## 2 Method  
### Governing Equation  
The borehole-to-surface electromagnetic response is governed by the diffusive form of [PERSON]'s equations, in which the displacement current is neglected and anisotropy is used to characterize the resistivity of complex subsurface media. Specifically, a three-dimensional principal-axis anisotropic resistivity model is considered, and a vertical finite-length-wire transmitter is employed as the excitation transmitter. By assuming a time-harmonic factor of \(e^{-i\omega t}\), [PERSON]'s equations can be expressed as follows:  
\[\
abla\times\mathbf{E}=i\omega\mu_{0}\mathbf{H} \tag{1}\]  
\[\
abla\times\mathbf{H}=\mathbf{J_{S}}+\mathbf{\sigma}^{*}\mathbf{E} \tag{2}\]  
where \(\mathbf{E}\) represents the electric field, \(\mathbf{H}\) denotes the magnetic field, \(\omega\) is the angular frequency, \(\mu_{0}\) is the magnetic permeability of free space, \(\mathbf{J_{S}}\) represents the distribution of the transmitter current, and \(\mathbf{\sigma}^{*}\mathbf{E}\) denotes the induced current in the conductive Earth; here, \(\mathbf{\sigma}^{*}\) represents the resistivity tensor, which is given by Equation (3):  
\[\mathbf{\sigma}^{*}= \begin{pmatrix}\sigma_{xx}&0&0\\ 0&\sigma_{yy}&0\\ 0&0&\sigma_{zz}\end{pmatrix}  \tag{3}\]  
In electromagnetic exploration, anisotropy is primarily manifested in electrical conductivity. For any anisotropic medium, the conductivity \(\mathbf{\sigma}^{*}\) is a \(3\times 3\) symmetric and positive-definite tensor, which can generally be obtained by applying a triple Euler rotation to a principal conductivity tensor \(\mathbf{\sigma}^{*}\)[41]. This transformation allows the representation of directional dependencies in conductivity, and it is this tensorial form that distinguishes anisotropic media from isotropic ones, where conductivity is a scalar. Equation (3) explicitly illustrates how anisotropy is introduced into the formulation through this tensor representation. By substituting Equation (1) into Equation (2), we obtain the following:  
\[\
abla\times\
abla\times\mathbf{E}+i\omega\mu_{0}\mathbf{\sigma}^{*}\mathbf{E}=i\omega\mu_{0} \mathbf{J}_{s} \tag{4}\]  
This approach is known as the curl formulation of the problem [42]. Typically, a primary/secondary field formulation is employed to capture the rapid variations in the primary field without requiring extensive mesh refinement [43]. In this context, it is also referred to as the scattering formulation [44], and the total electric field, \(\mathbf{E}\), is expressed as follows:  
\[\mathbf{E}=\mathbf{E}_{p}+\mathbf{E}_{s} \tag{5}\]  
\[\mathbf{\sigma}^{*}=\mathbf{\sigma}_{0}+\Delta\mathbf{\sigma} \tag{6}\]  
Here, \(\mathbf{E}_{p}\) and \(\mathbf{E}_{s}\) represent the primary and secondary fields, respectively, and \(\mathbf{\sigma}_{0}\) denotes the background resistivity. For layered models, minor effects are generally neglected in favor of considering only the dominant response \(\mathbf{\sigma}_{0}\). The primary field, \(\mathbf{E}_{p}\), can be obtained via a semi-analytical solution using Hankel transform filtering [45]. Therefore, we only need to make solutions for the secondary field, which is governed by the following equation:  
\[\
abla\times\
abla\times\mathbf{E}_{s}+i\omega\mu_{0}\mathbf{\sigma}^{*}\mathbf{E}_{s}=-i \omega\mu_{0}\Delta\mathbf{\sigma}\mathbf{E}_{p} \tag{7}\]  
Meanwhile, homogeneous Dirichlet boundary conditions are imposed, meaning that \(\mathbf{E}=0\) on the boundaries.