## 5 3-D TEM Inversion Algorithm Considering Magnetic Permeability  
The objective function for the 3-D TEM Inversion is [21]:  
\[\Phi(\mathbf{m})=\frac{1}{2}\|\mathbf{W}_{\mathrm{d}}[\mathbf{d}-\mathbf{d}_{ \mathrm{obs}}]\|_{2}^{2}+\frac{\lambda}{2}\|\mathbf{W}_{\mathbf{m}}(\mathbf{ m}-\mathbf{m}_{\mathrm{ref}})\|_{2}^{2} \tag{6}\]  
where \(\mathbf{m}\) denotes the model vector, which includes both resistivity and magnetic permeability; \(\mathbf{d}_{\mathrm{obs}}\) is the observation data vector, representing the vertical component of the induced electromotive force (\(\mathrm{d}\mathbf{b}_{\mathrm{z}}/\mathrm{d}t\)) response corresponding to multiple time gates for each measurement site; \(\mathbf{W}_{\mathrm{d}}\) represents a diagonal matrix of the inverse of the data uncertainties, and \(\mathbf{W}_{\mathrm{m}}\) is the model regularization matrix; \(\mathbf{d}\) denotes the forward modeling response of \(\mathbf{m}\); and \(\mathbf{m}_{\mathrm{ref}}\) represents the reference model. Figure 4: Comparison of forward responses for the anomaly with different parameters. \[\frac{\partial\mathbf{A}}{\partial m_{\mu}}=\left( \begin{array}{cccc}\frac{ \partial(\mathbf{M}+2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&&\\ \frac{\partial(-\mathbf{M})}{\partial m_{\mu}}&\frac{\partial(\mathbf{M}+2 \Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&\cdots&\\ \frac{\partial\mathbf{M}}{\partial m_{\mu}}&\frac{\partial(-\mathbf{M})}{ \partial m_{\mu}}&\frac{\partial(\mathbf{M}+2\Delta\mathbf{I}\mathbf{S})}{ \partial m_{\mu}}&&\\ &&\vdots&&\ddots&\vdots\\ &&&\cdots&\frac{\partial(\mathbf{M}+2\Delta\mathbf{I}\mathbf{S})}{\partial m _{\mu}}\end{array} \right), \tag{12}\]  
where \(\mathbf{M}\) is only related to the conductivity, and \(\mathbf{S}\) is only related to the magnetic permeability. Therefore, we have:  
\[\frac{\partial\mathbf{A}}{\partial m_{\sigma}}=\left( \begin{array}{cccc} \frac{\partial(\mathbf{M})}{\partial m_{\sigma}}&&&\\ \frac{\partial(-\mathbf{M})}{\partial m_{\sigma}}&\frac{\partial(\mathbf{M} )}{\partial m_{\sigma}}&&\cdots&\\ \frac{\partial\mathbf{M}}{\partial m_{\sigma}}&\frac{\partial(-\mathbf{M})}{ \partial m_{\sigma}}&\frac{\partial(\mathbf{M})}{\partial m_{\sigma}}&&\\ &\vdots&&\ddots&\vdots\\ &&&\cdots&\frac{\partial(\mathbf{M})}{\partial m_{\sigma}}\end{array} \right), \tag{13}\]  
\[\frac{\partial\mathbf{A}}{\partial m_{\mu}}=\left( \begin{array}{cccc}\frac{ \partial(2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&&\\ \frac{\partial(2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&\cdots&\\ &\frac{\partial(2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&\\ &\frac{\partial(2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}&&\\ &\vdots&&\ddots&\vdots\\ &&&\cdots&\frac{\partial(2\Delta\mathbf{I}\mathbf{S})}{\partial m_{\mu}}\end{array}  \right), \tag{14}\]  
Based on the above formulations, the two matrix-vector products \(\mathbf{J}_{\sigma}^{\mathrm{T}}\mathbf{r}\) and \(\mathbf{J}_{\mu}^{\mathrm{T}}\mathbf{r}\) can be computed separately. This allows the gradient vectors of the objective function with respect to the magnetic permeability and conductivity models to be obtained. Subsequently, based on the L-BFGS algorithm, the iterative updates for the model parameters are progressively completed. The regularization matrix \(\mathbf{W}_{m}\) is normally required to stabilize the inversion. For using the traditional structured grids, \(\mathbf{W}_{m}\) can be chosen as a discrete gradient operator [22]. However, for the unstructured tetrahedral grids, the model roughness matrix is influenced by factors such as the irregular shape, distance, and volume of the tetrahedron, etc. Therefore, additional constraints need to be imposed to make the regularization constraint work in this case. [PERSON] [23] presented the definition of \(\mathbf{W}_{m}\) for the 2-D grids using irregular triangular elements.