SSO Diagram Algorithm

The statistical slope orientation diagram of Chapman (1952) provides a tool for this analysis; the calculation and display of the orientation for an entire DEM takes only a few seconds. The diagram contours the normals to the earth's surface. The data are portrayed on an equal area net, averaged to show the concentration per 1% area on the map. This removes artifacts from the digitizing process (steps in the slope values calculated) and in plotting to the net.

In addition to the SSO diagram, the program will compute statistics about the distribution.  The organization parameter (vector length) and eigenvector S₃ direction (vector direction) can be plotted on a map to graphically show the strength and orientation of the terrain fabric.

Algorithm:

• At each interior point in the DEM, calculate the slope and aspect which define a vector normal to the earth’s surface. This will use the Preferred slope algorithm
• Compute direction cosines of the normal vector.
• Create a 3x3 matrix of the sums of the cross products of the direction cosines.
• Extract eigenvalues and eigenvectors from the 3x3 matrix
• Normalize three eigenvalues, and for terrain S₁ >> S₂ >S₃.  S₁ will be close to vertical
• Pseudocode for the preceding steps
• Eigenvector S₃ points in direction of dominant topographic fabric.
• Compute and graph the ratios ln(S₁/S₂ ) (flatness) and ln(S₂/S₃) (organization) (follows Woodcock, 1977).
  • s1s2 := ln(v[1]/v[2]);
  • s2s3 := ln(v[2]/v[3]);
• Can also calculate the shape and strength parameters from Fisher and others, 1987
  • Shape := ln(v[1]/v[2])/ln(v[2]/v[3]);
  • Strength := ln(v[1]/v[3]);
• Plot and contour the normal vectors on an equal area, lower hemisphere projection.

The diagram above shows the lower hemisphere, equal area projection. Slopes in both percent and slope angle are indicated inside the diagram, and the uphill direction around the margin. A 1% circle (used for the contouring) is also shown.  Contouring.

Select in MICRODEM

• Topographic grain submenu of Calculate menu.  Topographic fabric vector overlay.  This will plot vectors on the map, and is probably the most useful product.  It combines both the orientation and strength, and lets you rapidly assess if the algorithm is working.  If not, you should experiment with the spacing parameters.  This should be relatively fast.
• Calculate Menu,  or Raster GIS menu, Create derivative grid.  This will create grids of the different parameters, and will be very slow.

The box size to pick depends on the resolution of the DEM (how much detail it captures), and the characteristics of the terrain.  1000 m is probably reasonable for 10-30-90 m DEMs.

Because this will change only slowly, you don't need to compute it at every point.  Thus you can subsample.  If you don't care about time, or have a small DEM, 1 is OK.

If you do the fabric overlay (which provides a visual clue about the effectiveness of your parameters), you can plot the database it creates and have the same effect as the grid creation.

Example SSO analysis.

Terrain organization references.

Last revision 2/5/2017