## 2 Materials and Methods  
### Basic Theory  
A time harmonic dependence of \(e^{-wt}\) is assumed for EM fields, and the SI unit system is adopted. In the frequency domain, EM fields in the space are distributed according to [PERSON]'s equations:  
\[\
abla\times\mathbf{E}=i\omega\mu\mathbf{H}, \tag{1}\]  
\[\
abla\times\mathbf{H}=\left(\sigma\text{-}i\omega\varepsilon\right)\mathbf{E}. \tag{2}\]  
where \(\mathbf{E}\) is the vector electric field, \(\mathbf{H}\) is the vector magnetic field, \(\omega\) is the angular frequency, \(\mu\) is the magnetic permeability, \(\varepsilon\) is the dielectric constant, and \(\sigma\) is the electrical conductivity. Substituting Equation (1) into Equation (2), we can then get the second order differential equation for electric field \(\mathbf{E}\) as:  
\[\
abla\times\left(\
abla\times\mathbf{E}\right)-k^{2}\mathbf{E}=0 \tag{3}\]  
with  
\[k^{2}=i\omega\mu\sigma+\omega^{2}\varepsilon\mu \tag{4}\]  
where \(k\) is the wavenumber. The first term in Equation (4) accounts for the induction current (imaginary part of \(k^{2}\)), and the second term (real part of \(k^{2}\)) represents the displacement currents in the medium. In some applications, either the induction part or the displacement part is typically considered, and we refer to the wavenumber including both the induction and displacement terms as the complete wavenumber in this paper for convenience. As indicated in Equation (4), the wavenumber is a function of physical properties of the medium (i.e., magnetic permeability, dielectric constant, and electrical conductivity) and the operating frequency.  
The behavior of EM field is dependent on the wavenumber (the loss factor, \(\sigma\)/\(\omega\varepsilon\)), in which frequency is typically used as a design parameter, defining different geophysical applications. At low frequencies (typically less than \(10^{5}\) Hz), the wavenumber can be simplified to:  
\[k^{2}=i\omega\mu\sigma, \tag{5}\]  
where the displacement current is neglected, and only the induction current exists in the medium (namely, the quasi-static approximation). The EM fields are said to be diffusive, as used in MT applications and audio-magnetotelluric (AMT) surveying. At high frequencies (e.g., greater than \(10^{5}\) Hz), the wavenumber can be reduced to:  
\[k^{2}=\omega^{2}\varepsilon\mu. \tag{6}\]  
In such a scenario, the EM field propagates in space without significant attenuation, acting as waves, as used in ground penetrating radar (GPR) applications. However, in real applications, the critical frequency to use in Equation (5) or Equation (6) is usually not known a priori.  
In our work, we mainly focus on the geophysical application of the EM diffusion, in which the simplified wavenumber in Equation (5) is routinely applied. For the EM forward modeling based on the electric-field formulation, the air layer needs to be included in the computational domain with conductivities typically set to be small values (e.g., \(10^{-6}\)-\(10^{-14}\) S/m) and the displacement current is typically ignored. We calculate the loss factor \(\sigma\)/\(\omega\varepsilon\) for typical conductivity encountered in geophysical applications, as listed in Table 1. In the earth (normally greater than \(10^{-4}\) S/m), the second term of Equation (4) is negligibly small, and EM fields are dominantly diffusive. However, in the air, the conductivity is close to zero, which makes simplification of Equation (4) a problem.