# Propagation Attenuation Maps Based on Parabolic Equation Method +
## 3 Propagation Attenuation Map
### Parabolic Equation Method  
For modeling tropospheric ducting propagation (TDP), the mode theory, RTM, and PEM are usually used. The first two methods require greater computational effort and have significant problems with modeling radio wave propagation in the troposphere, analysis of range-dependent environments, and higher frequency ranges. Moreover, the RTM does not allow for precise estimating of the field strength [36]. Hence, the PEM is widely used as an effective tool for modeling the TDP considering the terrain topography. Therefore, we used the PEM in the proposed PAM algorithm. The development of the PEM and its widespread use took place in the early 1980s. However, the origins of the PEM as applied to the TDP modeling date back to 1946. Then, [PERSON] and [PERSON] [39] presented an analysis of TDP based on a solution to the parabolic equation. The first solutions were based on the first-order approximation of Taylor expansion, considering only the first two terms. This resulted in a low accuracy that allowed only analyzing a narrow beam in the elevation plane. This method was called narrow-angle PEM [36]. In 1973, [PERSON] and [PERSON] [40] proposed using the split-step Fourier transform (SSFT) algorithm to sequentially solve a parabolic equation along a given radius. This approach is one of the commonly used methods alongside two other numerical methods for solving a parabolic equation, i.e., a finite-difference method (FDM) and finite-element method (FEM). From the TDP viewpoint, the PEM allows considering the terrain irregularity, the earth conductivity, and air refractivity for determining distributions of the electric field strength or signal attenuation. The phenomenon of air refraction is considered by using different refractivity profiles, which depend on changes in air parameters (i.e., temperature, pressure, and humidity) at different altitudes. The earth's surface is also an important factor influencing the shape of these profiles, i.e., different nature of changes in the air refractivity index is above the surface of a sea, forest, mountains, desert, etc. [41]. This factor is also important in the analysis of the earth's electrical parameters. These parameters are considered in the lower impedance boundary conditions for the parabolic equation [42; 43]. The lower boundary conditions are applied at the contact level of the air and soil layers. In practice, when analyzing the TDP at greater distances, the earth's curvature radius should be additionally taken into account [41]. Generally, the PEM is based on a numerical solution of the 2D parabolic equation in the Cartesian system:  
\[\frac{\partial^{2}}{\partial z^{2}}E(x,z)-2 ik_{0}\frac{\partial}{\partial x}E (x,z)+k_{0}^{2}\Big{(}n^{2}-1\Big{)}E(x,z)=0, \tag{1}\]  
where \(E(x,z)\) is electric field strength for the assumed radius--terrain profile (generalized \(x\) coordinate) and for any height (\(z\) coordinate), \(i=\sqrt{-1},\ k_{0}=2\pi/\lambda\) is the wavenumber in the vacuum, \(\lambda\) is the wavelength, \(n=\sqrt{\mu_{r}\epsilon_{r}}\) is the refractivity index of air, \(\mu_{r}\) and \(\epsilon_{r}\) are the relative magnetic and electric permeability of air, respectively. Equation (1) is a parabolic approximation of the Helmholtz wave equation, whose full-wave solution is provided by the PEM [43]. This equation is solved numerically using one of the three mentioned methods (i.e., SFFT, FDM, or FEM) in a sequential manner from the TX along an analyzed terrain profile determined on a given azimuth direction of a propagation radius. Detailed descriptions of the algorithms are presented in [34; 35]. A free FEM implementation in the MATLAB environment is also available [44; 45]. We used the PEM implementation in MATLAB based on the SSFT approach, which considers the refractivity profile, terrain topography, impedance boundary conditions, and the earth curvature radius. In this case, we use the digital terrain elevation data (DTED) for determining the terrain profiles [46; 47].