# Terrestrial refraction

As terrestrial refraction (also refraction or atmospheric refraction which is called) of refraction of a light beam in the lowest Earth's atmosphere called. This is caused by the change in the refractive index of the air along the beam path as a result of the air density decreasing with altitude and causes an arcuate curvature of the beam, which must be applied as a correction ("reduction") to every measured vertical angle in more precise measurements or in the physical laboratory . This beam curvature averages 12-14% of the earth's curvature and increases the horizontal visibility slightly.

The astronomical refraction is a special case of terrestrial refraction.

## Visible effect of the refraction of rays

Oval sun disk at setting (simulation). A narrow band is missing on the right because the vertical gradient of the air (see mapping function ) is discontinuous.

Since the light rays are mostly curved towards the surface of the earth, they make distant objects appear higher than with a straight path. The rising or setting sun , which exactly touches the horizon of the sea, is geometrically still or completely below it. Its (flat) oval shape is due to the fact that its lower edge is more strongly deflected upwards than its upper edge; because the gradient of the air density normally decreases with the density upwards. Warm air layers close to the ground can, however, reduce the gradient, and in extreme cases even reverse it. Then the light rays are curved upwards, which appears as a mirage at a shallow angle of incidence . Conversely, the gradient is increased in an inversion slice.

## Regular refraction

Carl Friedrich Gauß already examined the relatively stable conditions of the refraction of rays when he was commissioned to survey the state of Hanover around 1800 . The refraction has a particular effect on height measurements over long distances: With constant curvature of the light rays, the refraction angle (ρ) increases linearly and the height error increases quadratically with the distance. Over a few hundred meters it is in the range of one millimeter, over 10 kilometers it is about one meter. The Gaussian "big triangle" has edge lengths of 68, 84 and 106 kilometers.

From his measurements he derived a mean refraction coefficient of k  = 0.13, that is, the mean curvature of the light rays is around 13 percent of the earth’s curvature (mean earth radius R  = 6371 km). This value fits well with the density gradient of the standard atmosphere and has been used for the reduction of most geodetic height measurements for 200 years .

Light beam curvature at different air temperatures

The radius of the beam curvature (with large ground clearance it varies from 40,000 to 50,000 km) depends on the gradient of the humidity , the temperature and the pressure of the air, so that it can be calculated using meteorological measurements along the beam path.

For high-precision measurement projects, it is necessary to examine the beam path more precisely, which can be done in several ways: by means of detailed measurement of the radiation balance , by setting up a 3D measuring field for the air temperature or with two-color laser measuring devices.

With leveling , the height error falls out, provided that the same target ranges are maintained when looking forwards and backwards and that the influence of refraction is symmetrical over both sights. Due to the limited by the terrestrial refraction accuracy, especially for large target distances ( trigonometric leveling ) plays in modern land surveying the Satellite Geodesy an important role. Here too, however, the terrestrial refraction acts in the lower atmosphere, and here, too, an attempt is made to eliminate similar components using differential methods.

Astronomical refraction is a special case of terrestrial refraction in the sense that the targeted object is outside of the atmosphere and is usually practically infinitely far away.

## Variation of the refraction coefficient near the ground

Time course of the refraction coefficient near the ground on a sunny (black curve) and cloudy day (red curve)

Near the ground (up to a few meters above the surface) the refraction coefficient often deviates from 0.13. When exposed to sunlight, it can assume values ​​of −2 to −4 during the day, for example over hot, reflective asphalt surfaces. At sunset even amplitudes of around +15 were observed in extreme cases. When it is cloudy , the variations in the refraction coefficient are lower. They depend not only on solar radiation, but also essentially on the nature of the soil and the observation height. The refraction coefficient is closely linked to the vertical temperature gradient and thus to the energy balance of the atmosphere .

## Terrestrial refraction in astronomical navigation

In astronomical navigation , a sextant is used to measure the height of a star above the horizon . Due to the refraction of the rays, the true height of the star is always lower than the height measured with the sextant, whereby the effect is more striking for small heights. The measured height must be corrected. The table below shows the number of arc minutes for 10 ° C and standard pressure (1013.2 hPa) that must be subtracted from the height measured with the sextant. The correction is particularly important at low altitudes, because every minute of arc of the altitude measurement in the location determination leads to a shift in the position of one nautical mile . The correction of the refraction of rays described here is one of the four charges that must be carried out when measuring the height of the star.

 Apparent height 0 ° 1 ° 2 ° 3 ° 4 ° 5 ° 6 ° 7 ° 8 ° 9 ° 10 ° refraction 35.4 ' 24.6 ' 18.3 ' 14.4 ' 11.8 ' 9.9 ' 8.5 ' 7.4 ' 6.6 ' 5.9 ' 5.3 ' Apparent height 12 ° 14 ° 16 ° 20 ° 30 ° 40 ° 50 ° 60 ° 70 ° 80 ° 90 ° refraction 4.4 ' 3.8 ' 3.3 ' 2.6 ' 1.7 ' 1.2 ' 0.8 ' 0.6 ' 0.3 ' 0.2 ' 0.0 '

## literature

• Heribert Kahmen : Applied Geodesy - Surveying . 20th edition, de Gruyter-Verlag, 2005.
• Karl Ramsayer : Geodetic Astronomy . Volume IIa of the Handbuch der Vermessungskunde JEK , JB Metzler, Stuttgart 1969 (see Refraction: Chapter 6, pp. 107-140, and Appendix II, pp. 832 ff.).
• Gottfried Gerstbach , M. Schrefl, W. Rössler: Determination of the integral refractive index by flying the measuring beam. In: Austrian Journal for Surveying and Photogrammetry . Baden 1981 and 1982.
• Maria Hennes: On the influence of refraction on terrestrial geodetic measurements in the context of measurement technology and instrument development . 2002, FuB, Heft 2, pp. 73–86 ( PDF; 758 kB ).
• Christian Hirt, Sebastian Guillaume, Annemarie Wisbar, Beat Bürki, Harald Sternberg: Monitoring of the refraction coefficient of the lower atmosphere using a controlled set-up of simultaneous reciprocal vertical angle measurements. Journal of Geophysical Research 115, 2010, doi : 10.1029 / 2010JD014067 ( full text at Researchgate ).