Due to a high sensitivity of terrain analysis algorithms to local conditions, any single realisation represents only one view on terrain morphology. This is especially important for the calculation of hydrological parameters where we are more interested in the general picture of the processes. Even for the perfectly adjusted DEM, the location of the stream network can differ up to 3-4 cells from the true location (Burrough and McDonnell, 1998). A statistically robust approach to reduce the errors in terrain parameters is to average a set of possible realisations given the uncertainty in elevation values (Burrough et al., 2000; Raaflaub and Collins, 2002). This is also referred to as the Monte Carlo method of error propagation (Heuvelink, 1994). The elevation values can be simulated using the inverse normal probability function (Banks, 1998):

where A and B are the independent random numbers within the 0-0.99... range, zi is the original value at ith location, z* is the simulated elevation with induced error and RMSE(z) is the standard deviation of elevation values. In order to produce a realisation of DEM with similar spatial dependence structure (i.e. similar smoothness), point simulation needs to be used (Holmes2000). It will produce a set of equiprobable realistic DEMs, each showing a similar histogram and variogram. Assuming gaussian spatial distribution of errors and for given RMSE(z) and covariance function (C0, C1 and R), the realisation with same internal properties as the original DEM can be produced by simulating a point sample, inducing the error at point locations and then re-interpolating it over the whole area. For each of the m simulated DEMs, terrain parameters are derived m times and then averaged per pixel:

This gives a more smoother, i.e. more realistic picture of the terrain. See changes in the plan curvature (PLANC) calculated by averaging 1 to 50 realisations. For more details on the algorithm see:

Hengl, T., Gruber, S. and Shrestha, D.P., 2004. Reduction of errors in digital terrain parameters used in soil-landscape modelling. International Journal of Applied Earth Observation and Geoinformation (JAG), 5:97-112.

Figure: Simulated error in DEM using conditional geostatistical simulations.

Figure: PLANC calculated by averaging up to 50 realisations.

 

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