## 2 MT Forward Modeling and Parameter Setting
### Finite Element Modeling  
The propagation of electromagnetic waves in subsurface media follows [PERSON]'s equations, which form the basis for deriving magnetotelluric forward modeling equations. In a two-dimensional medium, the electromagnetic field can be divided into two independent wave types: one includes the components \(E_{x}\), \(H_{y}\), and \(H_{z}\), known as transverse electric (TE) polarization mode, while the other includes \(H_{x}\), \(E_{y}\), and \(E_{z}\), referred to as transverse magnetic (TM) polarization mode. When solving the numerical solution of the magnetotelluric field in a two-dimensional medium, [PERSON]'s equations can be transformed into the following second-order partial differential equations that satisfy \(E_{x}\) and \(H_{x}\) (Equations (1) and (2)) [17,33,34]:  
\[\frac{\partial}{\partial y}\left(\frac{1}{i\omega\mu}\frac{\partial E_{x}}{ \partial y}\right)+(\frac{1}{i\omega\mu}\frac{\partial E_{x}}{\partial z})+( \sigma-i\omega\varepsilon)E_{x}=0 \tag{1}\]  
\[\frac{\partial}{\partial y}\left(\frac{1}{\sigma-i\omega\varepsilon}\frac{ \partial H_{x}}{\partial y}\right)+(\frac{1}{\sigma-i\omega\varepsilon}\frac {\partial H_{x}}{\partial z})+i\omega\mu H_{x}=0 \tag{2}\]  
Here, \(E_{x}\) represents the horizontal component of the electric field in the x-direction, \(H_{x}\) is the horizontal component of the magnetic field in the x-direction, \(\sigma\) is the medium's electrical conductivity, \(\varepsilon\) is the dielectric constant, \(\mu\) is the magnetic permeability, and \(\omega=2\pi f\) is the angular frequency. Appropriate boundary conditions must be applied to solve Equations (1) and (2). For example, in TM mode, the top boundary AB is typically set at the surface, where the field value \(u=H_{x}\) is set to 1 unit:  
\[u\big{|}_{AB}=1 \tag{3}\]  
Below the bottom boundary CD, homogeneous rock is assumed at a sufficient depth, where the anomalous field generated by local inhomogeneous rock is negligible on CD. This assumption holds when the lower boundary is deep enough and the medium below is  
Figure 1: Geological model depicting three faults (F1, F2, and F3) with varying dip angles and resistivity distribution used for forward modeling. homogeneous, allowing the influence of local inhomogeneities to be ignored. At this depth, the propagation equation for electromagnetic waves below the lower boundary is:  
\[u\big{|}_{CD}=u_{0}e^{-ky} \tag{4}\]  
Here, \(u_{0}\) is a constant, \(k=\sqrt{-i\omega\mu\sigma}\) is the propagation coefficient, and \(\sigma\) is the rock's electrical conductivity. Taking the derivative of Equation (4), we get \(\frac{\partial u}{\partial z}=-ku\). At the lower boundary \(\frac{\partial}{\partial z}=-\frac{\partial}{\partial u}\), the boundary condition at CD is given as:  
\[\frac{\partial u}{\partial n}+ku=0 \tag{5}\]  
When the left and right boundaries are sufficiently distant from the inhomogeneity, the anomalous field becomes zero, leading to symmetry in the electromagnetic field:  
\[\frac{\partial u}{\partial n}=0 \tag{6}\]  
Based on the above analysis, solving the magnetotelluric field reduces to solving the system of equations formed by partial differential Equation (1) to Equation (2), and the boundary conditions Equations (3)-(6). \[ \begin{array}{ll}\
abla\cdot(\tau\
abla u)+\lambda u=0&\in\Omega\\ u=1&\in\text{AB}\\ \frac{\partial u}{\partial n}=0&\in\text{AD, BC}\\ \frac{\partial u}{\partial n}+ku=0&\in\text{CD}\end{array}  \tag{7}\]  
For the TE polarization mode: \(u=E_{x}\), \(\tau=i\omega\mu\), and \(\lambda=\frac{1}{\sigma-i\omega\varepsilon}\)  
For the TM polarization mode: \(u=H_{x}\), \(\tau=\frac{1}{\sigma-i\omega\varepsilon}\), and \(\lambda=i\omega\mu\). The finite element method (FEM) is employed to numerically solve this system of equations. To calculate the numerical solution of equation system (7) using the finite element method, we consider its equivalent variational problem:  
\[