Electromagnetic Wave Propagation From Low-Earth Orbit Satellite to Ground Station Considering Interpolated Atmospheric Environments

We propose a novel method to calculate the electromagnetic (EM) wave propagation from low-earth orbit satellite (LEO) to a ground station based on the physical optics (PO), ray tracing technique, and geometric optics (GO) considering interpolated atmospheric environments. Our method includes the reflector antenna analysis using PO, the interpolation of the meteorological data using PCHIP and Kriging interpolation, transmission analysis using ray tracing and geometrical optics. Tropospheric and stratospheric environments are modeled using meteorological data—air pressure and temperature, relative humidity, and rain rate—measured at 9 different radiosonde observatories in and around South Korea. Furthermore, we utilize Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) and Kriging-exponential methods for vertical and horizontal interpolations of the raw meteorological data, respectively. Hence, the interpolated atmospheric environments are amenable to the best use of ray tracing technique and GO. Subsequently, effective refractive indices of the stratified media can be extracted via millimeter-propagation-model93. The simplified Appleton-Hartree equation characterizes the ionospheric environment. Considering a sun-synchronous orbit satellite passing through South Korea, we calculate atmospheric attenuation, boresight error, received power, and compensation angle of satellite antenna for various conditions.


I. INTRODUCTION
Low-earth orbit (LEO) satellites [1]- [4], revolving at an altitude between 160 to 2,000 km, can offer faster communications with low latency and higher bandwidth at low cost. In addition, a constellation of LEO satellites-Starlink, LeoSat, Boeing, CYGNSS, and Telesat LEO which provide various satellite services such as remote sensing, weather monitoring, internet of things (IoT), and satellite navigation [5], [6]-can attain continuous and global The associate editor coordinating the review of this manuscript and approving it for publication was Giorgio Montisci . coverage for various communication systems. Especially, a sun-synchronous orbit (SSO) satellite, which passes through the perihelion of its orbit at the same local time, is useful for imaging, intelligence, and weather satellites [7].
For the development of reliable LEO satellite links, it is crucial to predict the electromagnetic (EM) wave propagation in highly variable atmospheric environments which can significantly deteriorate the performance of communication systems. In general, EM waves propagate through the atmosphere while experiencing refraction, reflection, and retardation. In particular, the troposphere, stratosphere, and ionosphere are the main areas in which atmospheric effects VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ take place. Both the tropospheric and stratospheric effects are caused by the spectral absorption of oxygen and water vapor molecules, though, meteorological phenomena occur only in the troposphere [8]. The spectral absorption of oxygen and water vapor molecule is considered as complex refractive index calculated by millimeter-propagation-model 93 (MPM93) [9]. Thus, such atmospheric environments can be quantified by measurements from meteorological observatories and transformed into an effective refractive index for the subsequent EM analysis. As such, one can study how atmospheric effects, such as attenuation, boresight error, and time delay, vary with the elevation angle and distance between satellites and ground stations. The attenuation and time delay are important in satellite link margin and distance measurements. The boresight error is an indicator that allows the beam to arrive the observation point by adjusting the antenna steering angle when the tracking area is fixed on the ground.
On the other hand, in the ionosphere being an anisotropic cold/magnetized plasma in the range of 60 to 1000 km above sea level, the atmospheric effects primarily depend on the extent of ionization and geomagnetic fields [10]. In order to consider the ionosphere effects on the EM propagation, the refractive index of the ionosphere is calculated by Appleton-Hartree equation [11]. The ionospheric effects are frequency dependent and can be neglected when operating frequencies are larger than 10 GHz.
There have been a number of previous studies that estimated atmospheric effects to the EM wave propagation in a phenomenological sense [12]- [16]. For example, the attenuation induced by water vapor was estimated for slant paths [12], the oxygen attenuation model was developed by taking the advantage of an extensive set of high-resolution radiosonde observations (RAOBS) collected in several sites worldwide [13], the geometric positioning method was suggested to the atmospheric refraction calibration for optical remote sensing satellite [14], and the experimental transmittance data were presented from urban and rural environments in the troposphere [15]. In addition, the EM wave propagation in the ionosphere modeled by anisotropic magnetized plasmas was numerically analyzed by using finitedifference time-domain (FDTD) with perfectly matched layers [16].
However, to our knowledge, theoretical analyses for the EM wave propagation from a LEO satellite to a ground station are still lacking. In our previous work, we have shown a new approach to analyze the EM wave propagation in a well-defined single radio link, which is from a geostationary orbit (GEO) to a ground station, using high frequency approximation such as geometric optics (GO), ray tracing technique, and physical optics (PO) [17]. Unlike GEO satellites, LEO satellites orbit Earth much faster, approximately more than 12 times a day, such that the radio links can be significantly affected by atmospheric effects. Therefore, one needs to collect meteorological data over a quite larger area to capture correct atmospheric effects.
