Geoid Height Calculator

The geoid is the equipotential surface of Earth's gravity field that best matches undisturbed mean sea level and continues notionally beneath the continents. Geoid height, or undulation N, is its separation from the reference ellipsoid, and the three heights are linked by h = H + N, where h is the ellipsoidal height a GNSS receiver reports and H is the orthometric height along which water actually flows downhill. Because the ellipsoid is a smooth mathematical figure - WGS84 uses a semi-major axis of 6,378,137 m and inverse flattening 298.257223563 - while the geoid follows real mass distribution, N ranges globally from about -106 m south of India to about +85 m near New Guinea, and from roughly -8 m to -53 m across the contiguous United States. N is computed by summing spherical harmonic coefficients of the gravitational potential. The dominant term is J2 at -484.17 x 10^-6, the oblateness signal, followed by rapidly diminishing zonal and tesseral terms. A low-degree expansion such as the one used here reproduces only the broadest planetary pattern and can be tens of metres from the true value at any specific site, so it is a teaching tool rather than a survey instrument. Serious work uses EGM2008, accurate to roughly 10 to 20 cm globally, or a regional hybrid model such as NOAA GEOID18 for North America. Related quantities follow from the ellipsoid alone: the prime-vertical and meridian radii of curvature, and normal gravity from the Somigliana formula, which gives 9.780325 m/s2 at the equator and 9.832185 m/s2 at the poles.

Carl Friedrich Gauss described in 1828 the idea of a mathematical figure of the Earth defined by a level surface of gravity rather than by a geometric shape, and Johann Benedict Listing gave that surface the name geoid in 1873. George Gabriel Stokes published in 1849 the integral that converts gravity anomalies observed over the whole Earth into geoid undulations, and Heinrich Bruns supplied in 1878 the simple relation between the disturbing potential and N that underpins modern practice. Mikhail Molodensky proposed in the 1940s the normal-height alternative, which avoids having to assume a density for the rock between the geoid and the surface. Satellites transformed the field. Tracking of the first artificial satellites in the late 1950s revealed low-degree terms that gave the Earth its famous slightly pear-shaped description, and successive global models grew in resolution until the National Geospatial-Intelligence Agency released EGM96 in 1996 and EGM2008 in 2008, the latter complete to spherical harmonic degree 2159. The dedicated gravity missions GRACE, launched in 2002, and GOCE, flown from 2009 to 2013, mapped the long- and medium-wavelength field with unprecedented accuracy. Practical demand came from satellite positioning: receivers deliver ellipsoidal heights directly, so national geodetic agencies now publish geoid models, and the United States is moving its vertical datum onto a geoid-based framework so that GNSS observations convert straightforwardly into usable elevations.