## 2 Materials and Methods  
### Multi-Resolution Analysis  
We applied the method described in [PERSON], [PERSON] and Mauser [7] to determine irrigated area. It does not explicitly use spatial resolution as a parameter. The basis of the mapping method is annual NDVI time series. They are used together with parameters such as land suitability for agriculture, a land use classification, NDVI data and official national statistics to determine global irrigated area. The annual course of NDVI is analyzed, interpreted and compared with agricultural suitability evaluations [7; 33]. The method analyzes the NDVI time series using parameters such as amplitude of the NDVI, position of NDVI peaks and shape of the NDVI annual temporal course. If the course of NDVI suggests active vegetation growth with typical characteristics of agriculture and, simultaneously, the agricultural suitability is low due to a rainfall deficit, we assume a high likeliness of irrigation. In our case, the original mapping-method [7] is modified in two points to be applicable to the finer spatial resolutions: (1) the information about irrigated area derived from the official statistics are not used to avoid a biased result and (2) the restriction of the approach to only process the land-use cropland is lifted because using an external (coarser resolution) land use classification at this fine spatial resolution would lead to a predetermination of the result. The result of the threshold mapping method is a map containing Boolean information of the status of the field: irrigated or not irrigated. We derive scaling relation of irrigation extent vs. spatial resolution in three different regions: central Sudan around Khartoum, in northwestern China in the Uighur province Xinjiang and in Colorado, southeast of Denver (Figure 1). These three regions were selected based on the following criteria:  
* The region's agricultural suitability is low due to rainfall deficit to avoid both confusion between irrigated and rain-fed areas and high cloud cover. * The region should be dominated by irrigated agriculture. * The selected regions should cover a broad range of agricultural systems--from subsistence to high-intensity agriculture. Figure 1: Global overview of the selected regions including the Sentinel\(-\)2 tile name. All three regions are characterized by low annual precipitation values. The region in the USA (303 mm/year) is the wettest, followed by Sudan (202 mm/year) and China (173 mm/year), based on the ERA5 data of the year 2016 [34]. Each study area covers one Sentinel-2 tile of approx. 100 \(\times\) 100 km. According to [PERSON] et al. [35], the field sizes in the three tiles ranges from \"very small\" to \"very large\". The field sizes in the USA are categorized as \"large\", in China as \"medium\" and in Sudan from \"very small\" to \"small\". A visual pre-analysis shows that the sizes and shapes of the fields in China and Sudan vary strongly whereas the fields in the USA are homogeneous and only differ in shape: squared or the typical circular fields shaped by center pivot irrigation. The cultivated crops range from alfalfa and cereals to groundnut and fruits. The area in the USA is characterized by alfalfa (66%) and maize (31%); the remaining agricultural areas are used mainly for fruits and vegetables [36]. The agricultural areas in Sudan are mainly used for the cultivation of groundnut (71%), cotton (8%) and millet (6%). The remaining area of 15% is used for the cultivation of crop types such as maize, cassava, beans, dates and fruits. In the selected China tile, mainly cotton (37%) and maize (32%) are cultivated. Permanent crops such as grapes (4%) and apple trees (3%) are also cultivated, as well as vegetables and fruits. The study of the scaling relations is carried out for the year 2016.