, to the immediate vicinity of home and residency, and the fact that we live on a near spherical… …   Wikipedia

  • International Union of Geodesy and Geophysics — Die Internationale Union für Geodäsie und Geophysik (IUGG) ist eine der großen wissenschaftlichen Dachverbände der Welt und gliedert sich in acht fachliche Assoziationen. Über 100 Staaten sind Mitglieder der Union oder ihrer Teile, ferner etwa 60 …   Deutsch Wikipedia

  • International Union of Geodesy and Geophysics — The International Union of Geodesy and Geophysics, or IUGG, is a non governmental organisation dedicated to the scientific study of the Earth using geophysical and geodesic techniques. The IUGG was established in 1919. Some areas within its scope …   Wikipedia

  • University of Architecture, Civil Engineering and Geodesy — The University of Architecture, Civil Engineering and Geodesy is located in Sofia, Bulgaria. Faculties and departments * Faculty of Architecture * Faculty of Geodesy * Faculty of Hydrotechnics * Faculty of Structural engineering * Faculty of… …   Wikipedia

  • Spherical Earth — Medieval artistic representation of a spherical Earth with compartments representing earth, air, and water (c. 1400). The concept of a spherical Earth dates back to ancient Greek philosophy from around the 6th century BC …   Wikipedia

  • Figure of the Earth — An oblate spheroid Geodesy …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.