# Remote Sensing of the Water Quality Parameters for a Shallow Dam Reservoir
## 2 Data, Methods, and Techniques
### Model  
To compute concentrations of chlorophyll a (C\({}_{\rm chl}\)) from reflectance R obtained from satellite images for different wavelengths, a mathematical relationship has to be developed that meets the rules for logical concentration values for extremely high or low reflectance values. Empirical models, which are a linear combination of partial formulas, are risky to use because they cannot determine model parameters for all possible R values. The risk arises from a finite number of observational data, which usually does not include information on extreme values of R. Therefore, it may happen that C\({}_{\rm chl}\) calculated from the R values obtained from the new photos will be illogical, or even negative. Generally, extrapolation of such a model beyond the R values used to determine the model parameters yields highly ambiguous results; that is why models originating from the theory of dimensional analysis are considered to be safer. These types of models are usually the product of function modules at the appropriate power. They may also generate poor results outside of the R values used in the model estimation, but at least they guarantee that the results are positive. The initial model was defined as:  
\[C_{chl}=\alpha_{0}R_{443}^{a_{1}}R_{490}^{a_{2}}R_{560}^{a_{3}}R_{665}^{a_{4} }R_{705}^{a_{5}}R_{740}^{a_{6}}R_{783}^{a_{6}}R_{842}^{a_{6}}R_{865}^{a_{9}}R_{ 945}^{a_{10}}R_{1375}^{a_{11}}R_{1610}^{a_{12}}R_{2190}^{a_{13}} \tag{1}\]  
where:  
\(\alpha_{0}\) to \(\alpha_{13}\)--model coefficients;  
\(R\)....--radiation reflectance for wavelengths: 443, 490, 560, 665, 705, 740, 783, 842, 865, 945, 1375, 1610, and 2190 nm, determined from the satellite data; and  
\(C_{chl}\)--concentration of chlorophyll a [mg/m\({}^{3}\)]. The logarithmic Equation (1) leads to a linear relationship for the logarithms of reflectance R. The spectral curves of chlorophyll a (Figure 1) and the suspensions (Figures 2-4) show that the reflectance for wavelengths over 865 nm and shorter than 490 nm will not provide significant information on the concentration of chlorophyll a. Therefore, the logarithmic form of Equation (1), after simplification, takes the form:  
\[ \begin{array}{l}{\bf{ln}}C_{chl}=a_{0}+a_{1}{\bf{ln}}R_{490}+a_{2}{\bf{ln}}R_ {560}+a_{3}{\bf{ln}}R_{665}+a_{4}{\bf{ln}}R_{705}+\\ a_{5}{\bf{ln}}R_{740}+a_{6}{\bf{ln}}R_{783}+a_{7}{\bf{ln}}R_{842}+a_{8}{\bf{ln} }R_{865}\end{array}  \tag{2}\]  
Figure 10: Spectral curve of chlorophyll in water (7.609 mg chl/m\({}^{3}\)), USGS Spectral Library Version 7, [[https://crustal.usgs.gov/speclab/QueryAll07a.php?quick_filter=water](https://crustal.usgs.gov/speclab/QueryAll07a.php?quick_filter=water)]([https://crustal.usgs.gov/speclab/QueryAll07a.php?quick_filter=water](https://crustal.usgs.gov/speclab/QueryAll07a.php?quick_filter=water)) (accessed on 1 January 2022). Reprinted with permission from [39]. where:  
\(a_{0}\) to \(a_{8}\)--model coefficients,  
\(R\ldots\) --radiation reflectance for wavelengths: 490, 560, 665, 705, 740, 783, 842, and 865 nm, determined from the satellite data. To make the model more general, additional components, like the squares of the reflectance log, were introduced:  
\[ \begin{array}{l}\mathbf{ln}C_{chl}=a_{0}+a_{1}\mathbf{ln}R_{490}+a_{2} \mathbf{ln}R_{650}+a_{3}\mathbf{ln}R_{665}+a_{4}\mathbf{ln}R_{705}+\\ a_{5}\mathbf{ln}R_{740}+a_{6}\mathbf{ln}R_{783}+a_{7}\mathbf{ln}R_{842}+a_{8} \mathbf{ln}R_{865}+a_{9}(\mathbf{ln}R_{490})^{2}\\ a_{10}(\mathbf{ln}R_{560})^{2}+a_{11}(\mathbf{ln}R_{665})^{2}+a_{12}(\mathbf{ ln}R_{705})^{2}+a_{13}(\mathbf{ln}R_{740})^{2}+\\ a_{14}(\mathbf{ln}R_{783})^{2}+a_{15}(\mathbf{ln}R_{842})^{2}+a_{16}(\mathbf{ ln}R_{865})^{2}\end{array}  \tag{3}\]  
The regression analysis of Equation (3) enabled the selection of such coefficients (from \(a_{1}\) to \(a_{16}\)) that showed a probability lower than the significance coefficient 0.05 in the Student's t-distribution (significance test of the equation coefficients).