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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[edit]

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[edit]

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[edit]

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 per second (32 ft/s) 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

^ 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. S2CID 195290884. 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.

@@ Line 1: / Line 1: @@
+{{Short description|Study of the physical properties of the Earth's gravity field}}
-[[Image:Ocean gravity map.gif|right|thumb|350px|Ocean basins mapped with satellite altimetry. Seafloor features larger than 10 km are detected by resulting gravitational distortion of sea surface. (1995, [[NOAA]])]]
+{{Merge to |Geodesy|discuss=Talk:Geodesy#Merge Physical geodesy into Geodesy |date=March 2024 }}
+{{more citations needed|date=October 2021}}
+[[Image:Ocean gravity map.gif|right|thumb|350px|Ocean basins mapped gravitationally. Seafloor features larger than 10 km are detected by resulting gravitational distortion of sea surface. (1995, [[NOAA]])]]
 {{Geodesy}}
-'''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]].
+'''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==
@@ Line 8: / Line 11: @@
 [[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===
-== The geopotential ==
+Gravity is commonly measured in units of m·s<sup>−2</sup> ([[metre]]s per [[second]] squared). This also can be expressed (multiplying by the [[gravitational constant]] '''G''' in order to change units) as [[newton (unit)|newton]]s per [[kilogram]] of attracted mass.
-The Earth's gravity field can be described by a [[potential]] as follows:
+Potential is expressed as gravity times distance, m<sup>2</sup>·s<sup>−2</sup>.  Travelling one metre in the direction of a gravity vector of strength 1 m·s<sup>−2</sup> will increase your potential by 1 m<sup>2</sup>·s<sup>−2</sup>. Again employing G as a multiplier, the units can be changed to [[joule]]s per kilogram of attracted mass.
-:<math>
-\mathbf{g} = \nabla W = \mathrm{grad}\ W = \frac{\partial W}{\partial X}\mathbf{i}
-+\frac{\partial W}{\partial Y}\mathbf{j}+\frac{\partial W}{\partial Z}\mathbf{k}
-</math>
-which expresses the gravitational acceleration vector as the gradient of <math>W</math>, the potential of gravity. The vector triad <math>\{\mathbf{i},\mathbf{j},\mathbf{k}\}</math> is the orthonormal set of base vectors in space, pointing along the <math>X,Y,Z</math> coordinate axes.
-Note that both gravity and its potential contain a contribution from the [[centrifugal force|centrifugal pseudo-force]] due to the Earth's rotation. We can write
-:<math>
-W = V + \Phi\,
-</math>
-where <math>V</math> is the potential of the ''gravitational'' field, <math>W</math> that of the ''gravity'' field, and <math>\Phi</math> that of the centrifugal force field.
-The centrifugal force -- per unit of mass, i.e., acceleration -- is given by
-:<math>
-\mathbf{g}_c = \omega^2 \mathbf{p},
-</math>
-where
-:<math>
-\mathbf{p} = X\mathbf{i}+Y\mathbf{j}+0\cdot\mathbf{k}
-</math>
-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:
-:<math>
-\Phi = \frac{1}{2} \omega^2 (X^2+Y^2).
-</math>
-This can be verified by taking the gradient (<math>\nabla</math>) operator of this expression.
-Here, <math>X</math>, <math>Y</math> and <math>Z</math> are [[geocentric coordinates]].
-==Units of gravity and geopotential==
-Gravity is commonly measured in units of m·s<sup>−2</sup> ([[metre]]s per [[second]] squared). This also can be expressed (multiplying by the [[gravitational constant]] '''G''' in order to change units) as [[newton (unit)|newton]]s per [[kilogram]] of attracted mass.
-Potential is expressed as gravity times distance, m<sup>2</sup>·s<sup>−2</sup>.  Travelling one metre in the direction of a gravity vector of strength 1 m·s<sup>−2</sup> will increase your potential by 1 m<sup>2</sup>·s<sup>−2</sup>. Again employing G as a multipier, the units can be changed to [[joule]]s per kilogram of attracted mass.
 A more convenient unit is the GPU, or geopotential unit: it equals 10 m<sup>2</sup>·s<sup>−2</sup>. This means that travelling one metre in the vertical direction, i.e., the direction of the 9.8 m·s<sup>−2</sup> 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==
-== The normal potential ==
+{{excerpt|Earth's gravity}}
+==Potential fields==
-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:
+{{excerpt|Geopotential}}
+==Geoid==
-:<math>
+{{main|Geoid}}
-W \approx \frac{GM}{R}
+[[File:Earth_Gravitational_Model_1996.png|thumb|Map of the undulation of the geoid in meters (based on the [[EGM96]])]]
-</math>
+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 <math>N</math>.
+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.
-of which the ''equipotential surfaces''—the surfaces of constant potential value—are concentric spheres.
