Physical geodesy

Physical geodesy examines the physical characteristics of Earth’s gravity and its associated potential field, known as the geopotential, and applies this knowledge to geodesy.

Measurement procedure

Traditional surveying tools like theodolites use the gravity field to align their vertical axis with the local plumb line, typically with a spirit level. From this setup, vertical angles (zenith or elevation angles) are measured relative to the local vertical, while horizontal angles are taken in the plane of the local horizon, perpendicular to that vertical. Leveling instruments measure differences in geopotential between points on Earth’s surface, which can then be converted into height differences using metric units.

Units

Gravity is usually measured in metres per second squared (m·s⁻²). This can also be expressed as newtons per kilogram of attracted mass by multiplying by the gravitational constant G. Potential is expressed as gravity times distance, in m²·s⁻². Moving one metre in the direction of a gravity vector with a strength of 1 m·s⁻² increases potential by 1 m²·s⁻². Again using G as a multiplier, the units become joules per kilogram.

A more practical unit is the geopotential unit (GPU), equal to 10 m²·s⁻². Moving one metre vertically—the direction of the ambient gravity of about 9.8 m·s⁻²—changes potential by roughly 1 GPU. Thus, the difference in geopotential (in GPU) between a point and sea level gives a rough measure of height above sea level in metres.

Geoid

Because Earth’s true gravity field is irregular, the equilibrium shape of sea water—the geoid—is also irregular. For instance, west of Ireland the geoid (mathematical mean sea level) rises about 100 metres above the regular, rotationally symmetric GRS80 reference ellipsoid, while near Sri Lanka it dips below the ellipsoid by a similar amount. The separation between the geoid and the reference ellipsoid is called the geoid undulation, denoted N. The geoid is defined not only over oceans but also beneath land; it is the water surface that would exist if seawater could flow freely through tunnels under the continents. Technically, it is an equipotential surface of the true geopotential, chosen to coincide on average with mean sea level.

Since mean sea level is physically realized by tide gauge bench marks on various coasts, slightly incompatible “near-geoids” arise, differing by decimetres to over a metre due to dynamic sea surface topography. These are called vertical or height datums. At any point on Earth, the local direction of gravity (the vertical), realized by a plumb line, is perpendicular to the geoid.

Gravity anomalies

Gravity anomalies, denoted Δg, are computed as the difference between true observed gravity (g = ‖g⃗‖) and calculated normal gravity (γ = ‖γ⃗‖ = ‖∇U‖). (In practice, the location where γ is evaluated differs slightly from where g is measured.) This gives Δg = g − γ. These are free-air anomalies and are used in the Stokes equation. In geophysics, these anomalies are often further reduced by removing the attraction of the topography.

For a flat, horizontal plate (Bouguer plate) of thickness H, this attraction is a_B = 2πGρH. The Bouguer reduction is applied as Δg_B = Δg_FA − a_B, yielding Bouguer anomalies, where Δg_FA is the free-air anomaly. When the terrain is not flat, the local terrain height is used for H, and an additional terrain correction is applied.