1. Introduction
The design and operation of wireless transmission systems require a reasonable and effective evaluation and prediction of propagation losses within their coverage area. Among these systems, wave propagation modeling based on the parabolic equation is widely used in radar, communication, broadcast systems, and remote sensing inversion research [
1,
2]. In the process of PE modeling, the construction of the initial antenna field is a crucial factor that affects simulation accuracy. The initial field corresponds to the transmitting antenna radiation field in the actual communication system, and the antenna beam approximation method is used to construct the initial field in the research and application of radio wave propagation. The accuracy of PE modeling decreases with an increase in the beam angle due to the difference in the beamwidth of the actual transmitting antenna. To address this issue, PE modeling has been developed from narrow-angle parabolic equations to wide-angle parabolic equations. Hardin and Tappert first used Taylor approximation to construct narrow-angle parabolic equations for differential operators, and as the application scenarios of PE expanded, the wide keratalization of differential operators was also developed [
3]. The Claerbout approximation was initially applied in geophysics and later introduced into the field of radio wave propagation due to its universality [
4]. The Feit–Fleck approximation originated from the study of Feit and Fleck in optical fiber sensing [
5] and was later applied to the study of radio wave propagation by Thomson and Chapman [
6], which is suitable for the parabolic equation split-step Fourier transform method. Greene’s approximation, developed later, significantly broadens the beam range of PE, with an angle of up to 40° relative to the horizontal plane, and effectively controls the relative error caused by the approximation of differential operators [
7]. The development of these differential operator approximation methods has increased the application scenarios of PE wave propagation modeling. However, it has also reduced the accuracy of the Gaussian beam method, which was originally suitable for narrow-angle initial field approximation.
When using a Gaussian approximate narrow-beam transmitting antenna, the main beamwidth is generally within 15°. For such beamwidths, even considering the influence of sidelobes, the accuracy of adopting a Gaussian approximate transmitting antenna main beam is very high, and relevant research has been widely applied [
8,
9,
10,
11]. However, in actual wireless transmission systems, the main beamwidth of the transmitting antenna is often more than 15°, especially in scenes larger than 40°. Therefore, it is necessary to evaluate the impact of increasing the main beam of the Gaussian approximation to actual transmitting antenna. In addition to studying the error of the PE equation with different beamwidths, the error analysis of Gaussian approximation has also been reported [
12,
13]. Particularly, in the beam scanning of MIMO and other applications [
14], the main beam of the antenna will deviate from the zero-degree direction of the PE model at a larger angle, and the propagation loss of space radio waves within the coverage range will pose new challenges to the angle problem in PE modeling. Therefore, it is necessary to explain the difference between the PE model and the real antenna after using Gaussian approximation. Currently, there have been relevant studies on the construction of the initial field of the transmitting antenna using Gaussian approximation, but they are limited to the error analysis of the applicable angle range of the PE method [
15]. This paper presents detailed results of simulation calculations and actual measurements to verify the radio wave propagation loss caused by the Gaussian approximation of the transmitting antenna involving a wider angle.
Based on the above analysis, this paper focuses on four research aspects: (1) the difference between the actual antenna and its Gaussian approximate antenna pattern; (2) the difference between PE initial fields constructed based on antenna patterns; (3) the influence of two PE initial field construction methods on the simulation calculation of radio wave propagation losses within the coverage area; (4) the accuracy of the propagation model constructed by the two methods for retrieving the tropospheric duct structure. The specifics of the different sections of the article are listed below:
Section 2 presents the wide-angle PE equation theory, a comparison of Gaussian beam approximation, and the half-wave dipole antenna pattern. In
Section 3, the initial fields corresponding to the two types of antenna patterns and the propagation simulation in the PE model are calculated and compared, and the differences in the initial fields and their influence on the radio wave propagation calculation are explained.
Section 4 describes the actual test scenarios and test results, comparing the measured results with the calculated results of the propagation loss in the PE model using the two initial fields.
Section 5 employs particle swarm optimization (PSO) for a comparative analysis of atmospheric profile inversion. Finally,
Section 6 provides the research conclusions.
6. Conclusions
This paper analyzed the disparity between the Gaussian beam simulation and the actual antenna pattern in constructing the initial field of the parabolic equation and investigated its impact on the propagation loss of the radio wave model. Based on the simulation results of radio wave propagation and the analysis of measured data of broadcast signals in the Wuhan area, the findings indicate that the broadcast signal propagation model, which incorporates the actual antenna pattern, aligns more closely with the actual spatial propagation characteristics of radio waves, with a correlation coefficient of 0.873. Lastly, particle swarm optimization is employed to invert the tropospheric refractive index profile of radio wave propagation. The results demonstrate that the model utilizing the actual transmitting antenna pattern achieves higher inversion accuracy, with a root-mean-square error of 2.03.
In conclusion, to enhance the accuracy of the wave propagation model for wide-beam transmitting antennas, it is recommended to construct the initial field of the parabolic equation based on the actual antenna pattern instead of simulating the transmitting antenna using the Gaussian beam. Additionally, considering the interdependence among tropospheric duct parameters [
2], historical atmospheric environment information can be statistically analyzed to optimize the multi-parameter variation range of the inversion algorithm and improve its accuracy and efficiency.