Physical geodesy

Ocean basins mapped gravitationally. Seafloor features larger than 10 km are detected by resulting gravitational distortion of sea surface. (1995, NOAA)

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

Earth's gravity measured by NASA GRACE mission, showing deviations from the theoretical gravity of an idealized, smooth Earth, the so-called Earth ellipsoid. Red shows the areas where gravity is stronger than the smooth, standard value, and blue reveals areas where gravity is weaker (Animated version).^[1]

The gravity of Earth, denoted by $g$ , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).^[2]^[3] It is a vector quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm $g=\|{\mathit {\mathbf {g} }}\|$ .

In SI units, this acceleration is expressed in metres per second squared (in symbols, m/s² or m·s⁻²) or equivalently in newtons per kilogram (N/kg or N·kg⁻¹). Near Earth's surface, the acceleration due to gravity, accurate to 2 significant figures, is 9.8 m/s² (32 ft/s²). This means that, ignoring the effects of air resistance, the speed of an object falling freely will increase by about 9.8 metres (32 ft) per second every second. This quantity is sometimes referred to informally as little $g$ (in contrast, the gravitational constant $G$ is referred to as big $G$ ).

The precise strength of Earth's gravity varies with location. The agreed upon value for standard gravity is 9.80665 m/s² (32.1740 ft/s²) by definition.^[4] This quantity is denoted variously as $g n$ , $g e$ (though this sometimes means the normal gravity at the equator, 9.7803267715 m/s² (32.087686258 ft/s²)),^[5] $g 0$ , or simply $g$ (which is also used for the variable local value).

The weight of an object on Earth's surface is the downwards force on that object, given by Newton's second law of motion, or

F = m a

(force = mass × acceleration). Gravitational acceleration contributes to the total gravity acceleration, but other factors, such as the rotation of Earth, also contribute, and, therefore, affect the weight of the object. Gravity does not normally include the gravitational pull of the Moon and Sun, which are accounted for in terms of tidal effects.

Potential fields

Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation. In addition to the actual potential (the geopotential), a hypothetical normal potential and their difference, the disturbing potential, can also be defined.

Geoid

Map of the undulation of the geoid in meters (based on the EGM96)

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 $N$ .

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).

Geoid determination

The undulation of the geoid is closely related to the disturbing potential according to the famous Bruns' formula:

N=T/\gamma \,,

where $\gamma$ is the force of gravity computed from the normal field potential $U$ .

In 1849, the mathematician George Gabriel Stokes published the following formula, named after him:

N={\frac {R}{4\pi \gamma _{0}}}\iint _{\sigma }\Delta g\,S(\psi )\,d\sigma .

In Stokes' formula or Stokes' integral, $\Delta g$ stands for gravity anomaly, differences between true and normal (reference) gravity, and S is the Stokes function, a kernel function derived by Stokes in closed analytical form.^[6]

Note that determining $N$ anywhere on Earth by this formula requires $\Delta g$ to be known everywhere on Earth, including oceans, polar areas, and deserts. For terrestrial gravimetric measurements this is a near-impossibility, in spite of close international co-operation within the International Association of Geodesy (IAG), e.g., through the International Gravity Bureau (BGI, Bureau Gravimétrique International).

Another approach is to combine multiple information sources: not just terrestrial gravimetry, but also satellite geodetic data on the figure of the Earth, from analysis of satellite orbital perturbations, and lately from satellite gravity missions such as GOCE and GRACE. In such combination solutions, the low-resolution part of the geoid solution is provided by the satellite data, while a 'tuned' version of the above Stokes equation is used to calculate the high-resolution part, from terrestrial gravimetric data from a neighbourhood of the evaluation point only.

Gravity anomalies

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

\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\pi G\rho H,\,

The Bouguer reduction to be applied as follows:

\Delta g_{B}=\Delta g_{FA}-a_{B},\,

so-called Bouguer anomalies. Here, $\Delta g_{FA}$ is our earlier $\Delta g$ , 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.

References

↑ NASA/JPL/University of Texas Center for Space Research. "PIA12146: GRACE Global Gravity Animation". Photojournal. NASA Jet Propulsion Laboratory. Retrieved 30 December 2013.
↑ Boynton, Richard (2001). "Precise Measurement of Mass" (PDF). Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Archived from the original (PDF) on 27 February 2007. Retrieved 22 December 2023.
↑ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). Springer. ISBN 978-3-211-33544-4. § 2.1: "The total force acting on a body at rest on the earth's surface is the resultant of gravitational force and the centrifugal force of the earth's rotation and is called gravity."
↑ Bureau International des Poids et Mesures (1901). "Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de $g n$ ". Comptes Rendus des Séances de la Troisième Conférence· Générale des Poids et Mesures (in French). Paris: Gauthier-Villars. p. 68. Le nombre adopté dans le Service international des Poids et Mesures pour la valeur de l'accélération normale de la pesanteur est 980,665 cm/sec², nombre sanctionné déjà par quelques législations. Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de $g n$ .
↑ Moritz, Helmut (2000). "Geodetic Reference System 1980". Journal of Geodesy. 74 (1): 128–133. doi:10.1007/s001900050278. Retrieved 2023-07-26. γe = 9.780 326 7715 m/s² normal gravity at equator
↑ Wang, Yan Ming (2016). "Geodetic Boundary Value Problems". Encyclopedia of Geodesy. Cham: Springer International Publishing. pp. 1–8. doi:10.1007/978-3-319-02370-0_42-1. ISBN 978-3-319-02370-0.