- the closer the surface is perpendicular to sun direction, the more energy it receives per surface unit (we talk about irradiance). If the surface is parallel to sun direction, it does not receive direct sunlight. We can model it this way :
- For an horizontal surface :
- For a sloped surface
- is the Top of Atmosphere irradiance, and is the downward direct transmission, i.e. the proportion of the light that reaches directly the surface without being scattered by the atmosphere.
- Assuming that all the irradiance is direct, the measured reflectance if the surface was horizontal is calculated from the following formula: . However, the above assumption is not true and this formula tends to over correct terrain effects
- The surfaces also receive a diffuse sun irradiance scattered by the atmosphere. If the surface is not horizontal, a part of the sky is obscured by the slope reducing the diffuse irradiance. Moreover, the diffuse irradiance depends on the amount of aerosols (and clouds) in the atmosphere. In addition, the surrounding terrain can also hide a part of the sky, but we do not take this effect into account here in our modelling. We use the following approximation, which is equivalent to assuming that the slope is alone in a horizontal region.
- If surface is horizontal, the visible sky fraction is 1, if it is vertical, this fraction is 1/2
- Finally, the slope can receive light from surrounding surfaces, which become directly visible. In our simplified model, we always assume the entire environment of our slope is flat and,for instance, we do not take the effect of the opposite side in a valley into account :
- If surface is horizontal, the visible ground fraction is 0, if it is vertical, it is 1/2.
Finally, we use the following formula to compute the reflectance that would be observed if the surface was horizontal , as a function of the slope (inclined) reflectance :
où is the downward transmission, sum of direct and diffuse irradiances : , and is the average reflectance of the neighbourhood.
Finally, we can also account for bidirectional reflectance effects, but this correction is tricky since directional effects depend on the surface cover type. See for instance : Dymond, J.R.; Shepherd, J.D. 1999: Correction of the topographic effect in remote sensing. IEEE Trans. Geosci. Remote Sens. 37(5): 2618-2620.
It is very difficult to validate the correction of directional effects : , we could compare the correction results for satellite overpasses at different times in the day. But all the satellite optical imagers have nearly the same overpass time. A qualitative way of estimating the accuracy is to check that similar land covers on opposite slopes in a valley ( a meadow, a forest) have a similar reflectance after correction. The most suitable points are North-South valleys. Here are some examples of terrain effects correction results.
Finally, an essential part of the method's accuracy is the availability of a highly accurate digital elevation model (DEM), up to now, only the SRTM DEM is available globally, and it only has a 90 meter resolution. Its accuracy is somewhat inadequate and sometimes leaves artefacts if the slope changes are poorly located.