Mapping function
In some geosciences and astronomy, a mapping function or projection function is understood to be a mathematical model with which the astronomical refraction is represented as a function of the elevation angle .
The influence of the earth's atmosphere on the distance measurement to extraterrestrial objects is around 2.3 to 2.5 meters, depending on the weather , when the celestial object is at the zenith of the observer, i.e. H. when the measuring beam takes the shortest possible path through the atmosphere.
If, on the other hand, the direction to the object is closer to the horizon , so that the beam travels a longer path through the atmosphere, then the required reduction can be 10 to 50 times: the above. Average amount of 2.4 m is to be divided approximately by the sine of the elevation angle.
However, the sine function only applies to small pieces of the beam and neglecting the curvature of the earth . This requires a much more complicated mathematical-meteorological model, especially for the influence of the high atmosphere (which is far away from the observer) .
In the 1950s the Finnish geodesist Saastamoinen developed such a formula for satellite geodesy , which contains several angle functions and atmospheric constants depending on the desired accuracy . The Saastamoine formula gives the refraction amount accurate to about 1–2 percent; H. to a few centimeters.
State of research
Theory and scientific practice can only meet modern measuring accuracies of a few millimeters with further developments . Newer very precise models for GPS and VLBI measurements originate from a. from NASA and from European universities , e.g. B. the "Vienna Mapping Function" (VMF) by Harald Schuh and Johannes Böhm at the TU Wien (see also GNSS and atmospheric density function ).
The altitude of the astronomical station or the satellite station also plays a role.
Reduction of the observations
In order to achieve accuracies of better than a few meters, all electronic distance measurements to artificial earth satellites ( GPS , geodetic and navigation satellites ) and radio astronomical observations ( Very Long Baseline Interferometry ) must be reduced (adjusted) for the influence of the terrestrial atmosphere . The density of the air must also be precisely modeled.
This does not affect the mapping function, but the other parameters contained in the formulas - specifically the (assumed) integrated air temperature (average value through the entire atmosphere) and its vertical gradient , the formula for the altitude-related decrease in air pressure and humidity , etc. Therefore the development of such mathematical-physical models is only possible in interdisciplinary cooperation.