Math 2.0

Geophysics Definition

Geoid

Builds on GeopotentialLevel setReference ellipsoid

Definition

Let WW be the geopotential of the Earth. The geoid is the level set {š±āˆˆā„3:W(š±)=W0},\set{\mathbf{x} \in \R^3 : W(\mathbf{x}) = W_0}, where the constant W0W_0 is chosen so that this surface best fits mean sea level. The fit is in the least squares sense. The geoid undulation NN at a point of a reference ellipsoid is the height of the geoid above the ellipsoid, measured along the ellipsoid normal.

Remarks

Example: for a non-rotating Earth with spherically symmetric density, W=GM/rW = GM/r outside the Earth, so every level set there is a sphere r=GM/W0r = GM/W_0; the geoid would be a sphere. Non-example: the instantaneous sea surface is not the geoid, since tides, winds and currents move it away from a level surface by up to a few metres. The geoid is everywhere perpendicular to the gravity vector āˆ‡W\nabla W. The orthometric height HH of a point, its height above the geoid, satisfies hā‰ˆH+Nh \approx H + N, where hh is the ellipsoidal height. On the Earth, NN ranges roughly between āˆ’106-106 m and +85+85 m.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGeopotential, concept, no evidence levelGeopotentialconceptReference ellipsoid, concept, no evidence levelReference ellipsoidconceptFunction of several variables, concept, no evidence levelFunction of several vā€¦conceptLevel set, concept, no evidence levelLevel setconceptGeoid, definition, no evidence levelGeoiddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Geopotential concept
  4. Reference ellipsoid concept
  5. Function of several variables concept
  6. Level set concept
  7. Geoid definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Geoid
About this page
Tag2AB1XN3JQ4
URIm2://2AB1XN3JQ4
Kinddefinition
WritingW0 stub
MSC86A30, 86A20
DefinesGeoid
UsesGeopotential, Level set, Reference ellipsoid
Changed3 November 2026

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