# Insights into Segmentation Methods Applied to Remote Sensing SAR Images for Wet Snow Detection
## 2 Data and Method
### BERA Bulletins
#### 2.4.2 Scores  
To evaluate wet snow outputs from the segmentation methods, we compared them to total snow masks from optical measurements using several scores. These scores could measure disagreements between binary images, consider the distribution of pixels, and consider the presence of noise. The list of the used scores is as follows:  
* Correlation: This measures any possible statistical link between our two variables (total snow and wet snow). Correlations were computed using the following formula: \[r=\frac{\sum_{i=1}^{n}(S_{i}-\bar{S})(O_{i}-\bar{O})}{\sqrt{\sum_{i=1}^{n}(S_ {i}-\bar{S})^{2}}\sqrt{\sum_{i=1}^{n}(O_{i}-\bar{O})^{2}}}\] (3) where \(S\) is the binary wet snow product (output of segmentation), \(O\) is the binary total snow product from Sentinel-2, \(n\) is the number of pixels of \(S\) and \(O\)
* Contingency table: This was used to count the different possible combinations of pixels of the optical snow image and those of the \"predicted\" one (output of SAR segmentation). Hereafter, the following denotations apply: \(a\) the number of cases \"observed\" by optical data and predicted by the SAR segmentation method, \(b\) refers to the cases not observed and predicted, \(c\) refers to the cases observed and not predicted, \(d\) to the cases not observed and not predicted, and \(n\) is the total number of pixels. Figure 9: RGB SAR image composite using VH backscatters over the Belledonne massif (R: 20170825, G: 20180427, B: 20170825) showing areas with wet snow in magenta. These quantities are summarized in Table 1 and were used to obtain a variety of other scores listed below. \begin{table} \begin{tabular}{c c c c} \hline \hline
**Wet Snow by SAR** & \multicolumn{2}{c}{**Total Snow by**} \\  & & **Optical Image** & \\ \hline  & Yes & No & Total \\ \hline Yes & \(a\) & \(b\) & \(a+b\) \\ No & \(c\) & \(d\) & \(c+d\) \\ Total & \(a+c\) & \(b+d\) & \(a+b+c+d=n\) \\ \hline \hline \end{tabular}  a- Hamming Distance: This is a mathematical distance that corresponds to the proportion of pixels out of agreement between two binary images. It varies between 0 and 1, 0 being a perfect match between the binary images. The Hamming distance is calculated as follows:  
\[h=\frac{b+c}{n} \tag{4}\]  
b- False Alarm Rate (FAR): This score measures the proportion of false snow detection cases with respect to the number of snow pixels detected. It is defined by:  
\[FAR=\frac{b}{a+b} \tag{5}\]  
c- True Detection Rate (Hit Rate): This score measures the proportion of snow detection cases that are true with respect to the number of snow pixels observed. It is defined by:  
\[HR=\frac{a}{a+c} \tag{6}\]  
d- Heidke Skill Score (HSS): This score measures the benefit of SAR image segmentation compared to a random snow pixel distribution. It varies between \(-\infty\) and 1, with a negative value corresponding to a degraded detection compared to a random distribution, a null value corresponding to no benefit from the segmentation of the SAR images and a value of 1 being equivalent to perfect correspondence between the two images. HSS is defined as follows:  
\[HSS=\frac{2(ad-bc)}{(a+c)(c+d)+(a+b)(b+d)} \tag{7}\]  
* Structural Similarity Index (SSI): This score takes into account the variations, and the differences of structure in two images. It was built by [40], who pointed out that the Mean Square Error, commonly used to differentiate two images, does not take into account the similarities of structure between images. \(SSI\) takes the form \[SSI=l(S,O)\cdot c(S,O)\cdot s(S,O),\] (8) where \(l\) is based on mean images, \(c\) on standard deviation of images and \(s\) on correlations between images. * **Probabilistic Score for Satellite Products (PSSP):** The authors of [41] introduced this new score, which is particularly useful for assessing snowpack characteristics in mountainous areas, where factors such as altitude, slopes, and terrain orientation are critical. The PSSP score facilitates the integration of satellite products with varying spatial resolutions and types.