In this paper, we propose a novel method to calculate the EM wave propagation from LEO to a ground station based on the PO, ray tracing technique, and GO considering interpolated atmospheric environments. Tropospheric and stratospheric environments are modeled by using meteorological data-air pressure and temperature, relative humidity, and rain rate-which are measured at 9 different radiosonde observatories in and around South Korea. Furthermore, we utilize piecewise cubic Hermite interpolating polynomial (PCHIP) and Kriging-exponential methods for vertical and horizontal interpolations of the raw meteorological data, respectively; interpolated atmospheric environments are amenable to the best use of ray tracing technique and GO. Subsequently, effective refractive indices of the stratified media can be extracted via MPM93. The ionospheric environment is characterized by the simplified Appleton-Hartree equation. Considering a sun-synchronous orbit satellite passing through South Korea, we calculate and discuss rain attenuation, atmospheric attenuation, boresight error, and received power for various conditions. Also, steering angle of the satellite antenna is calculated using boresight error and our analysis procedure so that the maximum received power is maintained at the observation point. To check the validity of the present method, our rain attenuation calculation results are compared with ITU-R. P. 618 model, physical model based on the rain-cell concept, and rain attenuation model in tropical regions [18]- [20].

II. ATMOSPHERIC ENVIRONMENTS
Meteorological data-air pressure and temperature, relative humidity, and rain rate-were measured at 9 different radiosonde observatories in and around South Korea, as shown in Fig. 1. A list of the radiosonde observatories and their geographic coordinates are provided in Table 1. 9 different radiosondes measured meteorological data from their observatories' altitudes (provided in Table 1) up to 30 km above the sea level with different vertical intervals. Therefore, aggregate meteorological data do not have an altitude match in general, as illustrated in Fig. 2. The vertical profiles of two sets of meteorological data measured at different observatories have no altitude matching. Moreover, horizontal locations of the observatories are highly irregular. Consequently, the raw meteorological data collected from all observatories may not be compatible with the use of the ray tracing technique and GO.
To solve this, the raw meteorological data are PCHIPinterpolated along altitude first. The resulting interpolated atmosphere consists of stratified layers with a fixed interval H 0 , as depicted in Fig. 2. The use of PCHIP, which defines an interpolating function by means of polynomial pieces, ensures that some regularity conditions hold at the breakpoint [21].
Figs. 3-5 illustrate results of interpolated air pressure, air temperature, and relative humidity at 240 m above sea level by inverse distance weighting (IDW) and Kriging-exponential interpolation, respectively. The IDW  Table 1. and Kriging-exponential interpolation method is applied to the PCHIP-interpolated meteorological data for the horizontal interpolation. We determined that the weighting of IDW is inversely proportional to the 1st power of the distance. We also approximated the semivariogram of the Kriging interpolation as an exponential function. The disadvantage of IDW interpolation is that it is sensitive to outliers and sampling configuration. Note that we adopt the Kriging-exponential model which has shown the highest accuracy against IDW, Gaussian and spline models in temperature interpolation from finite interpolation sites [22]. Fig. 6 illustrates the refractivity, denoted by N , which was obtained after applying MPM93 to the interpolated   meteorological data. The MPM93 calculates the complex refractivity N for given atmospheric conditions, such as air pressure, air temperature, and relative humidity.     Table 2 shows the Pearson's correlation coefficient between refractivity and meteorological data. The Pearson's correlation coefficient is calculated from the standard deviation of data X, σ X , the standard deviation of data Y, σ Y , and the expected value of X, Y and XY via (1). Each calculation is based on the data interpolated at 0.1 latitude and longitude intervals. The two-dimensional profiles of the refractivity are the most correlated with the air temperature of meteorological data. The correlation coefficient between real refractivity and imaginary refractivity is 0.9949.
