Geoid

1. Ocean
2. Ellipsoid
3. Local plumb line
4. Continent
5. Geoid

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.[3]

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 to which the force of gravity is everywhere perpendicular. 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.

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.

The deviation ${\displaystyle N}$ between the ellipsoidal height ${\displaystyle h}$ and the orthometric height ${\displaystyle H}$ can be calculated by

$${\displaystyle N=h-H}$$

(An analogous relationship exists between normal heights and the quasigeoid.)

So a GPS receiver on a ship may, during the course of a long voyage, indicate height variations, even though the ship will always be at sea level (neglecting the effects of tides). That is because GPS satellites, orbiting about the center of gravity of the Earth, can measure heights only relative to a geocentric reference ellipsoid. To obtain one's orthometric height, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a tide gauge, as in traditional land surveying, is closer to orthometric height. Modern GPS receivers have a grid implemented in their software by which they obtain, from the current position, the height of the geoid (e.g. the EGM-96 geoid) over the World Geodetic System (WGS) ellipsoid. They are then able to correct the height above the WGS ellipsoid to the height above the EGM96 geoid. When height is not zero on a ship, the discrepancy is due to other factors such as ocean tides, atmospheric pressure (meteorological effects), local sea surface topography and measurement uncertainties.

Gravity and Geoid anomalies caused by various crustal and lithospheric thickness changes relative to a reference configuration. All settings are under local isostatic compensation.

Variations in the height of the geoidal surface are related to anomalous density distributions within the Earth. Geoid measures thus help understanding the internal structure of the planet. Synthetic calculations show that the geoidal signature of a thickened crust (for example, in orogenic belts produced by continental collision) is positive, opposite to what should be expected if the thickening affects the entire lithosphere. Mantle convection also changes the shape of the geoid over time.[8]

Three-dimensional visualization of gravity anomalies in units of Gal., using pseudo color and shaded relief.

Calculating the undulation is mathematically challenging.[9][10] This is why many handheld GPS receivers have built-in undulation lookup tables[11] to determine the height above sea level.

The precise geoid solution by Vaníček and co-workers improved on the Stokesian approach to geoid computation.[12] Their solution enables millimetre-to-centimetre accuracy in geoid computation, an order-of-magnitude improvement from previous classical solutions.[13][14][15][16]

Geoid undulations display uncertainties which can be estimated by using several methods, e.g. least-squares collocation (LSC), fuzzy logic, artificial neutral networks, radial basis functions (RBF), and geostatistical techniques. Geostatistical approach has been defined as the most improved technique in prediction of geoid undulation.[17]

Recent satellite missions, such as the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and GRACE, have enabled the study of time-variable geoid signals. The first products based on GOCE satellite data became available online in June 2010, through the European Space Agency (ESA)'s Earth observation user services tools.[18][19] ESA launched the satellite in March 2009 on a mission to map Earth's gravity with unprecedented accuracy and spatial resolution. On 31 March 2011, the new geoid model was unveiled at the Fourth International GOCE User Workshop hosted at the Technical University of Munich, Germany.[20] Studies using the time-variable geoid computed from GRACE data have provided information on global hydrologic cycles,[21] mass balances of ice sheets,[22] and postglacial rebound.[23] From postglacial rebound measurements, time-variable GRACE data can be used to deduce the viscosity of Earth's mantle.[24]

Geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also: Earth's pear shape.

Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM2020 (Earth Gravity Model 2020), determined in an international collaborative project led by the National Imagery and Mapping Agency (now the National Geospatial-Intelligence Agency, or NGA). The mathematical description of the non-rotating part of the potential function in this model is:[25]

$${\displaystyle V={\frac {GM}{r}}\left(1+{\sum _{n=2}^{n_{\text{max}}}}\left({\frac {a}{r}}\right)^{n}{\sum _{m=0}^{n}}{\overline {P}}_{nm}(\sin \phi )\left[{\overline {C}}_{nm}\cos m\lambda +{\overline {S}}_{nm}\sin m\lambda \right]\right),}$$

where ${\displaystyle \phi \ }$ and ${\displaystyle \lambda \ }$ are geocentric (spherical) latitude and longitude respectively, ${\displaystyle {\overline {P}}_{nm}}$ are the fully normalized associated Legendre polynomials of degree ${\displaystyle n\ }$ and order ${\displaystyle m\ }$, and ${\displaystyle {\overline {C}}_{nm}}$ and ${\displaystyle {\overline {S}}_{nm}}$ are the numerical coefficients of the model based on measured data. Note that the above equation describes the Earth's gravitational potential ${\displaystyle V\ }$, not the geoid itself, at location ${\displaystyle \phi ,\;\lambda ,\;r,\ }$ the co-ordinate ${\displaystyle r\ }$ being the geocentric radius, i.e., distance from the Earth's centre. The geoid is a particular equipotential surface,[25] and is somewhat involved to compute. The gradient of this potential also provides a model of the gravitational acceleration. The most commonly used EGM96 contains a full set of coefficients to degree and order 360 (i.e. ${\displaystyle n_{\text{max}}=360}$), describing details in the global geoid as small as 55 km (or 110 km, depending on your definition of resolution). The number of coefficients, ${\displaystyle {\overline {C}}_{nm}}$ and ${\displaystyle {\overline {S}}_{nm}}$, can be determined by first observing in the equation for V that for a specific value of n there are two coefficients for every value of m except for m = 0. There is only one coefficient when m=0 since ${\displaystyle \sin(0\lambda )=0}$. There are thus (2n+1) coefficients for every value of n. Using these facts and the formula, ${\textstyle \sum _{I=1}^{L}I={\frac {1}{2}}L(L+1)}$, it follows that the total number of coefficients is given by

$${\displaystyle \sum _{n=2}^{n_{\text{max}}}(2n+1)=n_{\text{max}}(n_{\text{max}}+1)+n_{\text{max}}-3=130317}$$

using the EGM96 value of ${\displaystyle n_{\text{max}}=360}$.

For many applications the complete series is unnecessarily complex and is truncated after a few (perhaps several dozen) terms.

Still, even high resolution models have been developed. Many of the authors of EGM96 have published EGM2008. It incorporates much of the new satellite gravity data (e.g., the Gravity Recovery and Climate Experiment), and supports up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients),[26] with additional coefficients extending to degree 2190 and order 2159.[27] EGM2020 is the planned follow-up of 2020 (now overdue), containing the same number of harmonics generated with better data.[28]

• Deflection of the vertical
• Geodetic datum
• Geopotential
• Planetary geoid
  • Areoid (Mars' geoid)
  • Selenoid (Moon's geoid)
• Physical geodesy
• International Terrestrial Reference Frame

Look up geoid in Wiktionary, the free dictionary.

• Main NGA (was NIMA) page on Earth gravity models Archived 20 June 2006 at the Wayback Machine
• International Geoid Service (IGeS) Archived 5 April 2014 at the Wayback Machine
• EGM96 NASA GSFC Earth gravity model
• Earth Gravitational Model 2008 (EGM2008, Released in July 2008) Archived 8 May 2010 at the Wayback Machine
• NOAA Geoid webpage
• International Centre for Global Earth Models (ICGEM)
• Kiamehr's Geoid Home Page Archived 20 July 2019 at the Wayback Machine
• Geoid tutorial from Li and Gotze (964KB pdf file)
• Geoid tutorial at GRACE website
• Precise Geoid Determination Based on the Least-Squares Modification of Stokes’ Formula (PhD Thesis PDF)