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Tom,
I want to react first to that quotation. It's a nice definition of something, but that something definitely is not what most people would consider "roughness". By this measure a basketball would be considered exceedingly "rough" while a hunk of conglomerate would be rather smooth! The reason is that it takes no account of the size of the "bumps" it is measuring. The tiny little bumps on a basketball would seem exceedingly rough according to this solely angle-based definition. Moreover, this definition would be exceedingly scale-dependent. At one scale, a polished marble slab would be very smooth but at another (microscopic) scale it would be exceptionally rough. Topographic features would be pretty smooth at small scales (say, 1:100,000 and smaller), rougher and rougher at intermediate scales (say, 1:10 to 1:100,000), and then get smoother again (as rocks and vegetation get resolved into their smoother surface details) at extremely large scales.
As for the rest, it appears you are interested in relating errors in estimated terrain characteristics (presumably these errors are assessed in some independent way, such as by comparison to reality) to some measure of "roughness." This suggests you adopt a measure of roughness that is related to the estimator in question.
Take, for instance, the slope estimate. The usual one is based on a 3 x 3 local neighborhood. Spatial Analyst fits a plane to the nine heights using ordinary least squares. The plane's slope is the slope estimator and its aspect is the aspect estimator.
Here is a part of ESRI's description (taken from the Spatial Analyst 1.1 help page for "slope"), slightly modified to suit this forum's formatting limitations. Let the values in the 3 x 3 window be
a b c
d e f
g h i
Then
Slope = sqrt((dz/dx)^2 + (dz/dy)^2) / (8 * cellsize)
where
dz/dx = ((a + 2d + g) - (c + 2f + i))
dz/dy = ((a + 2b + c) - (g + 2h + i)).
From these formulas, by taking the partial derivatives of the slope with respect to a, b, ..., i, you can readily establish that:
(1) The central value, e, does not influence the slope estimate at all;
(2) The edge values {b, d, f, h} have twice the influence of the corner values {a, c, g, i};
(3) The influence is inversely proportional to the slope.
This suggests that you develop some measure of surface roughness (loosely thought of as variation throughout local neighborhoods) that weights the edge values twice as much as the corner values but does not consider the central value at all. One such would be the square root of (twice the variance of (b, d, f, h) plus the variance of (a, c, g, i)), all divided by the slope estimate itself. (I don't like that too much, because it becomes infinite at zero slopes, but it's a start. If you avoid dividing by the slope, you will produce a "roughness" related to the sensitivity in estimating the square of the slope. More sophisticated variants of this procedure are available, to be selected according to the kinds of errors you can tolerate in the slope estimates.)
In a similar vein you could develop surface roughness indices appropriate for assessing the potential for errors in aspect, elevation, etc.
I hope this clarifies my point that your definition of "roughness" might best be determined by the analysis you propose to do with it.
Finally, you ask about neighborhood size. Using different cell sizes for the different DEM products is sure to give incommensurable results: the larger the cell size, the more smoothing is going on and the less rough the result will be. At a minimum, resample one (or both) of the grids to a common cellsize using cubic convolution, which has a hope of retaining some of the local statistical variation in the data. |