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The geoid (/di.d/) 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.[1] 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.[2]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).

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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.

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.

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.

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.

The surface of the geoid is higher than the reference ellipsoid wherever there is a positive gravity anomaly (mass excess) and lower than the reference ellipsoid wherever there is a negative gravity anomaly (mass deficit).[5]

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 and away a denser or lighter body, respectively, causing a dimple or a bump in the equipotential surface.[6]

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]

The precise geoid solution by Vanek 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 neural 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]

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]

Do you have access to a 3D printer? Print your own scale model of the geoid with a free plan created by a NOAA geodesist! This model exaggerates the bumpy surface of the geoid so it is easy to see the irregular shape of the planet's global mean sea level and reduces the diameter of the Earth to just a few inches.

This irregular shape is called "the geoid," a surface which defines zero elevation. Using complex math and gravity readings on land, surveyors extend this imaginary line through the continents. This model is used to measure surface elevations with a high degree of accuracy.

Is it possible to add a local geoid to ArcGIS Pro? I created a DEM in Agisoft which uses NAD83(CSRS)/UTM Zone 8N as a coordinate system, but is using a local geoid. I have a few versions of the geoid, one of which is a tiff but can not figure out how to import it.

@BenPearse, I presume you would like to use your local geoids to transform your DEM data to some other vertical coordinate system. Unfortunately, ArcGIS Pro currently does not support custom vertical transformations, but it is part of our plans for the future releases.

I have not worked with geoid models and need to obtain the orthometric height from the ellipsoidal height provided by reach RS2. I am also working in Asia where it is difficult to locate the geoid models in some countries. Can soomeone point me to some resources on obtaining, interpreting and applying the geoid models to my ellipsoidal data?

Thanks. I had looked at the data but found no model for Lao. I have a lao geoid model file in bin format, but not sure what to do with it. As well, I am not sure of the accuracy or provenance of the model. I know Reachview3 offers EGM96 which I think is a global model geoid and was considering that as a last resort. But really, I need a better understanding of how to practically use geoid models, their different formats, software etc ff782bc1db