# Earth's curvature

Hi Micha. In my network, elevations are measured by trigonometric leveling. Can I take into account the curvature of the Earth in the program by setting the value k = -1.
I need to consider only the curvature of the Earth
(I do not take into account the refraction index in the calculations).

## Earth's curvature

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.

regards
Micha

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applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

## Earth's curvature

Thanks Micha. The results are consistent with the Credo_Dat program, which is popular with us in Belarus.

## Earth's curvature

Hi Lyoyha,

The results are consistent with the Credo_Dat program, which is popular with us in Belarus.

Thank you for highlighting the identities between both adjustment tools!

Kind reagrds
Micha

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applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

## Earth's curvature

Hi,

I reorganise the reduction support in JAG3D. Now, it is possible to enable Earth's curvature reduction (without using the refraction index k). The latest version is available here.

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applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

## Earth's curvature

Hi, Micha!

Thanks. Perfectly!!!

## Earth's curvature

Now, it is possible to enable Earth's curvature reduction (without using the refraction index k).

I can enable the amendment, but turn it off again - the program does not allow ....

## Earth's curvature

Hi Lyoyha,

I can enable the amendment, but turn it off again - the program does not allow ....

Thank you for you reply and, yes, your are right! I found the reason and I'll provide a new release within this day.

regards
Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

## Earth's curvature

Hi,

I can enable the amendment, but turn it off again - the program does not allow ....

I changed some SQL statements and code parts related to the problem. Can you try the new JAG3Dv20200220 release?

Kind regards
Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

Tags:
JAG3Dv20200220, SQL

## Earth's curvature

Can you try the new JAG3Dv20200220 release?

I already tried it. Everything worked. Your program is becoming more powerful and better every day! )

## Earth's curvature

Hello Lyoha,

I already tried it. Everything worked.

Thank you for testing, comparing and reporting!

Have a nice weekend
Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences