**2. Wave Field Transformation Theory Based on Fractional-Order Cole–Cole Model**

#### *2.1. Transformation from Real Di*ff*usion Field to Fictitious Wave Field*

The Maxwell equation in the frequency domain of the real diffusion field is written as follows:

$$-\nabla \times \mathbf{H}(\mathbf{x}, \omega) + \sigma(\omega)\mathbf{E}(\mathbf{x}, \omega) = -\mathbf{J}(\mathbf{x}, \omega), \tag{1}$$

$$\nabla \times \mathbf{E}(\mathbf{x}, \omega) - i\omega\mu \mathbf{H}(\mathbf{x}, \omega) = -\mathbf{K}(\mathbf{x}, \omega), \tag{2}$$

where E and H are the electric and magnetic fields in the real diffusion field, respectively, J denotes the current density, and K denotes the magnetic current density. ω is the angular frequency, μ is permeability tensor, and σ is conductivity, while x is the direction.

The IP effect can be described by the Cole–Cole model [16].

$$
\sigma(\omega) = \sigma\_{\infty} (1 - \frac{\eta}{1 + \left(-i\omega\tau\right)^{\varepsilon}}),
\tag{3}
$$

where σ<sup>∞</sup> is the conductivity corresponding to infinite frequency, τ is the characteristic time constant, and c and η are the frequency dependence and chargeability, respectively.

Applying Equation (3) to the frequency-domain Maxwell equation (Equation (1)), we can get

$$-\nabla \times \mathbf{H}(\mathbf{x}, \omega) + \sigma\_{\text{os}} \mathbf{E}(\mathbf{x}, \omega) - \frac{\sigma\_{\text{os}} \eta}{1 + \left(-i\omega \tau\right)^{\text{c}}} \mathbf{E}(\mathbf{x}, \omega) = -\mathbf{J}(\mathbf{x}, \omega). \tag{4}$$

A fictitious dielectric constant is defined by the conductivity tensor (Mittet, 2010).

σ<sup>∞</sup> = 2ω0ε∞ , (5)

where ω<sup>0</sup> = 2π × *f*<sup>0</sup> is the scaling parameter, *f*<sup>0</sup> = 1*H*z, and ε∞ is the fictitious dielectric permittivity tensor. Substituting Equation (5) into Equation (4), we can get

$$-\nabla \times \mathbf{H}(\mathbf{x}, \omega) + 2\omega\eta \varepsilon\_{\rm os} \,' \mathbf{E}(\mathbf{x}, \omega) - \frac{\eta}{1 + (-i\omega \tau)^{\rm c}} 2\omega\eta \varepsilon\_{\rm os} \,' \mathbf{E}(\mathbf{x}, \omega) = -\mathbf{J}(\mathbf{x}, \omega). \tag{6}$$

Multiplying both sides of Equation (6) by -−iω <sup>2</sup>ω<sup>0</sup> gives

$$-\nabla \times \left[\sqrt{\frac{-i\omega}{2\omega\_0}} \mathbf{H}(\mathbf{x}, \omega)\right] + \left[\sqrt{\frac{-i\omega}{2\omega\_0}} 2\omega\wp\varepsilon\_{\rm obs} \,' \mathbf{E}(\mathbf{x}, \omega)\right] - \left[\frac{\eta}{1 + (-i\omega \varepsilon)^2} \sqrt{\frac{-i\omega}{2\omega\_0}} 2\omega\wp\varepsilon\_{\rm obs} \,' \mathbf{E}(\mathbf{x}, \omega)\right] = -\sqrt{\frac{-i\omega}{2\omega\wp}} \mathbf{J}(\mathbf{x}, \omega). \tag{7}$$

Equation (2) can be rewritten as

$$
\nabla \times \mathbf{E}(\mathbf{x}, \omega) - \sqrt{-2i\omega\omega\_0} \mu \sqrt{\frac{-i\omega}{2\omega\_0}} H(\mathbf{x}, \omega) = \mathbf{K}(\mathbf{x}, \omega). \tag{8}
$$

The relationship between the real diffusion field and fictitious wave field was given by Reference [23].

$$-i\omega' = \sqrt{-2i\omega\omega\_0} \tag{9}$$

$$E'(\mathbf{x}, \omega') = E(\mathbf{x}, \omega),\tag{10}$$

$$H'(\mathbf{x}, \omega') = \sqrt{\frac{-i\omega}{2\omega\_0}} H(\mathbf{x}, \omega), \tag{11}$$

$$J'(\mathbf{x}, \omega') = \sqrt{\frac{-i\omega}{2\omega\_0}} I(\mathbf{x}, \omega),\tag{12}$$

$$K'(\mathbf{x}, \omega') = K(\mathbf{x}, \omega), \tag{13}$$

where ω<sup>0</sup> = 2π is the scaling parameter, ω is the angular frequency in the fictitious wave field, *J* and *K* are the current and magnetic current densities in the fictitious wave field, and E and H are the electric and magnetic fields in the fictitious wave domain.

Using Equations (9)–(13), Equations (7) and (8) can be written in terms of fictitious wave fields.

$$-\nabla \times \mathbf{H}'(\mathbf{x}, \omega') - i\omega' \varepsilon\_{\infty} \,' \mathbf{E}'(\mathbf{x}, \omega') + \frac{\eta \dot{\mu} \omega' \varepsilon\_{\infty} \,'}{1 + \left(\frac{\tau}{2\omega\_0}\right)^{\varepsilon} (i\omega')^{2\varepsilon}} \mathbf{E}'(\mathbf{x}, \omega') = -\mathbf{J}'(\mathbf{x}, \omega'), \tag{14}$$

*Appl. Sci.* **2020**, *10*, 1027

$$\nabla \times \mathbf{E}'(\mathbf{x}, \omega') + i\omega'\mu \mathbf{H}'(\mathbf{x}, \omega') = \mathbf{K}'(\mathbf{x}, \omega'). \tag{15}$$

Because the appropriate kernel used to decompose the spectra is closer to a Cole–Cole function with an exponent c of 0.5 [26], then

$$-\nabla \times \mathbf{H}'(\mathbf{x}, \alpha') - i\omega' \varepsilon\_{\alpha \nu} \mathbf{'} \mathbf{E}'(\mathbf{x}, \alpha') + \frac{\eta \mathbf{i} \alpha' \varepsilon\_{\alpha \nu} \mathbf{'}}{1 + \sqrt{\frac{\tau}{2\omega\_0}} \mathbf{i} \alpha'} \mathbf{E}'(\mathbf{x}, \alpha') = -\mathbf{J}'(\mathbf{x}, \alpha'), \tag{16}$$

$$\nabla \times \mathbf{E}'(\mathbf{x}, \omega') + i\omega'\mu \mathbf{H}'(\mathbf{x}, \omega') = \mathbf{K}'(\mathbf{x}, \omega'). \tag{17}$$

Assuming *A* (**x**, ω ) = <sup>η</sup>*i*ω ε∞ 1+ - τ 2ω0 *i*ω **E** (**x**, ω ), the time-domain expressions (Equations (16) and (17))

can be obtained through inverse Fourier transform.

$$-\nabla \times \mathbf{H}'(\mathbf{x}, t') - \varepsilon\_{\alpha \gamma} \, ^\prime(\mathbf{x}) \partial\_{t'} \mathbf{E}'(\mathbf{x}, t') + A'(\mathbf{x}, t') = -\mathbf{J}'(\mathbf{x}, t'), \tag{18}$$

$$
\nabla \times \mathbb{E}'(\mathbf{x}, \mathbf{t}') + \mu \partial\_{\mathbf{t}'} \mathbb{H}'(\mathbf{x}, \mathbf{t}') = \mathbb{K}'(\mathbf{x}, \mathbf{t}'). \tag{19}
$$

By combining Equations (18) and (19), the electric and magnetic field x-components can be found as

$$E\_{x\_{i+1/2,j,k}}^{n+1} = E\_{x\_{i+1/2,j,k}}^n + \Delta t \frac{2\omega\_0}{\sigma\_{00}} (\partial\_y^- H\_{z\_{1+1/2,j+1/2,k}}^{n+1/2} - \partial\_z^- H\_{y\_{i+1/2,j,k+1/2}}^{n+1/2}) + \Delta t \frac{2\omega\_0}{\sigma\_{00}} A'(\mathbf{x}\_r \ \mathbf{t}') - \Delta t \frac{2\omega\_0}{\sigma\_{00}} I\_{\mathbf{x}r} \tag{20}$$

$$H\_{\mathbf{x}\_{i,j+1/2,k+1/2}}^{n+1/2} = H\_{\mathbf{x}\_{i,j+1/2,k+1/2}}^{n-1/2} - \frac{\Delta t}{\mu} (\partial\_y^+ E\_{\mathbf{z}\_{i,j,k+1/2}}^n - \partial\_z^+ E\_{y\_{i,j+1/2,k}}^n),\tag{21}$$

where

$$A'(\mathbf{x}, t') = \sqrt{\frac{2\omega\_0}{\tau}} e^{-\sqrt{\frac{2\omega\_0}{\tau}}} t' \int\_0^{t'} e^{\sqrt{\frac{2\omega\_0}{\tau}}} \eta \varepsilon\_{\infty} \left[\partial\_{I'} \mathbb{E}'(\mathbf{x}, t'')\right] dt''. \tag{22}$$

Equation (22) can be solved by iteration, as shown in Equation (23).

$$A'(\mathbf{x}, t^{n+1}) = A'(\mathbf{x}, t^n) e^{\sqrt{\frac{2\omega\_0}{\tau}}\Lambda t} - \eta \varepsilon\_{\infty} {}'E'(\mathbf{x}, t^N) (1 - e^{\sqrt{\frac{2\omega\_0}{\tau}}\Lambda t}),\tag{23}$$

where *t <sup>n</sup>* = *n*Δ*t*, and the forward staggered time step is *t <sup>N</sup>* = (*n* + <sup>1</sup> <sup>2</sup> )Δ*t*.

#### *2.2. Electrical Source Loading in Fictitious Wave Field*

As shown in Figure 1, Tx is the long wire source and Rx is the receiver. The fictitious emission source is only related to x- and y-components of the electric field, and the z-component is zero.

Here, the first-order Gaussian pulse is selected as the fictitious emitter, as shown in Equation (24).

$$\mathbf{J}'(\mathbf{t}') = -2\beta(\mathbf{t}'-\mathbf{t}\_0)\sqrt{\frac{\beta}{\pi}}\mathbf{e}^{-\beta(\ \mathbf{t}'-\mathbf{t}\_0)^2},\tag{24}$$

where β = π*f* <sup>2</sup> max, *f*max is the maximum frequency in the fictitious wave field.

**Figure 1.** Loading mode of fictitious wave field long wire source.
