# SMARTerra, a High-Resolution Decision Support System for Monitoring Plant Pests and Diseases
## 2 Methods
### Back-End
#### 2.2.3 Geostatistical Interpolation of Measured Data  
To interpolate agrometeorological values measured at stations, which are typically distributed in an irregular pattern, Kriging with External Drift (KED) is used. KED belongs to a well-known family of hybrid geostatistical interpolation methods that estimate the values \(z(s_{0})\) of a variable \(z\) at unsampled locations \(s_{0}\) based on its known measurements \(z(s_{i})\) at \(n\) locations \(s_{1},\ldots,s_{n}\), i.e., the stations. The underlying idea of such hybrid approaches is to merge the deterministic contribution of attribute values, known both at the stations and at the unsampled points, with a spatially autocorrelated stochastic component. As the formulation of KED is not straightforward, we resort to a closely related method, Regression Kriging (RK).  
RK originates from Ordinary Kriging (OK), which fundamentally involves a linear combination of the observed values.  
\[\hat{z}_{\text{OK}}(s_{0})=\sum_{i=1}^{n}\lambda_{i}z(s_{i}) \tag{1}\]  
with the constraint \(\sum_{i=1}^{n}\lambda_{i}=1\). The weights \(\lambda_{i}\) minimize the estimation variance and depend on the spatial correlation pattern of \(z\) as described by the variogram (see [19] and the references therein).  
RK combines the Ordinary Kriging method described above with standard regression analysis to achieve improved spatial predictions. In our case, five features \(q_{1},\ldots,q_{5}\) (longitude, latitude, distance from the sea, elevation above sea level EASL, and elevation above the valley floor EAVF) are used to estimate  
\[\hat{z}(s)=\beta_{0}+\beta_{1}q_{1}+\beta_{2}q_{2}+\beta_{3}q_{3}+\beta_{4}q_{ 4}+\beta_{5}q_{5} \tag{2}\]  
The RK approximation is obtained by merging Equation (2) with the Kriging contribution:  
\[\hat{z}_{\text{RK}}(s_{0})=\hat{z}(s_{0})+\sum_{i=1}^{n}\lambda_{i}(z(s_{i})- \hat{z}(s_{i}))=\hat{z}(s_{0})+\sum_{i=1}^{n}\lambda_{i}\epsilon(s_{i})=\hat{ z}(s_{0})+\hat{\epsilon}(s_{0}) \tag{3}\]  
Kriging with External Drift is similar to RK, but instead of explicitly separating the regression and Kriging contributions as in RK, it incorporates the \"external drift\" (i.e., the information from the features \(q_{k}\)) directly into the Kriging process. We do not provide mathematical details here and refer interested readers to [20]. The reason we prefer KED over RK is its slightly better performance in our numerical tests and more robust expected convergence properties.  
For an extensive application of KED to temperature interpolation, see [21]. The use of KED and RK for the mapping of soil properties is illustrated in [22; 23], respectively.  
A comprehensive analysis of different Kriging methods, along with considerations about their accuracy for the problem at hand, is a demanding task and cannot be performed here. Furthermore, Kriging methods are not the only possible approach; machine learning (ML) models can also be exploited for the optimal interpolation of agrometeorological parameters. A dedicated study on the mathematical aspects of different methods, including those from the Kriging family and various ML algorithms, with exhaustive assessments and benchmarks, is currently in preparation [7].