Editing Vertical deflection

{{Short description|Measure of the downward gravitational force's shift due to nearby mass}}
{{distinguish|Cathode-ray tube#Deflection}}
[[Image:GRAVIMETRIC DATUM ORIENTATION.SVG|thumb|upright=1.75|Earth's ellipsoid, geoid, and two types of vertical deflection.
{{legend-line|solid cyan|Ellipsoid of [[Geodetic astronomy|astro-geodetically]] oriented datum}}
{{legend-line|solid lime|[[Geoid]]}}
{{legend-line|solid red|[[Gravimetry|Gravimetrically]]-oriented ellipsoid}}
]]

The '''vertical deflection''' ('''VD''') or '''deflection of the vertical''' ('''DoV'''), also known as '''deflection of the plumb line''' and '''astro-geodetic deflection''', is a measure of how far the [[gravity direction]] at a given point of interest is rotated by local mass anomalies such as nearby mountains. They are widely used in [[geodesy]], for [[surveying]] networks and for [[geophysics|geophysical]] purposes.

The vertical deflection are the angular components between the true [[zenith]]–[[nadir]] [[curve]] ([[plumb line]]) [[tangent line]] and the [[normal vector]] to the surface of the [[reference ellipsoid]] (chosen to approximate the Earth's [[Sea level|sea-level]] surface). VDs are caused by [[mountain]]s and by underground [[geology|geological]] irregularities. Typically angle values amount to less than 10 [[arc-second]]s in flat areas or up to 1 [[arc-minute]] in mountainous [[terrain]].<ref name="t119">{{cite web | title=DEFLEC18 | website=National Geodetic Survey | date=2019-02-26 | url=https://geodesy.noaa.gov/GEOID/DEFLEC18/ | access-date=2025-03-23}}</ref>

==Components==
The deflection of the vertical has a [[meridian (astronomy)|north–south]] component ''ξ'' ([[Xi (letter)|xi]]) and an [[prime vertical|east–west]] component&nbsp;''η'' ([[Eta (letter)|eta]]). The value of ''ξ'' is the difference between the ''[[astronomic latitude]]'' and the ''[[geodetic latitude]]'' (taking north latitudes to be positive and south latitudes to be negative); the latter is usually calculated by geodetic network [[coordinate]]s. The value of ''η is'' the product of cosine of latitude and the difference between the ''[[astronomic longitude]]'' and the [[longitude]] (taking east longitudes to be positive and west longitudes to be negative). When a new [[mapping datum]] replaces the old, with new geodetic latitudes and longitudes on a new ellipsoid, the calculated vertical deflections will also change.

==Determination==
{{see also|Geoid determination}}

The deflections reflect the [[undulation of the geoid]] and [[gravity anomaly|gravity anomalies]], for they depend on the [[gravity field]] and its inhomogeneities.

Vertical deflections are usually determined astronomically. The ''true zenith'' is observed astronomically with respect to the [[star]]s, and the ''ellipsoidal zenith'' (theoretical vertical) by geodetic network computation, which always takes place on a [[reference ellipsoid]]. Additionally, the very local variations of the vertical deflection can be computed from gravimetric survey data and by means of [[digital terrain model]]s (DTM), using a theory originally developed by [[Felix Andries Vening Meinesz|Vening-Meinesz]].

VDs are used in [[astrogeodetic levelling]]: as a vertical deflection describes the difference between the geoidal vertical direction and ellipsoidal normal direction, it represents the horizontal [[spatial gradient]] of the [[geoid undulation]]s, i.e., the geoid [[Grade (slope)|slope]] or the inclination between geoid and reference ellipsoid.<ref name="r251">{{cite web | title=Geoid evaluation | website=National Geodetic Survey | url=https://www.ngs.noaa.gov/research/geopotential-datums/evaluation-dov.shtml | access-date=2025-03-23}}</ref>

In practice, the deflections are observed at special points with spacings of 20 or 50 kilometers. The densification is done by a combination of DTM models and areal [[gravimetry]]. Precise vertical deflection observations have accuracies of ±0.2″ (on high mountains ±0.5″), calculated values of about 1–2″.

The maximal vertical deflection of [[Central Europe]] seems to be a point near the [[Großglockner]] (3,798&nbsp;m), the highest peak of the [[Austria]]n [[Alps]]. The approx. values are ξ = +50″ and η = −30″. In the [[Himalaya]] region, very asymmetric peaks may have vertical deflections up to 100″ (0.03°). In the rather flat area between [[Vienna]] and [[Hungary]] the values are less than 15", but scatter by ±10″ for irregular rock densities in the subsurface.

