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
G
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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