In recent years, advancements in reusable launch vehicles and satellite communication technologies have led to a significant increase in orbital satellite deployment and operations [1, 2]. As a result, the number of artificial space objects, such as satellites and space debris, is growing rapidly each year, significantly increasing the risk of collisions and the possibility of these objects re-entering the atmosphere [3]. To identify and address these risks in advance, the positions of space objects must be accurately determined.

Space surveillance radars emit electromagnetic (EM) waves into space and analyze the reflected signals to estimate the positions of space objects [4]. However, EM waves passing through the atmosphere are affected by atmospheric effects, which alter both their propagation path and speed. Consequently, radars receive refracted and delayed signals, leading to errors in the estimated positions of space objects. To accurately predict and correct these errors, atmospheric effects on EM wave propagation must be accounted for by considering the atmospheric conditions at the time of radar operation.

For such an analysis, numerical methods that incorporate various environmental factors, such as the spatially and temporally varying atmospheric refractive index, ionospheric refractive index, and ionospheric electron density, are required. In this regard, the parabolic equation—a numerical method that approximates and solves the Helmholtz wave equation—is commonly used for modeling long-range wave propagation by accounting for the spatially varying atmospheric refractive index [5, 6]. However, it does not consider the ionospheric refractive index and electron density, which significantly affect wave propagation at higher altitudes. Ray tracing is a numerical method that models EM wave propagation based on wave theory and optics using Snell’s law, which is derived from the eikonal approximation of the wave equation under high-frequency conditions [7]. For ray tracing, the atmosphere is divided into multiple layers based on variations in the atmospheric refractive index, ionospheric refractive index, and ionospheric electron density with regard to altitude to account for its inhomogeneity. By analyzing the characteristics of EM waves at the interface between these layers, their propagation path and speed can be calculated while also factoring in atmospheric effects. Since EM waves propagating at high angles over distances of thousands of kilometers in space surveillance radar systems are affected not only by the atmospheric refractive index but also by the ionospheric refractive index and ionospheric electron density, the ray tracing method is appropriate for predicting EM wave propagation characteristics because it can comprehensively account for these environmental factors.

Previous studies have analyzed radar errors caused by the atmosphere using the vertical profile of the refractive index and the total electron content (TEC) in the zenith direction [8, 9]. However, to accurately determine the positions of space objects, it is also necessary to consider the horizontal profile. In addition, the three-dimensional ionospheric electron density is integral to calculating the TEC along the refracted propagation path.

In this paper, complex atmospheric environments are modeled by simultaneously accounting for the vertical and horizontal profiles of the atmospheric refractive index, as well as the three-dimensional ionospheric electron density, by drawing on meteorological data, interpolation techniques, and an ionospheric model. Using the modeled atmosphere, we analyze the propagation path and speed of EM waves using the ray tracing method and then calculate the range and angle errors of space objects.

This paper is organized as follows: Section II introduces the process of modeling the refractive index in the troposphere and ionosphere. Section III describes the calculation and analysis of the propagation characteristics of EM waves in the atmosphere based on the refractive index and ionospheric electron density using ray tracing. Section IV explains the process of calculating radar errors by accounting for atmospheric effects, providing a detailed analysis. Finally, Section V concludes the study.

After determining the refractive index, the propagation paths of EM waves were calculated using ray tracing. First, the atmosphere was divided into multiple layers based on the refractive index, which changes with altitude, latitude, and longitude. Subsequently, the propagation path of the EM waves passing through the boundary of each layer was calculated using Snell’s law.

Fig. 5 depicts the distance deviation in the actual propagation path of the EM waves on accounting for atmospheric effects as compared to the propagation path under the free-space assumption, based on the meteorological variables and ionospheric electron density data at 9:00 AM on January 12, 2023. This analysis covered an elevation angle range of 30°–90°, corresponding to the scan range of a space surveillance radar [22]. The positive distance deviation indicates that EM waves refract downward in the atmosphere compared to antenna boresight. Due to this characteristic, the actual space objects will always be located at a lower altitude than that estimated by the space surveillance radar.

