## 3 MT Method
### Three-Dimensional MT Inversion  
The 3D inversion is used to perform the information about the resistivity distribution of the Earth's subsurface by generating the Earth model. Three-dimensional inversion algorithm that is used in this study employs ModeEM software ([[https://www.mtnet.info/main/source.html](https://www.mtnet.info/main/source.html)]([https://www.mtnet.info/main/source.html](https://www.mtnet.info/main/source.html)) (accessed on 1 June 2022)) [51; 52]. The ModeEM 3D inversion code was used to create a 3D model of the geoelectric structure through gradient-based minimization of the objective function \(\mathbf{\Phi}\) as Equation (17) [52]:  
\[\mathbf{\Phi}(\mathbf{m},\mathbf{d})=(\mathbf{d}-\mathbf{f}(\mathbf{m}))^{T}\mathbf{C}_{\mathbf{d}}^{-1}(\mathbf{d }-\mathbf{f}(\mathbf{m}))+\
u(\mathbf{m}-\mathbf{m}_{0})\mathbf{C}_{\mathbf{m}}^{-1}(\mathbf{m}-\mathbf{m}_{0}) \tag{17}\]  
with \(\mathbf{C}_{\mathbf{d}}\) as the covariance of data errors, \(\mathbf{f}(\mathbf{m})\) as the forward mapping, \(\mathbf{d}\) as a data vector, \(\mathbf{m}\) as an M-dimensional Earth conductivity model parameter vector, \(\mathbf{C}_{\mathbf{m}}\) as the model covariance, \(\
u\) as the damping parameter, and \(\mathbf{m}_{0}\) as the prior model. The algorithm of the ModeEM software uses a gradient-based optimization algorithm such as the Nonlinear Conjugate Gradient (NLCG) with minimal structure to minimize the objective function, which is solved by the Data-Space Conjugate Gradient Scheme (DCG) solver. The NLCG was designed to minimize the difference between the modeled and observed data. Using the NLCG approach, the gradient of Equation (17) with respect to variations in model parameters \(\mathbf{m}\) must be evaluated, as shown in Equation (18):  
\[\left.\frac{\partial\mathbf{\Phi}}{\partial\mathbf{m}}\right|_{\mathbf{m}_{n}}=-2\mathbf{J}^{ T}\mathbf{r}+2\
u\mathbf{m}_{n} \tag{18}\]  
where \(\mathbf{J}\) is the sensitivity matrices, \(\mathbf{m}_{n}\) is a \(n\)-th of \(m\), \(\mathbf{r}\) is the data residual that is represented by \(\mathbf{r}=\mathbf{d}-\mathbf{f}(\mathbf{m}_{n})\), and \(\mathbf{f}(\mathbf{m})\) is the \(n\)-th forward mapping. The gradient is then used to calculate a new 'conjugate' search direction in the model space. The NLCG also essentially uses the same basic computational steps as required for solving the linearized equation [52]. The search for minimization error in the NCLG of the ModeEM software is based on a quadratic approach. If the solution does not fulfill the reduction conditions, then backtracking of the minimal error will use cubic interpolation. This step only needs one gradient evaluation, which is quite efficient for computation. The NLCG also implements automatic criteria to reduce the damping parameter (\(\
u\)) when the convergence stops [51]. This makes the ModeEM software more efficient for use. Using the DCG, the linearized objective function in Equation (17) transforms into that shown in Equation (19) [51]:  
\[\left(\mathbf{IC}_{\mathbf{m}}\mathbf{J}^{T}+\
u\mathbf{I}\right)\mathbf{b}_{n}=\hat{\mathbf{\lambda}} _{n} \tag{19}\]  
The model parameter update is computed by Equation (20) [51]:  
\[\mathbf{m}_{n+1}=\mathbf{C}_{\mathbf{m}}\mathbf{J}^{T}\mathbf{b}_{n} \tag{20}\]  
where \(\mathbf{b}_{n}\) is the boundary condition, and \(\mathbf{J}\) is the sensitivity matrices. The electrical resistivity of the Earth's subsurface structure is characterized by electromagnetic field components measured at the Earth's surface using a transfer functions relationship. Inversion is used to obtain the actual resistivity distribution of the Earth's subsurface in the form of the resistivity model. Electrical resistivity is a crucial physical property of rocks that aids in distinguishing geological formations with different petrophysical characteristics. The range of electrical resistivity varies from fractions of \(\Omega\)m in sedimentary rocks to hundreds of thousands of \(\Omega\)m in crystalline rocks [53; 54; 55].