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.

 Point Campo10m ArcGIS(S&E) Bau:AP2 Zeve:AP2 Herr_AP2 Max_pen Max_pen_tri Pl_ajuste Hara_AP3 Hick_mpab 8 1.146 14.897 3.694 7.341 3.694 4.864 5.527 6.857 6.592 11.251 21 1.146 1.459 2.388 3.11 2.388 3.628 5.092 2.919 3.013 1.403 22 1.146 1.922 3.212 2.545 3.212 3.448 5.092 2.647 2.58 2.143 24 1.146 1.348 3.932 2.048 3.932 2.495 6.757 2.008 1.924 1.107 5 1.718 4.424 3.694 3.595 3.694 3.054 2.485 3.473 3.655 2.438 17 1.718 15.041 16.237 12.647 16.237 15.023 19.56 13.009 12.613 20.119 20 2.291 0.754 2.912 0.754 2.912 0.952 4.477 0.717 0.596 0.852 25 4.004 5.618 9.704 6.903 9.704 7.595 14.438 6.804 6.075 5.551 15 4.574 13.12 14.447 19.636 14.447 21.765 15.923 19.767 20.545 15.724 26 5.143 3.825 10.018 6.232 10.018 8.003 15.321 6.073 5.983 5.046 14 6.277 15.133 15.914 12.086 15.914 11.106 19.803 12.218 12.367 13.359 18 6.277 15.275 12.702 12.459 12.702 13.038 18.692 12.827 13.081 14.533 19 6.843 1.025 6.121 4.85 6.121 6.325 10.834 4.925 4.532 1.085 29 6.843 7.789 10.492 9.778 10.492 13.099 20.832 9.8 9.897 10.279 2 7.407 8.514 8.715 8.432 8.715 6.838 10.38 8.118 8.677 9.524 27 7.407 14.015 10.581 11.878 10.581 11.535 10.988 11.618 11.716 12.022 28 7.407 7.455 10.581 8.412 10.581 7.333 20.832 8.22 7.853 5.573 30 7.97 12.828 12.662 15.093 12.662 14.105 20.737 14.472 14.651 11.234 23 9.09 6.766 8.388 10.121 8.388 14.023 11.443 10.476 11.276 11.157 32 11.31 24.858 21.889 27.826 21.889 29.487 24.385 27.317 27.633 26.406 16 11.86 11.161 14.447 11.526 14.447 10.969 19.56 11.623 11.104 10.503 6 12.407 15.283 8.715 12.448 8.715 10.582 9.254 11.605 11.767 14.684 4 14.036 15.444 15.979 14.244 15.979 8.609 14.196 13.564 13.343 11.411 13 14.036 6.675 15.873 7.885 15.873 9.634 17.122 8.042 7.424 6.021 31 14.574 16.128 13.56 15.612 13.56 20.503 24.385 15.609 15.795 23.584 3 16.699 10.179 12.885 10.228 12.885 5.84 8.358 9.613 9.694 5.439 12 19.29 17.582 20.322 19.889 20.322 25.407 21.287 20.002 20.488 28.066 10 20.807 23.926 19.742 23.495 19.742 21.35 21.762 23.086 23.099 26.044 11 20.807 25.91 20.322 27.015 20.322 28.018 22.864 27.072 27.615 24.754 1 22.294 14.864 16.931 15.205 16.931 13.278 13.609 15.128 15.365 14.332 7 30.964 15.64 15.651 14.331 15.651 12.12 9.254 14.263 14.72 15.025 9 30.964 23.494 12.663 24.581 12.663 20.5 20.384 23.822 24.179 19.684 MAE 4.930 4.585 4.556 4.585 5.731 7.530 4.571 4.675 5.811

(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.