Calculating Topographic Ruggedness Index in ArcGIS Desktop?

Question

Does anyone know how to calculate Topographic Ruggedness Index (a.k.a Terrain R.I.) in ArcGIS Desktop without access to command line ArcInfo Workstation?

"The topographic ruggedness index (TRI) is a measurement developed by Riley, et al. (1999) to express the amount of elevation difference between adjacent cells of a digital elevation grid. The process essentially calculates the difference in elevation values from a center cell and the eight cells immediately surrounding it. Then it squares each of the eight elevation difference values to make them all positive and averages the squares. The topographic ruggedness index is then derived by taking the square root of this average, and corresponds to average elevation change between any point on a grid and it’s surrounding area." -- from an aml arcscript by Jeffrey Evans

Answer 1

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).

Answer 2

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

Answer 3

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.

Answer 4

-Edit: the information below is incorrect. Please see the post by whuber explaining the correct process.....

TRI (Riley 1999) and TPI (Jenness 2002) are similar, but different.

To calculate TRI and TPI using ArcGIS 10.x...

Step 1: Use the Focal Statistics tool to make 2 new raster datasets from a DEM.

Raster 1 "MAX") Neighborhood: Rectangle, Height: 3, Width: 3, Units: Cell, Statistics type: Maximum

Raster 2 "MIN") Neighborhood: Rectangle, Height: 3, Width: 3, Units: Cell, Statistics type: Minimum

Step 2: Use the Raster Calculator to perform the following functions on the 2 raster datasets you just created.

For TRI: SquareRoot(Abs((Square("%MAX%") - Square("%MIN%"))))

For TPI: ("%Input DEM%" - "%MIN%") / ("%MAX%" - "%MIN%")

Here is sample Python code exported from a model I built for TRI....

# -*- coding: utf-8 -*-
# ---------------------------------------------------------------------------
# script.py
# Created on: 2014-03-06 08:56:13.00000
#   (generated by ArcGIS/ModelBuilder)
# Usage: script <Input_raster> <TRI_Raster> 
# Description: 
# ---------------------------------------------------------------------------

# Import arcpy module
import arcpy

# Check out any necessary licenses
arcpy.CheckOutExtension("spatial")

# Script arguments
Input_raster = arcpy.GetParameterAsText(0)

TRI_Raster = arcpy.GetParameterAsText(1)
if TRI_Raster == '#' or not TRI_Raster:
    TRI_Raster = "C:\\Users\\Documents\\ArcGIS\\Default.gdb\\rastercalc1" # provide a default value if unspecified

# Local variables:
MIN = Input_raster
MAX = Input_raster

# Process: 3x3Max
arcpy.gp.FocalStatistics_sa(Input_raster, MAX, "Rectangle 3 3 CELL", "MAXIMUM", "DATA")

# Process: 3x3Min
arcpy.gp.FocalStatistics_sa(Input_raster, MIN, "Rectangle 3 3 CELL", "MINIMUM", "DATA")

# Process: Raster Calculator
arcpy.gp.RasterCalculator_sa("SquareRoot(Abs((Square(\"%MAX%\") - Square(\"%MIN%\"))))", TRI_Raster)

Answer 5

This sounds very much like the Topographic Position Index, a process I used recently for one of my projects. There's an ArcScript on the ESRI support page, a Topography toolbox on the ESRI Resource Center page, and some more info on the process on the Jenness Enterprises page.