## 3 Methodology  
The oil-spill classification is split into three subsequent steps: (1) dark, in SAR and optical data, and bright, in optical data, object detection; (2) feature extraction; and (3) oil-spill and look-alike separation. Below, in Figure 5, we detail how our method approaches these steps in order to detect oil slicks across three distinct datasets, including SAR and optical imagery). (1) The first task can be framed as a classification problem, distinguishing between the target class (oil-slick and look-alike) and the background (oil-free sea). The oil films on the sea surface are observed as dark objects in Sentinel-1/ICYE-X images (T1), dark and bright pixels (T2 and T3) on Sentinel-2/Landsat-8 images, and dark objects on Sentinel-3 images (T865 variable) (T4). This involves binary classification, with the goal of distinguishing the target (T1-T4) from the background (BG). One often-used method for this task is histogram thresholding, which involves selecting an appropriate threshold. This step has a direct impact on the classification outcomes [43]. Assuming both T and BG follow a Gaussian distribution, parametric methods involve estimating parameters of the following Gaussian mixture density [44], or Probability Density Function (PDF):  
\[PDF(x) =\omega_{T}PDF_{T}(x)+\omega_{BG}PDF_{BG}(x)\] \[=\omega_{T}\frac{1}{\sqrt{2\pi\delta_{T}^{2}}}e^{-\frac{(x-y_{T} )^{2}}{2\delta_{T}^{2}}}+\omega_{BG}\frac{1}{\sqrt{2\pi\delta_{BG}^{2}}}e^{- \frac{(x-y_{BG})^{2}}{2\delta_{BG}^{2}}} \tag{2}\]  
\(\omega_{T}\) and \(\omega_{BG}\) are weights of the two distributions, \(\mu_{T}\) and \(\mu_{BG}\) are means, and \(\delta_{T}^{2}\) and \(\delta_{BG}^{2}\) are standard deviations of two distributions, respectively. Figure 5: Flowchart of oil-slick detection from Sentinel-1/ICYE-X SAR, Sentinel-2/Landsat-8 optical, and Sentinel-3 visible/near-infrared data. The accuracy of classification is significantly impacted by the proportions of classes present in the scene and the degree of overlap between \(PDF_{T}\) and \(PDF_{BG}\). When there is a significant imbalance between T and BG, it becomes challenging to accurately parameterize their PDFs. The degree of overlap between the PDFs directly impacts the accuracy of under- and over-detection. To address the two mentioned limitations, we employed an adaptive threshold method that was previously established for mapping floodwater [33]. The technique consists of two key steps. The algorithm begins by parameterizing the \(PDF_{T}\) and \(PDF_{BG}\). It then iteratively uses thresholding and region-growing techniques to determine the optimal threshold for seeds (THS) and the threshold for stopping the region growth process (THSRG). The parameterization of the PDFs is carried out using HSBA [33], which finds specific regions, or tiles, of the image where the \(PDF_{T}\) and \(PDF_{BG}\) can be fitted with more reliability, meaning that the histogram of a specific tile is clearly bimodal. The size of the tiles is determined by the ability to parameterize the PDFs associated with two distinct classes. HSBA begins by analyzing the entire image and subsequently decreases the size of the tiles by a quad-tree decomposition of the image (Figure 6). In the second stage, spatial information is incorporated into the selection of the optimal threshold to mitigate the impact of class overlap on the final classification. The latter is achieved by a region-growing method, assuming that the pixels forming T are clustered rather than distributed randomly over the entire image. Thus, we initially categorize as T those pixels that exhibit a high probability of belonging to T. Subsequently, we include in T those pixels that have a lower degree of belonging to T but are spatially connected to the initial predictions. In order to accomplish this, we employ the region growing method, in which the \(PDF_{T}\) and \(PDF_{BG}\) will determine the selection of THS and THSRG. Figure 6: An example of the HSBA algorithm’s results for a Sentinel-1 SAR image, 5 July 2021, 02:23:25.