The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended through the continents (such as with very narrow hypothetical canals). According to Gauss, who first described it, it is the "mathematical figure of the Earth", a smooth but irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of geodesy and geophysics, it has been defined to high precision only since advances in satellite geodesy in the late 20th century.
All points on a geoid surface have the same geopotential (the sum of gravitational potential energy and centrifugal potential energy). The force of gravity acts everywhere perpendicular to the geoid, meaning that plumb lines point perpendicular and bubble levels are parallel to the geoid. Being an equigeopotential means the geoid corresponds to the free surface of water at rest (if only gravity and rotational acceleration were at work); this is also a sufficient condition for a ball to remain at rest instead of rolling over the geoid. Earth's gravity acceleration (the vertical derivative of geopotential) is thus non-uniform over the geoid. The geoid undulation or geoidal height is the height of the geoid relative to a given reference ellipsoid. The geoid serves as a coordinate surface for various vertical coordinates, such as orthometric heights, geopotential heights, and dynamic heights (see Geodesy#Heights).
Description
The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as an ellipsoid), but is considerably smoother than Earth's physical surface. Although the "ground" of the Earth has excursions on the order of +8,800 m (Mount Everest) and −11,000 m (Marianas Trench), the geoid's deviation from an ellipsoid ranges from +85 m (Iceland) to −106 m (southern India), less than 200 m total.
If the ocean were isopycnic (of constant density) and undisturbed by tides, currents or weather, its surface would resemble the geoid. The permanent deviation between the geoid and mean sea level is called ocean surface topography. If the continental land masses were crisscrossed by a series of tunnels or canals, the sea level in those canals would also very nearly coincide with the geoid. In reality, the geoid does not have a physical meaning under the continents, but geodesists are able to derive the heights of continental points above this imaginary, yet physically defined, surface by spirit leveling.
Being an equipotential surface, the geoid is, by definition, a surface upon which the force of gravity is perpendicular everywhere. That means that when traveling by ship, one does not notice the undulations of the geoid; the local vertical (plumb line) is always perpendicular to the geoid and the local horizon tangential to it. Likewise, spirit levels will always be parallel to the geoid.
Simplified example
Earth's gravitational field is not uniform. An oblate spheroid is typically used as the idealized Earth, but even if the Earth were spherical and did not rotate, the strength of gravity would not be the same everywhere because density varies throughout the planet. This is due to magma distributions, the density and weight of different geological compositions in the Earth's crust, mountain ranges, deep sea trenches, crust compaction due to glaciers, and so on.
If that sphere were then covered in water, the water would not be the same height everywhere. Instead, the water level would be higher or lower with respect to Earth's center, depending on the integral of the strength of gravity from the center of the Earth to that location. The geoid level coincides with where the water would be. Generally the geoid rises where the earth material is locally more dense, which is where the Earth exerts greater gravitational pull.
Shape
The geoid undulation, geoid height, or geoid anomaly is the height of the geoid relative to a given ellipsoid of reference. The undulation is not standardized, as different countries use different mean sea levels as reference, but most commonly refers to the EGM96 geoid.
Relationship to GPS/GNSS
In maps and common use the height over the mean sea level (such as orthometric height) is used to indicate the height of elevations while the ellipsoidal height results from the GPS system and similar GNSS.