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By Mikhail Kanevski; Michel Maignan

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From this representation it is evident, that using the same and even simple method but changing the model dependent parameters it is possible to obtain qualitatively different results. 6. Because of the small search radius (I 0 km) there are some underestimated regions corresponding to white zones. The smoothing effect in this case is smaller (and even some "overfitting" can be observed) in comparison with the search radius I OO km. 7. An independent validation data set was used with the same two different search radii.

The box counting method of the fractal dimension. Draw a grid on the fractal that consist of (N1)2 squares, and determine the number of squares N1 that are needed to cover the fractal. Next we choose finer and finer grids with (N1 )2 < (N )2 <.. < (Nm)2 2 squares and calculate the corresponding numbers of squares N1 , ... , Nm needed to cover the fractal. Since S(N) scales as S(n) oc N (-dfl We obtain the fractal dimension by plotting S(N) versus plot. The asymptotic slope, for large N, gives df 1/N in a double logarithmic Of course, the finest grid size must be larger that the pixel size, so that many pixels can fall into the smallest square.

Tuning of the corresponding covariance matrix parameters can be carried out with the help of cross-validation. g. rescaling of the spatial coordinates can have the same effect but it is more difficult to tune if there is no external information about how to pre-process data. e. spatial interpolations with several variables. � � • � r i + I • + • -+- • ... + DATA Fig. 3. 7 Validation of inverse square distance model. "+" : R = I O; "oo" : R = I 00. 3 Multiquadric equations Another well-known "deterministic" method is the multiquadric equation model [Hardy, 1 986].

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