## 3 Results
### Geostatistical Analysis of Residuals  
After selecting the shortest separations from all 89,676 distances between the 424 sample points, the obtained mean of 3.684 km served as a starting point to choose the lag increment. For practical reasons, e.g., plotting the variogram, the lag increment was finally set to 4.0 km. In the range 0 to 200 km, the mean number of observations contained in each of the 51 lags was 1752. Only at lags > 150 km did the numbers of cases fall below 1000. The experimental autocorrelation coefficients \(\rho(h)\) are shown in Figure 4. The autocorrelation shows an immediate decline of autocorrelation coefficients within the first lag interval. At all subsequent lag distances \(h\geq 2\) km, the autocorrelation coefficients show a narrow range of scatter around zero. As a consequence of the considerable number  
Figure 3: Box-and-whisker plot showing the results of model evaluation by one hundred runs of tenfold cross-validation: coefficient of determination \(r_{\text{cv}}^{2}\), root mean square error \(\text{RMSE}_{\text{cv}}\), least squares analysis of slope \(b_{1\text{cv}}\), and the concordance correlation coefficient of [PERSON] (2017)CCC\({}_{\text{cv}}\). of negative coefficients, it is not possible to compute the autocorrelation length with the exponential autocorrelation function (Equation (4)) commonly used for this purpose [90]. \[\sum_{n-1}^{i=1}\left[\rho_{i}(0)-\mathrm{e}^{(-h/\lambda)}\right]=0 \tag{4}\]  
Even as the autocorrelation length reaches close to zero, the sum of coefficients cannot be balanced by the exponential autocorrelation function. Figure 4 shows an alternative fit that assumes all negative coefficients as zero, yielding an autocorrelation length of 1.1 km. These findings imply that autocorrelation is of minor importance at the scale analysed. Likewise, this can be taken as an indication that the main influences affecting SOC in the study area are already captured by the regression models. This is very similar to the findings of [PERSON] and [PERSON] [91] on the regionalisation of SOC in the forest soil of Baden-Wuerttemberg (Germany). They also observed only very weak autocorrelation coefficients for residuals remaining after regression analysis. The values of the sample variogram (Figure 4) reach the sill within a range of 100 km. After reaching this maximum, the semivariance shows a hole effect, which typically arises from regular repetitions [75] or spatial variation occurring at several scales [90]. The observed regularly oscillating semivariances are possibly caused by the periodically arranged glacial deposits in the north-eastern lowlands resulting from successive ice advances. Especially within the study area, the distance between moraines and related landscape elements left by the successive glacial stages is quite similar to the observed periodicity of the semivariogram [92; 93]. Although variables like \(z_{\text{CaCO}_{3}}\) and \(CF_{0...90}\) were entered into the regression equations, these effects may be enhanced by a slight under-representation of the corresponding predictors (e.g., particle size classes) within the regression models. For modelling the variogram, we selected an exponential function with an additional nugget component (Figure 4). The range for fitting the function to the experimental values was restricted to 100 km, thus neglecting the hole effect at larger lag distances. Following [PERSON] and [PERSON] [75], we did not fit the variogram model to the hole effect, as there was no clear theory for the observed periodicity. In addition, the use of increased weight factors at high distances in the kriging procedure would not support the desired incorporation of unknown large-scale effects. The combined variogram model achieved a high goodness of fit (R\({}^{2}\) = 0.874). The parameters estimated using the Levenberg-Marquardt  
Figure 4: Autocorrelogram, sample variogram, and fitted variongram model (nugget + exponential) for residuals of the regression equations.