## 2 Materials and Methods
### Kriging  
Kriging (KG) is the generic name given to a family of generalized least-squares regression algorithms, used in geostatistics for data spatial interpolation. The spatial correlation was analyzed in  
Figure 2: The maxima series recorded at the main stations. geostatistics using the variogram. The main ideas behind the methodology for Kriging are shortly presented below, based upon the bibliographic resources [19; 20]. Given a random function \(Z(s)\) that is stationary, with a constant mean \(\mathrm{E}(Z(s))=\mu\), the semivariogram is defined by:  
\[\gamma(h)=\frac{1}{2}\mathrm{E}(Z(s_{i})-Z(s_{i}+h))^{2}, \tag{3}\]  
where \(Z(s_{i})\) and \(Z(s_{i}+h)\) are the variable values at the study station and at a location situated at the distance \(h\) from the study location. When assuming the direction independence of the semivariance (isotropy), the variogram can be estimated using \(\sum\) the sample variogram, defined by  
\[\hat{\gamma}(h)=\frac{1}{2N_{h}}\sum_{i=1}^{N_{h}}\left(Z(s_{i})-Z(s_{i}+h) \right)^{2} \tag{4}\]  
where \(N_{h}\) is the number of sample pairs \((Z(s_{i}),Z(s_{i}+h))\) used in estimation and where \(N(h)\) is the number of data pairs, which are approximately separated by the lag \(h\). After computing the sample variogram, one should determine the empirical semivariogram by fitting a parametric model to it. This can be done by the Generalized Least Squares (GLS), Maximum Likelihood or Bayesian methods [19]. GLS is the method used in this article. The nugget, sill and range are the parameters used to describe a variogram (Figure 3). The nugget is the random error process, shown by the height of the jump of the semivariogram at the discontinuity at the origin. The sill is the variogram limit, when the lag tends to infinity. The range is the minimum lag at which the difference between the variogram and sill becomes negligible [21]. Different types of variograms may be used, as function of the data series, for example spherical, exponential, Gaussian, Matern and power [19]. At the final modeling stage of ordinary KG, the predictions are based on the model:  
\[Z(s)=\mu+\varepsilon(s) \tag{5}\]  
where \(\mu\) is the mean and \(\varepsilon(s)\) is the spatially correlated stochastic part of variation. The predictions are obtained by formula:  
\[z_{0}(s_{0})=\sum_{i=1}^{n}w_{i}(s_{0})z(s_{i})=\lambda_{0}^{T}z \tag{6}\]  
where \(\lambda_{0}^{T}\) is the transposed KG weights vector \((w_{i})\), and \(z\) is the vector containing the observations at \(n\) neighbor locations. The Kriging predictions are obtained by:  
\[z_{0}(s_{0})=\sum_{i=1}^{n}w_{i}(s_{0})z(s_{i})=\mathbf{w_{0}^{T}z} \tag{7}\]  
Figure 3: A variogram and its parameters [21]. where \(\mathbf{w_{0}^{T}}\) is the transposed kriging weights vector (\(w_{i}\)), and \(\mathbf{z}\) is the vector containing the observations at \(n\) neighbor locations. The weights are based on the covariances among points in the sample and the covariances between sample points and the point to be predicted. The Kriging estimator should be unbiased and the error variance is minimized. The weights for ordinary Kriging can be found by solving the Kriging equations:  
\[w=\left[ \begin{array}{cccc}C_{11}&\ldots&C_{1n}&1\\ \vdots&\ddots&\vdots&\vdots\\ C_{n1}&\ldots&C_{nn}&1\\ 1&\ldots&1&0\end{array} \right]^{-1}\left[ \begin{array}{c}C_{10}\\ \vdots\\ C_{n0}\\ 1\end{array} \right] \tag{8}\]  
where \(C_{ij}=\text{Cov}\Big{(}Z_{i},Z_{j}\Big{)}\), \(C_{i0}=\text{Cov}(Z_{i},Z_{0})\) and \(\lambda\) is the Lagrange multiplier that appears due to the constraint \(\sum_{i=1}^{n}w_{i}=1\). Since the first step of the KG procedure is building a variogram (not a covariogram), one should want to determine the Kriging equation, in terms of variogram. Under the hypothesis of second-order stationarity:  
\[C_{ij}=\sigma^{2}-\ \gamma_{ij}, \tag{9}\]  
\(\sigma^{2}\) is the variance and \(\gamma_{ij}\) is the semivariance. Therefore, (8) can be written in the equivalent form:  
\[\left[