Trigonometric height measurement

The trigonometric height measurement is a method of Geodesy , wherein the height difference between two points by measuring the vertical angle (zenith distance z and elevation angle h ) is determined. To do this, the inclined distance s between the measuring points must also be known. For small distances - i.e. H. without the influence of refraction and the curvature of the earth - the difference in altitude is

${\displaystyle \Delta H=s\cdot \sin(h)+(I-Z)=s\cdot \cos(z)+(I-Z)}$,

where the term means the difference between the height of the instrument and the target height above the ground points. ${\displaystyle (I-Z)}$

If instead the slope distance s horizontal distance s ₀ given, the difference in height results with

${\displaystyle \Delta H=s_{0}\cdot \tan(h)+(I-Z)=s_{0}\cdot \cos(z)+(I-Z)}$.

From a distance of about 100 m, the curvature of the earth must be taken into account, which is 0.8 mm here. It is added to Δ H as correction c and increases quadratically with the distance:

${\displaystyle c={\frac {s_{0}^{2}}{2R_{\mathrm {E} }}}}$ with earth radius ${\displaystyle R_{\mathrm {E} }=6370\,\mathrm {km} }$

and reaches around 8 cm for 1 km.

In most cases, the formula also takes into account terrestrial refraction ( refraction of rays in the atmosphere), which reduces the influence of the earth's curvature by about an eighth (mean refraction coefficient k = 0.13):

${\displaystyle c'={\frac {s_{0}^{2}}{2R_{\mathrm {E} }}}\cdot (1-k)=c\cdot (1-k)}$

The height difference becomes more precise when it is determined from both points. Then some small sources of error also fall out. There are special formulas for simultaneous mutual measurements in which the influence of a refraction coefficient other than 0.13 is eliminated.

literature

• Heribert Kahmen : Applied Geodesy - Surveying. 20th edition. Verlag de Gruyter, Berlin 2006, ISBN 978-3-11-018464-8 .
• Boris Resnik, Ralf Bill: Surveying for the planning, construction and environmental sectors. Wichmann Verlag, 2003, ISBN 3-87907-399-6 .