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Quadratic Method for Slope

by donn.rodekohr on ‎12-11-2015 09:12 AM - edited on ‎04-21-2016 06:09 AM by Technical Evangelist (1,137 Views)

Download Models and Sample Data

Quadratic Slope in Percent


Quadratic Slope in Degrees




  1. X and Y cell dimensions are equal, i.e., square pixels. Longitudinal ground measurements vary as one proceeds away from the equator. This is illustrated by longitudinal lines of convergence as one approaches the poles. All geographic coordinate system graticules are trapezoids.
  2. X, Y, and Z units are equal, i.e., a projected coordinate system. If the projection units are expressed as meters, then the elevation units must also be expressed in meters.  
  3. NoData values have been assigned to background values. Check the Metadata on the input elevation data to ensure this action has been accomplished.

DEM coefficients computed using the DEM Derivative Toolkit (the Chassis_Kit.gmdx) can be used to compute various elevation model products.  Field and statistical studies have demonstrated that the products of the quadratic curvatures more exactly represent slopes and slope aspect on landscape surfaces than “standard” GIS slope models (Florinsky, 1998, Jones, 1998; Weih, 2004).

Product:  Percent Slope — Qslope(%).   This specific Model computes the slope of a DEM and express the results in percent slopes (floating point values.)   We used formula 1.01 to accomplish the task and formula 1.02 to generate Qslope in degrees.



Where G and H refer to the quadratic coefficients as defined by Haegl, et al.


Model Operators and Assumptions

In an effort to make the models run as smoothly and efficiently as possible with a minimum of new coding, operators from the Chassis_Kit model transferred to a new model.  The Chassis_Kit computes the Coefficients D through H but  Slope% requires only coefficients G and H so D through F were deleted.  Formula 1.01 was translated into a Spatial Modeler operator sequence.

To express the output slope values as Degrees, two operators were added that converted the percent slope to Degrees.


Products and Quality Control

To test the results I first ran Terrain :: Surface Slope using Percent as a floating point value.  I used a 5 m resolution DEM from the USGS, projected into UTM zone 16N, Z values in meters, NAVD 88.  I extracted an oval shaped area to test the error checking along the edges.


The important stats to note are:

Slope value statistics

Standard Slope command

QSlope model output













Std. Dev.



A test on a USGS DEM, 30m resolution, projected to State Plane Coordinates (feet), Elevation units in feet (NAVD 88).

Slope value statistics

Standard Slope command

QSlope model output













Std. Dev.



Differences are to be expected but the values should all be in the same realm, which they are. The differences in the Max values can be traced to erroneous slope values on the edges. 



  • Florinsky, Igor V. 1998.  Accuracy of local topographic variables derived from digital elevation models.  Int. Journal of Geographical Information Science.  Vol. 12, No. 1, pp. 47-61.
  • Hengl, T, Gruber S., and D.P Shrestha.  2003.  Digital Terrain Analysis In ILWIS.  Integrated Land and Water Information System Academic Users Guide, Enshede, 2001.
  • Jones, Kevin H.  1998.  A Comparison of Algorithms Used to Compute Hill Slope as a Property of  the DEM.  Computers and Geosciences Vol 24, No. 4, pp. 315-323.
  • Smith, Michael P., Zhu, A-Xing, Burt, J.E., and Stiles, C.  2006.  The effects of DEM resolution and neighborhood size on digital soil survey.  Geoderma.  Vol. 137, Issues 1-2, pp. 58-69.
  • Weih, Robert C., Jr., Mattson, T.  2004.  Modeling Slope in a Geographic Information System.  Journal of Arkansas Academy of Science, Vol. 58. pp. 100-108.

Input parameters:

Digital Elevation Model filename: Digital Terrain Model (DTM) that is projected to a planar coordinate system where the X and Y dimensions of the cells are equal.  Also, the X and Y units of measurement must be the same as the Z units of measurement, eg Meters and meters.


Example data:

Sample USGS DEM from the USGS National Map.

33086_aea_snip.img – a small extract of a 30m resolution USGS DEM projected to Albers Conical Equal Area with the elevation unit expressed in meters.