External effects

Earth’s curvature

Effect of the earth’s curvature.

The effect due to the earth's curvature is well known from ships disappearing behind the horizon. It is illustrated schematically in the figure above. Because of the earth's curvature, points are observed at lower elevations. In the figure, point B which is seen from viewpoint A and has height H, has disappeared beneath the horizon by the difference e. This effect can be approximated as follows (Yoeli 1985):

where d is the distance between the viewpoint and the observed point and R is the radius of the earth (R = 6370 km). Of course the effect increases with growing distance d.

Atmospheric refraction

Atmospheric refraction.

Atmospheric refraction makes objects appear at higher angles than would be expected. This effect is caused by the fact that rays of light passing through the atmosphere are refracted or bent from a straight path in a direction towards the earth's surface under normal conditions of pressure gradient and temperature. Thus, as shown in the figure, point B is observed at B', with the height difference r. This effect can be approximated as follows (Yoeli 1985):

where k is an empirical coefficient (k = 0.13) and R is the radius of the earth (R = 6370 km). Again the effect increases with growing distance d.

Correcting the effects of the earth’s curvature and atmospheric refraction

The combination of the opposing effects of the earth’s curvature and of atmospheric refraction leads to the correction formula

Where d is the distance between viewpoint and observed object and R is the radius of the earth (R=6370 km). This difference, of course, is negligible for short viewing distances. However, for longer distances it becomes quite considerable. Therefore all heights of the DTM should first be corrected depending on their distances to the viewpoint, yielding a corrected DTM' with heights $H\text{'}=H–\Delta h$. In the table below the equation above was used to calculate the combined effect of the earth’s radius and atmospheric refraction for various distances d. The earth’s radius is assumed 6’370’000 m. Thus for a distance of 5 km the effect is 1.71 m, whereas for a distance of 50 km it is 170.72 m.
Some examples for the combined effect of the earth’s curvature and the atmospheric refraction for various distances d:

Other atmospheric interferences

Effects such as haze, fog, clouds, or smoke, obviously can have a great influence on the visibility of terrain. Reasons why these effects are often ignored is that they are not trivial to model, and also subject to rapid change. For some applications like radio signal transmission, atmospheric haze and fog do not matter that much as radio waves can pass through.

Various visualization software tools allow the modelling of atmospheric influences. Using this software the visual appearance of a landscape may be simulated if you have suitable terrain data at your disposal. In the next figure the landscape around Niesen and Bernese Oberland is modelled under various whether conditions and with different plant coverages. See how haze darkens the mountains in the background.

Niesen and the Alps of the Bernese Oberland under different atmospheric conditions at different times of the year. (Swisstopo 1991)

Obstacles and land cover

Another factor that influences the visibility of terrain is the land cover. Certain land use classes, such as forest or built-up areas, can form visual obstacles of large spatial extent and considerable height. For a more realistic solution of the visibility problem, the heights of relevant land cover types – as well as other potential natural and man-made obstacles – have to be taken into account. If digital land cover data are available, an estimated average object height can be added to each DTM point depending on which land cover types they fall in. Certain photogrammetric methods tend to capture the top of trees and buildings rather than the ground surface. Obviously, this presents a certain advantage in the use for visibility applications.

Effect of land covers on visibility analyses.