Ring Plateau Problem in the DEM's

This section looks at another problem found with the elevation data in the DEM's.  In some places where the terrain should roughly be a steady slope, there appears periodic flat stripes or plateaus.  These spurious flat shoulders follow around the elevation contour, giving the appearance of  concentric 'rings' in the data.  The following image shows a close up of the Santa Catalina Mtns NE of Tucson. 


 

 Note the periodic lines in the plot above, on the east side of the mountains.  Periodic lines are suggested on the northwest side as well.  These edges are different than those discussed in the section on the Edge Mis-match Problem.  Those edges were strictly vertical or horizontal, whereas the problems seen above, appear to follow the actual terrain.

The rings seen in the above image might be overlooked as some sort of artifact of the plotting program.  Perhaps places where the color scale changes more abruptly, although the color bar legend doesn't appear to have the sharp edges seen in the plot.    To complicate matters, the rings may not be seen in a 16.7 million color *.bmp or *.jpg file but when reduced to a 256 color *.gif, there are numerous rings.  This can be seen in many of the topographic maps on this site and these are a computer artifact.  There is a real problem with rings in the orignal number data and other kinds of plots do show it more clearly.  The next image covers the same region and is at the same 12 arcsec resolution.  The values plotted here are not the elevation, but the average slope in that grid rectangle.  We were looking at plots of this sort to investigate correlations between the slope and orientation of the terrain with lightning strike patterns. 


 

The slope values are rise divided by run in parts per thousand.  A value of 10 corresponds to a 1% slope, a value of 250 is a 25 % grade, etc.  The plotted value is the slope in the direction of greatest change, in other words the magnitude of the gradient vector.   The numerical method used to obtain these values from the DEM's is described at the end of this section.

The image above clearly shows concentric rings of data where the slope changes in sudden steps.  It can be seen that these rings cover a larger percentage of the image than is obvious from the first image of the actual elevation.  A look at the actual elevation values yields the image below. 


 

A horizontal, east-west, strip of grid rectangles has been highlighted and the elevation values plotted.  Each data point on the plot is the elevation of a 12 arcsec latitude by 12 arcsec longitude bin.  The plots shows fictitious plateaus at a very uniform spacing of about 70 m in altitude.

The USGS website (see link at the top of this page) has detailed information on how the data in the DEM's was obtained.  A mixture of techniques was used, including digitizing from existing contour maps and the use of stereo aerial photography.  The rings here have the appearance of representative contour lines being digitized first and then the terrain in between being filled in by some numerical algorithm.

These last comments explain how the slope values plotted above were obtained.  The gradient vector for the elevation field was calculated for each of the 12 arcsec grid rectangles.  Numerical approximations were used for the derivatives involved.  As discussed elsewhere, the 12 arcsec resolution data was obtained by taking the 3 arcsec resolution DEM's from the USGS and then averaging the values in a 4x4 block of these smaller rectangles.  The gradient calculation also started with the finer 3 arcsec grid.  The change in elevation with x was found by using the average elevation in the first and last columns of the 4x4 grid matrix.  The change in y used the average elevation in the top and bottom rows of the 4x4 grid matrix.  The graphic below shows a sample: 


 

 Note that the slope values for the 12 arcsec grid rectangles were calculated by using only values from within that rectangle and not values from neighboring strips.  This was done so that all the values for a quadrangle could be obtained without needing to read in bordering strips from the quads above, below, to the left and to the right.