Concept of the Geoid

The geoid is the shape taken by the ocean surface which would take under the influence of gravity and rotation of the earth, without any presence of winds and tides. The geoid surface is not only limited to the ocean but is extended through the continents as well through very narrow hypothetical canals. John Carl Friedrich Gauss was the first to describe it, and said that the geoid surface was the ‘mathematical figure of the Earth’. In the concept of the geoid a smooth but irregular surface the shape of which is the result of the uneven distribution of the mass within and on the Earth’s surface.

Concept of the Geoid

According to the concept of the geoid, which defines the shape taken by the surface of the ocean and also extends through the continents, the geoid earth surface is irregular but smoother than the actual physical surface of the Earth. Typically, the surface of the Earth is represented by the idealized mathematical representation of an ellipsoid. The difference between ellipsoid and geoid surface is that the ellipsoid is the smoothest representation of the surface while the geoid is an irregular representation of the surface of the Earth. 

This is because the irregular shape of the geoid earth takes into account the influence of gravitation and the rotation of the Earth but does not take into account the forces of winds and tides. Because of this, the geoid surface of the Earth deviates from the smooth ellipsoid within the range of +85 m (physical surface height of Iceland) to -106 m (physical surface depth of southern India). But this isn't as relevant to the physical feature of the Earth’s surface as the actual range is from + 8848 m height of Mount Everest to -11,034 m depth of Marianas Trench. 

As the parameters used to define Geoid, a Geoid surface in common parlance is said to be an earth-like surface. All the points on the geoid earth surface are said to have the same effective potential. In the process to define geoid, as the gravitational force is taken into consideration, the sum of the gravitational potential energy and the centrifugal potential energy is defined as the effective potential which is the same on all points of the geoid. Hence, it is an equipotential geoid surface with respect to gravitational potential. 

As per the concept of the gepoid the force of gravity on the surface of the geoid is perpendicular to all the points. On the surface of the geoid, the plumb lines point perpendicular and the water levels are parallel to the geoid only when gravitational and rotational acceleration are working together. The difference between ellipsoid and geoid is clearly visible when comparing between the two surfaces, it is clearly visible that the surface level of the geoid is higher than the reference surface of ellipsoid when the gravity anomaly is positive, and when the gravity anomaly is negative, the surface level of the geoid is lower than the reference surface of the ellipsoid. 

The difference between the geoid surface and the other surfaces is clearly visible in the following diagram: 

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Another representation of the geoid surface is shown below in which the colours represent the rise in the height and fall in the depth of the geoid surface:

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Determination of the Geoid Surface

There are many mathematical methods that help to determine and calculate the surface of the geoid. The undulations, i.e. the smooth up and down movement of the geoid surface, are calculated by various approaches combined together. Calculation of the undulating surface is challenging mathematically. Hence, because of this many of the GPS receivers that are handheld have built-in undulation lookup tables which are helpful in determining the height above sea level. The stokesian approach is the one that is widely used for the calculation of undulating surfaces. The precise solution of the geoid given by Vanicek and co-workers improved on the Stokesian approach for the computation. The solution obtained as such enables the accuracy to be in the range of millimetre-to-centimetre which is an order-of-magnitude better than the previous classical situations. 

Many of the calculations of the undulations of the geoid surface use approaches such as least-squares collocation, fuzzy logic, artificial neural networks, radial basis functions, and geostatistical techniques to estimate the uncertainties displayed in the undulations of the geoid. Out of all of them, the geostatistical approach has been defined as the most improved technique while utilizing for the prediction of the geoid undulation. 

Determination of the Surface of Other Celestial Bodies

The concept of the geoid has been applied not only to Earth but also to the other celestial bodies present in the Solar system. It has been extended to other planets and moons and also to the asteroids. An example of such a case is the ‘areoid’ which is the geoid equivalent of Mars. The areoid of Mars has been measured by the use of flight paths of human-made satellite missions such as the Mariner 9 and Viking. From the data collected for the areoid, the main difference from the reference ellipsoid is found at the Tharsis volcanic plateau which is an elevated region of terrain and is of the size of the continent. The same difference is visible for the antipodes found in the region.