Suitable digital elevation model


Hints for the DEM interpolation
DTM (digital terrain model) is mathematically defined as spatial distribution of heights that are described by continuous and regionally (within the segments) smooth surface. In the praxis good approximation of the DTM is digital elevation model (DEM) that is recorded as two-dimensional discrete matrix of data heights that is more common known as grid structure. In the next, we are going to centre to DEM.

DEM is usually produced form sampled data that are used as its source. Ideally the data sources would be used without of interpolation. For example, only contour lines themselves may represent a model of terrain. They can be acquired directly, for instance photogrammetrically from stereo model or indirectly, for example from analogue cartographic data, satellite images, by surveying, etc. Interpolation is also not necessary in the cases if data source is very precise and high density, and especially if the data is acquired directly into regular grid (DEM). But interpolation of data sources to produce DEM is necessary if the data sources themselves do not predict treated landscape phenomena.

Interpolation techniques base on the principles of spatial autocorrelation, which assumes that objects close together are more similar than objects far apart. On the edges of the interpolated area extrapolation is also reasonable. Unfortunately no one of the interpolation techniques is universal for all data sources, geomorphologic phenomenon or purposes. We should be aware that in the praxis, different interpolation methods and interpolation parameters on the same data sources lead to different results. The best chosen algorithms on fair data sources should not differentiate much from nominal ground that is idealisation of our desired model and which is commonly similar to actual Earth's surface. Divergences between results of interpolation and from nominal ground are especially consequences of the following circumstances:
  • available data sources do not approximate terrain (distribution, density, accuracy, etc. of the sources is not appropriate)
  • selected interpolation algorithm is labile (is not enough robust) on the employed data sources
  • chosen interpolation algorithms or data structure are not suitable for selected terrain geomorphology or application
  • perception or interpretation of Earth's surface (better: nominal ground) is not the same when more DEM operators work on the same problem; operator's own imagination is common and reasonable problem in DEM production.
Application requirements play important role to expected characteristics of the used DEM. For example, we do not need high geomorphologic quality of DEM for regional, small scale analyses and for calculating average altitudes. But geomorphologic accuracy is more sensitive for visibility analyses and even more for analyses that uses algorithms bases on derivates like slope, aspect, cost surface, drainage, path simulation, etc.

In the most cases, a very high quality DEM should cover all application demands. So it is preferable to find a good and robust interpolation algorithm, what unfortunately difficult task is. Even if more generalised surface is required, DEM with high detail can be simplified to the required quality. It should be noticed that appropriate generalisation methods are very important for producing required DEM. Commonly these methods are complex.

DEM modelling with common interpolation algorithms
We were tested some most common interpolation algorithms based on inverse distance weighted (IDW), kriging and spline using the same data sources. The IDW methods apply the idea that influence decreases with increasing the distance from particular points. The method could be good for interpolation of geomorphologically smooth areas. Kriging methods take into consideration autocorrelation structures of elevations in order to define optimal weights for different distances from a point and then automatically evaluate the results. The method requires a skilled user with geostatistical knowledge. Spline-based methods fit a minimum-curvature surface through the input points. The interpolation ensures continuous and differentiable (smooth) surface. Rapid changes in gradient or slope may occur in vicinity of the data points.

We employed all of three described algorithms using contour lines data sources on the study area, which is geomorphologically variable (see Fig 1). All of the algorithms were used on standardised way and with default parameters. First of all we decided to asses the results with visual approach, which is suitable for general overview of consequences of the interpolation methods.


Fig 1: Contours with interval of 10 m and lake of Bled in the western Slovenia (a). DEM is produced with IDW - smooth (b), kriging - more details (c) and spline based method - smooth but with recognisable characteristic features (d) (area of 5000 by 5000 m).

For general purpose it is difficult to decide which algorithm produces the best DEM form contour lines. IDW algorithm is optimal if we need results produced in a short time and if the real terrain is smooth. Kriging method is useful in this case but some problems occur mainly on the areas with low density of data sources. Spline-based algorithm produces smooth surface and fortunately without many of badly interpolated areas. If we would like decide to use only one of three basic methods of the contour lines interpolation, then we can think on following way:
  • to get optimal result without much effort: use spline algorithm
  • to get the best general result for more advanced analyses and visualisation: use kriging algorithm
  • to get the fastest result: use IDW.
We can stress that there are no bad DEM interpolation algorithms. Some of them have simply more advantages in certain circumstances. The algorithms are actually the most flexible part of the whole modelling process. It is because usually nobody has opportunity to use the ideal data and one can therefore only select the algorithm that is the most suitable for the used data sources and application.

If operator or user knows a purpose of the DEM's application, then he can decide about importance of particular quality parameters. Generally, the optimal is the DEM that requires good results after evaluation of many geomorphologic and statistical quality parameters. Let's propose to allow combination of the three proposed basic algorithms. Then the best DEM from contour lines would be produced as combination of kriging and spline. The kriging would be applied for the areas around the characteristical features like peaks, sinks, valleys, ridges, edges, etc., but the spline algorithm would be preferred on the other areas.

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