Prediction of Target Detection Probability Based on Air-to-Air Long-Range Scenarios in Anomalous Atmospheric Environments

Abstract

We investigate a target detection probability (TDP) using path loss of an airborne radar based on air-to-air scenarios in anomalous atmospheric and weather environments. In the process of calculating the TDP, it is necessary to obtain the overall path loss including the anomalous atmospheric environment, gas attenuation, rainfall attenuation, and beam scanning loss. The path loss including the quad-linear refractivity model and other radar input parameters is simulated using the Advanced Refractive Effects Prediction System (AREPS) software along the range and the altitude. For the gas and rainfall attenuations, ITU-R models are used to consider the weather environment. In addition, the radar beam scan loss and a radar cross section (RCS) of the target are considered to estimate the TDP of the airborne long-range radar. The TDP performance is examined by employing the threshold evaluations of the total path loss derived from the detectability factor and the free-space radar range equation. Finally, the TDPs are obtained by assuming various air-to-air scenarios for the airborne radar in anomalous atmospheric and weather environments.

1. Introduction

With the dramatic advances of radar system design technologies, the use of long-range airborne radar systems, such as synthetic aperture radars (SARs), airborne early warning (AEW) radars, and active electronically scanned array (AESA) radars, has been growing extensively [1,2,3,4,5,6]. Such airborne radar systems essentially require high-performance specifications for each component to increase the detection probability of long-range targets. However, the TDP can often be degraded by environmental or external factors including noises, clutter, atmospheric gas attenuation, multipath interference, atmospheric refraction, and rainfall attenuation. To be specific, an anomalous atmospheric refractive index including an unusual or abnormal distributions of temperature and relative humidity along the altitude can cause the refraction of the electromagnetic (EM) wave propagation different from that in normal conditions, i.e., sub-refraction, super-refraction, and ducting [7,8,9]. These wave refractions can mislead the accuracy of target position predictions, and in the worst case, long-range targets can be missed under anomalous atmospheric conditions. Therefore, it is important to model the detailed atmospheric refractive index in order for the precise predictions of the propagation path loss and propagation factor. Many studies have been conducted to model the refractive index under anomalous atmospheric conditions through the radar signal measurement [10,11,12], global positioning system tropospheric delay observation [13], and statistical analysis of the stored meteorological observatory data [14,15]. In addition, various studies have been conducted to analyze the wave propagation characteristics in consideration of low-altitude actual atmospheric data in ground-to-ground and ground-to-air scenarios. For example, the real atmospheric data of specific coastal areas in the UK [16], United States [17], Greece [18], China [19], and Korea [20] are used to calculate the long-range path loss using the propagation models, i.e., the parabolic equation model, ray optics model, waveguide mode model, and hybrid model [21,22,23,24,25]. These aforementioned studies have demonstrated the high accuracy of the path loss estimation in low-altitude situations. However, their path loss calculation processes need more in-depth consideration of the weather or atmospheric environment to observe the air-to-air airborne radar propagation characteristics at a high altitude over 5 km.

In this paper, we propose a TDP calculation process for the airborne long-range radar based on air-to-air scenarios in anomalous atmospheric environments. In the proposed process, it is necessary to obtain the overall path loss including the anomalous atmospheric environment, gas attenuation, rainfall attenuation, and beam scanning loss. To observe the airborne radar path loss in an anomalous atmospheric environment, the refractive index as a function of altitude is modeled using four linear lines to represent various wave refractions. Then, the path loss, including the refractivity model along the range and the altitude, is simulated using the AREPS software [26], which is a commercial software based on a hybrid model with a parabolic equation model and raytracing model. In this AREPS simulation, the radar antenna beam pattern and the digital terrain elevation data (DTED) are also employed as input parameters to accurately predict the airborne radar propagation. The 32 × 32 airborne radar array antenna with a triangular array configuration is used to calculate the radiation pattern, and the DTED of the southwest region in South Korea from the National Geographic Information Institute is employed. For gas and rainfall attenuations, ITU-R models are used to consider the weather environment in accordance with the water vapor pressure and precipitation. In addition, the radar beam scan loss and RCS of targets are considered to estimate the TDP of the airborne long-range radar. Finally, the TDPs are examined by assuming various air-to-air scenarios in anomalous atmospheric and weather environments.

