# Research on Space-Based Visible Detection for Conical Space Targets
## 4 Determination of Threshold Caliber with Signal-to-Noise Ratio as Evaluation Criteria  
The visible detection system's signal-to-noise ratio (SNR) is a key image quality metric that influences the system's detection capacity. SNR will be affected by different detecting calibers. As a result, it is critical to investigate the method for estimating the detecting system's caliber using the SNR as a reference index. Setting the radiation intensity received by the detection system to \(I\), the distance between the target and the detection system to \(L\), the entrance pupil caliber of the detection system to \(D\), the system transmittance to \(\tau_{0}\), the quantum efficiency of the image sensor to \(\eta\), the pixel fill factor to \(\varepsilon\), the integration time to \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\sigma\) & \(\boldsymbol{\varphi}\) & **Case** & \(\boldsymbol{\Theta_{s}}(^{\circ})\) & \(\boldsymbol{\Theta_{d}}(^{\circ})\) & \(\boldsymbol{I_{\min}}(\text{W/sr})\) & \(\boldsymbol{I_{\max}}(\text{W/sr})\) \\ \hline \multirow{6}{*}{60\({}^{\circ}\)} & & 1 & 30.0000–30.2136 & 77.2608–90.0000 & 1016.3 & 2054.0 \\  & & 3 & 77.2608–90.0000 & 30.0000–30.2136 & 805.1 & 1613.4 \\  & & 4 & 30.0000–65.8539 & 30.0000–65.8539 & 1035.3 & 3638.1 \\ \cline{2-8}  & & 1 & 30.0000–30.0947 & 81.4156–90.0000 & 689.4 & 1393.7 \\  & & 3 & 81.4156–90.0000 & 30.0000–30.9497 & 554.6 & 1096.1 \\  & & 4 & 30.0000–73.2604 & 30.0000–73.2604 & 616.3 & 3440.7 \\ \cline{2-8}  & & 1 & 30.0000–30.0236 & 85.6795–90.0000 & 350.3 & 704.4 \\  & & 3 & 85.6795–90.0000 & 30.0000–30.0236 & 297.2 & 565.9 \\  & & 4 & 30.0000–81.4156 & 30.0000–81.4156 & 266.6 & 3245.9 \\ \hline \multirow{6}{*}{72\({}^{\circ}\)} & & 1 & 18.0001–18.1452 & 68.1138–90.0000 & 1227.3 & 2354.2 \\  & & 3 & 68.1138–90.0000 & 18.0001–18.1452 & 972.3 & 1849.9 \\  & & 4 & 18.0001–51.4605 & 18.0001–51.4605 & 1484.4 & 2708.0 \\ \cline{2-8}  & & 1 & 18.0001–18.0643 & 74.9846–90.0000 & 831.9 & 1640.2 \\ \cline{2-8}  & 80\({}^{\circ}\) & 3 & 74.9846–90.0000 & 18.0001–18.0643 & 669.2 & 1290.4 \\  & & 4 & 18.0001–61.8784 & 18.0001–61.8784 & 1020 & 2561 \\ \cline{2-8}  & & 1 & 18.0001–18.0160 & 82.3539–90.0000 & 422.5 & 843.6 \\ \cline{2-8}  & 85\({}^{\circ}\) & 3 & 82.3539–90.0000 & 18.0001–18.0160 & 358.4 & 677.9 \\  & & 4 & 18.0001–74.9846 & 18.0001–74.9846 & 496.5 & 2416.0 \\ \hline \end{tabular}
\end{table} 
Table 1: The extreme values of radiation intensity of space targets received by detectors under different conditions. Figure 4: The minimum values of radiation intensity of space targets received by detectors under different conditions. \(t_{0}\), the center wavelength to \(\overline{\lambda}\), and the number of pixels occupied by target imaging to \(n\), the number of signal electrons generated by the system during the integration time \(t_{0}\) is expressed as follows:  
\[S_{e}=\frac{I}{L^{2}}\times\frac{\pi D^{2}\overline{\lambda}\eta\varepsilon t_{0} \tau_{0}}{4 hctn}, \tag{6}\]  
Photon noise, dark current noise, readout noise, and deep space background noise are all common causes of noise in the detecting system. The variance of the overall system noise is equal to the sum of the variances of all noise contributions, since each noise is independent of the others. The system's \(SNR\) may be calculated using the following definition of \(SNR\):  
\[SNR=\frac{S_{e}}{\sqrt{S_{e}+D_{e}+R_{e}+B_{e}}}, \tag{7}\]  
where \(D_{e}\) is the number of electrons of dark current noise, \(R_{e}\) is the number of electrons of readout noise, and \(B_{e}\) is the number of electrons generated in the deep space background. The sky is a smooth backdrop whose irradiance outside the Earth's atmosphere is about similar to a 23 rd magnitude star, and most of the detection background of the space-based surveillance system is a deep space background. As a result, the total number of electrons produced in the deep space background may be calculated as follows:  
\[B_{e}=\frac{E_{B}L^{2}}{I}S_{e}, \tag{8}\]  
where \(E_{B}\) is the irradiance generated by the deep space background.