Since the propagation speed of EM waves varies according to the different layers of the atmosphere, the propagation time must be defined differently for each layer. The total propagation time of EM waves passing through the atmosphere can be calculated using the following equation:

Table 2 presents the formulas used to calculate the propagation time of EM waves in different layers of the atmosphere [23]. In the troposphere, which extends from 0 to 30 km in altitude, the propagation time, denoted by τ₁, is defined as the summation of the propagation times for each of its subdivided layers. The propagation time for each layer was calculated as the product of the propagation path length R_i through the layer and the reduced propagation speed caused by the refractive index n_i. Notably, the multiple layers of the troposphere were divided based on the vertical and horizontal distribution of the refractive index. In Table 2, k represents the number of layers through which the EM wave passes. After passing through the troposphere, the waves travel through the region between 30 km and 90 km, which is approximated as free space (n = 1). In this region, the propagation time is denoted as τ₄, calculated by dividing the propagation path length R₁ by the speed of light c.

In the ionosphere, which extends from 90 to 1,000 km, the propagation time τ₁ is defined as the summation of the propagation times for each of its subdivided layers and the TEC along the refracted propagation path in the ionosphere. The TEC is calculated as the summation of the product of the propagation path length R_i and the electron density N_i for each layer, as shown in Table 2. Notably, the ionosphere was divided into multiple layers based on the vertical and horizontal distributions of electron density. In the given equation, m represents the number of layers through which the EM wave passes, and f is the frequency. After passing through the ionosphere, when EM waves travel through the region above 1,000 km, the propagation time, denoted as τ₄, is calculated by dividing the propagation path length R₄ by the speed of light c.

Overall, for calculating the time taken for EM waves to pass through the atmosphere, the refractive index in the troposphere and the electron density in the ionosphere are the main factors that slow down the propagation speed and increase the propagation time of EM waves.

Radar systems calculate the target range by transmitting EM waves and then receiving the echo reflected from the target. In free space, EM waves travel at the speed of light, so the range can be calculated from the time taken by the EM wave to propagate between the radar and the target, which is half the radar echo time. For example, if it takes 10 ms to receive a radar echo, the EM wave would have traveled for 5 ms to reach the target. Therefore, the range can be calculated by multiplying the one-way travel time of the EM wave by the speed of light, resulting in a range of 1,500 km.

However, in actual radar operations, the radar receives EM waves that have been delayed and refracted by the atmosphere. As a result, the target position estimated under the assumption of free space deviates from the actual target position. For instance, when the radar receives radar echo time τ with the beam steered at an elevation angle of, the actual target position corresponds to the location from which the EM wave transmitted at an elevation angle of θ traveled for τ/2 in the atmosphere. In other words, the radar receives radar echo time τ at an elevation angle of θ when the target is located at that position.

Fig. 6 illustrates the process of calculating the target’s position when an EM wave transmitted at an angle of θ has traveled for τ/2 in the atmosphere. The propagation characteristics of EM waves are determined by the atmospheric environment, which changes across time and space. Thus, the atmospheric parameters, such as temperature, pressure, humidity, and ionospheric electron density, at the time when the radar echo is received must be accounted for. For this purpose, the atmosphere is divided into multiple layers, and the refractive index n_i for each layer is calculated based on the atmospheric parameters. Therefore, considering an EM wave transmitted from the radar site p₀ at an initial elevation angle θ and azimuth angle ϕ, its arrival point at the next layer is determined, and the departure angle at that point is calculated using Snell’s law. The point where the EM wave arrives at the i-th layer is defined as p_i (i=1,2,…), and the time taken by the EM wave to reach p_i from p₀ is defined as τ_i. The propagation path is continuously traced until τ_i–τ/2<ε, with i incremented by 1 for each step and ε is determined based on the thickness of the stratified layers. If ε is too high relative to the layer thickness, the value of τ_i–τ/2 increases, resulting in an error proportional to this difference. On the other hand, if it is too low, it is not possible to obtain i such that τ_i–τ/2<ε. In this study, an optimal value for ε is selected, taking the aforementioned trade-off into account. Once τ_i–τ/2<ε is reached, the tracing stops, and point p_i is estimated as the actual position of the target.

Assuming a radar measurement with an echo time τ of 18.42405 ms and an elevation angle θ of 10.00734°, the target position estimated by the radar is calculated to be at an altitude of 1,000.000 km, with the same elevation angle of 10.00734°. Using the aforementioned procedure, the estimated target position is determined by accounting for atmospheric effects based on the position of the EM wave after it has traveled through the atmosphere for 9.212024 ms.