An effective refractive index n eff in the troposphere and stratosphere can be calculated from the complex refractivity N via (2) where N' and N'' are real and imaginary values of N [17]. In the ionosphere (i.e., cold/magnetized plasmas), the effective refractive index can be calculated by using Appleton-Hartree equation. Appleton-Hartree equation can be simplified to (3) because our target frequency is 35.75 GHz, where the Earth's magnetic field and electron collision are negligible when the electron density is at a typical concentration. The effective refractive index n eff in the ionosphere is calculated from electron density N e , electron charge e, electron mass m, permittivity of free space, and angular frequency ω. Then, propagation constant γ and the effective intrinsic impedance η eff can be calculated by (4) and (5) [17]. Fig. 7 depicts the overall analysis procedure to predict wave propagation in the atmosphere. PCHIP and Kriging-exponential interpolation methods, ray tracing technique, and GO are used to calculate the EM wave propagation between LEO satellite and ground station. We first stratify the atmosphere into more than 100 layers at regular intervals and utilize the PCHIP interpolation methods to have the raw meteorological data matched to regular altitude intervals. Then, the ray is shot with the initial angle of incidence from LEO satellite to the ground station. When the ray encounters the boundary of the stratified layer in the atmosphere, the transmission path is calculated from the difference of refractive index using ray tracing technique.
The effective refractive index in the stratified layer is calculated from the interpolated meteorological data using MPM93 and simplified Appleton-Hartree equation. The horizontal interpolation of the meteorological data is conducted using Kriging-exponential method whenever a ray reaches to boundary of stratified layer to consider the horizontal variations in reflexivity in the ray-tracing phase. The ray path and refractive index calculations are repeated until the ray arrives the last stratified layer. Then distance difference is evaluated between arrival point and observation point. The process of searching for the ray path is repeated by changing the angle of incidence until the distance error is less than 0.01 wavelengths. The angle of incidence is expressed in zenith angle and azimuth angle. Every iteration decreases the maximum error halved, making the rapid convergence of the zenith and azimuth angles. One can refer to [17] for more details about the ray tracing technique applied here.
If the ray path is fixed, the EM field is calculated at the boundary of the stratified layer. The reflection and transmission of the EM wave at an oblique angle upon a dielectric-dielectric interface is calculated by considering the polarization, direction, normal vector of the tangent plane at the boundary [23]. Incident EM fields at the boundary are calculated from transmitted EM fields using GO. Atmospheric effects such as attenuation, boresight error, and dispersion can be calculated from the EM field at the observation point.

III. CALCULATION RESULT
We calculate the atmospheric attenuation and refraction of the EM wave between LEO satellite and ground station using ray tracing technique, GO, and interpolation. The accuracy of the proposed method is validated by comparing our rain attenuation calculation with the rain attenuation of ITU-R. P. 618-13 model, physical model based on the rain-cell concept, and rain attenuation model in tropical region, as shown in Figs. 8 and 9 [18]- [20]. Time percentage is defined as a cumulative distribution of descending data. In our method and physical model based on the rain-cell concept, rain attenuation is calculated from rain rate instead of time percentage. Also, the algorithm of our method uses the refractive indices calculated from rain rates using MPM93. Therefore, the rain rates are calculated by applying time percentage to the model in ITU-R. P. 837-7 [25]. The time percentages of 5, 1, 0.5, 0.1, 0.05, 0.01, and 0.005% correspond to the rain rates 0.187, 3.819, 7.201, 20.236, 29.276, 61.419, 81.479 mm/h, respectively in South Korea. We used a fixed rain height, and the atmosphere was stratified into 200-meters-intervals.
The atmospheric effects, including atmospheric attenuation, excess atmospheric path length, dispersive effects, and boresight error, calculated by the proposed method and ITU-R. P.676-12 are shown in Table 3 [26]. Note that the proposed method can consider the boresight error through the EM analysis whereas ITU-R. P.676-12 cannot.   Consider the path of the SSO satellite, as described in Fig. 10. The SSO satellite passes through points A, B, C, and D over time. The altitude of orbit is 567.033 km above sea level, the inclination is 97.65754, and the period of orbit is 1.6 hours. The observation point is located at 240 m above sea level, 37.283 degrees north and 127.043 degrees east, equal to the latitude and longitude of point D. The location of point A was determined to have an elevation angle of 10 degrees and a coverage area of 1.76% [27].   Figs. 11(a)-11(d) illustrate the atmospheric attenuation from LEO satellite locations A, B, C and D to the ground station in the absence of rain by latitude and longitude, respectively. Atmospheric attenuation corresponds to the power difference between the received EM fields in the vacuum and the presence of atmospheric environments. The atmospheric attenuations for the satellite points A, B, C, and D are 3.274 dB, 1.218 dB, 0.876 dB, and 0.835 dB, respectively. In general, the atmospheric attenuation from LEO satellite to ground stations decreases as the satellite's location is closer to the observation point, though, the decreasing extent depends on the distribution of the refractive index in the atmosphere.
