Geoid

Geodesy
Fundamentals Geodesy Geodynamics Geomatics History
Concepts Geographical distance Geoid Figure of the Earth (radius and circumference) Geodetic coordinates Geodetic datum Geodesic Horizontal position representation Latitude / Longitude Map projection Reference ellipsoid Satellite geodesy Spatial reference system Spatial relations Vertical positions
Technologies Global Nav. Sat. Systems (GNSSs) Global Pos. System (GPS) GLONASS (Russia) BeiDou (BDS) (China) Galileo (Europe) NAVIC (India) Quasi-Zenith Sat. Sys. (QZSS) (Japan) Discrete Global Grid and Geocoding
Standards (history) GRS 80 Geodetic Reference System 1980 ISO 6709 Geographic point coord. 1983 NAD 83 North American Datum 1983 WGS 84 World Geodetic System 1984 NAVD 88 N. American Vertical Datum 1988 ETRS89 European Terrestrial Ref. Sys. 1989 GCJ-02 Chinese obfuscated datum 2002 Geo URI Internet link to a point 2010 International Terrestrial Reference System Spatial Reference System Identifier (SRID) Universal Transverse Mercator (UTM)
v t e

The geoid (

Description

The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as an

Earth's gravitational field is not uniform. An

Relationship to GPS/GNSS

In maps and common use, the height over the mean sea level (such as

The deviation ${\displaystyle N}$ between the

${\displaystyle h}$

${\displaystyle H}$

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

(An analogous relationship exists between normal heights and the "quasigeoid", which disregards local density variations.)

So a

Relationship to mass density

The surface of the geoid is higher than the

This relationship can be understood by recalling that gravity potential is defined so that it has negative values and is inversely proportional to distance from the body. So, while a mass excess will strengthen the gravity acceleration, it will decrease the gravity potential. As a consequence, the geoid's defining equipotential surface will be found displaced away from the mass excess. Analogously, a mass deficit will weaken the gravity pull but will increase the geopotential at a given distance, causing the geoid to move towards the mass deficit. The presence of a localized inclusion in the background medium will rotate the gravity acceleration vectors slightly towards or away from a denser or lighter body, respectively, causing a bump or dimple in the equipotential surface.[6]

The largest absolute deviation can be found in the Indian Ocean Geoid Low, 106 meters below the average sea level.^[7]

Gravity anomalies

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

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

Recent satellite missions, such as the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and

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

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. 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 the 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 ${\displaystyle V}$ that for a specific value of ${\displaystyle n}$ there are two coefficients for every value of ${\displaystyle m}$ except for ${\displaystyle m=0}$. There is only one coefficient when ${\displaystyle m=0}$ since ${\displaystyle \sin(0\lambda )=0}$. There are thus ${\displaystyle (2n+1)}$ coefficients for every value of ${\displaystyle n}$. 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

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

${\displaystyle n_{\text{max}}=360}$

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

Still, even higher 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

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

References

Further reading

• Li, Xiong; Götze, Hans-Jürgen (November 2001). "Ellipsoid, geoid, gravity, geodesy, and geophysics" (PDF). Geophysics. 66 (6): 1660–1668.
  doi:10.1190/1.1487109
  .
• Moritz, H. (March 2011). "A contemporary perspective of geoid structure". Journal of Geodetic Science. 1 (1): 82–87.
  doi:10.2478/v10156-010-0010-7
  .
• "Physical Geodesy". Geodesy for the Layman. NOAA. 1984.

External links

Look up geoid in Wiktionary, the free dictionary.

• NGA webpage on Earth Gravitational Models
• NASA webpage on EGM96
• NOAA webpage on Geoid Models
• International Centre for Global Earth Models (ICGEM)
• International Service for the Geoid (ISG)