### Comparative analysis of interpolation methods to generate the DEM (Digital Terrain Model)

As you must have already reckoned, we are
particularly interested in the transition of the territorial data to 3D.
The ArcGis Pro output with two side-by-side windows (2D
window ArcMap type and 3D window ArcScene type) is a novelty that must be
studied. In the long run it will become either the standard for most
GIS, or a vague memory of a failed attempt. This
will depend largely on you, whether you find that it adds a plus to your job,
or that it is, just, another gadget you can do without.
Therefore,
you have to try it, but for that, of course, you must have 3D displayable data.
The first being the DEM (Digital Terrain Model), where all
your other data will be displayed.
To
generate a DEM you have to start with vector data (points, lines, surfaces)
containing information about height (elevation), which one interpolates to have
a continuous XYZ surface.
The
accuracy of the generated terrain model depends on the interpolation mechanism
used. It is, therefore, necessary to study the comparative
performance of different methods in this context. We have
compared the general interpolation techniques, namely the inverse of the
weighted distance (IDW), the kriging, the ANUDEM method (topo to raster in
ArcGis), the natural neighbour, and the Spline method.

The different types of
interpolation methods.

Different interpolation methods applied to the same data can
produce different results. Therefore, it is necessary to
evaluate the comparative relevance of these methods.
The
interpolation methods are based on the principle of spatial autocorrelation,
which assumes that the closer the points, the more similar they are.
In the literature you will find many methods of interpolation,
they
are generally classified into two categories: local and global methods.

Local
methods predict the value of a point based on the values ​​of points in the
neighborhood. The most used local methods
are:


the inverse of the weighted distance (IDW) method,


the local polynomial method,


the natural neighbour method (NN), and



On the other hand, global interpolation methods, such as
polynomial interpolation functions, use all available sampling points to
generate forecasts for a particular point. These
methods facilitate the evaluation and elimination of global phenomena (such as
trend) in physical data.
All
these methods are also called deterministic as opposed to geostatistical
methods.

Kriging

It is a geostatistical interpolation method that uses a
variogram ( analysis of the variability of the data according to the distance
between them). The variogram depends on the
spatial distribution of the data rather than the actual values. When
applying the kriging method one can see results for input points different from
the input value.

The IDW method

It is a local deterministic interpolation technique that
calculates the value of a point by averaging the values ​​of the neighbouring points
weighted by the inverse of the distance at the calculated point: the closer the
points, the more the affected weighting is strong.
It
considers that the points closer to the location to be calculated will have
more influence.

The natural neighbour method

This method searches for the subset of samples closest to a
point and applies a weighting based on the area where they are located.
It is a local deterministic method and the interpolated
heights are necessarily within the range of values ​​used. It
does not produce peaks, pits, ridges or valleys that are not already present in
the input samples and adapts locally to the input data structure.
It does not require any configuration by the user and works
equally well for data distributed regularly as well as irregularly.

The Spline interpolation
method

This method uses a mathematical function to minimize the
curvature of the surface and produces a smooth surface that matches exactly the
entry points.

The ANUDEM method (topo to
raster of ArcGis 3D Analyst)

This method uses an interpolation technique specifically
designed to create a surface that best represents a natural drainage surface
and preserves both ridge lines and stream networks.

Results of the comparison
of interpolation methods

We compared the different
methods with the IGN repository for several projects, calculating the mean
square error.
In
order to study the sensitivity of the interpolation methods according to the
nature of the terrain, we grouped the results according to three types of
slope: zones with little slope, zones of steep slope, and zones having a
mixture of the two. The results are presented in
qualitative form (stars) but it can be said that the best results have an EQM
of the order of 50 cm and the least good an EQM of the order of 2m.

In general, we can say that IDW methods
and
Kriging adapt quite well, whatever the variations of
terrain. Other methods are generally more sensitive to variations.
The ANUDEM method has excellent performance when it
was a question of the calculation of ridges and areas of flow stream ..

In
low slope areas, Kriging and the Natural Neighbor
give very good results, and we advise to adopt them in this case.
In areas of steep slope, the average differences are less
marked and must analyze them more on a case-by-case basis. The Natural
Neighbor is
best applied on small areas of study. For
the calculation of the flow zones the ANUDEM method gives
the best results.
The
natural neighbour method showed almost optimal values ​​on smooth
surfaces.
Spline- based methods fit a minimum curved surface
through the entry points. It preserves trends in the