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United States Department of Agriculture

Agricultural Research Service

Title: Scaling Terrain Attributes By Fractal Methods

Authors
item Erskine, Robert
item Green, Timothy
item Niemann, Jeffrey - COLORADO STATE UNIVERSITY
item Ramirez, Jorge - COLORADO STATE UNIVERSITY

Submitted to: Annual American Geophysical Union Hydrology Days
Publication Type: Abstract Only
Publication Acceptance Date: February 15, 2008
Publication Date: March 26, 2008
Citation: Erskine, R.H., Green, T.R., Niemann, J.D., Ramirez, J.A. 2008. Scaling Terrain Attributes By Fractal Methods. Annual American Geophysical Union Hydrology Days.

Interpretive Summary: Terrain attributes derived from grid digital elevation models (DEMs) are commonly used in distributed hydrologic models. However, many attribute estimations are biased by DEM grid cell size. For example, land surface slopes estimated from 30-m DEMs are, on average, less than slopes estimated from 5-m DEMs. Since land surfaces display similar patterns at different scales of reference, they are considered to follow fractal behavior. This particular fractal behavior, self-affine, can be characterized by a scaling parameter known as the Hurst exponent (H). Therefore, scaling relationships for terrain attributes can be derived based on H. In this study, centimeter-level accurate global positioning system (GPS) data were collected at 5-m spacing on 12 agricultural fields (3 to 109 ha) in Colorado and Maine. By fitting a line to the log-log plot of a variogram, one can estimate H from the slope of this line. Using this method, H was estimated from the elevation data on each field over various ranges of separation distances suggesting multifractal characteristics of land surface terrain. For the range of 5 to 30-m separation distances, a single H value ranging from 0.73 to 0.98 was estimated (r2 > 0.99) for each field. Fields with greater mean slopes tended to produce higher H, while fine-scale surface roughness seen only on some of the relatively flat fields produced lower H. These estimates of H were incorporated into a slope scaling equation, and field mean slopes derived from 5-m DEMs were predicted from the field mean slopes derived from 30-m DEMs.

Technical Abstract: Terrain attributes derived from grid digital elevation models (DEMs) are commonly used in distributed hydrologic models. However, many attribute estimations are biased by DEM grid cell size. For example, land surface slopes estimated from 30-m DEMs are, on average, less than slopes estimated from 5-m DEMs. Since land surfaces generally obey self-affine fractal behavior, scaling relationships for terrain attributes can be derived based on the Hurst exponent (H). In this study, centimeter-level accurate global positioning system (GPS) data were collected at 5-m spacing on 12 agricultural fields (3 to 109 ha) in Colorado and Maine. Using the variogram method, H was estimated from the elevation data on each field over various ranges of separation distances suggesting multifractal characteristics of land surface terrain. For the range of 5 to 30-m separation distances, a single H value ranging from 0.73 to 0.98 was estimated (r2 > 0.99) for each field. Fields with greater mean slopes tended to produce higher H, while fine-scale surface roughness seen only on some of the relatively flat fields produced lower H. These estimates of H were incorporated into a slope scaling equation, and field mean slopes derived from 5-m DEMs were predicted from the field mean slopes derived from 30-m DEMs.

Last Modified: 10/20/2014
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