抛物线方程法提升Loran-C在复杂地形的ASF预测效率

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本文主要探讨了在不规则地形上应用抛物线方程法（Parabolic Equation Method, PEM）来预测Loran-C（Long Range Navigation - C）附加二次因子（Additional Secondary Factors, ASFs）的问题。Loran-C是一种早期的全球导航系统，其信号传播受到地表复杂性的影响，特别是粗糙地形和变化的电参数会引入额外的延迟，即ASF，这直接影响定位精度。作者们采用了一种基于分步傅立叶变换（Split-Step Fourier Transform, SSFT）算法的PE方法来解决这一问题。该方法具有数值效率高的优点，通过将复杂的地形分解为一系列简单的步骤，能够在保持准确性的同时显著降低计算时间。与积分方程（Integral Equation, IE）方法和有限差分时域（Finite Difference Time Domain, FDTD）方法相比，PE方法的计算速度更快，通常节省几个数量级的时间。研究结果表明，抛物线方程法对ASF的预测与IE和FDTD方法的结果有很好的一致性，证实了其在处理不规则地形下的适用性。然而，与IE方法相比，PE方法在内存需求上更为节省，而且比FDTD方法更轻量级。这对于需要高效处理大量数据和计算密集型任务的实时或嵌入式系统设计来说是一个重要的优势。本文的工作对于优化Loran-C系统的性能、提高导航精度以及在复杂地形环境中进行精确信号传播建模具有重要意义。未来的研究可以进一步探索如何优化算法以适应更广泛的地形条件，并可能扩展到其他类似的无线通信技术中，如GPS或卫星通信。此外，PE方法的并行化和硬件加速也是值得探讨的方向，以进一步提升计算效率。

734 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 15, 2016

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, and Li-Li Zhou

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

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

regular 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 equa-

tion (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—Additional secondary factors (ASFs), irregular

terrain, parabolic equation (PE), split-step Fourier transform

(SSFT).

I. INTRODUCTION

ROUND 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 long-wave

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 equa-

tion (IE) [8], [9] and ﬁnite-difference time-domain (FDTD)

Manuscript received July 07, 2015; accepted August 12, 2015. Date of pub-

lication August 20, 2015; date of current version March 16, 2016. This work

was supported in part by the National Natural Science Foundation of China

under Grants No. 61271091 and No. 61401261 and the Scientiﬁc Research Pro-

gram funded by Shaanxi Provincial Education Department under Program No.

14JK1513. (Corresponding author: X iao-Li Xi.)

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

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

710048, China (e-mail: xixiaoli@xaut.edu.cn).

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

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

Color versions of one or more of the ﬁgures in this letter are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/LAWP.2015.2471079

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 large error for steeper

terrain and choppy electric parameters and is hard to be used for

a three-dimensional (3-D) 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 algo-

rithm 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 pre-

diction 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 remaining propaga-

tion path. Using the SSFT algorithm, the PE method has higher

efﬁciency and accuracy when compared to 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 M

ODEL

A. Scenario of the PE Model

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

, assuming the ﬁelds are independent of the azimuth ( )

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