Physical geodesy

Physical geodesy is the study of the physical properties of Earth's gravity and its potential field (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.

Units Gravity is commonly measured in units of m·s−2 (metres per second squared). This also can be expressed (multiplying by the gravitational constant G in order to change units) 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. Again employing G as a multiplier, the units can be changed to joules per kilogram of attracted mass. 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.

Gravity

Potential fields

Geoid

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 Sri Lanka, it dives under the ellipsoid by nearly the same amount. The separation between the geoid and the reference ellipsoid is called the undulation of the geoid, symbol N {\displaystyle N} . 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 datums 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 (see astrogeodetic leveling).

Gravity anomalies

Above we already made use of gravity anomalies Δ g {\displaystyle \Delta g} . These are computed as the differences between true (observed) gravity g = ‖ g → ‖ {\displaystyle g=\|{\vec {g}}\|} , and calculated (normal) gravity γ = ‖ γ → ‖ = ‖ ∇ U ‖ {\displaystyle \gamma =\|{\vec {\gamma }}\|=\|\nabla 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

Δ g = g − γ . {\displaystyle \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 = 2 π G ρ H , {\displaystyle a_{B}=2\pi G\rho H,\,}

The Bouguer reduction to be applied as follows:

Δ g B = Δ g F A − a B , {\displaystyle \Delta g_{B}=\Delta g_{FA}-a_{B},\,}

so-called Bouguer anomalies. Here, Δ g F A {\displaystyle \Delta g_{FA}} is our earlier Δ g {\displaystyle \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.

See also Deflection of the vertical Dynamic height Friedrich Robert Helmert Geophysics Gravity of Earth Gravimetry LAGEOS Mikhail Molodenskii Normal height Orthometric height Satellite geodesy

References

Further reading 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).