# A Hybrid Bio-Optical Transformation for Satellite Bathymetry Modeling Using Sentinel-2 Imagery
## 2 Materials and Methods
### Methodology--Model Development
#### 2.3.4 Computation of Bio-Optical Parameters  
The adopted approach in this research took into account the local conditions of the study region with the aim of classifying the image according to its bio-optical water parameters. In order to model these water properties within the study area, the approach of [PERSON] et al. [23] was adopted. Firstly, the back-scattering coefficient (\(b_{bp}(\lambda)\)) related to suspended particles in the water column is computed using Equation (8); its application is based on the fact that since the tested area is entirely coastal, then the water optical properties are wavelength dependent. Hence, the most representative reference wavelength (\(\lambda_{\rm o}\)) for the areas covered mainly by coastal suspended sediments is the Sentinel-2A band B4 at 665 nm:  
\[b_{bp}(B4)\approx\frac{\alpha_{w}(B4)}{0.05}\star R_{rs}^{BOA}(B4) \tag{8}\]  
where \(\alpha_{w}\)(B4) = \(\alpha\)(665) = 0.428 m\({}^{-1}\) is the absorption coefficient of pure seawater given by [PERSON] et al. [24], and \(R_{rs}^{BOA}(B4)\) is the corrected Bottom-of-Atmosphere remote sensing reflectance of the Sentinel-2A red band (B4) over water areas. Knowing the sea surface reflectance \(R_{rs}^{BOA}(B_{i})\) from the multispectral imagery, it is possible to model the remote sensing reflectance below the sea surface (\(r_{rs}(\lambda)\)) (Equation (9)) for Sentinel-2A bands B2, B3, and B4 (at 10 m resolution):  
\[r_{rs}(\lambda)=\frac{R_{rs}^{BOA}(\lambda)}{0.52+1.7\star R_{rs}^{BOA}( \lambda)} \tag{9}\]  
The subsurface reflectance (\(r_{rs}(\lambda)\)) is a parameter of the \(u\)-ratio (Equation (10)), which in turn is a function of water IOPs, namely, the back-scattering coefficient (\(b_{b}(\lambda)\)) and the absorption coefficient (\(\alpha(\lambda)\)):  
\[u(\lambda)=\frac{b_{b}(\lambda)}{b_{b}(\lambda)+\alpha(\lambda)}=\frac{-0.089 5+\sqrt{\left(0.0895\right)^{2}+4\ast 0.1247\ast r_{rs}(\lambda)}}{2\ast 0.1247} \tag{10}\]  
The \(u\)(B2), \(u\)(B3), and \(u\)(B4) ratios are then calculated in order to connect the AOPs (i.e., reflectances \(K_{d}(\lambda)\)) with the IOPs (i.e., \(b_{b}(\lambda)\) and \(\alpha(\lambda)\) coefficients) in the equivalent spectral bands of the Sentinel-2A satellite. Subsequently, the power parameter \(Y\) is estimated (Equation (11)), which usually takes values between 0.1 and 2.5:  
\[Y=2.2\star\left[1-1.2\ast exp\left(-0.9\ast\frac{r_{rs}(B1)}{r_{rs}(B4)} \right)\right] \tag{11}\]  
where \(r_{rs}\)(B1) and \(r_{rs}\)(B4) are the subsurface remote sensing reflectance for bands (B1) and (B4), respectively. The total backscattering coefficient \(b_{b}(\lambda)\) (Equation (12)) can then be computed using the following equation:  
\[b_{b}(\lambda)=b_{bw}(\lambda)+b_{bp}(\lambda)=b_{bw}(\lambda)+b_{bp}(B4) \star\left(\frac{B4}{\lambda}\right)^{Y} \tag{12}\]  
In Equation (12), the backscattering coefficient (\(b_{bp}\)(B4)) of the red band due to suspended particles is determined by Equation (8), the power factor \(Y\) is provided by Equation (11), and the backscattering coefficient of pure seawater (\(b_{bw}(\lambda)\)) is obtained by interpolation from [PERSON] measurements ([PERSON], 1974, [25]. The wavelength (\(\lambda\)) in Equation (9) corresponds to the wavelengths of B2 and B3 Sentinel-2A spectral bands at 490 and 560 nm, respectively. Next, the total absorption coefficient \(\alpha(\lambda)\) is computed using Equation (10) with Equation (13):  
\[u(\lambda)=\frac{b_{b}(\lambda)}{b_{b}(\lambda)+\alpha(\lambda)} \Leftrightarrow\alpha(\lambda)=\frac{(1-u(\lambda))*b_{b}(\lambda)}{u(\lambda)} \tag{13}\]  
where the total backscattering coefficient \(b_{b}(\lambda)\) is given by Equation (12).