Earth's curvature

by Micha ⌂, Bad Vilbel, Tuesday, January 21, 2020, 15:59 (317 days ago) @ Lyoha

Hej Lyoha,

Can I take into account the curvature of the Earth in the program by setting the value k = -1.

Let's check it. The observation equation for zenith angles reads:

$z = \arctan{ \frac{\sqrt{ \Delta x^2 + \Delta y^2}} {\Delta z}} - k \frac{s_{2D}} {2R}$

From trigonometric leveling, one knows two corrections: the Earth's curvature correction and the refraction correction. The well-known relation is given by

$dH = s \cdot \cos{z} + \left( \frac{s^2}{2R} - \frac{ks^2}{2R}\right) \sin{z}$

where $s$ is the distance, $R$ the radius of the Earth and $k$ the refraction index. Since $z$ is almost about 100 gon, $\sin{z} \approx 1$ and the last term is vanished.

The first term $\frac{s^2}{2R}$ corrects for the Earth's curvature and the second term $-\frac{ks^2}{2R}$ corrects for the refraction index. Converting both terms into angles (using the approximation of a circular arc), one gets $\frac{\frac{s^2}{2R}}{s}=\frac{s}{2R}$ and $\frac{-\frac{ks^2}{2R}}{s}=-\frac{ks}{2R}$, respectively. Since JAG3D compensate for $k$, only the second term is applied. As you suggested, however, to compensate for the Earth's curvature, one must set $k = -1$. Yes, I think you are right.


-- - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

JAG3D, Earth's curvature, refraction index, correction, zenith angle

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