A novel detection performance modular evaluation metric of space-based infrared system

In order to reflect the space-based full chain information of the detection process comprehensively and objectively, we proposed a novel modular evaluation metric to discuss the target, background, and system independently. It takes the equivalent radiation intensity as the parameter, which can evaluate the detection performance of the system quantitatively. In this paper, taking the fifth-generation American stealth fighter F22 as an example, the mathematical detection model of the space-based infrared system to aircraft targets in the Earth background is described. A modular evaluation metric is proposed. The simulation analyzes the impact of different detection scenes and system parameters on system equivalent irradiance. Furthermore, recommendations for the optimization of the detection system are given. The research results provide a new idea for the analysis of the detection performance of highly maneuverable targets under dynamic backgrounds and have guiding significance for the performance evaluation and parameter design of the infrared detection system.


Introduction
Space-based infrared imaging platform is an effective method to achieve wide-area and high-time-sensitive detection of high maneuverability air targets such as stealth aircrafts. Compared with ground-based and air-based platforms, space-based remote sensing platforms have unique advantages such as wide-area detection, continuous monitoring, short atmospheric paths, and good detection direction. When a space-based detection platform detects air targets such as a stealth aircraft, target radiation, earth background radiation, and detection system parameters will all contribute to its detection performance (Rao and Mahulikar 2005).
Existing evaluation methods use a single evaluation index, which is difficult to reflect the full chains including dynamic target, complex background and different system parameters objectively and comprehensively. The signal-to clutter ratio (SCR) and signal-to-noise ratio (SNR) are the main indicators to evaluate the ability of space-based detection systems to detect targets. (Huang et al. 2012;Zhang et al. 2021;Hu et al. 2019;Zavvari et al. 2015;Wang et al. 2018;Ma et al. 2021) Caroline, S. used SCR as a criterion to analyze the detectability of missiles in different infrared bands and discussed the impact of different observation scenes and spatial resolution on SCR (Schweitzer et al. 2012). Yuan H calculated the local SCR at the aircraft plume position in three bands to analyze the detectability under the sea/cloud background from space-based platform (Yuan et al. 2019b;Yuan et al. 2020). Lv, W. P. used SNR as the evaluation standard to discuss the effect of several kinds of interference on the detectability of stealth aircraft (Lv and Li 2011). In air-based and ground-based detection, the maximum detectable distance is usually used to evaluate the system's detectability of air targets (Fei et al. 2014;Shen et al. 2017;Wang et al. 2011). Although the above evaluation metrics can reflect the detection performance of the system to targets, they cannot directly tell the influence of infrared radiation characteristics of the target, the background, and the system performance.
In order to reflect the space-based full chain information of the detection process comprehensively and objectively, we proposed modular evaluation metric to discuss the target, background and system independently. It takes the equivalent radiation intensity as parameters, which can evaluate the detection capability of the system quantitatively. Taking F22 as an example, the mathematical detection model of the space-based infrared system to aircraft targets in the Earth background is described. A modular evaluation metric is proposed. The simulation analyzes the impact of different detection scenes and system parameters on SEI. Furthermore, recommendations for optimization of the system are given. The research results provide a new idea for the analysis of the detection performance of highly maneuverable targets under dynamic backgrounds, and have guiding significance for the performance evaluation and parameter design of infrared detecting system.

