## 3 Results and Discussion
### Correlation of SAR Variables with Soybean Yield  
When correlating SAR variables derived from the processing of backscatter coefficients (DPSVI, RFDI, RVI, Pol, _VH_, and _VV_), we observed some indices with correlations equal to 1 and others below 0.2. With _VV_, except in the third development stage of the first growing harvest (Table 6), the correlations with DPSVI, RFDI, RVI, and Pol were above 0.8, while _VH_ correlations did not exceed 0.6. Additionally, studies by [PERSON] et al. [47] and [PERSON] et al. [48] show that _VH_ works better when plants do not grow straight up, such as maize, and are not affected by the direction the rows are planted. Given that soybean is an herbaceous plant, _VH_ is more suitable than _VV_ for this type of crop, and _VH_ showed a lower correlation with other SAR variables, while its correlation with yield remained within a similar range to _VV_. The SAR vegetation indices (RFDI, Pol, and RVI) showed an almost perfect correlation (r - 1), whereas with DPSVI, the correlation was frequently below 0.70. The RFDI, Pol, and RVI indices were developed for monitoring plant growth, crop classification, and land use but were not initially designed for agricultural applications [41; 47; 49; 50]. Thus, they may lack sensitivity to identify variability within the same agricultural crop. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Harvest 1—Stage 1**} & \multicolumn{10}{c}{**Harvest 2—Stage 1**} \\ \hline DPSVI & \(-0.71\) & \(0.71\) & \(-0.71\) & \(-0.22\) & \(-0.97\) & DPSVI & \(-0.72\) & \(-0.72\) & \(-0.35\) & \(-0.97\) \\ - & Pol & \(-1\) & \(1\) & \(-0.47\) & 0.86 & - & Pol & \(-1\) & \(1\) & \(-0.36\) & \(0.86\) \\ - & - & RFDI & \(-1\) & \(0.47\) & \(-0.86\) & - & - & RFDI & \(-1\) & \(0.36\) & \(-0.86\) \\ - & - & - & RVI & \(-0.47\) & 0.86 & - & - & - & RVI & \(-0.36\) & \(0.86\) \\ - & - & - & - & _VH_ & 0.021 & - & - & - & _VH_ & 0.13 \\ - & - & - & - & - & _VV_ & - & - & **-** & **-** & **-** & _VV_ \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Spearman correlation coefficient between SAR variables and vegetation indices derived from backscatter coefficients. Significant correlations at 5%. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Harvest 1**} \\ \hline Boxcar & 1 & - & - & - \\ Gamma & 0.999 & 1 & - & - \\ Lee & 0.999 & 0.999 & 1 & - \\ Refined & 0.914 & 0.907 & 0.911 & 1 \\ \hline \hline \multicolumn{4}{c}{**Harvest 2**} \\ \hline Boxcar & 1 & - & - & - \\ Gamma & 0.999 & 1 & - & - \\ Lee & 0.999 & 0.999 & 1 & - \\ Refined & 0.914 & 0.911 & 0.913 & 1 \\ \hline \hline \multicolumn{4}{c}{**Harvest 3**} \\ \hline Boxcar & 2 & - & - & - \\ Gamma & 0.999 & 1 & - & - \\ Lee & 1 & 0.999 & 1 & - \\ Refined & 0.830 & 0.907 & 0.911 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Spearman correlation coefficient between the tested filters based on _VH_ polarization backscatter coefficient values combined with SAR images across the three growing harvests and three development stages. Significant correlations at 5%. For the variables generated from polarimetric decomposition (alpha and entropy), both showed a high correlation with each other (r > 0.85--Table 7) across all development stages of the crop. Entropy reflects the disorder of the reflected energy from a target, and for soybeans, its value increases as the crop develops (Figure 2), indicating the presence of multiple scattering events, which is expected in a crop canopy [14]. Therefore, entropy is more appropriate in this context, as it better captures variations in crop growth over time [51] and is not affected by the direction in which the rows are planted. To complement the information within a prediction algorithm, SAR variables with less than 80% correlation with each other were used. This approach improves prediction performance by including non-collinear variables in a predictive model [52].