# Satellite-Derived Bathymetry for Selected Shallow Maltee Coastal Zones
## 3 Methods  
The intensity values of the ten spectral bands shown in Table 1 were extracted for each in situ vector point. The method proposed by [36] for deriving bathymetry from satellite imagery, which is based on a nonlinear solution and utilises band ratio calculations, was then used to compute the depth values. This method relies on the fact that the reflectance of two bands decrease exponentially with increasing depth [10]. The band with the shorter wavelength will attenuate less than the band with the longer wavelength. Thus, the ratio between the water reflectance values of the spectral bands should increase with depth [37]. Furthermore, this model reduces the effects caused by the dynamic substrate albedo in deriving accurate bathymetric data [2]. Changes in bottom albedo alters both band ratios, respectively, making the ratio substantially more sensitive to variations in depth rather than bottom albedo fluctuations [38]. When compared to the model presented by [PERSON] [39], this solution reduces the sensitivity to bottom changes and requires fewer empirical parameters to predict depth values. Most examples in the literature use the blue (490 nm) and the green (560 nm) wavelengths. However, in this study, to further investigate the best technique for SDB around the Maltese Islands, four other ratios were considered. These included the red wavelength (665 nm) and the infrared wavelength (842 nm). Such bands were chosen as they all have the same spatial resolution of 10 m. The empirical ratio model [36] is expressed by Equations (1) and (2). \[\mathrm{SDB}=\mathrm{m}_{1}(\mathrm{pSDB})-\mathrm{m}_{0} \tag{1}\]  
\[\mathrm{pSDB}=\frac{\mathrm{ln}(\mathrm{nR}_{\mathrm{w}}(\mathrm{\lambda}_{ \mathrm{i}}))}{\mathrm{ln}(\mathrm{nR}_{\mathrm{w}}(\mathrm{\lambda}_{\mathrm{ j}}))} \tag{2}\]  
Here, \(SDB\) is the Satellite-Derived Bathymetry in metres; \(pSDB\) is the dimensionless pseudo depth provided by the satellite. The constant \(n\) is fixed to 1000 to assure the logarithmic ratio with depth results in a linear response and that the ratio remains positive for all points. \(R_{\mathrm{w}}(\mathrm{\lambda}_{i})\) and \(R_{\mathrm{w}}(\mathrm{\lambda}_{j})\) are the water reflectance values for spectral bands \(i\) and \(j\), respectively. The scaling coefficient, \(m_{1}\), and the offset coefficient, \(m_{0}\), are derived through linear regression of the band ratios from a single satellite. Apart from generating the SDB values using traditional regression techniques, the applicability of machine learning methods was also investigated. A training set was generated from the raw band digital numbers and computed band ratios. The in situ depth measurements from the Golden Bay and the Ghajn Tuffieha embayments were used as the target values. Random forest, k-star, linear regression, and multilayer perceptron were used to build the models. Random forest (RF) is a machine learning technique that aggregates a large number of decision trees by sampling the dataset, randomly selects features of each subset to estimate the decision tree, and finally fuses the predicted outputs of each decision tree to enhance the accuracy of the modelled prediction [11]. The user must define two parameters to initialise \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sentinel-2 Bands** & **Central Wavelength (nm)** & **Resolution (m)** \\ \hline Band 2—Blue & 490 & 10 \\ Band 3—Green & 560 & 10 \\ Band 4—Red & 665 & 10 \\ Band 5—Vegetation Red Edge & 705 & 20 \\ Band 6—Vegetation Red Edge & 740 & 20 \\ Band 7—Vegetation Red Edge & 783 & 20 \\ Band 8—Near Infrared & 842 & 10 \\ Band 8—Vegetation Red Edge & 865 & 20 \\ Band 11—Shortwave Infrared & 1610 & 20 \\ Band 12—Shortwave Infrared & 2190 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spatial and spectral characteristics of the Sentinel-2 bands used in this study. the RF algorithm, the number of trees to grow, \(N\), and the number of variables used to split each node, \(m\).