2. TDP Simulation Process

2.1. Anomalous Atmospheric Refractivity and Measurement

Figure 1 shows a conceptual figure of the air-to-air airborne radar wave propagation in accordance with an anomalous atmospheric environment. The gray and red aircrafts indicate an airborne radar and a target, which are located at heights of h_r and h_t, having a distance of R_t (slant range) and a horizontal range of R_h. The target is placed under various atmospheric conditions according to the wave refraction. To observe the wave refraction of the airborne radar, the refractive index needs be calculated using the measured atmospheric data such as temperature, air pressure, and relative humidity based on Equations (1)–(3), which are usually obtained by employing a rawinsonde or a GPS sonde [27,28,29].

$$N=\left(n-1\right){10}^6=\frac{77.6p}{T}+\frac{e_{s}3.73\cdot {10}^5}{T^2}$$

(1)

$$e_s=\frac{rh6.105e^x}{100}$$

(2)

$$x=25.22\frac{T-273.2}{T}-5.31\ln \left(\frac{T}{273.2}\right)$$

(3)

where p, T, rh indicate air pressure in hPa, temperature in K, and relative humidity in %. Then, the modified refractive index M at a height of h needs to be calculated using Equation (4) [30]. An example for a calculation of the modified refractive index M based on the measured data from Heuksando Meteorological Observatory [31] is listed in

Table 1. Modified refractive index based on measured data (3 January 2017).

Height (m)	Air Pressure (hPa)	Temperature (°C)	Relative Humidity (%)	Refractivity (M-Unit)
80	1015	6	87	333.7
752.2	934.8	4.5	9.6	383.3
1477.2	855.1	4.7	14.5	476.7
2229.8	779.6	2.9	10.1	572.9
2999.6	708.2	–2.7	10.6	676.9
3793.1	640.3	–7.9	10.3	784.7

.

$$M=N+0.157h.$$

(4)

The gradient of the modified refractive index along the height ∇M can determine four common wave propagation refractions: normal refraction (79 < ∇M ≤ 157), super-refraction (0 < ∇M ≤ 79), sub-refraction (157 < ∇M), and ducting (∇M < 0). In accordance with these four refractions, the airborne radar wave propagation can become bent in the direction opposite or toward Earth’s surface, and it can be trapped in the ducting region. In particular, elevated ducts sometimes occur at high altitudes, which affect the wave propagation with a high path loss near the elevated ducting region. Thus, it is necessary to precisely model the refractivity of the elevated ducting in the air-to-air situation of the anomalous environment using a quad-linear model [32]. Figure 2a shows the conceptual figure of the quad-linear refractivity model. Four linear lines with slopes of m₁, m₂, m₃, and m₄ are used to model the refractivity, and the heights where the slope changes are h₁, h₂, and h₃. Figure 2b represents the modified refractivity according to height using the quad-linear refractivity model in comparison with the actual refractivity data from a meteorological observatory. The measured refractivity data is interpolated using the typical linear interpolation method due to the sparse data along the height. The solid and dashed lines indicate the comparisons between the actual data from Heuksando Meteorological Observatory [31] on 25 July 2017 and the quad-linear model.

