A geoid is a model of the figure of the Earth (i.e., its analogue in size and shape), which coincides with the average sea level, and in the continental regions it is determined by the alcohol level. Serves as a base surface from which topographic heights and depths of the ocean are measured. A scientific discipline about the exact shape of the Earth (geoid), its definition and significance is called geodesy. More information about this is presented in the article.
Constancy of potential
The geoid is everywhere perpendicular to the direction of gravity and in shape approaches a regular flattened spheroid. However, this is not always the case because of local concentrations of accumulated mass (deviations from uniformity at depth) and because of differences in height between the continents and the seabed. Mathematically speaking, a geoid is an equipotential surface, i.e., characterized by the constancy of the potential function. It describes the combined effects of gravitational attraction of the mass of the Earth and centrifugal repulsion caused by the rotation of the planet around its axis.
Simplified Models
Geoid due to the uneven distribution of mass and the resulting gravitational anomalies is not a simple mathematical surface. It is not quite suitable for the standard geometric figure of the Earth. For this (but not for topography), approximations are simply used. In most cases, a sufficient geometric representation of the Earth is a sphere for which only the radius should be indicated. When a more accurate approximation is required, an ellipsoid of revolution is used. This is the surface created by rotating the ellipse 360 ° about its minor axis. The ellipsoid used in geodetic calculations to represent the Earth is called a reference. This shape is often used as a simple base surface.
The rotation ellipsoid is defined by two parameters: the semi-major axis (equatorial radius of the Earth) and the semi-minor axis (polar radius). The flattening f is defined as the difference between the major and minor semi-axes divided by the larger f = (a - b) / a. The semi-axes of the Earth differ by about 21 km, and the ellipticity is about 1/300. Deviations of the geoid from the ellipsoid of revolution do not exceed 100 m. The difference between the two semiaxes of the equatorial ellipse in the case of the triaxial ellipsoid model of the Earth is only about 80 m.
Geoid concept
Sea level, even in the absence of the effects of waves, winds, currents and tides, does not form a simple mathematical figure. The undisturbed surface of the ocean should be the equipotential surface of the gravitational field, and since the latter reflects the density inhomogeneities inside the Earth, the same applies to equipotentials. Part of the geoid is the equipotential surface of the oceans, which coincides with the undisturbed mean sea level. Under the continents, the geoid is not directly accessible. Rather, it represents the level to which water rises if narrow channels are made through the continents from ocean to ocean. The local direction of gravity is perpendicular to the surface of the geoid, and the angle between this direction and the normal to the ellipsoid is called the deviation from the vertical.
Deviations
It may seem that a geoid is a theoretical concept that has little practical value, especially with respect to points on the land surface of continents, but this is not so. The heights of points on the ground are determined by geodetic alignment, at which the alcohol level sets the tangent to the equipotential surface, and the calibrated landmarks are aligned using a plumb line. Therefore, differences in height are determined with respect to the equipotential and therefore are very close to the geoid. Thus, the determination of the 3 coordinates of a point on the continental surface by classical methods required the knowledge of 4 values: latitude, longitude, height above the geoid of the Earth and deviations from the ellipsoid in this place. The deviation of the vertical played a large role, since its components in the orthogonal directions introduced the same errors as in the astronomical definitions of latitude and longitude.
Although geodesic triangulation provided relative horizontal positions with high accuracy, triangulation networks in each country or continent started from points with assumed astronomical positions. The only possibility of combining these networks into a global system was to calculate deviations at all starting points. Modern methods of geodetic positioning have changed this approach, but the geoid remains an important concept with certain practical benefits.
Shape definition
A geoid is essentially an equipotential surface of a real gravitational field. In the vicinity of a local excess of mass, which adds the potential ΔU to the normal potential of the Earth at a point, in order to maintain a constant potential, the surface must be deformed outward. The wave is given by the formula N = ΔU / g, where g is the local value of the acceleration of gravity. The mass effect over the geoid complicates the simple picture. This can be solved in practice, but it is convenient to consider a point at sea level. The first problem is the determination of N not through ΔU, which is not measured, but by the deviation of g from the normal value. The difference between local and theoretical gravity at the same latitude of an ellipsoidal Earth, free of density changes, is Δg. This anomaly occurs for two reasons. Firstly, due to the attraction of the excess mass, the influence of which on gravity is determined by the negative radial derivative -∂ (ΔU) / ∂r. Secondly, because of the effect of height N, since gravity is measured on a geoid, and the theoretical value refers to an ellipsoid. The vertical gradient g at sea level is -2g / a, where a is the radius of the Earth, so the height effect is determined by the expression (-2g / a) N = -2 ΔU / a. Thus, combining both expressions, Δg = -∂ / ∂r (ΔU) - 2ΔU / a.
Formally, the equation establishes a relationship between ΔU and the measurable value of Δg, and after determining ΔU, the equation N = ΔU / g gives the height. However, since Δg and ΔU contain the effects of mass anomalies over the entire uncertain region of the Earth, and not just under the station, the last equation cannot be solved at one point without reference to others.
The problem of the connection between N and Δ g was solved by the British physicist and mathematician Sir George Gabriel Stokes in 1849. He obtained an integral equation for N containing Δg values with a function of their spherical distance from the station. Before the launch of satellites in 1957, the Stokes formula was the main method for determining the shape of a geoid, but its application was very difficult. The spherical distance function contained in the integrand very slowly converges and when trying to calculate N at any point (even in countries where g were measured on a large scale), the uncertainty arises from the presence of unexplored areas that can be located at considerable distances from station.
Satellite contribution
The appearance of artificial satellites, whose orbits can be observed from the Earth, completely revolutionized the calculation of the shape of the planet and its gravitational field. A few weeks after the launch of the first Soviet satellite in 1957, an ellipticity value was obtained, which replaced all previous ones. Since then, scientists have repeatedly refined the geoid with near-Earth orbit observation programs.
The first geodetic satellite was Lageos, launched by the United States on May 4, 1976 into an almost circular orbit at an altitude of about 6 thousand km. It was an aluminum sphere with a diameter of 60 cm with 426 laser reflectors.
The shape of the Earth was established through a combination of Lageos observations and surface measurements of gravity. Deviations of the geoid from the ellipsoid reach 100 m, and the most pronounced internal deformation is located south of India. There is no obvious direct correlation between continents and oceans, but there is a connection with some of the main features of global tectonics.
Radar altimetry
The Earth’s geoid over the oceans coincides with the average sea level, provided that there are no dynamic effects of winds, tides and currents. Water reflects radar waves, so a satellite equipped with an altimeter radar can be used to measure the distance to the surface of the seas and oceans. The first such satellite was the Seasat 1, launched by the United States on June 26, 1978. Based on the data obtained, a map was compiled. Deviations from the result of calculations made by the previous method do not exceed 1 m.