+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 datum]]s'' or ''height [[datum (geodesy)|datum]]s''.
-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 ''a''&nbsp;= 6378137.0&nbsp;m, and flattening ''f''&nbsp;= 1/298.257222101.
+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]]).
-A geopotential field <math>U</math> is constructed, being the sum of a gravitational potential <math>\Psi</math> and the known centrifugal potential <math>\Phi</math>, 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 &times; 10<sup>8</sup> m<sup>3</sup>·s<sup>−2</sup>, we obtain for the ''potential at the reference ellipsoid:''
-:<math>
-U_0=62636860.850 \ \textrm m^2 \, \textrm s^{-2}
-</math>
-Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (<math>R\rightarrow\infty</math>), 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 geoid ==
-Once a clean, smooth geopotential field <math>U</math> 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 (measured) potential <math>W</math> of the real Earth. The result is defined as '''T''', the ''disturbing potential'':
-:<math>
-T = W-U
-</math>
-The disturbing potential '''T''' is numerically a great deal smaller than '''U''' or '''W''', and captures the detailed, complex variations of the true gravity field of the actually existing Earth from point-to-point, as distinguished from the overall global trend captured by the smooth mathematical ellipsoid of 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 <math>N</math>, and is closely related to the disturbing potential.
-According to the famous [[Heinrich Bruns|Bruns]] formula, we have
+===Geoid determination===
+{{main|Geoid determination}}
+The [[undulation of the geoid]] is closely related to the disturbing potential according to the famous ''[[Heinrich Bruns|Bruns]]' formula'':{{anchor|Bruns formula}}
 :<math>
 N=T/\gamma\,,
 </math>
 where <math>\gamma</math> is the force of gravity computed from the normal field potential <math>U</math>.
-In 1849, the mathematician [[George Gabriel Stokes]] published the following formula named after him:
+{{anchor|Stokes formula}}In 1849, the mathematician [[George Gabriel Stokes]] published the following formula, named after him:
 :<math>
@@ Line 104: / Line 51: @@
 </math>
-In this formula, <math>\Delta g</math> 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 <math>N</math> anywhere on Earth by this formula requires <math>\Delta g</math> to be known ''everywhere on Earth''. Welcome to the role of international co-operation in physical geodesy.)
+In ''Stokes' formula'' or ''Stokes' integral'', <math>\Delta g</math> 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.<ref name="Wang 2016 pp. 1–8">{{cite book | last=Wang | first=Yan Ming | title=Encyclopedia of Geodesy | chapter=Geodetic Boundary Value Problems | publisher=Springer International Publishing | publication-place=Cham | year=2016 | isbn=978-3-319-02370-0 | doi=10.1007/978-3-319-02370-0_42-1 | pages=1–8}}</ref>
+Note that determining <math>N</math> anywhere on Earth by this formula requires <math>\Delta g</math> 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).
-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.
+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 [[Gravity Recovery and Climate Experiment|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.
-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 [[datum (geodesy)|datum]]s''.
-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 [[leveling|levelling]]'', for deriving the local figure of the geoid by measuring ''[[vertical deflection|deflections of the vertical]]'' by astronomical means over an area.
 == Gravity anomalies ==<!-- This section is linked from [[Divergence theorem]] -->
@@ Line 116: / Line 61: @@
 {{main|Gravity anomaly}}
 Above we already made use of ''gravity anomalies'' <math>\Delta g</math>. These are computed as the differences between true (observed) gravity <math>g=\|\vec{g}\|</math>, and calculated (normal) gravity <math>\gamma=\|\vec{\gamma}\|=\|\nabla U\|</math>. (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
 :<math>
@@ Line 138: / Line 83: @@
 so-called [[Bouguer anomaly|Bouguer anomalies]]. Here, <math>\Delta g_{FA}</math> is our earlier <math>\Delta g</math>, 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]] (''TC'').
+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]].
 == See also ==
-* [[LAGEOS]]
+* [[Dynamic height]]
 * [[Friedrich Robert Helmert]]
 * [[Geophysics]]
 * [[Gravity of Earth]]
-* [[gravimetry]]
+* [[Gravimetry]]
-* [[satellite geodesy]]
+* [[LAGEOS]]
+* [[Mikhail Molodenskii]]
+* [[Normal height]]
+* [[Orthometric height]]
+* [[Satellite geodesy]]
 ==References==
+{{Reflist}}
+==Further reading==
 * 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).
+{{Authority control}}
 [[Category:Geodesy]]
-[[Category:Gravitation]]
+[[Category:Gravity]]
 [[Category:Geophysics]]
+[[Category:Gravimetry]]

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