International Journal of Antennas and Propagation

Volume 2018, Article ID 9353294, 10 pages

https://doi.org/10.1155/2018/9353294

## A New Approximate Method for Lightning-Radiated ELF/VLF Ground Wave Propagation over Intermediate Ranges

^{1}Key Laboratory of Meteorological Disaster, Ministry of Education (KLME)/Joint International Research Laboratory of Climate and Environment Change (ILCEC)/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disaster (CIC-FEMD)/Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration, Nanjing University of Information Science and Technology, Nanjing, China^{2}Yunnan Electric Power Test Institute (Group) Co., Ltd., Electric Power Research Institute, Kunming, China

Correspondence should be addressed to Qilin Zhang; moc.361@17niliqgnahz

Received 20 December 2017; Revised 28 April 2018; Accepted 9 May 2018; Published 11 July 2018

Academic Editor: Francesco D'Agostino

Copyright © 2018 Wenhao Hou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new approximate method for lightning-radiated extremely low-frequency (ELF) and very low-frequency (VLF) ground wave propagation over intermediate ranges is presented in this paper. In our approximate method, the original field attenuation function is divided into two factors in frequency domain representing the propagation effect of the ground conductivity and Earth’s curvature, and both of them have clearer formulations and can more easily be calculated rather than solving a complex differential equation related to Airy functions. The comparison results show that our new approximate method can predict the lightning-radiated field peak value over the intermediate range with a satisfactory accuracy within maximum errors of 0.0%, −3.3%, and −8.7% for the earth conductivity of 4 S/m, 0.01 S/m, and 0.001 S/m, respectively. We also find that Earth’s curvature has much more effect on the field propagation at the intermediate ranges than the finite ground conductivity, and the lightning-radiated ELF/VLF electric field peak value (V/m) at the intermediate ranges yields a propagation distance (km) dependence of .

#### 1. Introduction

Lightning-radiated very low-frequency (VLF) (3–30 kHz) and low-frequency (LF) (30–300 kHz) signals can propagate along a spherical earth with finite conductivity at an intermediate distance of hundreds to a couple of thousand kilometers, and the observed lightning-radiated electromagnetic field signal would be significantly attenuated and distorted due to the propagation effects (e.g., Wait [1–4], Shao and Jacobson [5], and Zhang et al. [6–9]). In the intermediate range (within 1500 km), a planar earth assumption will bring some obvious errors, and the propagation effect of a spherical earth should be considered.

The general problem of the radiation from an antenna and the propagation of ground wave over homogeneous earth has been studied since the 1910s (e.g., Watson [10, 11], Sommerfeld [12], Van der Pol [13], and Norton [14]) and has been discussed in detail by Wait [1–4]. In addition, the ground wave propagation attenuation function for spherical inhomogeneous earth has been derived by Wait [3, 4]. However, the formula of attenuation function involves the roots of a complex differential equation that is related to the highly oscillatory Airy functions, and the computation of the propagation attenuation function is very difficult and time consuming. In 1980, Hill and Wait [15] generalized the previous computation methods of the attenuation function and presented an approximate method to calculate the ground wave attenuation function for arbitrary surface impedance along the spherical earth surface. In their method, different approximate formulas were used according to the phases of normalized surface impedance and a fourth-order Runge-Kutta formula had to be used to obtain the roots of differential equations. In 1993, Maclean and Wu [16] presented a Taylor series method to compute the attenuation function, and the roots of the complex differential equation were approximated by a Taylor series in their method.

In 2009, in order to study the propagation effect of lightning-radiated VLF/LF field along the spherical earth with finite conductivity, based on the formulas presented by Wait [3, 4], Shao and Jacobson [5] made some modifications for their study. In their method, a complex Newton–Raphson’s root-finding method was used to solve the complex differential equation related to the highly oscillatory Airy functions. Although this method has high accuracy and strong applicability, the iteration process may be somewhat complicated and time consuming.

