# Insights into Segmentation Methods Applied to Remote Sensing SAR Images for Wet Snow Detection
## 2 Data and Method
### BERA Bulletins
#### 2.4.2 Scores
By calculating snow probability curves from Sentinel-1  
\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. By calculating snow probability curves from Sentinel-1  
\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:  
\[h=\frac{b+c}{n} \tag{9}\]  
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:  
\[FAR=\frac{b}{a+b} \tag{10}\]  
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:  
\[HR=\frac{a}{a+c} \tag{11}\]  
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.