Geoid is a model of the Earth's figure (i.e., its analogue in size and shape), which coincides with the mean sea level, and in continental regions is determined by the spirit level. Serves as a reference surface from which topographic heights and ocean depths are measured. The scientific discipline about the exact shape of the Earth (geoid), its definition and significance is called geodesy. More information about this is provided in the article.
Constancy of potential
The geoid is everywhere perpendicular to the direction of gravity and in shape approaches a regular oblate spheroid. However, this is not the case everywhere due to local concentrations of accumulated mass (deviations from uniformity at depth) and due to height differences between continents and the seafloor. Mathematically speaking, the geoid is an equipotential surface, i.e., characterized by the constancy of the potential function. It describes the combined effects of the gravitational pull of the Earth's mass and the centrifugal repulsion caused by the planet's rotation on its axis.
Simplified models
The geoid, due to the uneven distribution of mass and the resulting gravitational anomalies, does notis a simple mathematical surface. It is not quite suitable for the standard of the geometric figure of the Earth. For this (but not for topography), approximations are simply used. In most cases, a sphere is a sufficient geometric representation of the Earth, for which only the radius should be specified. When a more accurate approximation is required, an ellipsoid of revolution is used. This is the surface created by rotating an ellipse 360° about its minor axis. The ellipsoid used in geodetic calculations to represent the Earth is called the reference ellipsoid. This shape is often used as a simple base surface.
An ellipsoid of revolution is specified by two parameters: the semi-major axis (Equatorial radius of the Earth) and the minor semi-axis (polar radius). The flattening f is defined as the difference between the major and minor semiaxes divided by the major 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 semi-axes of the equatorial ellipse in the case of a three-axis 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 density inhomogeneities inside the Earth, the same applies to equipotentials. Part of the geoid is the equipotenti althe surface of the oceans, which coincides with the undisturbed mean sea level. Beneath the continents, the geoid is not directly accessible. Rather, it represents the level to which water will rise if narrow channels are made across 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
The geoid may seem like a theoretical concept with little practical value, especially in relation to points on the land surfaces of continents, but it is not. The heights of points on the ground are determined by geodetic alignment, in which a tangent to the equipotential surface is set with a spirit level, and calibrated poles are aligned with a plumb line. Therefore, the differences in height are determined with respect to the equipotential and therefore very close to the geoid. Thus, the determination of 3 coordinates of a point on the continental surface by classical methods required the knowledge of 4 values: latitude, longitude, height above the Earth's geoid and deviation from the ellipsoid at this place. The vertical deviation played a big role, since its components in orthogonal directions introduced the same errors as in the astronomical determinations of latitude and longitude.
Although geodetic triangulation provided relative horizontal positions with high accuracy, triangulation networks in each country or continent started from points with estimatedastronomical positions. The only way to combine these networks into a global system was to calculate the deviations at all starting points. Modern methods of geodetic positioning have changed this approach, but the geoid remains an important concept with some practical benefits.
Shape definition
Geoid is, in essence, 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 the point, in order to maintain a constant potential, the surface must deform outwardly. The wave is given by the formula N=ΔU/g, where g is the local value of the acceleration of gravity. The effect of mass over the geoid complicates a simple picture. This can be solved in practice, but it is convenient to consider a point at sea level. The first problem is to determine N not in terms of ΔU, which is not measured, but in terms of 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 excess mass, the effect of which on gravity is determined by the negative radial derivative -∂(ΔU) / ∂r. Secondly, due to the effect of height N, since gravity is measured on the geoid, and the theoretical value refers to the ellipsoid. The vertical gradient g at sea level is -2g/a, where a is the radius of the Earth, so the height effectis determined by the expression (-2g/a) N=-2 ΔU/a. Thus, combining both expressions, Δg=-∂/∂r(ΔU) - 2ΔU/a.
Formally, the equation establishes the relationship between ΔU and the measurable value Δg, and after determining ΔU, the equation N=ΔU/g will give the height. However, since Δg and ΔU contain the effects of mass anomalies throughout an undefined 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 relationship between N and Δg was solved by the British physicist and mathematician Sir George Gabriel Stokes in 1849. He obtained an integral equation for N containing the values of Δg as a function of their spherical distance from the station. Until the launch of satellites in 1957, the Stokes formula was the main method for determining the shape of the geoid, but its application presented great difficulties. The spherical distance function contained in the integrand converges very slowly, and when trying to calculate N at any point (even in countries where g has been measured on a large scale), uncertainty arises due to the presence of unexplored areas that may be at considerable distances from station.
Contribution of satellites
The advent of artificial satellites whose orbits can be observed from Earth has 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, the valueellipticity, which supplanted all previous ones. Since that time, scientists have repeatedly refined the geoid with observation programs from near-Earth orbit.
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 6,000 km. It was an aluminum sphere with a diameter of 60 cm with 426 reflectors of laser beams.
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 basic features of global tectonics.
Radar altimetry
The geoid of the Earth over the oceans coincides with the mean sea level, provided there are no dynamic effects of winds, tides and currents. Water reflects radar waves, so a satellite equipped with a radar altimeter can be used to measure the distance to the surface of the seas and oceans. The first such satellite was 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.