Calculation of Range and Angle Error for Space Surveillance Radar Using Two-Dimensional Refractive Indices of the Troposphere and Ionosphere

Article information

J. Electromagn. Eng. Sci. 2025;25(4):371-380

Publication date (electronic) : 2025 July 31

doi : https://doi.org/10.26866/jees.2025.4.r.308

Junmo Yang¹, Do Hyeon Lee¹, Sangho Lim², Ik Hwan Choi², Yong Bae Park^1,3,

¹Department of AI Convergence Network, Ajou University, Suwon, Korea

²The 3rd R&D Institute 4th Directorate, Agency for Defense Development, Daejeon, Korea

³Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea

^*Corresponding Author: Yong Bae Park (e-mail: yong@ajou.ac.kr)

Received 2024 October 11; Revised 2024 November 21; Accepted 2024 December 11.

Abstract

I. Introduction

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.

II. Atmosphere Modeling

1. Troposphere

EM waves are refracted and reflected at the interface between two media with different refractive indices [10]. In this regard, the atmosphere can be modeled as layered media, with each layer having a different refractive index. Consequently, the behavior of EM waves in the atmosphere can be predicted using the refractive index of each layer of the atmosphere. Furthermore, the atmosphere can be divided into the troposphere and the ionosphere based on its interaction with EM waves. In the case of the troposphere, which extends up to approximately 30 km, the propagation characteristics of EM waves passing through it are determined by meteorological variables, such as temperature, pressure, and humidity. Based on these variables, the atmospheric refractive index n can be calculated using the following equation [11]:

(1) N=(n-1)×10-6,

(2) N=77.6PT-5.6eT+3.75×105eT2.

In Eqs. (1) and (2), N represents the atmospheric refractivity in N-units, T is the temperature in Kelvin (K), P denotes the atmospheric pressure in hectopascals (hPa), and e refers to the water vapor pressure in hectopascals (hPa). Notably, N-units is a unit of refractivity used in atmospheric science to describe the refractive index of air in a more practical scaled form. Since n is very close to 1 (e.g., 1.0003), refractivity is expressed in terms of N-units to simplify its representation and calculations. The meteorological variables required to calculate the vertical profile of the atmospheric refractive index can be measured using radiosondes. Notably, these measurements are provided twice a day by various upper-air meteorological stations worldwide [12, 13].

Fig. 1 and Table 1 indicate the locations and information related to the upper-air meteorological observations made in South Korea and its neighboring country. For an accurate prediction of the propagation characteristics of EM waves in the atmosphere, a horizontal profile of the refractive index is required. However, obtaining meteorological variables for every point within the study area is not feasible. Therefore, to estimate the values for the unobserved locations, an interpolation technique using discrete measurements from meteorological observations was employed.

Fig. 1

Location of the meteorological observations in South Korea and its neighboring country.

Table 1

Information on meteorological observations

Notably, the heights at which meteorological variables are measured by the nine meteorological observation stations usually differ because radiosondes, although they measure these variables at regular time intervals, ascend through the atmosphere at varying rates due to differences in wind speed, wind direction, and other atmospheric conditions.

Fig. 2 illustrates the distribution of measurements at different altitudes for each observation station. Since direct measurements were not available at the specific altitude h, the value at h had to be estimated by interpolating the measurements taken at various altitudes at each observation station. Given the monotonic and smooth nature of the refractive index as it decreases with altitude, employing the piecewise cubic Hermite interpolating polynomial (PCHIP) interpolation method was considered appropriate for predicting its values. This method ensures the preservation of monotonicity and avoids oscillations that could distort the physical characteristics of the refractive index. Once the data for interpolation were prepared, spatial interpolation methods were applied.

Fig. 2

Distribution of measured and interpolated values at different altitudes for each observation station.

A straightforward spatial interpolation method is inverse distance weighting (IDW). IDW calculates the value of a point within a study area by assigning weights based on its distance from surrounding meteorological observations. Considering that the distance between each meteorological observation station and the point to be interpolated is d_i, the interpolated value considering the weights of each station can be calculated using the following equation:

(3) ui=∑k=1nwkuk∑k=1nwk,         wi=1di.

