Physical geodesy

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

Measurement procedure

To measure the Earth, traditional instruments like theodolites use the gravity field to align their vertical axis with the local plumb line—the true vertical—using a spirit level. From this alignment, 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 it. Levelling instruments determine geopotential differences between points on the surface, which can then be converted into height differences in metric units.

Units

Gravity is usually measured in metres per second squared (m·s⁻²). Multiplying by the gravitational constant G converts these units to newtons per kilogram of attracted mass. Potential, expressed as gravity times distance, uses units of m²·s⁻². Moving one metre in the direction of a gravity vector 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. This means moving one metre vertically—in the direction of Earth’s 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 reference ellipsoid of GRS80, while near Sri Lanka it dips below that 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 under land; it is the equilibrium water surface that would exist if sea water could move freely through tunnels beneath 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 benchmarks on different coasts, several slightly incompatible “near-geoids” arise, with differences of decimetres to over a metre due to dynamic sea surface topography. These are known as vertical or height datums. At any point on Earth, the local direction of gravity—the vertical, materialized by a plumb line—is perpendicular to the geoid (a concept used in astrogeodetic levelling).

Gravity anomalies

Gravity anomalies, denoted Δ*g, are the differences between observed gravity g and calculated normal gravity γ. (In practice, the location where γ is evaluated differs slightly from where g is measured.) This gives the free-air anomaly: Δ*g = g − γ. These anomalies are used in Stokes’ equation. In geophysics, they 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 = 2π*G*ρ*H. The Bouguer reduction is then Δ*g_B = Δ*g_FA − a_B, producing Bouguer anomalies. Here, Δ*g_FA is the free-air anomaly. If the terrain is not flat, the local terrain height is used for H, and an additional terrain correction is applied.