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The parabolic equation method based on digital elevation model (DEM) is applied on propagation predictions over irregular terrains. Starting from a parabolic approximation to the Helmholtz equation, a wide-angle parabolic equation is deduced under the assumption of forward propagation and the split-step Fourier transform algorithm is used to solve it. The application of DEM is extended to the Cartesian coordinate system and expected to provide a precise representation of a three-dimensional surface with high efficiency. In order to validate the accuracy, a perfectly conducting Gaussian terrain profile is simulated and the results are compared with the shift map. As a consequence, a good agreement is observed. Besides, another example is given to provide a theoretical basis and reference for DEM selection. The simulation results demonstrate that the prediction errors will be obvious only when the resolution of the DEM used is much larger than the range step in the PE method.

For many years, the parabolic equation (PE) method has been widely used to model the propagation of electromagnetic waves in complex atmospheric, hydrological, and geographical environments. As a forward full-wave analysis method, PE modeling has attracted a lot of interest since it was first proposed by Leontovich and Fock [

So far, the three commonly used algorithms of PE are SSFT, the finite-difference method (FDM) [

In the past decades, a number of scholars [

In recent years, the vectorial version of the PE method has been employed to analyze the field propagation processes in road and railway tunnels where the presence of dissipative and field depolarization processes cannot be neglected. Bernardi et al

In this paper, we address the terrain problems in PE modeling and a WAPE method based on the digital elevation model (DEM) is developed using the SSFT algorithm. In fact, as a more advanced modeling approach, the DEM has already been applied in some propagation software programs. However, in the previous models, all the DEMs used are in the geographic coordinate system while in our model, the application of DEM is extended to the 3D Cartesian coordinate system and the effect of the resolution of the DEM on propagation predictions is discussed in order to provide a theoretical basis and reference for DEM selection.

In the ensuing theory, a time convention

Paraxial propagation over irregular terrain.

We introduce the reduced function associated with the paraxial direction

If the backward propagation is ignored, the PE corresponding to forward propagating waves will be

According to the operator splitting proposed by Feit and Fleck [

The error is of the order of the coupled term

If we write

Thus, the Feit-Fleck WAPE is obtained.

The above WAPE has the formal solution

The Fourier transform of

Substitute (

It means that the forward propagating field can be obtained at a given range from the field at a previous range, which is quite suitable for numerical computation.

As a digital model was created from terrain elevation data, a DEM is able to provide a 3D representation of a terrain’s surface with high efficiency, especially when the environment is complex enough. The storage format of the DEM data in the 3D Cartesian coordinate system is shown in Figure

The storage format of the DEM data.

Assuming that the transmitting antenna is located at Tx and the receiving antenna Rx, the terrain elevation data between Tx and Rx in a 2D vertical plane can be easily obtained from the DEM, and according to the PLSM [

Two examples are given in this section. In the first one, the DEM-based modeling approach is compared with the shift map to validate its accuracy while the second one deals with the effect of DEM’s resolution on prediction errors. In the following simulations, the transmitting antenna is a horizontally polarized Gaussian antenna with

The 3D digital map of the Gaussian terrain model.

The DEM of the Gaussian terrain model with a resolution of 100 m.

In order to simplify the calculation process and compare the DEM-based method with the shift map, Tx is set to (0 m, 25 km, 50 m) and Rx is set to (50 km, 25 km, 50 m); then the 2D terrain profile between Tx and Rx is given by

As Figure

Propagation factor calculated by different methods: (a) the shit map; (b) the DEM-based method.

The propagation factor at range 50 km is also compared, as shown in Figure

Propagation factor versus height at range 50 km.

Another issue of interest is the dependence of the prediction errors on the resolution of the DEM used. As shown in Table

Average error and standard deviation.

Resolution (m) | Average error (dB) | Standard deviation (dB) |
---|---|---|

10 | 0.019 | 0.335 |

50 | 0.012 | 0.377 |

100 | 0.006 | 0.406 |

200 | 0.005 | 0.485 |

300 | 0.063 | 0.721 |

400 | 0.061 | 0.847 |

500 | 0.040 | 1.291 |

600 | 0.070 | 1.326 |

700 | 0.062 | 1.802 |

800 | 0.046 | 1.952 |

Average error and standard deviation versu resolution.

In this paper, under the assumption of forward propagation, a DEM-based PE method is applied on propagation predictions over irregular terrains using the SSFT algorithm. The terrain is assumed to be perfectly conducting, and a 3D Gaussian terrain profile is simulated by using the DEM-based method and the shift map, respectively. As a result, a good agreement is observed. Furthermore, the simulation results indicate that the prediction errors will be obvious only when the resolution of the DEM used is much larger than the range step in the PE method.

The authors declare that they have no conflicts of interest.

This work was supported by National Key Laboratory of Electromagnetic Environment and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant no. 61621005).