Koppe Accuracy

Height and position accuracy are an essential feature of topographic maps with contour lines or elevation layers. For cartography in high mountains the correlations between accuracy of recording, scale, equidistance are highly important.
The Koppe error formula was originated during alignment for long-distance train routes on the basis of large-scale contour maps. The same formula is used with Digital Elevation Models where similar detection methods, such as airborne laser scanning or digital stereo photogrammetry are used. Therefore the measurement error behaves equally for methods. With many attempts to improve the formula, it is still a valid definition with flexibility forms as you will learn further.

"Height error is the same for the same terrain, larger in the mountains and smaller in the lowlands."

Formula Expression

Accuracy carries geometrical meaning. It should be reduced or tested empirically. The method for this is the Koppe Formula. $$\sigma_Z$$ is the standard deviation in height of measurement: $\sigma_H = \sigma_Z + \sigma_G \cdot \tan \alpha$ $$\sigma_G$$ is the standard deviation in scale of measurement: $\sigma_L = \sigma_G + \sigma_Z \cdot \cot \alpha$
For simplification we assume the deviation of layer height as factor $$A$$ and the deviation of scale as $$B$$. In this table you can see the differences for some countries.

Country CodeScaleFactor AFactor B
A1:10.0001 m3 m
CH1:50.0001,5 m10 m
USA1:50.0001,8 m15 m

Many modifications can be made with those two factors. $$A$$ and $$B$$ have simple linear dependencies with the elevation slope. Asume you have the same elevation but in two landscapes: highland and lowland.

Height Deviation

The mean square error in height is higher with a steeper elevation slope. It stems with:
$mH = \pm(A + B \cdot \tan \alpha)$
This means, looking at the profile view on the right, for highlands a higher mean error than in lowlands.

Position Deviation

For horizontal accuracy the formula is reversed. The shallower the higher the mean sqaure error in postition:
$mL = \pm(B + A \cdot \cot \alpha)$
Thus follows for the top view on the right, in lowlands a higher mean error than in highlands.

Hands-On

First select a Country Code, where Factors $$A$$ and $$B$$ are used from the above table. Then click on the map for a Location with Elevation measurement.
The Mean Error in Height and Position will be calculated with the elevation angle at this Location. For comparison try different areas of steeper or shallower ground.
Map and Elevation layer
Latitude:
Longitude:
Elevation:
Mean Error in Height:
Mean Error in Position: