# Remote Sensing of the Water Quality Parameters for a Shallow Dam Reservoir
## 2 Data, Methods, and Techniques
### Model
Intensity of radiation \(I_{chl}\), \(I_{m}\) reflected from water at different depths is defined as the depth integration from the derivative of the radiation intensity at different depths, with respect to the depth defined by (5); intensity \(I_{0}\) must be reduced by the intensity of the radiation reflected from the water surface \(\alpha_{s}I_{0}\). Therefore:  
\[ \begin{array}{l}I_{chl}=-\frac{h}{0}\frac{d(\alpha_{chl}C_{chl}(I_{0}- \alpha_{s}I_{0})\mathbf{exp}(-k\cdot C\cdot 2l))}{dl}dl=\\ \alpha_{chl}C_{chl}(I_{0}-\alpha_{s}I_{0})(1-\mathbf{exp}(-k\cdot C\cdot 2 h))\end{array}  \tag{8}\]  
\[ \begin{array}{l}I_{m}=-\frac{h}{0}\frac{d(\alpha_{m}C_{m}(I_{0}-\alpha_{s}I_ {0})\mathbf{exp}(-k\cdot C\cdot 2l))}{dl}dl=\\ \alpha_{m}C_{m}(I_{0}-\alpha_{s}I_{0})(1-\mathbf{exp}(-k\cdot C\cdot 2 h))\end{array}  \tag{9}\]  
\(C_{chl}\)--concentration of chlorophyll a [mg/m\({}^{3}\)];  
Figure 11: Images of Lake Dobczyce at four wavelengths (color representation: blue, yellow, violet, with the same color saturation in each photo). \(\alpha_{chl}\)--coefficient of the radiation reflectance for chlorophyll a [m\({}^{3}\)/mg];  
\(C_{m}\)--concentration of non-chlorophyll a substances in water responsible for light reflection [g/m\({}^{3}\)];  
\(\alpha_{m}\)--coefficient of the radiation reflectance for substances other than chlorophyll a [m\({}^{3}\)/g]. If h and/or C are sufficiently high, then \(\exp(-k\cdot C\cdot 2h)<<1\) and:  
\[I_{chl}\approx\alpha_{chl}C_{chl}(I_{0}-\alpha_{s}I_{0})\ \Rightarrow\ R_{chl}\approx\alpha_{chl}C_{chl}(1-\alpha_{s}) \tag{10}\]  
\[I_{m}\approx\alpha_{m}C_{m}(I_{0}-\alpha_{s}I_{0})\ \Rightarrow\ R_{m}\approx\alpha_{m}C_{m}(1-\alpha_{s}) \tag{11}\]  
In model (4), the \(R\)... reflectance for different wavelengths is the computational reflectance \(R_{calc}\), which are formally sums \(R_{chl,\...}+R_{m,\...}\) for different wavelengths. Therefore, each reflectance in model (4) should be corrected and replaced with the computational reflectance for different wavelengths:  
\[ \begin{array}{l}R_{...}\gets R_{calc,\...}=R_{chl,\...}+R_{m,\...}=\\ R_{--}-\alpha_{s,\...}-\alpha_{b,\...}(1-\alpha_{s,\...})\exp(-k_{--} \cdot C\cdot 2h)\end{array}  \tag{12}\]  
\(\Box\)... --the index dots refer to different wavelengths;  
\(R\)... --radiation reflectance for different wavelengths determined from satellite;  
\(R_{calc,\...}\)--computational reflectance for different wavelengths. The coefficient \(\alpha_{s}\) (or reflectance) for different wavelengths is small if compared to other reflectance, and therefore should be neglected, so:  
\[ \begin{array}{l}R_{...}\gets R_{calc,\...}=R_{chl,\...}+R_{m,\...}=\\ R_{--}-\alpha_{b,\...}\exp(-k_{--}\cdot C\cdot 2h)\end{array}  \tag{13}\]  
A reflectance correction (13) in Equation (4) gives the equation describing the concentration of chlorophyll a as:  
\[ \begin{array}{l}C_{chl}=\exp[a_{0}+a_{3}\ln(R_{665}-\alpha_{b,720 mu665}\exp (-k_{665}\cdot C\cdot 2h))+\\ a_{4}\ln(R_{705}-\alpha_{b,720 mu705}\exp(-k_{705}\cdot C\cdot 2h))+\\ a_{5}\ln R_{740}+a_{7}\ln R_{842}+\\ a_{11}(\ln(R_{665}-\alpha_{b,720 mu665}\exp(-k_{665}\cdot C\cdot 2h)))^{2}+\\ a_{12}(\ln(R_{705}-\alpha_{b,720 mu705}\exp(-k_{705}\cdot C\cdot 2h)))^{2}+\\ a_{13}(\ln R_{740})^{2}+a_{15}(\ln R_{842})^{2}]\end{array}  \tag{14}\]  
\(C_{chl}\)--chlorophyll a concentration [mg/m\({}^{3}\)];  
\(\exp(a_{0})\)--model's coefficient [mg/m\({}^{3}\)]. Assume the following parameter values for Equations (7)-(9) relating to one wavelength: \(\alpha_{s}\) = 0.002, \(\alpha_{m}\) = 0.001, \(\alpha_{b}\) = 0.3, \(\alpha_{chl}\) = 0.001, \(k\) = 0.05 (NTU\(\cdot\)m)\({}^{-1}\), \(C_{m}\) = \(C\) \(\approx\) 5 NTU (in practice, water turbidity is assumed as C) and \(C_{chl}\) = {1, 5, 10, 20, 50} mg/m\({}^{3}\). From the Equations (7)-(9), the reflectance R can be determined, as a function of the depth h (Figure 12). Knowing R and the parameters of Equations (7)-(9), the concentration of chlorophyll a \(C_{chl}\) can be again determined.