## 2 Material and Methods
### Sampling Design Method Considering Both Good Spatial Coverage and Feature Space Coverage
#### 2.2.3 Determination of the Sampling Locations in Each Super-Grid  
The SOM content can influence crop growth and fertile accumulation within one super-grid [19]. We do not have a priori knowledge of SOM content before designing soil sampling. To determine the representative sample locations, some remote sensing productions that are highly correlated with the target soil variable were used to help discern representative sample locations. Many studies have shown high correlations between SOM and yield [65; 66; 67]. The relationship between SOM and yield can be restricted by the interacting factors related to management, climate, and soil type. However, the management of Sheltara farm involves mechanized unified crop cultivation, irrigation and fertilization, and this paper selected one natural field to validate the proposed method for farm field-level soil mapping. The tested field only grows one crop and its soil type is chemozem. It is believed that the climate, temperature, and disaster conditions are similar in a limited field space (i.e., super-grid). Thus, the SOM-yield relationship may have been less affected by the interacting factors related to management, climate, and soil type within one super-grid. In general, high SOM content results in high crop yield, whereas low SOM content leads to low crop yield within one super-grid. Hence, the spatial variation of the crop yield reflect the spatial variation of the farm's SOM content for a limited field space, since the differences in the soil physio-chemical properties (texture, drainage, SOC, carbonates, etc.) should be minimal in limited field space. The sampling locations could be determined based on the remote sensing crop yield data due to the high correlation between the crop yield and SOM content. First, the k-means algorithm [20; 68; 69; 70] was applied to the environmental variable of the remote sensing crop yield data to generate clusters within each super-grid. The number of clusters was equal to the number of samples within the super-grid. Note that if the number of samples was equal to 1, the operation of k-means clustering would be avoided.  
Second, the approach of determining the sample locations within each super-grid could be defined using the following rules. In the case of the number of sampling points being equal to 1, the mean (M1) and median (M2) of the crop yields in each super-grid are calculated, and then the mean (M3) of the crop yields between M1 and M2 is calculated. The candidate sampling point with a crop yield that is closest to M3 within the super-grid is recognized as the sampling point. In the case that the number of sampling points is greater than 1, the variance (v) of the crop yields of all the candidate sampling points within the super-grid is calculated using Equation (2). Then, the desired variance (DV) is calculated using the following formula:  
\[\mathrm{DV}=\frac{\mathrm{v}}{\mathrm{N}}\cdot\mathrm{N1} \tag{5}\]  
where N is the number of sampling points within the super-grid, and N1 is the number of the candidate sampling points within the super-grid. Next, one candidate sampling point is selected from each cluster to create the sampling point groups within the super-grid, and the sampling point group with a crop yield variance that is closest to the DV is regarded as the sampling point of the super-grid. The sampling design of the farm field soil is the output after determination of the sample locations within each super-grid.