Figs. 12(a)-12(d) illustrate the atmospheric attenuation from LEO satellite to ground station when it rains uniformly with a rain rate of 7.201 mm/h in the atmosphere   3.601 km by latitude and longitude. The atmospheric attenuations for the satellite points A, B, C, and D are 31.185 dB, 11.152 dB, 6.646 dB, and 5.842 dB, respectively. As the rain attenuation, which increases as the path length increases, dominates total attenuation, atmospheric attenuation decreases as the satellite gets closer to the observation point.
Figs. 13-15 show the normalized received power by latitude and longitude from LEO satellite to South Korea. The radiation of the parabolic reflector antenna is calculated using the PO on a reflector surface which is divided by less than 1/10 wavelength. The parabolic reflector antenna's size and focal length are the same as those of the CubeSat reflector antenna developed by the national aeronautics and space administration (NASA) jet propulsion laboratory (JPL) [28]. We assume that as the satellite moves the main lobe is adapted to the direction toward the observation point (the latitude of 37.283 degrees north and longitude of 127.043 degrees, east). In the absence of the atmosphere (viz., vacuum), the normalized received power is shown in Fig. 13(a)-13(d). Each normalized received power is calculated by the ratio of the received power to the maximum received power when the satellite is located at point D. Note that we account for the antenna's gain and elevation angle in the calculation of the received power. It is interesting to observe in Figs. 13(a), 13(b), 13(c), and 13(d) that the observation point and location for which the received power becomes maximized do not coincide even though the antenna is directing toward the observation point. The separation distance for the satellite points A, B, C, and D was recorded 8211 m, 456.8 m, 45.4 m, and 0 m, respectively. When the antenna's gain or elevation angle become higher, the separation distance gets smaller.
In the atmosphere without rain, the normalized received power from LEO satellite to South Korea is illustrated by latitude and longitude in Figs. 14(a)-14(d). With the decrease of the elevation angle, the boresight error increases, and the maximum received power decrease. The boresight error refers to the angular difference between the direct path and the path in the atmosphere, from the satellite to the observation point for the maximum received power.
In the atmosphere with a rainfall of 7.201 mm/h, the normalized received power from LEO satellite to South Korea  Table 4 lists atmospheric effects and compensation angle from LEO satellite to observation point shown in Fig. 14 and 15. The received power and location of maximum received power depend on the satellite location and rain rate. Atmospheric attenuation can be indirectly evaluated by taking the power difference between the maximum received powers in vacuum and each atmospheric environment. In Fig. 14(a)-14(d), the power difference is 14.84 dB, 6.75 dB, 2.09 dB, 0.83 dB, respectively. And in Fig. 15(a)-15(d), the power difference under rainfall is 48.62 dB, 17.74 dB, 8.44 dB, 6.33 dB, respectively. In all four cases, the power difference is larger than atmospheric attenuation evaluated in Figs. 11 and 12. In Figs. 11 and 12, we calculated the atmospheric attenuation assuming that the antenna is a complex point source which generates a single ray with the perfectly coherent phase characteristics. On the other hand, the reflector antenna, modeled by many complex point sources distributing over the reflector surface, creates a bundle of rays having a pseudo-random phase front. This leads to the power weakening of a main beam due to inhomogeneous atmospheric environments in each ray's path. The boresight error and power difference decrease as the satellite moves from point A to point D over time. The steering angle of the antenna is compensated by using the boresight error and repeating our analysis procedure until the maximum received power approaches the observation point sufficiently. In all case except for Fig. 15(d), the maximum received power approached sufficiently to observation point within 2 iterations. VOLUME 9, 2021

IV. CONCLUSION
A novel method to calculate the EM wave propagation from LEO satellite to ground station has been proposed. PO, Ray tracing technique, GO, and interpolation methods were used to consider inhomogeneous atmospheric environments. In the troposphere and stratosphere, the meteorological data are measured at 9 observatories in and around South Korea to calculate the refractive index of the atmosphere. PCHIP and Kriging-exponential interpolation methods were used to tailor the highly irregular meteorological data into the preferrable form of using ray tracing technique and geometric optics. The ionospheric environment was characterized by the simplified Appleton-Hartree equation. We validated the proposed method for the rain attenuation calculation compared with previous rain attenuation models. In addition, we calculated the atmospheric attenuation, boresight error, received power and confirmed that atmospheric effects strongly depend on the satellite location and rain rate. Atmospheric effects such as boresight errors and atmospheric attenuation tend to decrease as the satellite's position is closer to the observation point. The steering angle of satellite antenna can be compensated by using the boresight error and our analysis procedure so that the maximum received power is maintained at the observation point. Also, when it rains, the attenuation increases further and the position of the maximum received power shifts more. Our method is useful for satellite mission such as navigation, surveillance, and constellation networks that require the accurate estimation of the satellite links from LEO satellite to the ground station.