# Imaging Characteristics of Disturbance Flow Field Surrounding a Hypersonic Target
## 3 Results and Discussion  
### The Results of Imaging Blurring  
The initial position vector \(\mathbf{r}_{0}\) is fixed at the bottom of the disturbance flow field, and the incident rays are along the positive \(z\) axis. The phase difference after ray tracing is obtained by Equation (16). Figure 4 shows the distribution of the phase difference under different flight conditions.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline Orbit height (km) & 400 \\ Focal length (m) & 1.0 \\ Pixel size (\(\mu\)m) & 5 \\ Array size & \(1024\times 1024\) \\ Diameters of pupil (m) & 0.2 \\ Detect wavelength (nm) & 532 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The parameters of an ideal optical system for imaging.  
Figure 3: The schematic diagram of BOS.  
The phase difference exhibits a decreasing trend with an increasing flight altitude due to the thinning of the atmosphere at higher altitudes, resulting in a reduction of the refractive index and optical path difference of light. Similarly, as the Mach number progressively rises, the atmosphere experiences enhanced compression, leading to an augmentation in the phase difference. The phase difference is influenced by the density and refractive index distribution within the disturbance flow field. Specifically, a higher degree of fluctuation in the refractive index distribution along the propagation path corresponds to a larger optical path difference of the light. Remarkably larger values of the phase difference are symmetrically distributed downstream of the disturbance flow field, where the disturbance flow field extends extensively, consequently inducing larger optical path differences. The negative phase difference arises due to the comparatively lower atmospheric density in the wake of the target compared to free space, which aligns with the principles of fluid dynamics.  
Figures 5-7 show the results of imaging blurring, which were computed using Equation (21), against three typical backgrounds: desert, ocean, and city. In comparison to the corresponding original images Figures 5-7, the resultant images exhibit varying degrees of blurriness, and as the flight altitude increases, the background noise gradually diminishes. The stark contrast observed between the resultant images and their original counterparts significantly enhances the identification of space-based remote sensing images.  
Figure 4: The phase difference distribution of disturbance flow field: Refer to Table 1 for flight conditions from (**a**–**f**).  
Figure 5: The results of imaging blurring over desert.  
Figure 6: The results of imaging blurring over ocean.  
As presented in Table 3, the Peak Signal-to-Noise Ratio (PSNR) is computed for each image to provide a more accurate quantification of the degree of imaging blurring. The results reveal that across three typical backgrounds, the PSNR demonstrates an improvement with increasing flight altitude and Mach number. In reference to Figure 4, at a Mach number of 12, the phase difference attains its maximum value of 252.0 rad, accompanied by a wave aberration of 40.1\(\lambda\). However, the disturbance flow field area is found to be the smallest within the detection range, leading to results that contradict the phase difference trend. This discrepancy can be attributed to fluid dynamics, where the angle of the shock wave decreases as the Mach number increases, subsequently resulting in the smallest flow field area at 12 Ma within the detection range.