*2.1. Forward Modeling*

The MT method involves measuring orthogonal components of the electric field **E** and the magnetic field **H** at the Earth's surface (Figure 1a). The electromagnetic field we observe is excited by the natural magnetic field. The frequency is lower than 10<sup>5</sup> Hz so we could ignore the displacement current in the quasi-static approximation of electromagnetic field. When a magnetic field **H** is applied to the ground, it produces an electric field **E** through

electromagnetic induction. The impedance **Z** is used to express the relation between the electromagnetic fields as follows [2]:

$$
\begin{bmatrix}
\mathbf{E}\_{\mathbf{x}} \\
\mathbf{E}\_{\mathbf{y}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{Z}\_{xx} & \mathbf{Z}\_{xy} \\
\mathbf{Z}\_{yx} & \mathbf{Z}\_{yy}
\end{bmatrix} \begin{bmatrix}
\mathbf{H}\_{\mathbf{x}} \\
\mathbf{H}\_{\mathbf{y}}
\end{bmatrix} \tag{1}
$$

**Figure 1.** Basic introduction of the magnetotelluric (MT) method. (**a**) shows the layout of the MT signal acquisition system. The electrodes connected by wires is used to obtain electric field data, and the magnetic field probes are used for collecting magnetic field data. The host is used to record the signal at various frequencies. (**b**) The application of the optimization process in MT inversions.

For the one-dimensional case, **Z***xx* = 0, **<sup>Z</sup>***yy* = 0 and **<sup>Z</sup>***xy* = <sup>−</sup>**Z***yx*. The impedance tensor can be decomposed into two components, corresponding to the apparent resistivity and the phase. For an *N*-stratum geoelectric model, the apparent resistivity *ρω* and the phase *ϕ* can be derived from the impedance **Z** regardless of the orientations of the *x* and *y* axes as follows:

$$\begin{aligned} \rho\_{\omega^\*} &= \frac{|Z\_1|^2}{\omega\mu} & \boldsymbol{\varrho} &= \tan^{-1} \frac{\text{Im}(Z\_1)}{\text{Re}(Z\_1)}\\ Z\_{\text{m}} &= Z\_{\text{om}} \frac{1 - L\_{\text{m}+1}\boldsymbol{\varepsilon}^{-2k\_{\text{m}}h\_{\text{m}}}}{1 + L\_{\text{m}+1}\boldsymbol{\varepsilon}^{-2k\_{\text{m}}h\_{\text{m}}}} & L\_{\text{m}+1} &= \frac{Z\_{\text{sm}} + Z\_{\text{m}+1}}{Z\_{\text{sm}} + Z\_{\text{m}+1}}\\ Z\_N &= Z\_{\text{oN}} & Z\_{\text{conv}} &= -i\omega\mu/k\_{\text{m}} & k\_{\text{m}} &= \sqrt{-i\mu}\sigma\_{\text{m}}\omega \end{aligned} \tag{2}$$

where *Zm* is the impedance at the top of the *m*th stratum, *Zom* is the intrinsic impedance of the *m*th stratum, the magnetic permeability *μ* is assigned its free space value and *ω* is the angular frequency. For the *m*th stratum, *ρm* is the resistivity and *hm* is the thickness. Usually, the apparent resistivity is the observed response that is used to obtain the geoelectric model through inversion.
