Corresponding author: Q. Zhang, College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China. (email@example.com)
 In this paper we extend the Cooray-Rubinstein (C-R) formula into a mixed propagation path (vertically stratified conductivity) and estimate the lightning horizontal electric fields over the mixed path, and we have examined its accuracy at distances of 100 m to 1000 m from the lightning channel by using finite difference time domain (FDTD). When the ground conductivity near the observation point is less than that near the strike point, it is noted that the C-R formula predicts a satisfactory accuracy in the initial several microseconds after the beginning of the return stroke. However, when the ground conductivity near the observation point is larger than that near the strike point, the initial negative excursion predicted by the C-R formula is underestimated, and the error increases with the increase of the difference between the two sections conductivity.
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 The lightning horizontal electric fields are very important in the Agrawal's coupling model [Agrawal et al., 1980] for the field-power line-coupling model. In order to predict the lightning-radiated horizontal electric field above the homogeneous and finitely conducting ground, Cooray [1992, 2002] and Rubinstein  proposed a Cooray–Rubinstein (C-R) simplified formula, which has been examined to have a reasonably good accuracy by using numerical solution of Sommerfeld's integrals at distances of tens of meters to 1 km with a conductivity ranging from 0.1 to 0.001 S/m [Cooray, 2010; Delfino et al., 2008a, 2008b; Shoory et al., 2005]. For a horizontally stratified conducting ground, Shoory et al. [2011b] presented a new formula for evaluating the lightning horizontal electric field. However, their new formula is similar to the C-R formula, which can be viewed as the generalization of the C-R formula both for the homogeneous conducting and horizontally stratified conducting ground.
 For a vertically stratified ground (mixed-path), Cooray and Ming  and Shoory et al. [2011a] analyzed the propagation of lightning-radiated electromagnetic fields over a mixed propagation path. However, they only considered the vertical components of the electric field and did not discuss the horizontal electric field. To the best of our knowledge, no one has estimated the lightning horizontal electric field over the mixed propagation path by using any simplified approach. Therefore, in this paper we will attempt to estimate the lightning horizontal electric field over a mixed propagation path by using the C-R formula and test its accuracy by using finite difference time domain (FDTD).
2. The Simplified Formula for a Mixed Propagation Path
where, the first term is the surface impedance term (SIT) and the second term is the ideal field term (IFT). Es,σ(h, d, jω) and Es,∞(h, d, jω) are the lightning horizontal electric fields above the mixed path and perfectly conducting ground at a distance d over a height h, respectively. Hϕ,∞(0, d, jω) is the azimuthal magnetic field on the perfectly conducting ground level. Z2 is the surface impedance near the observation point with a conductivity σ2 and a dielectric constant ε2 = ε0εr2, ω is the angular frequency. W(0, d, jω) is the attenuation function along the mixed propagation path. For a mixed propagation path, two different attenuation functions are given by Wait  and Wait and Walters [1963a, 1963b]:
where W1(0, d, jω) and W2(0, d, jω) are the attenuation functions for the first and second sections, respectively:
where “erfc” is the complementary error function, c is the light speed, j = . Δn(n = 1, 2) is the normalized surface impedance corresponding to each section of the ground defined as [Hill and Wait, 1980; Wait, 1998]:
when |Δ2| < |Δ1|, we can use the expressions of equation (2), and use equation (3) when |Δ2| > |Δ1| [Hill and Wait, 1980]. Z1 and Z2 are the surface impedance functions for the first and second sections, respectively.
3. Validity Assessment of the C-R Formula
 In the following analysis, we calculate the ideal lightning electromagnetic fields over a perfectly conducting ground by using the formulas provided by Thottappillil and Rakov  and Thottappillil et al. , based on the modified transmission line model with a linear decay of current with height (MTLL) [Rakov and Dulzon, 1991]. The channel height is assumed to be H = 7.5 km and the return stroke speed is v = 1.5 × 108 m/s. The parameters of the return stroke discharging current for the first return stroke and subsequent return stroke current waveforms are referred to Rachidi et al. .
 The accuracy of the extended C-R formula is examined by using FDTD simulation technique. The simulation domain of the FDTD technique is shown in Figure 1. The working space is 2000 × 2000, which is divided into square cells of Δd × Δz = 1 m × 1 m, the time increment is set to 1.66 ns, and the first-order Mur absorbing boundary condition is employed in order to simulate unbounded space.
3.1. For the First Return Stroke
 Let us first consider the ocean-land mixed path and the lightning horizontal electric field at d = 100 m and 1000 m from the lightning channel, and the observation point is at a height of h = 2 m above the ground. We assume that the lightning flash strikes on the ocean surface (see Figure 1), the ocean electric parameters are assumed to be σ1 = 4 S/m,εr1 = 80, and the ground electric parameters (near the observation point) are assumed to be σ2 = 0.001 S/m, εr2 = 1 in Figures 2a and 2b, and σ2 = 0.01 S/m, εr2 = 10 in Figures 2c and 2d. From Figures 2a and 2b, it is noted that the horizontal electric field is characterized by an bipolar waveform with an obvious initial negative excursion followed by a positive overshoot at the late times, and with the increase of the observation distance, the contribution of the surface impedance term (SIT) in equation (1) becomes more predominant, resulting in more and more significant negative excursion characteristics. The C-R formula has a good accuracy at early times within several microseconds, but it may cause some errors at the late time response. As the conductivity of two sections is approaching, the accuracy of the C-R formula is improved (Figures 2c and 2d).
