## 2 Materials and Methods
### GBDT Model  
The Gradient-Boosting Decision Tree (GBDT) algorithm was proposed by [PERSON] in 1999 [PERSON] (1999). The algorithm restricts weak learners from using only the Classification and Regression Tree (CART) model, a widely used model for constructing decision trees for both classification and regression problems. When building a regression tree using the CART model, the feature selection index generally uses the node minimum sample variance. The larger the sample variance, the greater the node data scatter with low purity. The CART branches through the variance threshold of each node. When all the variance is lower than the threshold value of each node, or after reaching the set stop conditions, the CART decision tree is completed. In the GBDT algorithm, the CART decision tree is associated \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Dates**} & **Site** & \multirow{2}{*}{\begin{tabular}{c} **Concentration** \\ **(ng/L)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Relectance** \\ **B1** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Relectance** \\ **B2** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **B6** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **B7** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **B7** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **B8** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **B7** \\ \end{tabular} } \\
14 April 2015 & GX05 & 5.30 & 0.077 & 0.070 & 0.084 & 0.059 & 0.038 & 0.029 & 0.024 \\
23 October 2015 & GX04 & 2.80 & 0.085 & 0.075 & 0.083 & 0.054 & 0.030 & 0.012 & 0.006 \\
28 December 2016 & GX02 & 3.20 & 0.070 & 0.061 & 0.074 & 0.048 & 0.017 & 0.005 & 0.003 \\
14 February 2017 & GX02 & 2.00 & 0.078 & 0.068 & 0.070 & 0.040 & 0.023 & 0.008 & 0.005 \\
11 July 2018 & GX13 & 8.80 & 0.119 & 0.116 & 0.127 & 0.108 & 0.108 & 0.108 & 0.088 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Chl-a concentration and reflectance of some samples. Figure 3: Comparison before and after atmospheric correction. with the boosting algorithm. In general, the residuals are calculated and evaluated after each iteration and processed as the input by the next iteration, thus minimizing the loss function and improving the fitting accuracy of the model. When the residuals reach a lowest value, or the setting termination condition has been reached, the model is constructed, and the regression result will be exported. The specific step of the GBDT regression algorithm is shown in the following sequence of equations and in Figure 5.  
1. Initialize the cart learner:  
\[f_{0}(x)=argmin_{c}\sum_{i=1}^{n}L(y_{i},c) \tag{1}\]  
Figure 4: Reflectance boxplot of each band. Figure 5: Flow chart of the Gradient Descent Boosting Decision Tree algorithm. 2. In round \(t\), the negative gradient of each sample is calculated:  
\[r_{t,i}=-\left[\frac{\partial L(y_{i},f(x_{i}))}{\partial f(x_{i})}\right]_{f_{( x)}=f_{t-1}(x)} \tag{2}\]  
3. The CART regression tree \(T_{t}\) is obtained by fitting (\(x_{i}\), \(r_{t,i}\)), \(i=1,2,\ldots,m\), and the leaf node region is divided into \(R_{t,i}\), \(j=1,2,\ldots,J\);  
4. Traverse (referring to one visit to each node in the tree (or graph) along a certain search route) the node region, and calculate the output value of each leaf node \(R_{t}\), namely the best fitting value \(c_{t,i}\):  
\[c_{t,i}=argmin_{c}\sum_{x_{i}\in R_{t,i}}L(y_{i},f_{t-1}(x_{i})+c) \tag{3}\]  
5. Update the learner:  
\[f_{t}(x)=f_{t-1}(x)+\sum_{j=1}^{I}c_{t,j}I(x\in R) \tag{4}\]  
6. Repeat these steps until the termination condition is reached, and the final strong learner expression is obtained by adding the weak learners as follows:  
\[f(x)=f_{0}(x)+\sum_{t=1}^{T}\sum_{j=1}^{I}c_{t,j}I(x\in R) \tag{5}\]  
To avoid over- or underfitting of the model, the setting of the GBDT model was determined by grid searching with cross-validation. By 10-fold cross-validation, the method of cross validation used in this study, the dataset was separated into 10 subsamples.