## 1 Introduction

[2] The accurate estimation of the lightning-induced voltages on overhead line is very important for the lightning protection. According to the field-line coupling model [*Agrawal et al*., 1980], the total induced wave on the overhead line is composed of the two components named by the incident- and scatter-induced waves. The incident- and scatter-induced waves are caused by the vertical and horizontal electric field, respectively; however, it is noted that the dominance of any of these field components depends on the tilt of the lightning return stroke channel and the orientation of the overhead line with respect to the lightning channel. Within hundreds of meters from the lightning channel, the vertical electric field is less affected by the finitely conducting ground; however, the horizontal electric field is very sensitive to the finitely conductivity of the earth. There are several approximate formulas for predicting the lightning horizontal electric field in frequency domain and time domain. [*Barbosa and Paulino*, 2007; *Barbosa et al*., 2013; *Caligaris et al*., 2008; *Cooray*, 1992, 2002; *Delfino et al*., 2008; *Khosravi et al*., 2013; *Norton*, 1936; *Rachidi et al*., 1996; *Rubinstein*, 1996; *Wait*, 1997; *Zeddam and Degauque*, 1987]. Among them, the most remarkable one is the Cooray-Rubinstein (C-R) approximate formula in frequency domain [*Cooray*, 1992, 2002; *Rubinstein*, 1996], which has been proved to have a reasonably good accuracy at distances of tens of meters to 1 km with a conductivity ranging from 0.1 S/m to 0.001 S/m for homogeneously conducting ground [*Caligaris et al*., 2008; *Cooray*, 2010; *Shoory et al*., 2005]. In the last two years, the Cooray-Rubinstein (C-R) formula has been extended into the horizontally stratified ground and mixed propagation path. For instance, *Shoory et al*. [2011] have extended the C-R formula into the horizontally stratified conducting ground, and *Zhang et al*. [2012] extended it into the mixed path and estimated its accuracy at distances of 100 m to 1000 m from the lightning channel by using finite-difference time-domain (FDTD) method.

[3] Recently, *Zhang et al*. [2013] have further extended the C-R formula into a rough and ocean land mixed propagation surface and analyzed the propagation effect of the roughness of the ocean surface and land section on the lightning-radiated horizontal field. However, the accuracy of the extended C-R formula over a rough ground surface with homogeneous or vertically stratified (mixed) conductivity has not been validated by using other techniques, which will restrict the extensive applicability of the C-R formula. More recently, *Li et al*. [2013] have developed a three-dimensional (3-D) FDTD technique for simulating the lightning field over the two-dimensional (2-D) rough boundary condition. It is noted that the effect of the 2-D surface roughness on the horizontal field cannot be ignored even at a distance of 100 m from the lightning channel, and the increase of the land roughness results in a lower field magnitude because of the more propagation attenuation, compared with smooth ground surface.

[4] However, although the 3-D FDTD method can simulate the lightning field over the rough and conducting ground, the FDTD method is very complex and time consuming; in practical engineering applications, the approximate method maybe more efficient and valuable than the FDTD method. Therefore, in the following section, we will first extend C-R formula into a rough ground with the fractal geometry, and then validate its accuracy by using our 3-D FDTD method proposed by *Li et al*. [2013].