The dotted and dash-dotted lines denote those on September 26, 2017. To model the actual refractive index on July 25, 2017, the slopes of m₁, m₂, m₃, and m₄ are 125.9, 875, −25.6, and 142. The heights of h₁, h₂, and h₃ are set to be 6223, 6239, and 6399 m. The second refractivity data on September 26, 2017, are also modeled using the quad-linear model (m₁ = 140.8, m₂ = −18.9, m₃ = 142.6, m₄ = 136.7, h₁ = 5680 m, h₂ = 5786 m, and h₃ = 6459 m). These quad-linear refractivity models have 2001 data points along the altitude, which are employed as input parameters for the AREPS software to obtain the path loss according to the range and altitude. Furthermore, the other input parameters of the antenna beam pattern and DTED are also applied to the AREPS simulation to precisely estimate the air-to-air wave propagation path loss. The antenna beam pattern is derived by calculating the array factor of a 32 × 32 airborne radar array antenna using a triangular array configuration [31]. The array distances along the x- and y-axes are 0.475λ and 0.538λ at 10 GHz. This radiation pattern has a half-power beamwidth of 2.85° and a side lobe level of 13.5 dB. The DTED of the southwest region in South Korea are obtained from the Korean National Geographic Information Institute [33]. Then, the path loss computed using the AREPS software can be defined as L_AREPS (m₁, m₂, m₃, m₄, h₁, h₂, h₃, R_h, h_r, h_t) considering the anomalous environment. For example, with specific input parameters (m₁ = m₂ = m₃ = m₄ = 85, h₁ = h₂ = h₃ = 15 km, R_h = 150 km, h_r = 11 km, and h_t = 5 km), we can obtain the path loss L_AREPS of 163.5 dB in the elevated ducting of the anomalous atmospheric environment.

2.2. Weather Environment Models

Figure 3 illustrates the attenuation of the wave propagation by the atmospheric gases and rainfall. Such weather conditions can have a serious impact on the path loss, which is directly related to the TDP performance of the airborne radar. Hence, it is necessary to have the detailed attenuation models to increase the accuracy of the TDP estimation in the air-to-air airborne radar wave propagation. The atmospheric gas attenuation can be modeled based on ITU-R P.676-12 as follows [34]:

$${\gamma }_g=0.1820f\left(N_{Ox}^{″}\left(f\right)+N_{Wat}^{″}\left(f\right)\right)$$

(5)

$$N_{Ox}^{″}\left(f\right)={\displaystyle \sum _i{S}_{i,ox}{F}_i+N_D^{″}\left(f\right)}$$

(6)

$$N_{Wat}^{″}\left(f\right)={\displaystyle \sum _i{S}_{i,Wat}{F}_i}$$

(7)

$$S_{i,ox}=a_1{10}^{-7}{\left(\frac{300}{T}\right)}^3\exp \left[a_2\left(1-\frac{300}{T}\right)\right]P$$

(8)

$$S_{i,Wat}=b_1{10}^{-1}{\left(\frac{300}{T}\right)}^{3.5}\exp \left[b_2\left(1-\frac{300}{T}\right)\right]e$$

(9)

$$L_g={\gamma }_g{R}_t$$

(10)

where γ_g is the specific gas attenuation in dB/km, and f is the frequency in GHz. N^″_Ox and N^″_Wat are the imaginary parts of complex refractivities in terms of oxygen and water vapor. S_i,Ox and S_i,Wat are the strengths of the ith oxygen and water vapor lines, and N^″_D(f) is the dry continuum. P and e are dry air and water vapor partial pressures in hPa, and T is the absolute temperature in K. F_i is the oxygen or water vapor line shape. a₁, a₂, b₁, and b₂ are the line strength coefficients. The detailed coefficient values are listed in tables in [34]. Thus, the gas attenuation according to the range R_t can be defined as L_g in Equation (7).

In addition, the rainfall attenuation can be modeled based on ITU-R P.530-17, P.837-7, and P.838-3 as follows [35,36,37]:

$${\gamma }_r=\kappa {\rho }^{\alpha }$$

(11)

$$r=\frac{1}{0.477R_t^{0.633}{\rho }_{0.01}^{0.0173\alpha }{f}^{0.123}-10.579\left(1-\exp \left[-0.024R_t\right]\right)}$$

(12)

$$d_{eff}=r{R}_t$$

(13)

$$L_r={\gamma }_r{d}_{eff}$$

(14)

where γ_r is the specific rainfall attenuation in dB/km, and ρ is the rainfall precipitation in mm/h. α and κ are the rainfall attenuation parameters that depend on the frequency, polarization state, and angle of the signal path to the target. d_eff is the effective propagation distance considering the target range R_t multiplied by the scale factor of r. The rainfall attenuation in terms of the range can be defined as L_r in Equation (11).

