Editing Geoid (section)

==Determination==
{{See also|Earth ellipsoid#Determination}}

The undulation of the geoid ''N'' is closely related to the [[disturbing potential]] ''T'' according to '''Bruns' formula''' (named after [[Heinrich Bruns]]):{{anchor|Bruns formula}}
:<math>
N=T/\gamma\,,
</math>
where <math>\gamma</math> is the force of [[normal gravity]], computed from the normal field potential <math>U</math>.

Another way of determining ''N'' is using values of ''[[gravity anomaly]]'' <math>\Delta g</math>, differences between true and normal reference gravity, as per '''{{vanchor|Stokes formula}}''' (or '''Stokes' integral'''), published in 1849 by [[George Gabriel Stokes]]:

:<math>
N=\frac{R}{4\pi \gamma_0}\iint_\sigma \Delta g \,S(\psi)\, d\sigma.
</math>

The [[integral kernel]] ''S'', called ''Stokes function'', was 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).

Another approach for geoid determination 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.

Calculating the undulation is mathematically challenging.<ref>{{Cite book | doi=10.1007/978-90-481-8702-7_154| chapter=Geoid Determination, Theory and Principles| title=Encyclopedia of Solid Earth Geophysics| pages=356–362| series=Encyclopedia of Earth Sciences Series| year=2011| last1=Sideris| first1=Michael G.| isbn=978-90-481-8701-0| s2cid=241396148}}</ref><ref>{{Cite book |doi = 10.1007/978-90-481-8702-7_225|chapter = Geoid, Computational Method|title = Encyclopedia of Solid Earth Geophysics|pages = 366–371|series = Encyclopedia of Earth Sciences Series|year = 2011|last1 = Sideris|first1 = Michael G.|isbn = 978-90-481-8701-0}}</ref>
The precise geoid solution by [[Petr Vaníček]] and co-workers improved on the [[George Gabriel Stokes|Stokesian]] approach to geoid computation.<ref>{{cite web|url=http://www2.unb.ca/gge/Research/GRL/GeodesyGroup/SHGeo.html |title=UNB Precise Geoid Determination Package |access-date=2 October 2007 }}</ref> Their solution enables millimetre-to-centimetre [[accuracy]] in geoid [[computation]], an [[order of magnitude|order-of-magnitude]] improvement from previous classical solutions.<ref>{{cite journal |last=Vaníček |first=P. |author2=Kleusberg, A. |date=1987 |title=The Canadian geoid-Stokesian approach |journal=Manuscripta Geodaetica |volume=12 |issue=2 |pages=86–98 |doi=10.1007/BF03655117 }}</ref><ref>{{cite journal |last=Vaníček |first=P. |author2=Martinec, Z. |date=1994 |title=Compilation of a precise regional geoid |journal=Manuscripta Geodaetica |volume=19 |pages=119–128 |doi=10.1007/BF03655333 |url=http://gge.unb.ca/Personnel/Vanicek/StokesHelmert.pdf }}</ref><ref>{{cite report|first1=P.|last1=Vaníček|first2=A.|last2=Kleusberg|first3=Z.|last3=Martinec|first4=W.|last4=Sun|first5=P.|last5=Ong|first6=M.|last6=Najafi|first7=P.|last7=Vajda|first8=L.|last8=Harrie|first9=P.|last9=Tomasek|first10=B.|last10=ter Horst|title=Compilation of a Precise Regional Geoid|publisher=Department of Geodesy and Geomatics Engineering, University of New Brunswick |docket=184|url=http://gge.unb.ca/Personnel/Vanicek/GeoidReport950327.pdf|access-date=22 December 2016}}</ref><ref>{{cite book|last1=Kopeikin|first1=Sergei|last2=Efroimsky|first2=Michael|last3=Kaplan|first3=George|title=Relativistic celestial mechanics of the solar system|url=https://archive.org/details/relativisticcele00kope|url-access=limited|date=2009|publisher=[[Wiley-VCH]]|location=Weinheim|isbn=9783527408566|page=[https://archive.org/details/relativisticcele00kope/page/n735 704]}}</ref>

Geoid undulations display uncertainties which can be estimated by using several methods, e.g., [[Least squares|least-squares]] collocation (LSC), [[fuzzy logic]], [[Artificial neural network|artificial neural networks]], [[radial basis function]]s (RBF), and [[Geostatistics|geostatistical]] techniques. Geostatistical approach has been defined as the most-improved technique in prediction of geoid undulation.<ref>{{Cite journal|last1=Chicaiza|first1=E.G.|last2=Leiva|first2=C.A.|last3=Arranz|first3=J.J.|last4=Buenańo|first4=X.E.|date=2017-06-14|title=Spatial uncertainty of a geoid undulation model in Guayaquil, Ecuador|journal=Open Geosciences|volume=9|issue=1|pages=255–265|doi=10.1515/geo-2017-0021|issn=2391-5447|bibcode=2017OGeo....9...21C|doi-access=free}}</ref>