## 2 Materials and Methods
### Scaling Relation at Different Spatial Resolution  
The irrigation mapping results differ depending on the spatial resolution. Whereas a perfectly homogeneous image does not show differences in NDVI with changing resolution, the averaging of heterogeneous (with reference to the considered resolution) reflectances in the higher resolution images results in a tendency to homogenize the NDVI values in the coarser resolution images. Since the irrigation detection algorithm is non-linear with NDVI, this changes the amount of detected irrigation, with NDVIs averaged over heterogeneous areas. Therefore, we assume a relationship between the heterogeneity of the spatial position and formation of the irrigated area as it is shown in the fine resolution and the area changes when moving up to coarser resolutions. To measure the heterogeneity or homogeneity of the irrigation pattern, we use landscape metrics, a measure for the complexity of a landscape. To quantify the relation between landscape metrics and the areal change with resolution of the mapped irrigated area, we calculate landscape metrics for the three regions. To increase the number of samples, we split each region in 36 tiles to generate more stable statistics. A pre-analysis showed that at and above 6 by 6 pixels, the results remained constant. For the 36 tiles, the areal change between 300 m and 10 m is calculated as follows:  
\[\text{areal change}[\%]=\text{irrigated area 300 m }[\%]-\text{irrigated area 10 m }[\%] \tag{2}\]  
While the pixel at 300 m gives Boolean information (irrigated or not irrigated), the result at 10 m gives more precise information about the irrigated area at the corresponding 300 m pixel. This information is used to determine the difference between the mapping result at 300 m and 10 m. Depending on the position and spatial arrangement of the irrigated area, the change in spatial resolution from 10 m to 300 m can result in positive or negative areal change of irrigated area detected by the algorithm. Negative changes occur in case of a high heterogeneity of the considered area. Positives changes occur when the majority of the considered area is classified as irrigated, and the spectral reflectance is hardly affected by the upscaling process. Positive changes are rather theoretical and hardly ever occur. Therefore, this study will focus solely on the negative changes.  
To quantify the relation between landscape metrics and the areal change with resolution of the mapped irrigated area, we calculate landscape metrics on the same 36 tiles of the three regions using the R-package of [PERSON] et al. [40]. We assume that the position, the shape and the spatial arrangement are reasons for the areal change of the mapping results. To explain the negative changes of irrigated area with decreasing resolution, we calculate the Landscape Shape Index (LSI, see Equation (3)), which describes the ratio of the total edge length of a class, in our case, irrigated area, to the minimum edge length. LSImeasures the complexity of a selected class (irrigated) compared to the other classes (not irrigated) of a landscape.  
\[\begin{split}\text{LSI}=\frac{\text{E}}{\text{min}(\text{E})}\\ \text{E}=\text{total edge length of the class}\end{split} \tag{3}\]  
Thus, as a ratio between the actual class edge length and the minimum class edge length, the LSI is an 'aggregation metric'. In case of only one class in the landscape, the minimum length equals the edge length. The higher the ratio, the more complex the pattern of the irrigated area. The result is a high expected loss of mapped irrigated area at the coarser spatial resolution. The LSI is calculated for the 36 tiles of the irrigation mapping at 300 m and is correlated with the negative areal change for the mapping result between 300 m and 10 m.