Fig. 7 visualizes the target position estimated by the space surveillance radar and that estimated considering atmospheric effects. Table 3 presents the differences between these positions. Notably, the atmosphere was modeled using meteorological data and ionospheric electron density at the time the radar echo was received. Assuming the radar echo was received on April 30, 2024, at 9:00 AM in the Korean Peninsula region, the elevation angle error was calculated to be approximately 0.0980057° and the range difference was about 0.0130760 km. The Euclidean distance between the target locations before and after correction was 4.558590 km, indicating that the estimated target position considering atmospheric effects was approximately 0.0980057° below the target position estimated by the radar, 0.0130760 km closer to radar, and 4.558590 km away from the target position before calculation. Therefore, by accounting for atmospheric effects, the calculated target position is expected to be closer to the actual target position than that before calculation.

Furthermore, to validate the results obtained using the method proposed in this study, the elevation angle errors were compared with those from a previous study conducted under the same simulation conditions [24]. Fig. 8 shows the results obtained by a previous study [24] on calculating the elevation angle error for a target located at an altitude of 200 km and a frequency of 2 GHz using the National Bureau of Standards exponential model [25] for the refractive index, compared to the results calculated using the proposed algorithm considering the same simulation parameters. In the reference study [24], the calculation accuracy of the elevation angle error was verified using space vehicle measurements. The good agreement between the results in Fig. 8 suggests that our method accurately estimated the elevation angle error, thereby validating the effectiveness of our algorithm under different atmospheric conditions.

The accuracy of radar error calculations depends on the availability of atmospheric data pertaining to the time when the radar echo is received. In other words, when atmospheric data for specific time periods are available, it is possible to make accurate predictions. However, meteorological variables are provided only twice a day at 12-hour intervals by meteorological stations, while the ionospheric electron density is updated every 15 minutes in the IRI-2016 model. Therefore, atmospheric conditions at the time of radar operation are often difficult to accurately determine during most operational periods. In such a situation, where a real-time refractive index is not available, other refractive indices can be utilized. Supposing that the radar echo of a target was received by the space surveillance radar at 9:00 AM on April 30, 2024, some alternatives that can be considered for calculating the radar error are the refractive index from the previous afternoon or morning, the weekly average, the monthly average, the seasonal average, or last year’s seasonal average, as shown in Table 4. Notably, each of these refractive index models is calculated by averaging the refractive index for each altitude over the corresponding period.

Fig. 9 shows the distance between target positions before and after calculation, as well as the range and elevation angle error, for a target located at an altitude of 1,000 km with an elevation angle of 10° using each of the refractive index models in Table 4. The error on the right side of the graph indicates the difference from the error calculated using the current refractive index. Assuming that the refractive index at a current time is unknown, the calculated errors decreased in the order of the models constructed based on data from the previous afternoon, seasonal average, monthly average, last year’s seasonal average, weekly average, and previous morning.

Notably, the smallest error occurred when using the refractive index from the previous morning. When the error was calculated using the refractive index of the previous morning, the distance was 4,488.671 m, the range error was 12.65908 m, and the elevation angle error was 1.684251 mrad, indicating a difference of approximately 26.50046 m, 0.01699 m, and 0.009946 mrad, respectively, from the numbers obtained using the actual refractive index. This suggests that if the radar echo is obtained in the morning, it is appropriate to use the previous morning’s refractive index for error calculation, and if the radar echo is received in the evening, the previous afternoon’s refractive index should be employed. In this way, even when the refractive index is unknown, radar errors can be calculated in real time using the previous day’s refractive index.

In this paper, refractive indices for the troposphere and ionosphere were modeled to account for the spatial-varying and time-varying characteristics of the atmosphere. The vertical profile of the refractive index in the troposphere was obtained through the radiosondes used by meteorological stations, while the horizontal profile was predicted from these data using spatial interpolation techniques. For the vertical and horizontal profiles of the ionospheric refractive index, a three-dimensional profile of the ionospheric electron density was obtained using the IRI-2016 model. Based on the refractive index, propagation characteristics in the troposphere and ionosphere were analyzed using ray tracing methods, confirming that the propagation path refracts downward and the propagation time increases consistently. Subsequently, the positions of space objects were calculated using the propagation path and propagation time of EM waves in the atmosphere. Additionally, a radar error calculation method was proposed for situations where the refractive index is unknown. Several refractive index models are available for use at any given time. By analyzing the errors using various refractive index models available at a given time, it was found that the results obtained using the previous day’s refractive index were the closest to those obtained using the refractive index at the current time. Thus, it was concluded that when refractive index data are available, radar errors should be calculated using the provided meteorological data. However, in the absence of current refractive index data, the previous day’s refractive index can be effectively used for error calculation.