## 4 Modeling Disturbances of the Geoelectric Field  
Using information on the geoelectric structure of the underlaying crust, one may reproduce the dynamics of the telluric response for real events from geomagnetic observations, even in the absence of synchronous electric field records. The construction of a detailed global model of the Earth's electrical conductivity is a task for the future; therefore, nowadays, calculations have to use some approximate scheme [26]. A comparison of various methods showed that with a good accuracy for calculation of the telluric field, one can use the impedance relation in the plane wave approximation. This approximation, also named the strong skin-effect condition, is valid when the horizontal scale of the disturbance is much larger than the skin-depth \(d=\sqrt{2/\mu_{0}\omega\sigma}\) of the electromagnetic field with cyclic frequency \(\omega\) in the conducting Earth with electrical conductivity \(\sigma\) (Wait-Price criterion) [27; 28]. Under this condition, the actual spatial structure of geomagnetic disturbance does not influence the impedance relationship, so the primary disturbance should not necessarily be a \"plane vertically incident\" wave. For practically important calculations of GIC, the situation is greatly simplified by the fact that integral estimates of the potential difference throughout an extended system (at least several hundred kilometers) are needed, and therefore the necessary estimates can be performed with a sufficient accuracy with a relatively rare network of magnetometers and a rough model of conductivity. Figure 3: An example of localized magnetic disturbances (MPE) recorded on 9 September 2019 (**left-hand**) and 17 December 2019 (**right-hand**). The format of the figure is the same as in Figure 2.  
The geoelectric field in the plane wave approximation is coupled with geomagnetic variations through the surface MT impedance \(Z(\omega)\). In media with a low conductivity, the negative effects of strong magnetic disturbances are most probable, because the excited telluric fields and potential differences turn out to be larger, so the induced currents mainly flow through conductive elements of industrial networks. On the Earth's surface, the impedance relation holds between the spectral amplitudes of the horizontal electric \(\textbf{E}=\{Ex,\)\(\textit{Ey}\}\) and magnetic \(\textbf{B}=\{X,\)\(Y\}\) fields as follows: \(\textbf{E}=\mu_{o}^{-1}\textbf{ZB}\), where **Z** is the impedance matrix. In a simplified approach, the use of 1D impedance \(Z(\omega)\) in the impedance boundary condition may be sufficient, namely  
\[\left( \begin{array}{c}E_{x}(\omega)\\ E_{y}(\omega)\end{array} \right)=\mu_{0}^{-1}\left( \begin{array}{cc}0&Z( \omega)\\ -Z(\omega)&0\end{array} \right)\left( \begin{array}{c}X(\omega)\\ Y(\omega)\end{array} \right) \tag{1}\]  
For uniform conductivity of the Earth, the impedance \(Z(\omega)=\sqrt{-i\omega\mu_{0}\rho}\) (where \(\rho=1/\sigma\) is the electric resistivity) depends on the period of the incident wave as a power law \(Z(T)\propto T^{-1/2}\). In the vicinity of coastal areas with a sharp gradient of conductivity, the use of 3D impedances may be necessary to account for horizontal inhomogeneity [29]. The telluric field enhancement in such regions may be as high as several times. The voltage on a virtual transmission line modeled using a piecewise 1-D conductivity profile was estimated to differ by 10-20% (depending on orientation relative to the geoelectric field polarization) compared with modeling that accounted for 3-D conductivity structure [3]. Here, using the recorded time series of \(\textbf{B}(t)=\{H,\)\(E\}\) in geomagnetic coordinates and frequency domain MT impedance \(Z(\omega)\), we synthesized the temporal variations in the electric field \(\textbf{E}(t)=\{E_{\text{H}},\)\(E_{\text{E}}\}\) by a two-step procedure. First, a spectral transform of the recorded time series was used in the impedance condition (1), and then, the inverse Fourier transform was applied to the spectrum of the telluric field.