In general, the precipitation is measured using rain gauge, weather radar, and satellite, and the measured weather information can be employed to calculate the rainfall attenuation [38,39,40,41,42,43]. To verify these attenuation models, we have investigated the attenuation measurements [41] and then some of the measurements according to the frequency with different precipitations are compared to the ITU-R attenuation models, as shown in Figure 4a. Furthermore, Figure 4b depicts the attenuation results compared with those of the measurement in accordance with the precipitation [42,43]. These comparison results demonstrate that the ITU-R model agrees well with the measurement, and this model can be adopted to increase the accuracy of the TDP in real atmospheric and weather environments. To predict the TDP along the range, we calculate atmospheric gas and rainfall attenuation results according to the range from 0 to 190 km, as presented in Figure 4c. The solid and dashed lines indicate the atmospheric gas attenuations in dry and wet weathers, and the water vapor densities for the dry and wet weathers are 0.1 and 18 g/m3 without the rainfall. The maximum attenuation values of the dry and wet weather conditions are 1.4 and 4.9 dB in the range from 0 to 190 km. The dotted and dash-dotted lines denote the rainfall attenuations in accordance with the precipitations of 10 mm/h and 30 mm/h, which are defined as the normal and heavy rain conditions in this research. The maximum attenuations in the normal and heavy rainfalls are 5.7 dB at 46.5 km and 15.1 dB at 54 km in the range from 0 to 190 km.

In the atmospheric environment modeling, it is important to consider uncertainties of the measurement for the atmospheric refractivity [44] and rainfall attenuation, which can possibly affect the TDP results. For example, if we make an assumption that the measurement of the refractivity had the uniform uncertainty of 2.5 % along the height, then the deviations between the original measurement and the uncertainties of the TDP can be increased up to 4.7% in the long-range over 80 km.

2.3. TDP Calculation with Airborne Radar Parameters

In Section 2.1, the refractive index, gas attenuation, rainfall attenuation, and radar parameters were modeled to estimate the total path loss of the wave propagation in an anomalous atmospheric environment; however, more radar parameters and computations are needed to accurately predict the TDP of the airborne radar. We determine the TDP using the threshold evaluations derived from the free-space path loss according to the range. To examine the threshold for the total path loss, the detectability factor is required for a given probability of detection and false alarm as written in Equation (14). In addition, the fluctuation loss is modeled using Equation (15) to calculate the TDP performance for describing the fluctuating target. Finally, these equations are directly related to the radar range equation for the detectable range that is introduced by Blake as follows [45]:

$$t=0.9\left(2P_d-1\right),x_0={\left(g_d+g_{fa}\right)}^2$$

(15)

$$g_d=\left(\frac{1.23t}{\sqrt{1-{\left(t\right)}^2}}\right), g_{fa}=2.36\sqrt{-{\log }_{10}\left(P_{fa}\right)}-1.02$$

(16)

$$D\left(P_d\right)=\frac{L_f{x}_0}{4N_p}\left(1+\sqrt{1+\frac{16N_p}{x_0}}\right)$$

(17)

$$L_f\left(P_d\right)=\frac{1}{-\ln \left(P_d\right)\left(1+\frac{g_d}{g_{fa}}\right)}$$

(18)

$$R_{fs}\left(P_d\right)=\sqrt[4]{\frac{c_0^2{P}_t\sigma \tau G^2}{{\left(4\pi \right)}^{3}kT\left[D\left(P_d\right)\right]f^2{N}_f{L}_s}}$$

