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Topographic Roughness   Tom colson Dec 21, 2004
Re: Topographic Roughness   William Huber Dec 21, 2004
Re: Topographic Roughness   Tom colson Dec 21, 2004
Re: Topographic Roughness   William Huber Dec 22, 2004
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Subject Topographic Roughness 
Author Tom colson 
Date Dec 21, 2004 
Message I have seen a lot written on calculating topographic roughness, and I'm a bit confused as to what the "official" way of doing it is. I've seen it as "Slope Standard Deviation" (Miller, S. N., P. D. Guertin, et al. (1999). USING HIGH RESOLUTION SYNTHETIC APERTURE RADAR FOR TERRAIN MAPPING: INFLUENCES ON HYDROLOGIC AND GEOMORPHIC INVESTIGATION. 1999 Annual Summer Specialty Conference, Bozeman, Montana.) And I've seen it referred to as the standard deviation of the elevation values based on a 3 * 3 or 5 * 5 search window. From International Journal of Climatology, Volume 23, Issue 13 (p 1637-1654):
"Topographic rugosity, which is a value describing the topographic roughness. A square window is successively positioned on each pixel of the DEM. For each pixel, a local polynomial of the first degree is calculated to provide an adjustment surface in two dimensions. The differences between the elevations of this surface and the elevations given by the DEM are the residues of which the standard deviation is
calculated. This is what we define s ‘topographic rugosity’."



Any comments?
 
 
Tom Colson
North Carolina State University
College of Natural Resources
Center For Earth Observation
Raleigh, North Carolina, 27695
(919) 673 8023
 
   
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Subject Re: Topographic Roughness 
Author William Huber 
Date Dec 21, 2004 
Message Tom, there's no universal way. What definition, and what calculation, you use depends on what you need the "roughness" for. You might use different measures for air modeling, hydrologic modeling, cut and fill calculations, etc.

It's worth noting that the "topographic rugosity" measure you describe can depend heavily on the grid resolution and neighborhood size. For instance, a smooth grassy field might have a rugosity of zero based on the NED 10-meter representation, of 0.1 m based on a LIDAR dataset, and of many meters on a crude, small-scale DEM. 
  --Bill Huber
Quantitative Decisions (http://www.quantdec.com )
More GIS Q&A at http://gis.stackexchange.com/q/3083/664 
   
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Subject Re: Topographic Roughness 
Author Tom colson 
Date Dec 21, 2004 
Message Here's a definition from Geomorphology:

"One technique to
quantify local topographic surface roughness is to
measure the variability in slope and aspect in local
patches of the DEM (Fig. 4). In this approach, unit
vectors are constructed perpendicular to each cell in
the DEM. The vectors are defined in three dimensions
(using polar coordinates) by their direction
cosines: xi = sinhicos/i, yi = sinhisin/i and zi = coshi,
where hi is the colatitude and /i is the longitude of a
unit orientation vector. Local variability of vector
orientations is then evaluated statistically."

Here's the dillema: I need to quantify topographic roughness for two DEM products (20 foot grid LIDAR) and (30 meter grid NED, which is resampled 20 foot LIDAR). So...would it be appropriate to use a 5 cell window on the LIDAR DEM (100 feet) and a 1 cell window on the NED DEM (96.2 feet)...if I were to use std_dev of Z values by masking. Ultimate analysis here is determing what level of correlation (if any) exists between errors in estimated terrain characteristics (Slope, DTI, Aspect, Surveyed Height, CA, etc...) and topo.rug

Thanks for comments. 
 
Tom Colson
North Carolina State University
College of Natural Resources
Center For Earth Observation
Raleigh, North Carolina, 27695
(919) 673 8023
 
   
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Subject Re: Topographic Roughness 
Author William Huber 
Date Dec 22, 2004 
Message 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. 
  --Bill Huber
Quantitative Decisions (http://www.quantdec.com )
More GIS Q&A at http://gis.stackexchange.com/q/3083/664