Physical geodesy is the study of the physical properties of the
gravityfield of the Earth, the geopotential, with a view to their application in geodesy.
Traditional geodetic instruments such as
theodolites rely on the gravity field for orienting their vertical axis along the local plumb lineor local vertical directionwith the aid of a spirit level. After that, vertical angles ( zenithangles or, alternatively, elevationangles) are obtained with respect to this local vertical, and horizontal angles in the plane of the local horizon, perpendicular to the vertical. Levellinginstruments again are used to obtain geopotentialdifferences between points on the Earth's surface. These can then be expressed as "height" differences by conversion to metric units.
The Earth's gravity field can be described by a
which expresses the gravitational acceleration vector as the gradient of , the potential of gravity. The vector triad is the orthonormal set of base vectors in space, pointing along the 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
where is the potential of the "gravitational" field, that of the "gravity" field, and that of the centrifugal force field.
The centrifugal force is given by
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:
This can be verified by taking the gradient () operator of this expression.
Here, , and are geocentric coordinates.
Units of gravity and geopotential
Gravity is commonly measured in units of m s-2, (
metres per secondsquared). This also can be expressed as newtons per kilogramof 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:
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 m, and flattening .
A geopotential field is constructed, being the sum of a gravitational potential and the known centrifugal potential , 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:"
Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (), 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
Once a clean, smooth geopotential field 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 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 , and is closely related to the disturbing potential.
According to the famous Bruns formula, we have
where is the force of gravity computed from the normal field potential .
In 1849, the mathematician
George Gabriel Stokespublished the following formula named after him:
In this formula, 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 anywhere on Earth by this formula requires to be known "everywhere on Earth". Welcome to the role of international co-operation in physical geodesy.)
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.
Above we already made use of "gravity anomalies" . These are computed as the differences between true (observed) gravity , and calculated (normal) gravity . (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
These anomalies are called free-air anomalies, and are the ones to be used in the above Stokes equation.
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
Bouguer reductionto be applied as follows:
so-called Bouguer anomalies. Here, is our earlier , 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
Friedrich Robert Helmert
* 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).
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