 For other cases of the two-section mixed path, the C-R formula also reproduces in a satisfactory manner the horizontal field, as shown in Figure 3. However, the accuracy of the C-R formula is affected by the direction of the field propagation from one section to another. For example, when the field propagates from a section of lower conductivity to a higher one (Figure 3a), the C-R formula will cause a little error in the initial negative excursion. When passing from a section of higher conductivity to a lower one (Figure 3b), it predicts a good accuracy at initial times within several microseconds.
3.2. For the Subsequent Return Stroke
Figure 4 further shows the simulated results at d = 100 m and 1000 m from the lightning channel, and the observation point is at a height of h = 2 m above the ground. Note that the C-R formula still has a satisfactory accuracy at early times within several microseconds. However, compared with Figures 2 and 4, we can find that the C-R formula predicts a better result for the first return stroke than that for the subsequent return stroke, because the first return stroke has relatively more low frequency components which are less affected by the finitely conducting ground.
Table 1 shows the detail error analysis at a distance of d = 1000 m from the lightning channel for different values (e.g., the section width dl and the height of observation point h as shown in Figure 1). It is found that the parameter dl has no obvious effect of the accuracy of the C-R formula, however, the error increases with the increase of the height of the observation point h. Similar to the first return stroke, the accuracy of the C-R formula for the subsequent return stroke is also affected by the direction of the field propagation from one section to another. For example, when the field passes from one section with a lower conductivity of 0.001 S/m to another with a higher conductivity of 0.1 S/m, the field error over a height of 10 m is nearly two times larger than that over a height of h = 2 m. However, when crossing the boundary in the opposite direction, the error is nearly consistent.
Table 1. Effect of the Different Values (the Section Width dl and the Height of Observation Point h as Shown in Figure 1) on the Accuracy of C-R Formula at a Distance of d = 1000 m for the Subsequent Return Stroke
σ (S/m), εr
Initial Negative Peak (V/m)
σ1 = 0.001, σ2 = 0.1 εr1 = 1, εr2 = 30
σ1 = 0.1, σ2 = 0.001 εr1 = 30, εr2 = 1
σ1 = 0.001, σ2 = 0.1 εr1 = 1, εr2 = 30
σ1 = 0.1, σ2 = 0.001 εr1 = 30, εr2 = 1
 Also, it is well known that the temporal variation of the calculated electric field depends on many parameters such as current waveform, current decay along the channel as well as the return stroke velocity along with height [Cooray and Orville, 1990]. Figure 5 shows the effect of the different return stroke models on the accuracy of C-R formula for (a) TL and (b) MTLE, where λ is the decay height constant, λ = 2000 m [Nucci et al., 1988; Rachidi and Nucci, 1990]. Note that, although the different return stroke models predict different field values, they have little effect on the accuracy of the C-R formula.
 Compared with Figures 2 and 4, it is found that the initial negative excursion is sharper for the subsequent return stroke. In order to explain this case, the considered electric parameters are assumed to be the same as in Figure 2b. Figure 6 shows the comparison of the surface impedance term (SIT) and the ideal field term (IFT) for the first and subsequent return stroke from equation (1). It is found that the initial negative excursion is primarily dominated by SIT, and SIT is the ideal azimuthal magnetic field multiplied by the factor W(0, d, jω) · Z2, and the latter increases with the increase of the lightning frequency, as shown in Figure 7 in decibel (i.e., 20 log10|W(0, d, jω) · Z2|). As a consequence, because the subsequent return stroke has relatively more high frequency components, the corresponding initial negative excursion is sharper.
4. Conclusion and Discussion
 In this paper we extend the C-R formula into the case of a mixed propagation path, and have examined its accuracy by using FDTD technique at distances of 100 m to 1000 m from the lightning channel. When the ground conductivity near the observation point is less than that at the strike point, it is found that the C-R formula predicts a satisfactory accuracy in the initial several microseconds after the beginning of the return stroke. When the field passes from a section of lower conductivity to a higher one, the initial negative excursion predicted by the C-R formula is underestimated, and the error increases with the increase of the observation distance and the difference between the conductivity of two layers. Because the C-R formula is usually used to evaluate the horizontal electric field at close distances, and the magnetic field that appears in equation (1) is the close magnetic field, which is composed of the induction field component and radiation field component, but the attenuation functions given in equations (2) and (3) are correspond to radiation fields.
 Also, although the temporal variation of the calculated electric field depends on many parameters such as current waveform, current decay as well as the return stroke velocity along the height along the channel, the accuracy of the C-R formula seems not to be affected by the return stroke model.
 The research was supported by the National Natural Science Foundation of China (40975002), Commonwealth Industry Research Project of China (GYHY200806014), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).