**4. Electromagnetic field calculation using antenna model of the lightning channel at a lossy ground**

Based on experimentally measured characteristics of natural lightning (Berger et al., 1975; Lin et al. 1979; Anderson & Eriksson 1980) and also artificially initiated lightning discharges, LEMF can be estimated using some of the models from literature (Rakov & Uman, 1998, 2006), and lightning currents are assumed to propagate with attenuation and distortion while distributing charge along the channel. In the case of perfectly conducting ground calculations are simple in time domain, but full-wave approach in the case of a lossy ground is complex to implement, so Fourier transform is applied. After calculations are done in frequency domain, the conversion to time domain is performed by Inverse Fast Fourier transform (IFFT). A solution of the Sommerfeld's problem is required for the lossy ground. A few alternatives are proposed in literature.

## **4.1 Alternatives to full-wave time domain computation**

The problem of LEMF calculation can be easily solved directly in time domain if the ground is treated as perfectly conductive. In such a case there exist vertical electric and azimuthal magnetic field components in the observed point at the ground surface. Horizontal component of electric field is zero at the perfectly conducting ground surface, but non-zero above the ground surface. However, it exists above, under, and at the surface of the lossy ground. Vertical component of the lightning electric and azimuthal component of the magnetic field can be easily (but in that case approximately) determined at the distances greater than a kilometer under the assumption of perfectly conducting ground. For smaller distances, propagation above ground of finite conductivity results in the noticeable attenuation of the high-frequency components of electric and magnetic field, and thus in appearance of the horizontal electric field at the surface. Finite conductivity has greater impact on horizontal than on vertical electric field, so calculation of horizontal component requires rigorous computation or, at least, acceptable approximations.

Approximate formulas in frequency domain are often used for calculation of the horizontal electric field in air, up to heights of tens of meters above the ground surface. These formulas can be integrated in the calculation of LEMF in time domain, but the obtained expressions are much more complex. There are simple approximations: the assumption of perfectly conducting ground, "wavetilt" formula, Cooray's approach and Rubinstein's approach. Cooray, 1992, proposed the calculation of horizontal electric field at the surface of finitely conductive ground using azimuthal magnetic induction and the expression for ground surface impedance. He showed that this simple formula provides very accurate results at the distances of about 200m. Rubinstein, 1996, proposed expression for the horizontal electric field with two terms: 1) horizontal electric field calculated under the assumption of perfectly conducting ground, and 2) the correction factor, given as a product of the magnetic field calculated under the same assumption and the function similar as in "wavetilt" formula which represents the effect of finite conductivity. The basic assumption of Rubinstein's approximation is σ1>>ε0εr1, and that finite ground conductivity does not affect the horizontal magnetic field at the surface. If this is not the case, then more general formula can be written, known in literature as the Cooray-Rubinstein's formula (Cooray, 2002). Wait gave generalization of Cooray-Rubinstein's formula and the exact evaluation of horizontal electric field, showing under which circumstances this general expression reduces to Cooray-Rubinstein's formula (Wait, 1997). Cooray and Lindquist, 1983, and Cooray, 1987, using the attenuation function in time domain proposed by Wait, 1956, included effects of the finite conductivity, and obtained results for the electric field that are in better agreement with experiments. Terms for approximate formulas in time domain are complex, so the approach in frequency domain is preferred.