Fig. 3 depicts the horizontal profile of the refractive index at an altitude of 3,450 m on January 12, 2023, at 9:00 AM, obtained using IDW based on measurements from the nine meteorological stations. This proves that the horizontal profile of the refractive index can be predicted using the IDW. These predicted values can then be employed to calculate the propagation characteristics of EM waves in the troposphere.

Fig. 3

Example of the horizontal profile of the atmospheric refractive index.

2. Ionosphere

The ionosphere is a region extending from an altitude of 90 km to 1,000 km, where EM waves are influenced by the static magnetic field and plasma, composed of electrons and positive ions ionized by extreme ultraviolet radiation and X-rays from the Sun [14]. The propagation characteristics of EM waves in the ionosphere can be accurately calculated by accounting for electron density, the Earth’s magnetic field, and electron collisions [15, 16], while its refractive index can be calculated using the Appleton–Hartree equation. Moreover, for frequencies above 300 MHz, the simplified Appleton–Hartree equation, which disregards the effects of the Earth’s magnetic field and electron collisions [17], can be used, as expressed in Eq. (4):

(4) n=1-1ω02Ne2ɛ0m.

In Eq. (4)N represents the electron density of the ionosphere in electrons per cubic meter (electrons/m³), e is the charge of an electron in coulomb (C), ω₀ denotes the angular frequency in radians per second (rad/s), ɛ₀ refers to the permittivity of free space in farad per meter (F/m), and m is the mass of an electron in kilograms (kg). The ionospheric electron density, which is integral to the calculation of the refractive index of the ionosphere, can be obtained from the International Reference Ionosphere (IRI) 2016 model provided by the National Aeronautics and Space Administration by inputting a specific date, time, latitude, longitude, and altitude into the IRI-2016 [18]. Notably, the IRI model is an international standard ionospheric model developed jointly by the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). It accurately models the complex structure and variability of the ionosphere by compiling diverse observational data from around the world [19]. The model is continuously updated to reflect the latest scientific discoveries and data, with its latest version—the IRI-2016—recognized as the most accurate representation of the current state of the ionosphere [20].

For altitudes ranging from 0 to 30 km and 90 to 1,000 km, the refractive index can be calculated using meteorological data and ionospheric electron density models, respectively. However, for altitudes between 30 and 90 km or above 1,000 km, no data are available for estimating the refractive index, as illustrated in Fig. 4(a). This highlights the need for refractive index modeling at these altitudes.

Fig. 4

Refractive index in the atmosphere according to height above sea level: (a) before modeling and (b) after modeling.

In the lower ionosphere, the ionization caused by solar radiation decreases with a decline in altitude, leading to a reduction in electron density, which causes the refractive index to approach unity. Similarly, in the upper troposphere, atmospheric density decreases rapidly as altitude increases, causing the refractive index to approach unity. Therefore, the altitude range from 30 to 90 km can generally be assumed as free space (n = 1) [21].

In the ionosphere, the electron density increases with an increase in altitude due to more intense ionization from solar radiation at this stage. However, as the atmospheric density decreases at higher altitudes, the electron density in the ionosphere begins to decrease beyond a certain altitude. Consequently, in the upper ionosphere, the refractive index approach unity as the altitude increases. Thus, altitudes above 1,000 km can also be considered free space (n = 1). Consequently, when EM waves propagate toward space objects located at altitudes of thousands of kilometers and pass through the troposphere and ionosphere, the refractive index along their propagation path can be modeled as shown in Fig. 4(b).

III. Propagation Characteristics

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.

Fig. 5

Distance deviation in the propagation path of EM waves considering atmospheric effects compared to the path under free-space assumption by elevation angle.

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.

Table 2

Calculation of propagation time of EM waves

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.

IV. Radar Error Calculation

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.

Fig. 6

Flowchart for estimating the actual target position, considering atmospheric effects.

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.

Fig. 7

Before calculation vs. after calculation.

Table 3

Comparison of elevation angle and range of targets

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.

Fig. 8

Comparison of elevation angle error between the proposed method and the reference study [24].

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.

Table 4

Alternative refractive index models for radar error calculation

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.

Fig. 9

Calculations of target position error using different refractive index models from Table 4 for a target at an altitude of 1,000 km and an elevation angle of 10°, with errors between target positions predicted using the current time model (<1>) and those predicted using other models (<2> through <7>): (a) distance, (b) range error, and (c) elevation angle error.

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.

V. Conclusion

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.

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