# Research on Space-Based Visible Detection for Conical Space Targets
## 4 Determination of Threshold Caliber with Signal-to-Noise Ratio as Evaluation Criteria
When a visible detection system detects a space target, in order to effectively eliminate the interference of noise, it is usually required that the detection \(SNR\) is not less than the threshold \(SNR\) (\(SNR_{T}\)). \[SNR\geq SNR_{T}, \tag{9}\]  
The determination of \(SNR_{T}\) will play an important role in the selection of the entrance pupil caliber. Next, we will study the reasonable range of \(SNR_{T}\), based on the detection capability of the system. For the pixel in the image sensor, depending on whether it is imaging the target, there are two hypothetical situations, either that the pixel is not imaging target, or that the pixel is the imaging target. We assume that the detector noise follows a Gaussian distribution with a variance of \(\sigma_{n}^{2}\), while setting \(\overline{DN^{N}}\) to be the average gray value of the pixel not imaging target and \(\overline{DN^{S+N}}=\overline{DN^{S}}+\overline{DN^{N}}\) to be the average gray value of the pixel imaging target. Therefore, the detection probability and false alarm rate of space targets can be expressed as the following formulas:  
\[ \begin{split}& P_{d}=\frac{1}{2}\bigg{[}1-erf\bigg{(}\frac{DN_{th }-\overline{DN^{S+N}}}{\sqrt{2}\sigma_{n}}\bigg{)}\bigg{]},\\ & P_{fa}=\frac{1}{2}\bigg{[}1-erf\bigg{(}\frac{DN_{th}- \overline{DN}}{\sqrt{2}\sigma_{n}}\bigg{)}\bigg{]},\end{split}  \tag{10}\]  
where \(DN_{th}\) is expressed as the detection threshold, and \(erf(x)\) is expressed as the error function. Similarly, according to the definition of \(SNR\), we are able to obtain another representation of \(SNR\): \(SNR=\overline{DN^{s}}/\sigma_{n}\). By combining Formulas (10), the relationship between the detection probability, the false alarm rate and \(SNR\) is expressed as follows:  
\[erf^{-1}\Big{(}1-2P_{fa}\Big{)}+erf^{-1}(2P_{d}-1)=\frac{SNR}{\sqrt{2}}, \tag{11}\]  
We set the \(SNR\) to change from 1 to 10, and simulate the change of detection probability when the false alarm rate values are \"10\({}^{-2}\)\", \"10\({}^{-4}\)\", \"10\({}^{-6}\)\", \"10\({}^{-8}\)\" and \"10\({}^{-10}\)\", respectively. The simulation result is shown in Figure 5.  
The simulation findings reveal that increasing the \(SNR\) increases detection probability, and the greater the false alarm rate, the higher the rate of \(SNR\) growth. As a result, if the system's detection probability is 97% and the false alarm rate is 0.01%, the system's SNR must be at least 5.6. The \(SNR_{T}\) is set to 5.6 to fulfill the criteria of the aforementioned detection capability indicators. According to the Johnson criterion, the system can detect the target when it is imaging 2 pixels on the sensor front; it can roughly classify the target when imaging 3-5 pixels; it can identify the target when imaging 5-10 pixels; and it can identify and confirm the target when the image is larger than 15 pixels. We assume that the system can detect the target at the limit detection distance, and, thus, the number of pixels that can roughly classify the target is set to four. Simultaneous Formulas (6)-(9) can deduce the caliber expression form of the detection system:  
\[D\geq\sqrt{\frac{2 hcn}{\pi\overline{\lambda}e\eta t_{0}\tau_{0}}\times\left( \frac{SNR_{T}}{\frac{I}{L^{2}}}\right)^{2}\times\left(\frac{I}{L^{2}}+E_{B} \right)\times\left(1+\sqrt{1+\left(D_{e}+R_{e}\right)\left(\frac{I}{L^{2}} \frac{2}{SNR_{T}\left(\frac{I}{L^{2}}+E_{B}\right)}\right)^{2}}\right)}, \tag{12}\]  
We take the SBV sensor on the MSX satellite as an example for verification and analysis. The MSX satellite has an orbital height of 908 km. If the SBV sensor wants to meet the staring observation of the target on the geosynchronous orbit, the working distance of the detection system needs to reach 35,000 km, which is the limit detection distance. The SBV sensor's parameters were detailed in Table 2.