Therefore, in this paper, we will present an improved new approximate method for computing the propagation effect of lightning-produced extremely low-frequency (ELF) and very low-frequency (VLF) ground wave propagation over intermediate ranges. In our method, we will divide the ground wave attenuation function presented by Wait [3, 4] into two attenuation factors representing the propagation effects of the finite ground conductivity and Earth’s curvature, and these two attenuation factors have more clear and simple expressions in the frequency domain, which can be easily calculated by multiplying them rather than solving a complex differential equation related to Airy functions.

#### 2. Method Introduction

##### 2.1. General Equations

For a vertical electric dipole source located on the surface of a smooth spherical earth, the vertical electric field strength at a great circle distance can be expressed as [3, 4] where is the vertical electric field of the same dipole source located on a flat and perfectly conducting ground and is the attenuation function accounting for the Earth’s curvature () and finite ground conductivity (). For an electric dipole with current moment , the produced vertical electric field on the earth surface at a long distance can be approximately calculated by the following formula, which is derived from Uman [17]. where is the current moment, is the great circle distance between the source and the receiver, is the light speed, and is the dielectric constant.

For the electromagnetic field computation of the cloud-ground lightning, both the source and the receiver are assumed to be on the earth surface and the attenuation function in (1) can be approximately expressed as where , , the normalized earth surface impedance , and the wavenumber in the earth . is the Earth’s radius, and is the angular frequency. and are the dielectric constant and magnetic permeability of free space, respectively. and are the Earth’s relative dielectric constant and conductivity, respectively. are the roots of the complex differential equation and is expressed as where and are the Airy functions defined by Miller [18].

In order to obtain the roots , Shao and Jacobson [5] used the Newton–Raphson root-finding method, and the initial guess for the root is facilitated with the mean of the two roots from and corresponding to and , respectively. This method is very complicated because the Airy functions are highly oscillatory in the entire complex plane, and we have to iterate many times to obtain roots for different angular frequencies and different earth conductivities. However, because the Newton–Raphson root-finding method has higher accuracy, we will regard the results of this method as a “true value” to evaluate the accuracy of our proposed approximation algorithm.

##### 2.2. Our Approximate Method

The ground conductivity and Earth’s curvature are the two factors affecting the propagation of electromagnetic wave over the earth surface; in our approximate method, the propagation attenuation function in (1) is divided into two factors where and represent the propagation attenuation factors caused by the ground conductivity () and Earth’s curvature (), respectively.

The attenuation function just describes the propagation effect of planar earth with finite conductivity, without considering the curvature of the earth, and it can be expressed as below (e.g., Wait [19], Cooray [20], Zhang et al*.* [8, 9]):
where , is the complementary error function, is the light speed, and other symbols have the same expression and meaning as shown in (3).

Attenuation function describes the effect of Earth’s curvature; it corresponds to a perfectly conducting but spherical ground, which means in (3). Then (3) and (4) can be simplified as

The parameter in (8) is the roots of (9) and can be approximately expressed as below according to Sollfrey [21]:

The error of this approximate expression (10) is 0 for , 0.0024 for , 0.0012 for , and less than 0.0005 for all higher values of [21]. Note that we have slightly modified the original formulas in Sollfrey [21] in order to let the value of in (3), (8), (10), and (11) both start at 1.

#### 3. Results and Analysis

##### 3.1. Validation of Our Approximate Method

Firstly, according to (3), we compute Wait’s original formula by using the Newton–Raphson root-finding method and compare our result with that presented by Shao and Jacobson [5], as shown in Figure 1. The attenuation functions in the frequency domain ranges from 0 to 1000 kHz. The ground conductivity is assumed to be 0.02 S/m, and both source and observed sites are located on the ground. It is found that the calculated attenuation function by our Newton–Raphson root-finding codes is nearly the same as that presented in [5], which means that our Newton–Raphson root-finding codes are correct.