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One of the disadvantages of the IDW interpolation is the lack of direction-specific (anisotropic) information. Therefore, spatial correlations are ignored and are not integrated into the result of the estimation. This disadvantage is leveled out by geostatistical interpolations.
The name "geo" already points to the most important feature of these methods: spatial-statistical parameters constitute the main basis for these interpolation methods.
The variogram or the variography (i.e. the method to derive it from spatial point data) is the basis for a successful geostatistical interpolation.
Geostatistical interpolations are advanced and to some extent complicate methods. Their sensible application requires a large amount of knowledge and experience. At this point, a few keywords about their implementation will be sufficient.
The main procedures are the Kriging methods. They are named after a South African engineer, D.G. Krige. In his diploma thesis in 1951, he laid the foundations for kriging. However, the main developments come from the work of G. Matheron in the 1960s.
Using variography, we get indications of how similar or dissimilar the measurement values of adjacent data points are as a function of their distance from each other.
a) First, we constitute pairs of all the data points and compare their two values. Of each data pair we know the difference (semivariance) and the distance (h):
b) Second, we divide the distances (x-axis) into intervals (so-called lags) and we take the mean of the semivariances of the data pairs within (red dots). By connecting these red points of every lag, we get the experimental variogram. This curve describes how similar the values of two adjacent positions are as a function of their distance from each other.
c) To better handle this representation of spatial (dis-)similarity, we can construct simple curve functions onto the experimental variogram to match it as well as possible. This curve is called the theoretical variogram.
Now, we attempt Kriging by incorporating our data into this model of spatial continuity – the model that we have developed or found in the section about variogram modeling. Based on such a model we can calculate error variance for our estimations and seek their minimum.
Interpolation with Kriging is a kind of curve fitting: from our known data points we have derived a model of how the spatial relationships might be designed. Based on that model we now estimate the unknown points. If we consider it in two dimensions only (for simplicity), we work with a regression technique: a curve fitting.
You often hear the term "exact interpolator" in connection with Kriging, just like IDW and some other estimation methods. This means that a surface estimated with one of these methods is intersecting the known data points. If we perform the Kriging calculation at a position of a known value, Kriging will typically give us exactly that value in return.
In comparison, see below for the result of an inverse distance weighting compared to that of a Kriging interpolation:
All interpolation methods can additionally be controlled by the definition of a search neighborhood, i.e. how many or which known data points are used to calculate an unknown position. If we ignore this neighborhood, all available known data are used for the estimation of every point. In the case of the Swiss rainfall data, this would mean that for the calculation of a precipitation value in Ticino, the values of observation stations in Jura are included as well. This just does not make any sense.