More recently, a combination of [[digital camera]] and [[tiltmeter]] have also been used, see [[zenith camera]].<ref>{{Cite journal | doi = 10.1061/(ASCE)SU.1943-5428.0000009| title = Modern Determination of Vertical Deflections Using Digital Zenith Cameras| journal = Journal of Surveying Engineering| volume = 136| pages = 1–12| year = 2010| last1 = Hirt | first1 = C. | last2 = Bürki | first2 = B. | last3 = Somieski | first3 = A. | last4 = Seeber | first4 = G. N. | hdl = 20.500.11937/34194| url = https://espace.curtin.edu.au/bitstream/20.500.11937/34194/2/153379_153379.pdf | hdl-access = free }}</ref>

==Application==
Vertical deflections are principally used in four matters:
# For precise calculation of '''survey networks'''. The geodetic [[theodolite]]s and levelling [[measuring instrument|instrument]]s are oriented with respect to the true [[vertical direction|vertical]], but its [[Deflection (physics)|deflection]] exceeds the geodetic measuring accuracy by a factor of 5 to 50. Therefore, the data would have to be corrected exactly with respect to the global ellipsoid. Without these reductions, the surveys may be [[distortion|distorted]] by some centimeters or even decimeters per km.
# For the '''geoid determination''' (mean sea level) and for exact transformation of [[elevation]]s. The global geoidal [[Undulation of the geoid|undulation]]s amount to 50–100&nbsp;m, and their [[region]]al values to 10–50&nbsp;m. They are adequate to the [[integral]]s of VD components ξ,η and therefore can be calculated with cm accuracy over distances of many kilometers.
# For '''[[GPS]] surveys'''. The [[satellite]]s measurements refer to a pure [[geometry|geometrical]] system (usually the [[WGS84]] ellipsoid), whereas the terrestrial heights refer to the geoid. We need accurate geoid data to combine the different types of measurements.
# For '''[[geophysics]]'''. Because VD data are affected by the physical structure of the Earth's [[Crust (geology)|crust]] and mantle, [[geodesist]]s are engaged in [[model (abstract)|models]] to improve our knowledge of the Earth's interior. Additionally and similar to ''applied geophysics'', the VD data can support the future [[exploration]] of raw materials, [[Petroleum|oil]], gas or [[ore]]s.

==Historical implications==
Vertical deflections were used to measure [[Earth's density]] in the [[Schiehallion experiment]].

Vertical deflection is the reason why modern [[prime meridian]] passes more than 100&nbsp;m to the east of the [[Prime meridian (Greenwich)|historical astronomic prime meridian]] in Greenwich.<ref>{{cite journal| title = Why the Greenwich meridian moved | first1 = Stephen | last1 = Malys | first2 = John H. | last2 = Seago | first3 = Nikolaos K. | last3 = Palvis | first4 = P. Kenneth | last4 = Seidelmann | first5 = George H. | last5 = Kaplan | journal = Journal of Geodesy | volume = 89 | issue = 12 | pages = 1263 | date = 1 August 2015 | doi = 10.1007/s00190-015-0844-y|bibcode = 2015JGeod..89.1263M | doi-access = free }}</ref>

The [[meridian arc measurement]] made by [[Nicolas-Louis de Lacaille]] north of [[Cape Town]] in 1752 ([[de Lacaille's arc measurement]]) was affected by vertical deflection.<ref>{{cite web
 |url=https://assa.saao.ac.za/sections/history/expeditions/arc_meridian
 |title=Arc of the Meridian
 |publisher=Astronomical Society of South Africa
 |access-date= 27 August 2020
 }}</ref> The resulting discrepancy with Northern Hemisphere measurements was not explained until a visit to the area by [[George Everest]] in 1820; [[Maclear's arc measurement]] resurvey ultimately confirmed Everest's conjecture.<ref>{{cite journal
 |title=Lacaille 250 years on
 |last=Warner
 |first=Brian
 |journal=Astronomy and Geophysics
 |date= 1 April 2002
 |volume=43
 |issue=2
 |pages=2.25–2.26
 |doi= 10.1046/j.1468-4004.2002.43225.x
 |doi-access=free
 }}</ref>

