Formal weaknesses of some definitions of roughness, and a solution proposal

We have analyzed from a formal point of view two definitions of roughness. Both are applicable only for elevations modeled through a DEM. One (named α) states that roughness is computed as the standard deviation of the slope at the 9 grid points of a 3x3 window. Another similar one (named β), computes the same over a detrended version of the elevation surface. As defined, we prove that both roughness values vanish at the limit of the cell size going down to zero, irrespective of any other characteristic of the terrain. Despite both metrics attempt to compute a property of the topographical surface through the use of a regular sample of elevations (i.e. DEM) we can show that the value is still a function of the cell size. Thus, a seemingly intrinsic property of the topographic surface is dependent of one arbitrary characteristic of the sample: the DEM cell size. Through a cumbersome but otherwise straightforward formal manipulation we can prove that the squared standard deviation of the slope over a 3x3 window is directly proportional to the cell size h raised to a power 1 or 2. If we consider instead the slightly modified definition β where the surface is first detrended, then the power is always 2. The proposed adjusted definition of roughness requires using the standard deviation of the detrended slope over a 3x3 window divided by the squared cell size h. We will show the relationship of such magnitude to the set of partial derivatives of the topographic surface, thus offering a mathematical counterpart of the so defined roughness. In addition, a procedure to estimate its uncertainty under the assumption that the elevation values are error free is also provided.