# Remote Sensing of the Water Quality Parameters for a Shallow Dam Reservoir
## 3 Results
### Models Parameters for Turbidity  
The turbidity C model was developed similarly to the chlorophyll a model. At first, it was determined which factors from the range of \(a_{0}\)-\(a_{16}\) are significant for turbidity in Equation (3). Then, the significance test was used for the \(a_{1}\)-\(a_{16}\) coefficients at a significance level of 0.05 using the Student's t-distribution. It was found that the most important coefficients are related to the wavelengths: 705, 740, and 783 nm; these lengths approximately correspond to the local minimum and maximum on the spectral curve of suspensions (Figures 2-4). The initial form of the equation was as follows:  
\[\begin{array}{l}\mathbf{In}C=a_{C,0}+a_{C,4}\mathbf{In}R_{705}+a_{C,5} \mathbf{In}R_{740}+a_{C,6}\mathbf{In}R_{783}+\\ a_{C,12}\left(\mathbf{In}R_{705}\right)^{2}+a_{C,13}\left(\mathbf{In}R_{740} \right)^{2}+a_{C,14}\left(\mathbf{In}R_{783}\right)^{2}\end{array} \tag{16}\]  
\(C\)--turbidity [NTU];  
\(a\)....--model coefficients;  
\(R\)....--radiation reflectance for wavelengths: 705, 740, and 783 nm.  
Taking into account the reflectance of the lake bottom leads to a relationship:  
\[\begin{array}{l}C=\mathbf{exp}\quad a_{C,0}+a_{C,4}\mathbf{In}\left(R_{705} -a_{C,b,705}\mathbf{exp}(-k_{C,705}\text{\text{\text@underline{C}}}\cdot 2h) \right)+\\ a_{C,5}\mathbf{In}R_{740}+a_{C,6}\mathbf{In}R_{783}+\\ a_{C,12}\left(\mathbf{In}\left(R_{705}-a_{C,b,705}\mathbf{exp}\left(-k_{C,705} \text{\text{\text@underline{C}}}\cdot 2h\right)\right)\right)^{2}+\\ a_{C,13}\left(\mathbf{In}\mathbf{\text{\text@underline{R}}}_{740}\right)^{2} +a_{C,14}\left(\mathbf{In}R_{783}\right)^{2}\end{array} \tag{17}\]  
Figure 16: Chlorophyll concentration in Lake Dobczyce 9 May2021 (imagery by Sentinel 2).  
Figure 17: Two-dimensional field of the average vertical velocity in the main part of Lake Dobczyce (without the northern bay), with the total flow of 10 m\({}^{3}\)/s and no-wind conditions (model RMA2). Reprinted with permission from [40].  
The effect of light reflection by the lake bottom is shown in Figure 11. It appears that it is sufficient to consider the reflectance correction for 705 nm to obtain a satisfactory accuracy of model (17). Of course, it is also possible to correct the reflectance for 740 nm and 783 nm, but the overall improvement of the model quality would be negligible.  
Equation (17) is implicit due to turbidity C. The C value can be obtained by successive approximations, i.e., inserting in \(\textbf{exp}\left(-k_{C,705}\textbf{-}\mathcal{C}\textbf{-}2\mathcal{H}\right)\) values C from the previous approximation; after several attempts (e.g., 4), the value C becomes reasonably accurate. Another method involves searching for the function's zero, i.e., the difference of the right side of Equation (17) and C. Here, the regula falsi method of searching for the function's zero can be used.  
Based on the measurement data of: turbidity C, lake depth h, and the reflectance \(R\)... for wavelengths 705, 740, and 783 nm, the parameters of model (17) were determined. Due to the large number of parameters:  
\[a_{C,0}\;,\;a_{CA}\;,\;a_{C,b,\;705}\;,\;a_{C,5}\;,\;a_{C,6}\;,\;a_{C,12}\;,\; a_{C,13}\;,\;a_{C,14}\]  
the two-stage procedure was employed. First, the values of some parameters were initially estimated: \(a_{C,0}\;,\;a_{CA}\;,\;a_{C,5}\;,\;a_{C,6}\;,\;a_{C,12}\;,\;a_{C,13}\;,\;a_{C,14}\), and then all the others were estimated, while correcting the values of the pre-estimated parameters. The parameters were determined using the least squares method (the best fit) for turbidity C and turbidity from model (17). The model parameters are summarized in Table 3. The correlation coefficient was 0.939, and the model fit is shown in Figure 19. A good fit of the model to the measurement data was obtained.