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I was looking for an automated process to "identify" the toe of a slope from a DEM and ran across this Esri thread and technique:

I am at Step 7 but have no idea how this is accomplished. I get it is in Raster Calculator but how?

Re: Finding toe-of-slope in a DEM

Author William Huber

Date Nov 29, 2003

Message There are a lot of approaches that might be effective. Here's one. Implement it using the Raster Calculator.

(1) Create an indicator grid of "flat" areas. In the example (attached), I have begun with a DEM of part of Highland County, VA. The "flat" areas are those with a slope of less than 6 percent.

(2) Creat an indicator grid of "steep" areas. This is the complement of the flat areas.

(3), (4) Use the indicator grids to mask the original DEM, creating grids showing elevations only where the slope is flat or steep.

(5), (6) Compute neighborhood means of the steep elevations and flat elevations.

(7)Create a "valley edges" grid at cells where the mean of the steep elevations exceeds the mean of the flat elevations.

(8), (9) Simplify this valley edge grid by performing a Boundary Clean operation, then shrink it by a cell or two.

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Apply the Greater Than operator to the results of (5) and (6).

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whuber seems to have it covered, but an alternative approach might involve taking the slope of a slope raster. The slope function of a DEM gives us the rate of elevation change, and taking the slope function of that would give us in the rate of change in slope. Ideally, the toe of slope would be located in areas with high rates of slope change.

This probably wouldn't produce the same result as the method you outlined (because there are probably high rates of slope change in areas other than the ones you're actually interested in), but it would be an interesting check.

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  • This is an interesting idea, but it perhaps covers too much ground: slopes will be changing anywhere the surface is not closely approximated by a plane. This will include (sloping) valleys, ridges, peaks, rugged areas, and much more. Another difficulty is numerical: small amounts of noise in the data will be doubly magnified by a slope-of-slope calculation. You can do better on both counts, while keeping the spirit of your idea, by carrying out curvature calculations. Any effective ways to compute "steep" and "flat" areas can be incorporated early in this workflow to improve it.
    – whuber
    Commented Dec 4, 2013 at 22:42

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