## 2 3-D TEM Forward Modeling Based on the Unstructured Finite Element Method  
In TEM modeling, the effect of displacement current is generally negligible, which leads to the time-domain diffusion equation for the electric field [14]:  
\[\
abla\times\frac{1}{\mu(\mathbf{r})}\
abla\times\mathbf{e}(\mathbf{r},t)+ \sigma(\mathbf{r})\frac{\partial\mathbf{e}(\mathbf{r},t)}{\partial t}+\frac{ \partial\mathbf{j}_{\mathbf{s}}(\mathbf{r},t)}{\partial t}=0 \tag{1}\]  
where \(\mathbf{r}\) denotes the spatial coordinate; \(t\) is time; \(\mathbf{e}\) represents the electric field; \(\sigma\) and \(\mu\) represent the conductivity and magnetic permeability, respectively; and \(\mathbf{j}_{\mathbf{s}}\) denotes the source current density. As the artificial source is fixed during the measurement, we can separate the source current density term as \(\mathbf{j}_{\mathbf{s}}(\mathbf{r},t)=I(t)\mathbf{\overline{j}_{\mathbf{s}} (\mathbf{r})}\), where \(I(t)\) is the source current intensity that is employed for simulating the source waveform and \(\mathbf{\overline{j}_{\mathbf{s}}}\) denotes the spatial pattern of \(\mathbf{j}_{\mathbf{s}}\) using unit current intensity. In this study, forward modeling uses unstructured tetrahedral meshes for the 3-D model discretization and applies the edge-based finite element method for the spatial discretization of Equation (1) that discretize the electric fields on the grid edges. The electric field at any point within the \(k\)-th tetrahedral element of the mesh can be expressed as [15]:  
\[\mathbf{e}(\mathbf{r},t)=\sum_{i=1}^{6}e_{i}^{k}(t)\mathbf{n}_{i}^{k}(\mathbf{r}) \tag{2}\]  
where \(e_{i}^{k}(t)\) represents the magnitude of the electric field component along the \(i\)-th edge direction in the \(k\)-th element, and \(\mathbf{n}_{i}^{k}(\mathbf{r})\) is the first-order edge-based interpolation function. By applying the [PERSON] method, a semi-discrete system can be established [16]:  
\[\mathbf{M}\frac{\text{d}\mathbf{e}(t)}{\text{d}t}+\mathbf{S}\mathbf{e}(t)+ \frac{\text{d}I(t)}{\text{d}t}\mathbf{\bar{J}}=0 \tag{3}\]  
where \(\mathbf{M}\) denotes the mass matrix; \(\mathbf{S}\) is the stiffness matrix; \(\mathbf{e}(t)\) represents the solution vector of the discretized electric fields at time \(t\); and \(\mathbf{\bar{J}}\) denotes the source term vector [17]. \[ \begin{array}{c}\mathbf{M}_{i,j}^{c}=\sigma^{c}\iint_{\
u^{c}}\mathbf{n}_{ i}^{c}(\mathbf{r})\cdot\mathbf{n}_{j}^{c}(\mathbf{r})\text{d}\upsilon\\ \mathbf{S}_{i,j}^{c}=\frac{1}{\prod_{\
u\in\mathcal{H}}\iint_{\
u^{c}}\left[ \
abla\times\mathbf{n}_{i}^{c}(\mathbf{r})\right]\cdot\left[\
abla\times \mathbf{n}_{i}^{c}(\mathbf{r})\right]\text{d}\upsilon\\ \mathbf{\bar{J}}_{i}^{c}=\iint_{\
u^{c}}\mathbf{n}_{i}^{c}(\mathbf{r})\cdot \mathbf{\bar{J}}_{i}(\mathbf{r})\text{d}\upsilon\\ i=1,\ldots,6\\ j=1,\ldots,6\end{array}  \tag{4}\]  
Equation (4) gives the expansion forms of the local matrix elements for a specific tetrahedral element (\(e\)), where \(\
u_{r}^{c}\) denotes the relative magnetic permeability of the element, and different elements can be assigned with different values. Using the second-order implicit backward Euler method to discretize Equation (3) in the time domain [18], we can derive the unconditional stable time-domain iterative formulation:  
\[ \begin{array}{c}(3\mathbf{M}+2\Delta t_{k}\mathbf{S})\mathbf{e}_{k}=\mathbf{ M}(4\mathbf{e}_{k-1}-\mathbf{e}_{k-2})-(3I_{k}-4I_{k-1}+I_{k-2})\mathbf{\bar{J}}\\ k=1,\ldots,n_{\text{step}}\end{array}  \tag{5}\]  
where \(\mathbf{e}_{k}\) and \(I_{k}\) represent the discrete electric field solution vector and current intensity at the \(m\)-th time point, and \(\Delta t_{m}\) is the time step. The time iteration calculation requires a total of \(n_{\text{step}}\) steps to complete. In each time step, we solve the above system of linear equations to derive the \(\mathbf{e}_{k}\) at the new time point.