Errors in the [[meridian arc of Delambre and Méchain]] determination, which affected the original definition of the [[metre]],<ref name="Alder 2002 p. ">{{cite book | last=Alder | first=K. | title=The Measure of All Things: The Seven-year Odyssey and Hidden Error that Transformed the World | publisher=Free Press | year=2002 | isbn=978-0-7432-1675-3 | url=https://books.google.com/books?id=gQu6uMyYrB4C | access-date=2020-08-02 }}</ref> were long known to be mainly caused by an uncertain determination of [[Barcelona]]'s latitude later explained by vertical deflection.<ref>Jean-Étienne Duby, Rapport sur les travaux de la Société de Physique et d’Histoire naturelle de Genève de juillet 1860 à juin 1861 par M. le Pasteur Duby. Lu à la séance du 13 juin 1861, in Mémoires de la Société de physique et d’histoire naturelle de Genève, 16 (1861-1862), 196-197.</ref><ref name="VaníčekForoughi2019">{{cite journal|last1=Vaníček|first1=Petr|last2=Foroughi|first2=Ismael|title=How gravity field shortened our metre|journal=Journal of Geodesy|volume=93|issue=9|year=2019|pages=1821–1827|issn=0949-7714|doi=10.1007/s00190-019-01257-7|bibcode=2019JGeod..93.1821V|s2cid=146099564}}</ref><ref>{{Cite journal |last=Levallois |first=Jean-Jacques |title=La méridienne de Dunkerque à Barcelone et la déterminiation du mètre (1972-1799) |url=https://dx.doi.org/10.5169/seals-234595 |access-date=2022-12-23 |website=E-Periodica |year=1991 |language=fr |doi=10.5169/seals-234595}}</ref> When these errors where acknowledged in 1866,<ref>{{Cite journal |last=Hirsch |first=Adolphe |title=Sur les progrès des travaux géodésiques en Europe |url=https://dx.doi.org/10.5169/seals-88030 |access-date=2022-12-23 |website=E-Periodica |year=1865 |language=fr |doi=10.5169/seals-88030}}</ref> it became urgent to proceed to a new measurement of the French arc between Dunkirk and Perpignan. The operations concerning the revision of the French arc linked to Spanish triangulation were completed only in 1896. Meanwhile, the French geodesists had accomplished in 1879 the junction of Algeria to Spain, with the help of the geodesists of the Madrid Institute headed by the late [[Carlos Ibáñez e Ibáñez de Ibero|Carlos Ibañez Ibáñez de Ibero]] (1825–1891).{{efn|He had been president of the International Geodetic Association (now called [[International Association of Geodesy]]), first president of the [[General Conference on Weights and Measures|International Committee for Weights and Measures]], and one of the 81 initial members of the [[International Statistical Institute]].<ref>{{Cite EB1911|wstitle= Earth, Figure of the| volume= 8 |last1= Clarke |first1=  Alexander Ross |author1-link= Alexander Ross Clarke | |last2= Helmert |first2= Friedrich Robert |author2-link= Friedrich Robert Helmert| pages = 801&ndash;813|quote= see page 811 |short=1}}</ref>}}

Until [[Hayford ellipsoid]] was calculated in 1910, vertical deflections were considered as [[Observational error|random errors]].<ref name=":33">{{Cite book |title=Géodésie in Encyclopedia Universalis |publisher=Encyclopedia Universalis |year=1996 |isbn=978-2-85229-290-1 |pages=Vol 10, p. 302 |oclc=36747385}}</ref> Plumb line deviations were identified by [[Jean le Rond d'Alembert|Jean Le Rond d'Alembert]] as an important source of error in geodetic surveys as early as 1756. A few years later, in 1828, [[Carl Friedrich Gauss]] proposed the concept of [[geoid]].<ref>{{Cite web |last=d'Alembert |first=Jean Le Rond |date=1756 |title=Article Figure de la Terre, (Astron. Géog. Physiq. & Méch.), vol. VI (1756), p. 749b–761b |url=http://enccre.academie-sciences.fr/encyclopedie/article/v6-872-5/ |access-date=2022-12-23 |website=enccre.academie-sciences.fr}}</ref><ref>{{Cite web |last=US Department of Commerce |first=National Oceanic and Atmospheric Administration |title=What is the geoid? |url=https://geodesy.noaa.gov/GEOID/geoid_def.html |access-date=2022-12-23 |website=geodesy.noaa.gov |language=EN-US}}</ref>

==See also==
*[[Deviation survey]]
*[[Gravity anomaly]]
*[[Isostasy]]
*[[Vertical direction]]
*[[Zenith]]

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

== External links ==
* The NGS website gives vertical deflection anywhere in the United States [https://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/deflec09_prompt.prl here] and [https://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/deflec99_prompt.prl here].

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{{DEFAULTSORT:Vertical Deflection}}
[[Category:Geodesy]]
[[Category:Geophysics]]
[[Category:Gravity]]