(19)

where P_d is the detection probability ranging from 0 to 0.99, and P_fa indicates the false alarm rate of the airborne radar. L_f is the fluctuation loss in Swerling Case 1, and N_p is the number of pulses integrated by the detector depending on the radar system. D is the detectability factor, and R_fs is the free-space detectable range in meters. c₀ is the light speed of 3 × 10⁸ m/s, and σ is the RCS of the target in m². k is the Boltzmann constant, and τ is the pulse length of the radar in seconds. P_t and G are transmitting power in watts and the array antenna gain, respectively. N_f and L_s are the noise figure and miscellaneous system loss of the radar. In the radar range Equation (16), it is required to set the system parameters of the airborne radar to specific values: P_t = 1 MW, τ = 2 μs, P_fa = 10⁻⁸, G = 10³, k = 1.38 × 10⁻²³, f = 10 GHz, N_f = 10^0.5, and L_s = 10^0.3. Then, the total path loss of the threshold related to the TDP can be calculated by using the detectable range as follows:

$$L_{tot}\left(P_d\right)=20\log \left[R_{fs}\left(P_d\right)\right]+20\log \left(f\right)+20\log \left(\frac{4\pi }{c_0}\right)$$

(20)

Figure 5 shows the calculated path loss threshold and detectability factor results according to the detection probability at a fixed RCS value of 8.5 m². The path loss threshold varies from 151.4 to 159.2 dB when the P_d increases from 70% to 100%. The maximum detectability factor is 17.9. The total path loss threshold is significantly affected by the detectable range R_fs, and the detectability factor is dominantly decided by the fluctuation loss for the fluctuating type of the target. To further observe the TDP performance with the ranges and angles, we additionally compute the theoretical scan losses in terms of the scan angle ϕ_scan for the beam scanning of the radar array antenna, as written in Equation (18) [46,47].

$$L_{scan}={\cos }^n\left({\varphi }_{scan}\right)$$

(21)

where L_scan is the scan loss of the radar, and n is the power of the cosine function set to be −2.5 in this research. Figure 6a presents the scan loss according to the scan angle. Due to the beam scanning, the array antenna gain can be gradually attenuated until 6.8 dB at the scanning angle of 70°. We also model a target aircraft using FEKO full EM software to obtain the RCSs at the azimuth angles. Figure 6b presents the RCS for the rear direction of the target aircraft. In the scanning angle, the maximum and averaged RCS values are 67.2 and 2.8 dBsm at the scan angles of 0° and 70°. With the environment models and the radar parameters, we can predict the total wave propagation loss for the airborne radar in the air-to-air situation. This total loss L_tot is then calculated to examine the TDP using Equation (19). For example, in the normal atmospheric environment, if the total path loss L_tot is 147.5 dB at R_t of 50 km, then the TDP can be obtained over 60% regardless of the target RCS. On the other hand, the TDP for very small targets with the RCS of lower than 2 m² (typical military airborne) dramatically decrease for the long-range target over 100 km (L_tot > 154.2 dB).

$$L_{tot}=L_{AREPS}+L_g+L_r+L_{scan}\hspace{1em} [dB]$$

(22)

3. TDP Calculation with Airborne Radar Parameters

Figure 7 shows three air-to-air scenarios considered for the airborne radar TDP simulation in anomalous atmospheric environments. The detailed scenario explanations are as follows:

(1)

TDPs when azimuth beam scanning within 90° in an anomalous atmospheric environment of the refractivity, as shown in Figure 7a.

(2)

TDPs when azimuth beam scanning within 90° in a rainy weather environment, as shown in Figure 7b.

(3)

TDPs according to the distance (R_t < 190 km) to the target in a heavy rain environment, as shown in Figure 7c.

In all scenarios, the height h_r of the airborne radar is consistent at 11 km, and the target height h_t is set to be 5 km. Additionally, these scenarios are in the long-range radar propagation situations, which assumes that the slant range is almost same with the horizontal range (R_t ≈ R_h). The detailed explanations for the scenarios are listed in

Table 2. TDP simulation scenarios.

	Scenario I	Scenario II	Scenario III
Scan range (Rt)	0–190 km	0–190 km	0–190 km
Elevation steering angle	−4.3°	−4.3°	−1.8–31°
Scan angle (ϕscan)	−20–70°	−20–70°	0°
Atmospheric condition (∇M)	Normal (∇M = 85) Sub (∇M = 300) Super (∇M = 10) Ducting (∇M = −80)	Normal (∇M = 85) Super (∇M = 10)	Normal (∇M = 85)
Weather condition (e)	Dry (0.1 g/m3)	Dry (0.1 g/m3) Rainfall (12 mm/h)	Dry (0.1 g/m3) Heavy rainfall (22 mm/h)

.

