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Jim Kompanek
Centre County, Pennsylvania Elevation
This lesson involved multiple methods of interpolation to determine the elevation of Centre County, Pennsylvania based up approximately 300 randomly generated observation points (Figure 1). This data was then compared against the existing DEM to determine the accuracy of the various methods employed. Based upon Figure 1, it is apparent that Centre County is comprised of a series of mountain ridges that trend in a southwest-to-northeast manner at an approximately 45 degree angle. Taking this into account was a challenge for each of the methods. Having the complete data available for comparison is a great asset, and one not likely to always be available.
Figure 1. Digital Elevation Model of Centre County, PA showing randomly generated elevation observation points.
Inverse Distance Weighted Method
The first interpolation employed in this lesson was IDW. This method of interpolation attempts to determine the value of the unknown based upon the values of known control points, with closer points being weighed more heavily than distance points. One of the most important variables in the IDW method is the power, with a power of less than 1 resulting in more distant points having a greater weight, a power greater than 1 resulting in closer points having more weight, and a power of 1 resulting in both close and far points weighing equally. ArcMap also allows for the adjustment of other variables, including the number of points in the search radius and the cell size. For the purpose of this lesson, I used a search radius of 12 points and a cell size of 500 m. The number of points used for the search radius did not appear to significantly alter the resulting maps.
For my first attempt at interpolation (Figures 2 and 3), a power of 2 was used for the IDW because the terrain consisted of long, parallel mountains and this seemed like a good middle-of-the-road approach (it was also the default setting). Furthermore, I also experimented with a power of 1 (Figures 4 and 5) and a power of 3 (Figure 6 and 7). As can be seen in Figure 5, IDW with a power of 1 resulted in a rather jagged and unrealistic interpolation because all points were weighed equally, regardless of distance. A power of 3 resulted in the opposite problem, with closer points weighing too much, which in turn gives a rather spiky terrain model. An IDW interpolation with a power of 2 appeared to give the most realistic elevation representation; though, it still appeared somewhat unrealistic when compared against the actual county elevation.
Figure 2. Interpolated results using IDW based upon approximately 300 observation points (power = 2).
Figure 3. Settings used to generate Figure 2.
Figure 4. Interpolated results using IDW based upon approximately 300 observation points (power = 1).
Figure 5. Settings used to generate Figure 4.
Figure 6. Interpolated results using IDW based upon approximately 300 observation points (power = 3).
Figure 7. Settings used to generate Figure 6.
Inverse Distance Weighted Error
Based upon experimentation with different powers, an IDW power of 2 appeared to best model the terrain of the county. The next step was to determine just how well (or poorly) it did at this. This was accomplished with Raster Calculator, by subtracting the value of each interpolated cell from the existing DEM. Figure 8 depicts the interpolation error using a diverging color scheme. Based upon this map, it appears that this method does a rather poor job depicting the abrupt elevation changes on either side of the mountain ridges. Figure 9 displays the same data with contour lines overlaid on the existing DEM. The error generated surrounding the parallel ridges is even more pronounced.
This inaccuracy appears to be the result of how this interpolation method relies on a circular search radius which only takes distance into account. The relatively small number of control points may also play a big roll in the overall surface generated. I suspect IDW may make a better interpolation method for elevation in areas with much different topography, such as relatively flat areas with rolling hills; not areas with abrupt changes in topography.
Figure 8. Interpolation Error based upon a power of 2.
Figure 9. Interpolation Error contours overlaid on DEM.
Simple Kriging (Isotropic)
In the previous section, the interpolated surface was generated using IDW, which only takes distance between observation points into account, not the actual spatial arrangement. Because of the geography of Centre County, IDW produced rather inaccurate results. The next section involves Kriging, which takes the distribution of points into account. Figure 10 displays the settings used for generating the interpolated surface. The Semivariogram/Covariance Surface (lower left of Figure) seems to display the general trend of the county terrain. To determine the accuracy of the generated surface, I simply overlaid it on top of the existing DEM and observed the variation at the edge of the surface. Although this surface appears more accurate than what was created using IDW, it still leaves quite a bit to be desired. This is likely do to the linear pattern of mountains and valleys, as well as too few control points. Presumably, anisotropic Kriging will result in a better surface because it allows for greater control of the variables.
Figure 10. Settings for Simple Kriging (isotropic).
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Figure 11. Interpolated surface based on Simple Kriging (isotropic).
Simple Kriging (Anisotropic)
With anisotropy enabled, it was possible to modify the search radius from a circle to ellipse to better model the county terrain. Figures 12 and 13 display a surface (and settings) generated using the default settings. Though this surface appears to do a slightly better job representing the terrain (based upon the edges of the surface), it still appears somewhat over simplified and not significantly different than what was generated with isotropic Kriging.
By modifying the Minor Range and Direction, I was able to gradually work towards a better representation of the terrain by adjusting at what angle and how skewed the resulting search ellipses were. By trial and error, a Minor Range of 20,000 appeared to best depict the actual elevation of the county (Figures 14 - 16). This resulted in a very skewed ellipse at an angle of 47.4 degrees. Though, it should be noted that even though Figure 15 does the best job at depicting the general terrain of the study area, it still leaves a great deal to be desired and misses a significant deal of detail (Figure 17). It is unclear how additional control points may improve the process. It appears that an anisotropic semivariogram gives greater control in where and how control points are selected. Though, an ultimate limitation would be in real world interpolation based upon control points, it would not be possible to mimic the known real world surface. Modeling complete unknowns would likely be a great challenge, regardless of the method employed.
Figure 12. Settings for Simple Kriging (anisotropic default settings).
Figure 13. Interpolated surface based on Simple Kriging (anisotropic default settings).
Figure 14. Settings for Simple Kriging (minor range = 20,000).
Figure 15. Settings for Simple Kriging (Minor Range = 20,000).
Figure 16. Settings for Simple Kriging (Minor Range = 50,000).
Figure 17. Error surface based on Simple Kriging (Minor Range = 20,000).
This document is published in fulfillment of an assignment by a student enrolled in an educational offering of The Pennsylvania State University. The student, named above, retains all rights to the document and responsibility for its accuracy and originality.
