## 2 Material and Methods  
### Study Area and Dataset  
The study area was Sheltara Farm, which is located in the northeast part of the Nei Monggol Autonomous Region, China, where it covers an area of 595 km\({}^{2}\) (Figure 1). Sheltara Farm intensively exploits arable land, and commonly cultivated crops in this area include wheat, barley, rape, silymarin, and potatoes. Due to climate limitations, the area's cropping system is \"one crop a year.\" Crops are planted in May and harvested in September. The elevation ranges from 488 to 771 m, and the central and eastern areas have a relatively high relief. The climate is windy and rainless in spring, and it has concentrated precipitation in the summer, with a rapid cooling and early frost in the autumn, as well as a long duration of cold and snow cover in the winter. The average annual temperature ranges from \(-\)5.9 to 5.9 \({}^{\circ}\)C, and the average annual precipitation ranges from 125 to 542 mm.  
Farm field SOM content mapping was the main target of this paper, and the tested field was 70.2 ha in size. As a way to validate the proposed sampling design method, all the sampling and validation points were the extracted values from one SOM map during the autumn of 2018, which we used as the ground truth. The SOM map was obtained from the local soil experts of Sheltara Farm. Note that the local soil experts collected 883 soil samples over the whole area of Sheltara Farm in the autumn of 2018 to obtain the spatial distribution of the SOM content of the whole farm, and to form the cultivation plan for the next year. The SOM map of Sheltara Farm was produced using the ordinary kriging (OK) method and the 883 collected soil samples. The SOM content of these 883 soil samples ranged from 28.71 to 65.54 g/kg, and the mean and standard deviation were 44.59 g/kg and 7.38 g/kg, respectively. The prediction errors of the SOM mapping for the whole farm are displayed in Table 1. Subsequently, the SOM map of the tested farm field was clipped from the predicted SOM map of the whole farm, and it was used as the ground truth to test the proposed sampling design method. The descriptive statistics of the tested farm field SOM map are shown in Table 1.  
Figure 1: Sheltara farm (left) and the tested farm field (right) for validating the proposed method.  
The digital elevation model (DEM), slope, aspect, and crop yield could partly explain the spatial variation of the SOM content; thus, these four environmental variables were used to optimize the sample locations in the feature space. Table 2 displays the information on the SOM map taken in the autumn of 2018 and the four environment variables in detail. The DEM data with a 30 m resolution was downloaded from the Geospatial Data Cloud website ([[http://www.gscloud.cn/](http://www.gscloud.cn/)]([http://www.gscloud.cn/](http://www.gscloud.cn/))). For consistency with the resolution of the crop yield data, the DEM data was resampled to 16 m using the nearest neighbor assignment algorithm. The slope and aspect were calculated based on the resampled DEM data using the 3D Analyst Tools in ArcGIS 10.2. The crop yield data in 2018 was from the Aerospace Information Research Institute, Chinese Academy of Sciences. The remote sensing crop yield data was produced by Meng Group using the WOFOST Model [54].