抛物线方程法在不规则地形上优化Loran-C ASFP预测

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本文是一篇研究论文，主要关注的是在不规则地形上应用Loran-C（Long Range Navigation）的Addition Secondary Factors (ASFs)预测的抛物线方程法（Parabolic Equation Method, PEM）。Loran-C是一种早期的全球导航卫星系统，而ASFs是其在复杂地形环境下信号传播时要考虑的重要因素，它们影响了定位精度。PEM作为一种数值模拟方法，利用分步傅立叶变换（Split-Step Fourier Transform, SSFT）算法来解决这个问题，该算法在计算效率上表现出色。文章对比了PEM与两种常见的分析方法：积分方程（Integral Equation, IE）方法和有限差分时间域（Finite-Difference Time-Domain, FDTD）方法。结果显示，PEM在预测ASFs方面的结果与IE和FDTD方法有很好的一致性，表明其在准确性上是可靠的。然而，一个显著的优势在于PEM的计算时间远少于IE和FDTD方法，这使得它在处理大规模或实时地形分析时具有显著优势。尽管PEM在处理不规则地形的ASF问题时具有较高的计算效率和内存需求，但它的内存需求与IE方法相当，而小于FDTD方法，这进一步优化了资源使用。因此，抛物线方程法在处理Loran-C在复杂地理条件下信号传播的问题时，提供了一个实用且高效的解决方案，对于提高定位精度和降低计算负担有着重要的实际意义。对于研究者、工程师以及依赖Loran-C技术的领域（如航空、航海和地面通信），这篇论文提供了有价值的技术参考，特别是在寻求在有限时间内处理大量地形数据和保持定位精度的挑战时。同时，它也为未来的无线通信和信号传播模型的优化提供了新的视角和可能的改进方向。

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI

10.1109/LAWP.2015.2471079, IEEE Antennas and Wireless Propagation Letters

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

Abstract—The parabolic equation (PE) method is employed to

solve the Loran-C additional secondary factors (ASFs) over

irregular terrain. Based on the split-step Fourier transform

(SSFT) algorithm, the method has been proven to be numerically

efficient. The ASF results are compared to those of the integral

equation (IE) method and the finite-difference time-domain

(FDTD) method. Very good agreements are observed. The

computational time of the PE method is several orders less than

that of the other two. The memory requirement is similar to the IE,

and less than the FDTD method.

Index Terms—Parabolic Equation (PE), additional secondary

factors (ASFs), irregular terrain, split-step Fourier transform

(SSFT).

I. INTRODUCTION

GROUND wave propagating over complex paths with

varying electric parameters and rough terrains causes an

extra delay named additional secondary factor (ASF) [1]. It is a

significant factor ensuring the precision of longwave

navigation system and timing service. A small deviation of

ASF may result in large errors up to several kilometers [2].

Over the years, researchers have developed simplified

simulation models to calculate the ASF with different

assumptions and approximations. The three most commonly

used ones are the smooth-earth (SE), smooth-earth mixed-path

(SEMP), and irregular-earth mixed-path (IEMP) models where

the former two are very mature during the past years [3].

Formulas and calculation methods are built up based on these

models. For example, the flat-earth formula and residue Series

formula are developed based on the SE model with the

assumption that the propagation path is smooth, and the electric

parameters are homogeneous [4], [5]. The Wait integral

equation, Millington empirical formula [5], [6], and wave mode

conversion method are based on the SEMP model [7]. The

algorithms for IEMP model are usually numerical ones, such as

the integral equation (IE) [8], [9] and finite-difference

time-domain (FDTD) method [10], [11]. The IE method is

based on the assumption that the impacts from backscattering

waves are very small and can be neglected. As expected, it has a

This work was supported in part by the National Natural Science Foundation

of China (No. 61271091 and No. 61401261) and the Scientific Research

Program Funded by Shaanxi Provincial Education Department (Program No.

14JK1513 and No. 14JK1084).

Corresponding Author is X.L. Xi, e-mail: xixiaoli@xaut.edu.cn.

D. D. Wang, X. L. Xi, Y. R. Pu, and J. F. Liu are with Faculty of Automation

and Information Engineering, Xi’an University of Technology, Xi’an, 710048,

China.

L. L. Zhou is with College of Electrical and Information Engineering,

Shaanxi University of Science and Technology, Xi’an, 710021, China.

large error for steeper terrain and choppy electric parameters

and is hard to be used for a three-dimension (3D) case because

of the vulnerability of the algorithm itself. The FDTD method

has been proven to be most the precise one for complex paths

with irregular terrain, but the computational expenditures of

memory and time are huge for the large area prediction.

The parabolic equation (PE) is an approximation of the

Helmholtz equation assuming that the energy propagates in the

paraxial direction [12]. Numerical methods for solving the PE

can categorize into the finite-difference (FD) and the split-step

Fourier-transform (SSFT) algorithms [12]-[16]. The FD

algorithm is capable for complex terrains but needs fine

sampling of the range, leading to huge computational

expenditures [17]. The SSFT algorithm allows for a fairly large

range step, but the boundary condition must be enforced to flat

[12], [13]. The main issue with the PE method is its poor

accuracy for short propagation ranges with large angles due to

its paraxial approximation nature [18]. Therefore, the initial

field injection needs special attention. In [19]- [22], the PE

method has been successfully used to predict the path loss of

ground wave propagation in the HF/VHF band, where the

initial fields are injected using the near-field/far-field

transformation of the antenna pattern.

In this letter, we employ the PE method for Loran-C ASF

prediction over irregular terrain. We propose a hybrid solution

that uses analytical equations to calculate the initial field for the

PE method. The field distribution at a cross-sectional plane

with a certain distance to the radiation source is calculated

analytically, where the Earth surface must be flat between the

source and the plane. The PE method then takes the field on the

cross-sectional plane as input and continues to solve the rest

propagation path. Using the SSFT algorithm, the PE method

has higher efficiency and accuracy when compared with the IE

and FDTD method. This letter is organized as follows: Section

II outlines the derivations of the PE computational model.

Validation of the PE method is given in Section III via several

representative examples.

II. PE MODEL

A. Scenario of the PE model

We work with A two-dimensional (2D) cylindrical

coordinates

 

,uv

, assuming the fields being independent of

the azimuth (



) direction, where

and

correspond to the

range and height, respectively. Time dependence is assumed as





. The simulation model shown in Fig. 1 composes an

analytical computation domain (I) and a PE domain (II). The

Parabolic Equation Method for Loran-C ASF

Prediction over Irregular Terrain

Dan-Dan Wang, Xiao-Li Xi, Member, IEEE, Yu-Rong Pu, Jang-Fan Liu, Li-Li Zhou

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