## 3 Results
### Simple 1D Models  
To examine the effect of the quasi-static approximation on the EM diffusion modeling, we design two half-space models, named model I and model II, respectively. The conductivity is \(10^{-4}\) S/m for model I and \(10^{-1}\) S/m for model II. Six frequencies of \(10^{1}\), \(10^{2}\), \(10^{3}\), \(2\times 10^{3}\), \(5\times 10^{3}\), and \(10^{4}\) Hz are considered. For cases based on the quasi-static approximation, the conductivity in the air is set to be \(10^{-4}\), \(10^{-6}\), \(10^{-8}\), or \(10^{-10}\) S/m. The numerical modeling of electrical responses also using the finite-difference method are compared with the analytical solution [1]. The comparison between the x-component electrical fields calculated by treating air conductivities differently is shown in Figure 2a,b along with the analytical solution for model I. As indicated in Figure 2a,b, the numerical solution for the case using the complete wavenumber matches the analytical solution perfectly. For cases based on the quasi-static approximation, as the conductivity used for the air layer decreases, the differences between the numerical solution and the analytical solution become smaller for both real and imaginary parts of the x-component of the electrical field. Please note that a further decrease in the air conductivity cannot guarantee a better solution as indicated in the figure, called air conductivity saturation here for convenience. For results with the quasi-static approximation, an increase in frequency leads to a larger deviation between the numerical result and the analytical solution. For air conductivity of \(10^{-8}\) S/m, while the frequency is less than \(10^{3}\) Hz, they are generally in good agreement. A more intuitive comparison is indicated in Figure 2c,d, showing similar results as in Figure 2a,b. For cases based on the quasi-static approximation, the relative error of the numerical solution exclusively reaches more than 40% (e.g., real parts have relative errors of 100%, 79%, 45%, and 44% for air conductivity of \(10^{-4}\), \(10^{-6}\), \(10^{-8}\), and \(10^{-10}\) S/m, respectively) at \(10^{4}\) Hz. Large errors start to appear at frequencies of greater than \(10^{3}\) Hz for this simple model. Figure 2: Comparison of the \(E_{x}\) solutions from cases based on the quasi-static approximation with \(10^{-4}\) (orange dashed lines), \(10^{-6}\) (purple asterisk lines), \(10^{-8}\) (dark green square lines), and \(10^{-10}\) S/m (red circle lines) for air conductivity, and the one with the complete wavenumber (blue dotted cross lines) along with the analytical solution (solid lines) for model I. (**a**) the real part and (**b**) the imaginary part of \(E_{x}\) are indicated, and the relative errors of (**c**) the real part (**d**) the imaginary part of \(E_{x}\) are indicated. A similar comparison is carried out for model II. The real and imaginary results for the x-component of the electrical field are compared in Figure 3 along with the analytical solutions. The comparison results are generally similar to those indicated for model I. Using the complete wavenumber in the air produces accurate modeling results. For modeling based on the quasi-static approximation, using a conductivity less than or equal to \(10^{-6}\) S/m, the numerical results are in good agreement with the analytical solution over the range \(10^{1}\)-\(10^{4}\) Hz. The two modeling examples above indicate that the quasi-static approximation in the air can affect the accuracy of EM simulation based on the electric-field formulation. The extent of this impact on the modeling results is dependent on the conductivity used for the air and the conductivity of the earth, as well as the operating frequency. For a particular earth conductivity, saturation air conductivity exists. The value of saturation air conductivity is closely related to the earth conductivity (around four orders of magnitude more conductive).