Radiation signature of the aircraft and background
In general, the total infrared radiation signature from the aircraft can be attributed to skin infrared radiation and plume infrared radiation, both of which meet Planck's law. In addition, the solar radiation reflected by aircraft skin also affects its radiation intensity, especially at SWIR (Wei and Honghu 2015).
Therefore, the spectral infrared radiant intensity of the aircraft skin can be expressed as where is the wavelength, skin is the spectral infrared emissivity of the skin, T skin_ave is the average skin temperature, A skin is the projected area of the skin to the detector array. The infrared radiation intensity of the aircraft skin is related to the aerodynamic heating effect of the surrounding atmosphere environment, while the impact of the sun, sky and Earth radiation can be ignored (Mahulikar et al. 2009). In addition, the stealth aircraft incorporate specific surface heat dissipation materials to exchange heat with the surrounding low-temperature environment during flight, which reduces the average skin temperature (Sun and Wang 2019).
In summary, the average aircraft skin temperature can be expressed as where T atm is the ambient atmospheric temperature of the aircraft, is the temperature recovery coefficient, is the specific heat capacity ratio, v aircraft is the flight velocity of the aircraft, C is a constant ( C≈0.84) (Li et al. 2015). The high-temperature tail flame ejected from the aircraft exhaust nozzle is a mixture of many gaseous species produced by the combustion of hydrocarbon fuel, mainly CO 2 and H 2 O (Retief et al. 2011).
The tail flame is a selective radiator, and its infrared emissivity changes sharply with the change of wavelength, as shown in Fig. 1a (Yuan et al. 2019a;Rao and Mahulikar 2013;Yuan et al. 2019b).
Accurately simulating the temperature distribution of the aircraft plume is the main factor in studying the radiation characteristics of the plume. Retief, S. J. P. et al. proposed a radiation inversion technique that uses the aircraft plume radiation records to construct a three-dimensional radiation model of the plume in flight, which can be obtained projected plume temperature distribution at any observation angle (Retief 2012;Retief et al. 2014). Figure 1b is the temperature distribution diagram of the tail flame with length L and width W when viewed perpendicular to the plane of the aircraft. The scattered red high-temperature areas in the figure are the Mach disks generated during supersonic flight of the aircraft.
The plume temperature distribution diagram is divided into two-dimensional grids, with M × N pixels, and the temperature of each grid is T i (Ni et al. 2021). The plume spectral radiant intensity I plume can be acquired through the superposition method and expressed as where the projected area of each grid A grid can be expressed as In addition to thermal radiation, the aircraft skin also reflects solar radiation, which is mainly diffuse reflection due to the aircraft's special surface which is designed to not be detectable against the earth background.
Considering the change in the location of the sun, the radiation caused by the reflection of stealth aircraft to the sun can be expressed as where r diff is the spectral infrared reflectivity of skin, S sun ( , ) is the spectral solar irradiance to the aircraft. sun ( ) is the spectral atmospheric transmittance of the sun to the aircraft, A ref is the effective area of the aircraft skin reflecting solar radiation. Figure 2 is a schematic diagram of the space-based infrared detection system detecting aircraft targets and the background of the earth. Aircraft target radiation, including the skin's own radiation I skin , the skin's reflected radiation to the sun I reflect , and the tail flame's radiation I plume , will be attenuated during atmospheric transmission, but at the same time, there will be radiation I path from the atmospheric path, so the aircraft target reaches the system. The total radiation at the entrance pupil can be expressed as Besides targets, the system can collect the infrared radiation of the earth background where L q,back is the background radiance, which is simulated by MODTRAN3.7, S is the system spatial resolution.

Model of infrared detection system
The aircraft is considered a point source because the target solid angular subtense is much less than the detector solid angular subtense. As shown in Fig. 3, only part of the signal falls on the detector. This fraction is the point visibility factor (PVF) (Holst 2017). SNR is the primary index describing the performance of the system for aircraft detection. In infrared imaging systems, it is defined as the ratio of the difference between the target and background response electrons to the noise electrons. The expression of signal-to-noise ratio is where A aircraft is projected area of the skin to the detector array, A opt is the entrance pupil area, opt is the optical transmittance, is the quantum efficiency, T int is the integration time, R is the distance between the space-based system and the target.
noise is the number of noise electrons in the system, which can be expressed as where N tar and N dark contribute quantum noise, n read is the readout circuit noise of the detector.
(6) I q,aircraft = 2 ∫ 1 hc ((I skin ( ) + I plume ( ) + I reflect ( )) ⋅ atm + I path ( ))d Fig. 3 Aircraft target detection schematic in Earth background In space-based detection, stealth aircraft targets are usually considered as point targets, their area is much smaller than the entire pixel area, so the signal-to-noise ratio can be approximately expressed as where the number of target response electrons N aircraft can be expressed as It must be recognized that the integration time of the system is limited by three factors. First, the system response electron number cannot be oversaturated during the integration time, and second, the target flight distance of the aircraft cannot be greater than the ground resolution during the integration time. Finally, considering that the angular velocity of the low-orbit satellite operating at the sub-satellite point is much greater than the earth's rotation angular velocity, the travel distance of the sub-satellite point within the integration time cannot be greater than the ground resolution. Based on the above analysis, the integration time needs to satisfy the following formula.
where N full is full well capacity, v aircraft is the aircraft flying velocity, v sub is the subastral point velocity of the satellite. S is the system spatial resolution, expressed as S = pix ⋅ R∕f , as shown in Fig. 3.

Modular evaluation metric
Based on the radiation characteristics of the aircraft target, the earth background and the performance parameters of the space-based infrared detection system, a modular evaluation method is proposed to realize the target-background-system separation discussion, and intuitively reflect the influence of various factors on the detection performance.
(1) Noise equivalent irradiance (NEI) is used as an index to evaluate system performance, and its value is only related to the parameters of the space-based detection system. For fixed instrument parameters, NEI remains unchanged when the integration time is constant, (2) Clutter equivalent irradiance (CEI) is used as an index to evaluate background clutter noise, which reflects the radiation intensity of the detection background. For a fixed system, CEI remains unchanged under the same background, where is the clutter coefficient. The spatial noise brought by the atmospheric background clutter is easily mistaken for the target in the scene. Considering temporal noise brought by the fluctuation of radiation from the target and the background, is set to 0.95 .
(3) Since the response of the detection system mainly comes from the detection background and system noise, the system equivalent irradiance (SEI) is proposed to characterize the sensitivity of the detection system, The smaller value of the SEI means the higher sensitivity of the detection system.
(4) Combined with Eq. (12), SNR can be expressed as Therefore, the minimum target radiation intensity that can be detected can be calculated where TNR is the threshold signal-to-noise ratio.
In summary, when the system parameters and the detection background are unchanged, the detectability of the target can be quickly judged by Eq. (20). And by comparing the values of CEI and NEI to judge the main factors that affect the system's target detection efficiency. Figure 4 is a flowchart of the method,

