## 2 Material and Methods
### Sampling Design Method Considering Both Good Spatial Coverage and Feature Space Coverage
#### 2.2.1 Determination of the Minimum Sampling Unit, Sample Size, and Super-Grid Size  
First, the minimum sampling unit was determined. In this paper, a grid size of \(20\times 20\) m\({}^{2}\) was considered as the minimum sampling unit. Then, the fishnet with the minimum sampling unit was created using the Data Management Tools in ArcGIS 10.2, and the center of each fishnet was recognized as a candidate sampling point. The number of candidate sampling points in this tested field was 1759. Second, the limited sample size was determined. This paper set two percent of the number of candidate sampling points as the limited sample size, i.e., 36 sampling points. Finally, the super-grid size was determined. Note that the super-grid refers to one square region in which the length of its sides is larger than the interval between two candidate sampling points, and it is at least the double interval. To ensure good spatial coverage, one or more sampling points were required to be recognized in each super-grid. Thus, the number of super-grids could not exceed the sample size. In this study, the number of super-grids was between one and 36. In other words, the super-grid size ranged from \(140\times 140\) m\({}^{2}\) to \(800\times 800\) m\({}^{2}\). To analyze the influence of super-grid size on sampling design, this paper compared the sampling design results using super-grid sizes of \(200\times 200\) m\({}^{2}\), \(300\times 300\) m\({}^{2}\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{6}{c}{**The Prediction Errors of the SOM Mapping of the Whole Farm**} \\ \hline
**Mean** & **Root-Mean-Square** & **Mean Standardized** & **Root-Mean-Square** & **Average Standard Error** \\
**(g/kg)** & **(g/kg)** & **(g/kg)** & **Standardized (g/kg)** & **(g/kg)** \\ \hline
0.08 & 5.42 & 0.02 & 0.79 & 6.79 \\ \hline \multicolumn{6}{c}{**The Descriptive Statistics of the Tested Farm Field SOM Map**} \\ \hline
**SOM** & **Min (g/kg)** & **Max (g/kg)** & **Median (g/kg)** & **Mean (g/kg)** & **Standard Deviation** \\
**Content** & **Min (g/kg)** & **Max (g/kg)** & **Median (g/kg)** & **Mean (g/kg)** & **(g/kg)** \\ \hline Tested field & 42.86 & 44.75 & 43.94 & 43.89 & 0.45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The prediction errors of the soil organic matter (SOM) mapping of the whole farm and the descriptive statistics of the tested farm field SOM map.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Spatial Resolution** & **Time** \\ \hline Soil organic matter (SOM) map & Raster, 16 m & 2018 \\ Digital elevation model (DEM) & Raster, 16 m & 2009 \\ Slope & Raster, 16 m & 2009 \\ Aspect & Raster, 16 m & 2009 \\ Remote sensing crop yield data & Raster, 16 m & 2018 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The dataset characteristics of the tested field.  
\(400\times 400\) m\({}^{2}\), \(500\times 500\) m\({}^{2}\), \(600\times 600\) m\({}^{2}\), and \(700\times 700\) m\({}^{2}\). The approach to create a super-grid was the same as creating the minimum sampling unit, as shown in Figure 3.  
Figure 2: The schematic of the proposed sampling design method for farm field-level soil mapping.