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GIS Slope Angle Algorithms and Accuracy/Error

There are many different methods to derive slope angle in GIS from elevation data. Different methods can have very different results. How well derived slope angle correlates to actual measured slope angle is mainly a function of the accuracy and resolution of the elevation data as well as the slope algorithm used. 

Dunn and Hickey (1998; pdf) show a comparison between different techniques to calculate slope from a DEM in GIS and the benefits of applying the "maximum downhill slope" method.

The data below from Srinivasan and Engel (pdf) show that different GIS slope calculation methods can result in very different values for the same elevation data. Slope algorithms from top to bottom in each cell below are: 1 = Neighborhood Method [Spatial Analyst in ArcGIS]; 2 = Quadratic Surface Method; 3 = Best Fit Plane Method; 4 = Maximum Slope Method).  In this study it was found that the neighborhood (average maximum slope) method "most closely approximates observed slope values".

 An accuracy comparison of the above (Srinivasan and Engel) study is shown below

(For the steeper slopes on right, Observed and Neighborhood convert to 6.62° and 8.98°, respectively)

Data below is from Table 1 in Garcia Rodriguez and Gimenez Suarez (2010; pdf) (abbreviations for slope calculation methods are described below) but is sorted here from lowest to highest slope angle (degrees); Campo10m is the field measured slope degree value.

(MAE is the Mean Absolute Error and was calculated for this webpage based on the data in Rogriguez and Suarez [2009] Table 2 and is calculated as the average of the square root or the squared error)

Slope methods from data above from Rogriguez and Suarez (2009) (check article for full citations)

Slope methods corresponding to citation

USGS 10-meter DEM versus LiDAR-derived elevation slope angle uncertainty comparison based on the maximum downhill slope method

A description and comparison of slope angle uncertainty based on the maximum downhill slope method can found in the documents that can be accessed through the links below. The conclusion based on the maximum downhill slope method is that LiDAR derived slope angles only have 1° less uncertainty that USGS 10-meter derived slope angles (about 2 - 3° expected uncertainty for LiDAR to about 3 - 4° degrees uncertainty for USGS 10-meter DEMs for the range of slope angles shown in the plot below):

Haneberg (2006) - describes USGS 10-meter based slope angle uncertainty

Haneberg (2008) - describes LiDAR based slope angle uncertainty and concludes that LiDAR uncertainty is only 1° less than USGS 10-meter derived slope angles uncertainty

Below is the standard deviation or uncertainty plot from Haneberg (2006) based on 10-meter USGS DEM elevation and slope angle being calculated with the maximum slope method

Reference

Burrough, P. A. and McDonell, R.A. 1998. Principles of Geographical Information Systems (Oxford University Press, New York), p. 190.

Dunn, M. and R. Hickey. 1998. The effect of slope algorithms on slope estimates within a GIS. Cartography, v. 27, no. 1, pp. 9 – 15.

Garcia Rodriguez, J.L., and M.C. Gimenez Suarez. 2010. Comparison of mathematical algorithms for determining the slope angle in GIS environment. Aqua-LAC-Vol. 2 - No. 2 - Sep. 2010.