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Home > Physical geodesy

1 Definition

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.

Traditional geodetic instruments such as theodolites rely on the gravity field for orienting their vertical axis along the local plumb line or vertical 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.

2 The geopotential

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

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 co-ordinate 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

where

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 co-ordinates.

3 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 newtonThis article is about the SI unit of force. For other uses see Newton (disambiguation In physics, the newton (symbol: N) is the SI unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics. It was adopted by the Generals per kilogramThe kilogram (symbol: kg is the SI base unit of mass. A gram is defined as one thousandth of a kilogram. Conversion of units describes equivalent units of mass in other systems. Multiples SI prefixes are used to name multiples and subdivisions of the kilo 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 change 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.

4 The normal potential

To a rough approximation, the Earth is a sphereFor other uses, see sphere (disambiguation). A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball . But in mathematics, a sphere is the boundary of a ball,, or to a much better approximation, an ellipsoidDefinition In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid in an x ''y ''z Cartesian coordinate system is : where a b and c are fixed positive real numbers determi. 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.

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