Typical condition analysis
The background radiation of the space-based infrared system for earth observation includes the self-heat radiation of the earth's surface and atmospheric path, the reflected radiation of the earth's surface to the sun, and the scattered radiation of the ambient atmosphere to the sun. When the atmosphere is covered by clouds, the self-radiation of the clouds and their reflected radiation to the sun should also be considered. In this Fig. 4 Flow chart of the modular evaluation metric paper, three scenes with different seasons and cloud backgrounds in mid-latitude daytime are selected as typical scenes for simulation analysis, as shown in Table 1, and the irradiation of them is shown in Fig. 5.
The parameters of the detection system are shown in Table 2, According to the expressions given in Sect. 2, The NEI, CEI and SEI of three typical scenes are shown in Table 3.  Generally, the point targets are considered to be detectable when SNR > 6. According to Eq. (20), the target can be detected under the three scenes when its radiation intensity reaches 8.61 W/sr, 11.55 W/sr and 63.20 W/sr.
The fifth-generation American stealth fighter F22 is taken as a typical target for analysis. The structure parameters of F22 are shown in Table 4 The structural parameters of F22. The fuselage is flying parallel to the sea. Considering the F22's strong infrared stealth capability, this paper sets the infrared emissivity of the skin to 0.5, and the diffuse reflectance of the fuselage is 0.2 (Li et al. 2015;Baranwal and Mahulikar 2015;Cha et al. 2014;Pan et al. 2015;Wu et al. 2020).
Taking the afterburner and non-afterburner state of the aircraft in flight into account, the maximum temperature range of the aircraft tail flame is about 1000 ~ 1500 K. The characteristic temperature is 1300 K (Hang et al. 2019;Cha et al. 2014;Pan et al. 2015).
Considering the flight condition of the F22 (Wu et al. 2020;Jian et al. 2012), simulate the radiation intensity of the aircraft at the entrance pupil when F22 is in typical flight state at different altitudes.
The calculation results of the entrance pupil radiation intensity of the aircraft at different altitudes are shown in Fig. 6. The reference lines represent the minimum radiation intensity of the target that can be detected in different scenes. The figure intuitively reflects the detectable heights in different scenes.

Influence of system parameters and optimization suggestions
As a parameter for evaluating system performance, the smaller NEI means better system performance. Combining the analysis with Eq. (16), if you want to improve the performance of the detection system, it can be achieved by reducing the readout noise n read , increasing the optical aperture A opt , extending integration time T int or increasing the optical transmittance opt and quantum efficiency . Figure 7 shows the NEI, CEI and SEI in three scenes with different resolutions. As can be seen from Fig. 7 (a, b), for Scene I and Scene II, according to Eq. (14), the integration time is limited by the ground resolution and subsatellite point velocity now. When the ground resolution is improved, the NEI becomes larger and the CEI becomes smaller. In Scene I, under different ground resolutions, the main index affecting the detection sensitivity of the system is different. The improvement of ground resolution increases the SEI value and worsens the sensitivity of the system; In Scene II, the background radiation intensity is about 4 times that of Scene I. At this time, the background is the main factor affecting detection performance. When the ground resolution is 240 m, the SEI value is the smallest, and the system sensitivity is the best; For Scene III, the background radiation intensity is about 40 times that of Scene I, and the integration time is limited by the background. At this time, the background is the main factor affecting the detection performance. Improving the ground resolution can effectively enhance the system sensitivity.
Reducing readout noise n read can effectively decrease SEI and improve system sensitivity (as shown in Fig. 8), but for Scene III, background suppression is the primary consideration to improve detection efficiency, which can be solved through spectral segment optimization.
For Scene III, the background radiation intensity is about 40 times that of Scene I. At this time, due to the high background intensity, the system integration time is limited, and the NEI varies with the resolution. Increase the optical aperture, NEI and CEI will not change, the detection efficiency of the system is difficult to improve. At this time, background suppression is the primary consideration for improving detection performance, which can be solved by spectrum optimization.
Characterizing the detection performance of the system through the modular evaluation methods of NEI, CEI, and SEI can intuitively reflect the influence of different factors on the detection performance, which is helpful to put forward reasonable system improvement suggestions.

Conclusion
The development of a generalized and modular evaluation method is of great significance for realizing the simulation analysis of target detection effectiveness in complex backgrounds. This article takes the stealth aircraft as an example to analyze and establish a modular evaluation index system that discusses target-background-system independently.
This method realizes the rapid analysis of the detection system's target detection performance, intuitively reflects the full chain information in the space-based detection and has important reference significance for the simulation research of dynamic targets such as stealth aircraft and the design and performance evaluation of space-based infrared detection systems.
Funding National Natural Science Foundation of China (61975222).
Code availability All codes are fully available without restriction.

Declarations
Competing interests The authors have not disclosed any competing interests.