## 2 Material and Methods
### Sampling Design Method Considering Both Good Spatial Coverage and Feature Space Coverage
#### 2.2.2 Determination of the Number of Sampling Points in Each Super-Grid  
To achieve good coverage of feature space, we believed that more sampling points were required if the variation of the environment variables in one local field region was very high. This paper combined the four local variation information of the DEM, slope, aspect, and crop yield to assess the variation of the target soil attribute. First, the values of these four environmental variables were extracted using candidate sampling points using the Spatial Analyst Tools in ArcGIS 10.2. Then, a standardization procedure was applied to the environmental variables using the following formula:  
\[\overline{\overline{X_{i}}}=\frac{X_{i}-X_{i\_\_\min}}{X_{i\_\_\max}-X_{i\_\_ \min}} \tag{1}\]  
where \(X_{i}\) is environmental variable i and \(X_{i\_\_\min}\) and \(X_{i\_\_\max}\) are the lowest and highest values of environmental variable i, respectively. Next, the variances of the normalized environmental variables within one super-grid were calculated as follows:  
\[\mathrm{v_{i}}=\frac{\sum\limits_{j=1}^{n}\left(y_{ij}-\overline{y_{i}} \right)^{2}}{n} \tag{2}\]  
where \(y_{ij}\) is one of the values of environmental variable i within one super-grid, \(\overline{y}_{i}\) is the mean of the values of environmental variable i within the super-grid, and n is the number of candidate sampling points within the super-grid. Finally, the diversity index (DI) of the super-grid was calculated to assess the variation of the target soil attribute as follows:  
\[\mathrm{DI}=\sqrt{\sum\limits_{i=1}^{n}\mathrm{v_{i}^{2}}} \tag{3}\]  
where \(v_{i}\) is the variances of normalized environmental variable i within one super-grid. The area of the candidate sampling region in the super-grid is also the non-negligible element for determining the number of samples in the super-grid. In this paper, the number of candidate sampling points in the super-grid was regarded as the area (A). Then, the combination of DI and A was considered to calculate the sample size (\(N_{k}\)) in each super-grid, as follows:  
\[\mathrm{N_{k}}=\mathrm{m}\cdot\frac{\mathrm{A_{k}}\cdot\mathrm{DI_{k}}}{\sum \limits_{k=1}^{n}\mathrm{A_{k}}\cdot\mathrm{DI_{k}}} \tag{4}\]  
where m is the sample size discussed in Section 2.2.1, and \(A_{k}\) and \(DI_{k}\) represent the area and diversity index of super-grid k, respectively. Assume that one sampling point is enough to obtain the spatial distribution of the SOM content within one farm field and that one farm has 1000 fields. One sampling point is far from enough to achieve the reliable spatial distribution of the whole farm's SOM content. More sampling points are required for larger areas. In addition, two regions with the same area have different spatial variations of their SOM contents. The region with a high SOM content variation was called Region1, and the other was called Region2. Region1 required more sampling points than Region2 to obtain satisfactory soil mapping results. The high target variable variation corresponded to a high DI. Thus, more sampling points are necessary for a large area and high target variable variation, wherein this paper used the product of \(A_{k}\) and \(DI_{k}\) as the weight to determine the number of samples for super-grid k. Note that the number of sampling points in each super-grid may not be an integer when using Equation (4). In this situation, the number of sampling points is rounded up and should satisfy the condition that at least one sampling point is recognized within one super-grid. Then, the number of sampling points for all the super-grids are summed. If the result is greater than the given sample size (i.e., 36 sampling points), then 36 sampling points are randomly selected from the samplingdesign after the sampling locations are determined (the detailed information is given in Section 2.2.3).