Figure 8a illustrates the simulation results of the TDP along the range and the azimuth angle for the first scenario, where the deep red and blue colors indicate the TDP of 100% and 0%. The TDP over 90% is obtained at the scan angle of 0° due to the high RCS level of the target aircraft, and the target is detected over the probability of 80% in the scan angle between −20° and 60° and the range from 50.5 to 153.8 km. On the other hand, the range from 0 to 50.2 km within the whole scan angle has the low TDP because this is the area below the airborne radar beam that cannot be reached. In addition, the scan loss of the airborne radar increases the path loss, which reduces the TDP in the wide angle from 60° to 90°. Figure 8b depicts the contour plot of the TDP over the 90% region according to the different atmospheric conditions of the refractivity. The solid, dashed, dotted, and dash-dotted lines denote the normal refraction, super-refraction, sub-refraction, and elevated ducting cases. We defined a detectable area by calculating the area inside the contour region to intuitively examine the TDP. The resulting detectable area of the super-refraction is larger than those of the others, which are similar to each other. It is because the super-refraction makes the wave propagation bend toward Earth’s surface, which results in low path loss levels over a large area. The summary of the detectable area results for TDPs over 80% and 90% is shown in Figure 8c.

Figure 9 illustrates the contour plot of the TDP over 80% according to the different weather conditions in the sub- and normal refractions for the second scenario. The solid and dashed lines indicate the TDP results according to the dry and rainy weather in the normal atmospheric condition, while the dotted and dash-dotted lines denote the results in the amorous atmospheric condition (super-refraction). In the dry weather, the TDP for the normal and super-refractions is similar to the results of the first scenario, where the detectable areas are 4738 and 6298 km², respectively. In contrast, in rainy weather, the TDP of the airborne radar is extremely deteriorated because of the rainfall attenuation. The resulting detectable areas for the normal and super-refraction atmospheric environments are 1670 and 1734 km². Note that the rainfall loss considerably affects the TDP of the airborne radar. In the third scenario, we additionally simulate and calculate the TDP of the airborne radar using another wave propagation simulation software (PETOOL) based on one-way and two-way split-step parabolic equation [48] and the ITU-R P.528-4 model [49] to confirm the results of the proposed process. Figure 10 shows the comparison of the TDP results along the target range of R_t using the different propagation models for the third scenario when the target and the airborne radar are gradually getting closer to each other. The airborne radar TDP becomes significantly low (almost zero TDP) when the heavy rainfall occurs. In the target range of R_t below 100 km, the AREPS simulation result well agrees with those of the PETOOL and ITU-R model. These results demonstrate that the proposed process is feasible for observing the TDP of the airborne radar considering the anomalous atmospheric environments.

4. Conclusions

In this paper, we proposed the TDP calculation process using the airborne long-range radar path loss based on the air-to-air scenarios in anomalous atmospheric and weather environments. The refractive index along an altitude was modeled using a quad-linear model. Then, the path loss including the refractivity model and other radar input parameters was simulated using the AREPS software along the range and the altitude. ITU-R models were used to consider the weather environment for the gas and rainfall attenuations according to the range. In addition, the radar beam scan loss and RCS of the target were considered to estimate the TDP of the airborne long-range radar. Various air-to-air scenarios were assumed for the airborne radar TDP simulation in the anomalous atmospheric and weather environments. In the first scenario, the target was detected over the probability of 80% in the scan angle between −20° and 60° and the range from 50.5 to 153.8 km. In the second scenario, the TDP of the airborne radar was extremely deteriorated in the rainfall weather because of the rainfall attenuation. The resulting detectable areas for the normal and super-refraction atmospheric environments were 1670 and 1734 km². In the third scenario, the airborne radar TDP became significantly low due to the rainfall.