# Underwater Communications for Video Surveillance Systems at 2.4 GHz
## 6 Mathematical Model  
Finally, using data shown in Section 5, it is possible to model the signal behavior as a function of the three parameters under study, i.e., temperature, modulation and working frequency. This Section is going to show these mathematical models. The represented temperature range is from 16 to 27 \({}^{\circ}\)C except to CCK modulation which only shows coherent values up to 22 \({}^{\circ}\)C. For each modulation, we have  
Thus, the distance between antennas can be expressed as a function of the working frequency and the water temperature (see Equation (5)). \[\mathrm{d}=\mathrm{F}\left(\mathrm{f},\mathrm{t}\right), \tag{5}\]  
We have used Eureqa Formulize [34] to estimate the mathematical expression. Eureqa is a scientific data mining software package that searches for mathematical patterns hidden in your data. To use it, we have to import data from a *.CSV. After that, we should select the variable to be studied. The next step is to select the kind of blocks and kind of expression the program can use to make the expressions. These can be addition, subtraction, modulus, floor, etc. Finally, the program gives us a list of possible mathematical expression with the error. The last step is to select, the equations in order to check which one match with our values and is able to calculate intermediate points. Firstly, we have used all obtained values to extract the mathematical model for underwater communications using BPSK modulation. Equation (6) relates the distance with the working frequency and the environmental temperature using BPSK modulation:  
\[\mathrm{d}=669.9\mathrm{f}+360.8\mathrm{t}+0.393\mathrm{ft}^{3}-3341-8.921 \mathrm{t}^{2}-7.834\mathrm{ft}^{2}-0.004985\mathrm{ft}^{4}, \tag{6}\]  
where d represents the distance in cm, f refers to the working frequency in GHz and t is the water temperature in \({}^{\circ}\)C. The equation presents a correlation coefficient of 0.8676 and its average absolute error is 1.036 cm. Using Equation (6), it is possible to estimate the maximum distances, as a function of the working frequency and temperature, between antennae when using BPSK modulation (see Figure 20). As we can see, the maximum distance is obtained for 23.5 \({}^{\circ}\)C working at 2412 MHz. In addition, we can see that for temperatures of 27 \({}^{\circ}\)C, the maximum reached distance is about 9 cm, meanwhile for temperature around 16 \({}^{\circ}\)C, the maximum distances is 21 cm for 2412 MHz and less than 19 cm. for 2457 MHz. Equation (7) relates the distance with the working frequency and the environmental temperature using QPSK modulation:  
\[\mathrm{d}=594.9\mathrm{f}^{2}+335.9\mathrm{t}+0.3719\mathrm{ft}^{3}-3024-8.42 6\mathrm{ft}^{2}-7.336\mathrm{ft}^{2}-0.00475\mathrm{ft}^{4}, \tag{7}\]  
where d is the achieved distance in cm, f represents the frequency in MHz and t refers to the temperature in \({}^{\circ}\)C. Equation (7) shows a correlation coefficient of 0.8831 and an average absolute error of 0.961 cm. Figure 21 shows, through Equation (7), the estimation of maximum distances as a function of the working frequency and temperature when the devices are using QPSK modulation. The maximum distance (26 cm) is obtained for 23.5 \({}^{\circ}\)C working at 2412 MHz. Furthermore, we can see that for  
Figure 20: Estimated maximum distances for BPSK modulation. temperatures of 27 cm, the maximum reached distance is about 7 cm and for temperatures of 16 \({}^{\circ}\)C, we can obtain distances between 18.8 cm to 21.5 cm. Working with OFDM, we can estimate values between 16 to 22 \({}^{\circ}\)C. Equation (8) relates the distance with the working frequency and the environmental temperature using QPSK modulation:  
\[