Calculating Topographic Ruggedness Index in ArcGIS Desktop?

Let's do a little (just a little) algebra.

Let x be the value in the central square; let x_i, i = 1, .., 8 index the values in the neighboring squares; and let r be the topographic ruggedness index. This recipe says r^2 equals the sum of (x_i - x)^2. Two things we can compute easily are (i) the sum of the values in the neighborhood, equal to s = Sum{ x_i } + x; and (ii) the sum of squares of the values, equal to t = Sum{ x_i^2 } + x^2. (These are focal statistics for the original grid and for its square.)

Expanding the squares gives

r^2 = Sum{ (x_i - x)^2 }

= Sum{ x_i^2 + x^2 - 2*x*x_i }

= Sum{ x_i^2 } + 8*x^2 - 2*x*Sum{x_i}

= [Sum{ x_i^2 } + x^2] + 7*x^2 - 2*x*[Sum{ x_i } + x - x]

= t + 7*x^2 - 2*x*[Sum{ x_i } + x] + 2*x^2

= t + 9*x^2 - 2*x*s.

For example, consider a neighborhood

1 2 3
4 5 6
7 8 9

Here, x = 5, s = 1+2+...+9 = 45, and t = 1+4+9+...+81 = 285. Then

(1-5)^2 + (2-5)^2 + ... + (9-5)^2 = 16 + 9 + 4 + 1 + 1 + 4 + 9 + 16 = 60 = r^2

and the algebraic equivalence says

60 = r^2 = 285 + 9*5^2 -2*5*45 = 285 + 225 - 450 = 60, which checks.

The workflow therefore is:

Given a DEM.

• Compute s = Focal sum (over 3 x 3 square neighborhoods) of [DEM].

• Compute DEM2 = [DEM]*[DEM].

• Compute t = Focal sum (over 3 x 3 square neighborhoods) of [DEM2].

• Compute r2 = [t] + 9*[DEM2] - 2*[DEM]*[s].

Return r = Sqrt([r2]).

This consists of 9 grid operations in toto, all of which are fast. They are readily carried out in the raster calculator (ArcGIS 9.3 and earlier), the command line (all versions), and Model Builder (all versions).

BTW, this is not an "average elevation change" (because elevation changes can be positive and negative): it is a root mean square elevation change. It is not equal to the "topographic position index" described at http://arcscripts.esri.com/details.asp?dbid=14156 , which (according to the documentation) equals x - (s - x)/8. In the example above, the TPI equals 5 - (45-5)/8 = 0 whereas the TRI, as we saw, is Sqrt(60).

I would recommend to look outside ArcGIS) Very easy using the free gdal software: http://www.gdal.org/gdaldem.html

gdaldem TRI input_dem output_TRI_map

Or if you'd prefer it in saga gis: http://www.saga-gis.org/saga_modules_doc/ta_morphometry/ta_morphometry_16.html

The Riley et al., (1999) TRI is square root of the summed squared deviations. This is very close to unscaled variance. If you want an implementation of Riley's TRI then please follow the methodology outlined by @whuber (the methodology provided by @user3338736 generalized the metric to the maximum in the window and does not represent the cell by cell variation).

I have a variation of TRI in our Geomorphometry & Gradient Metrics ArcGIS Toolbox that is the variance of a specified window. I find this more flexible and justifiable. There are also some other surface configuration metrics including rugosity and dissection.