TRI and the Diagonal Neighbors

TRI purports to be a roughness (or ruggedness) measure, but is in fact a slope measure.

TRI averages the difference in elevation for the central point and its eight neighbors, and has meters as the unit.  If it divided by the distance between the two points, it would have a directional derivative.  Taking the average would lead to an "average adjacent neighbor" slope algorithm.  I cannot find a reference for this algorithm, but there is a steepest downhill neighbor (O'Callahan and Mark, 1987) and average adjacent neighbor (Sharpnack & Akin, 1969).

Because it lacks only the division by the horizontal distance, TRI correlates strongly with all the various slope algorithms and does not provide an independent measure of the landscape.  Figure 1 shows the correlation between slope and the TRI computed by 4 GIS programs (MICRODEM, WhiteBox, GDAL, and SAGA).  This uses a DTM with 1" spacing at latitude 36 with noticeably rectangular pixels.

Figure 1.  Slope correlations with TRI from 4 GIS programs

Figure 2 shows the correlations among the programs for TRI; note that all are very highly correlated.  The Wilson algorithm, introduced for marine cases with smoother terrain, has the lowest correlations with the others.

Figure 2.  TRI correlations from 4 GIS programs


MICRODEM introduced a number of algorithms to account for the different distances between the diagonal neighbors, and the difference between the EW and NS neighbors in geographic grids.  These compute what the elevation would be at a particular distance, assuming the slope was constant in that direction.
  • None.  This matches the other programs exactly.
  • EW. Normalize using the east west spacing.
  • NS.  Normalize using the NS spacing
  • 30 m.  Normalize using 30 m, the nominal spacing for the 1" geographic grids.
  • Interpolate.  Computes the elevation via bilinear interpolation at the point with grid coordinates COL + 0.707, ROW + 0.707.  This will not necessarily on the 45 degree line for geographic grids
Figure 3 shows the correlations for these algorithms in computing the NE neighbor for every pixel in the DEM.  The EW, NS, and 30m methods correlate exactly, because their only difference is the normalization factor.  Even the worst correlation is still very high, suggesting that the method used to compute the diagonal neighbors does not matter.

Figure 3.  Correlations among 5 algorithms for computing the elevation of the NE neighbor.

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