## 2 Basic Theory
### Theory of Sea Surface Infrared Calculations  
(1) Radiation from the sea surface target itself  
The radiant energy of a surface target is based on the temperature field and emissivity of the target surface. Therefore, after calculating the temperature field of the target surface, the surface IR flux of the target can be calculated from [PERSON]'s formula. The target surface can be assumed to be a diffuse gray body, and the spectral radiant flux of the target surface element \(i\) at a particular wavelength band is [17]  
\[E_{1}=\int_{\lambda_{1}}^{\lambda_{2}}\varepsilon_{1}\times\frac{C_{1}}{ \lambda^{5}\cdot\exp\Big{(}\frac{C_{2}}{\lambda\cdot T_{i}}-1\Big{)}}d\lambda \tag{5}\]  
where \(C_{1}\) is the first radiation constant, \(C_{1}=3.742\times 10^{-16}\) W\(\cdot\) m\({}^{2}\), \(C_{2}\) is the second radiation constant, \(C_{2}=1.439\times 10^{-2}\) m \(\cdot\) K, \(\lambda\) is the radiation wavelength, and \(T\) is the thermodynamic temperature. The \(\varepsilon_{1}\) is the emissivity of the target surface. The optimal launch rate of the ship at different times and speeds is about 0.8 and is taken as \(\varepsilon_{1}\) = 0.8 for the surface of the underwater vehicle [18].  
(2) Reflection of sky radiation by sea surface targets\[E_{2}=\rho_{t}(\lambda)\cdot\varepsilon_{sky}\sigma T_{a}^{4}\cdot\frac{1+\cos \sum}{2}\cdot\frac{\int_{\lambda_{1}}^{\lambda_{2}}\varepsilon_{1}\times\frac{ C_{1}}{\lambda^{5}\cdot\exp\left(\frac{C_{2}}{\pi T_{sky}}-1\right)}d\lambda}{ \sigma T_{sky}^{4}} \tag{6}\]  
where \(\rho\) is the average reflectance of the sea surface to solar radiation, \(\varepsilon_{sky}\) is the sky target surface emissivity, \(\sigma\) is the [PERSON] constant, \(T_{sky}\) is the equivalent medium temperature, \(T_{a}\) is the air temperature, and \(\sum\) is the inclination of the surface element of the target surface at sea. The horizontal surface \(\sum\) = 0.  
(3) Reflection of solar radiation by sea surface targets [19; 20]  
\[E_{3}=(1-\alpha_{sum})\cdot\frac{solar-heat-flux_{i}}{\alpha}\cdot\frac{\int_ {\lambda_{1}}^{\lambda_{2}}\varepsilon_{1}\times\frac{C_{1}}{\lambda^{5}\exp \left(\frac{C_{2}}{\pi T_{sun}}-1\right)}d\lambda}{\sigma T_{sun}^{4}} \tag{7}\]  
where \(solar-heat-flux_{i}\) is the density of solar heat flow received by the small surface element on the surface of the sea surface target, \(\alpha_{sum}\) is the absorption rate on the surface of the sea surface target, and \(T_{sum}\) is the solar temperature, taken as 5800 K. Again, as the calculation state in this paper is nighttime, this part does not need to be calculated.  
(4) Reflection of sea surface radiation by sea surface targets  
\[E_{4}=\rho_{i}\cdot\varepsilon_{sea}\sigma T_{sea}^{4}\frac{1-\cos\sum}{2} \cdot\frac{\int_{\lambda_{1}}^{\lambda_{2}}\varepsilon_{1}\times\frac{C_{1}}{ \lambda^{5}\cdot\exp\left(\frac{C_{2}}{\pi T_{sky}}-1\right)}d\lambda}{ \varepsilon_{sea}\sigma T_{sea}^{4}}\cos^{-1}\theta \tag{8}\]  
where \(\theta\) is the angle between the line connecting the detector and the center of the surface element and the normal of the surface element and \(\sum\) is the target surface inclination of the sea surface. The horizontal surface \(\sum=0\).  
(5) Infrared radiation characteristics model of a sea surface target  
Similarly, as the distance between the sea surface target and the detector studied in this paper was less than 1 km and the calculated weather environment is a typical bright night without the direct influence of clouds, frost, rain, and fog, ignoring the effect of atmospheric path radiation and atmospheric transmission attenuation, through the above analysis, the total spectral radiation of the sea surface target is expressed as Equation (9)  
\[E(\lambda)=E_{1}+E_{2}+E_{3}+E_{4} \tag{9}\]