## 2 Differential Pulse Spectrum Analysis and 3D Forward Modeling
### 3D Forward Modeling of Differential Pulse  
For the 3D forward modeling of differential pulse, this study adopts the unstructured tetrahedral vector finite element method. To achieve efficient numerical simulations, the second-order backward differentiation formula (BDF2) is employed during the ON-Time stage, while the shift-and-invert Krylov (SAI-Krylov) subspace method is used during the OFF-Time stage.  
For a given computational domain \(\Omega\), neglecting the displacement current, [PERSON]'s equations can be expressed in the following form [34]:  
\[\
abla\times\mathbf{e}=-\frac{\partial\mathbf{b}}{\partial t}, \tag{1}\]  
\[\
abla\times\mu_{0}^{-1}\mathbf{b}-\sigma\mathbf{e}=\mathbf{J}_{s}(t). \tag{2}\]  
where \(\mathbf{e}\) represents the electric field intensity (V\(\cdot\)m\({}^{-1}\)), \(\mathbf{b}\) denotes the magnetic flux density (T), \(\mu_{0}\) is the magnetic permeability of free space (H\(\cdot\)m\({}^{-1}\)), \(\mathbf{J}_{s}\) represents the current density (A\(\cdot\)m\({}^{-2}\)), and \(\sigma\) is the electrical conductivity of the subsurface (S\(\cdot\)m\({}^{-1}\)). At infinity, the first-type boundary condition is applied in this study.  
By discretizing the space using an unstructured tetrahedral mesh, eliminating the magnetic flux density \(\mathbf{b}\) from Equation (1) and (2), and applying the [PERSON] method, the governing equation for the electric field can be derived as follows:  
\[\int\
u\ \left(\
abla\times\mathbf{N}_{i}\right)\cdot\big{(}\
abla\times\mathbf{N}_{j} \big{)}dV\mathbf{e}+\int\
u\ \mu_{0}\sigma\mathbf{N}_{i}\cdot\mathbf{N}_{j}dV\frac{\partial\mathbf{e}}{\partial t}= \int\
u\ \mu_{0}\mathbf{N}_{i}\cdot\mathbf{N}_{j}dV\frac{\partial\mathbf{J}_{s}}{\partial t}. \tag{3}\]  
where \(\mathbf{N}_{i}\) and \(\mathbf{N}_{j}\) are the interpolation basis functions within the element. Applying the BDF2 to the derivative terms in Equation (3) yields the following:  
\[\int\
u\ \left(\
abla\times\mathbf{N}_{i}\right)\cdot\big{(}\
abla\times\mathbf{N}_{j} \big{)}dV\mathbf{e}^{n}+\int\
u\ \mu_{0}\sigma\mathbf{N}_{i}\cdot\mathbf{N}_{j}dV\big{(}\frac{\partial\mathbf{e}^{n}-4\mathbf{ e}^{n-1}+\mathbf{e}^{n-2}}{2\Delta t}\big{)} \tag{4}\]  
where \(\mathbf{e}^{n}\) and \(\mathbf{J}_{s}^{n}\) represent the electric field intensity and current density at time step n, respectively. In this study, the matrices are defined as follows: \(\mathbf{M}_{curl}=\int\
u\ \left(\
abla\times\mathbf{N}_{i}\right)\cdot\big{(}\
abla \times\mathbf{N}_{j}\big{)}dV\), \(\mathbf{M}_{\sigma}=\int\
u\ \mu_{0}\sigma\mathbf{N}_{i}\cdot\mathbf{N}_{j}dV\), \(\mathbf{M}_{inner}=\int\
u\ \mu_{0}\mathbf{N}_{i}\cdot\mathbf{N}_{j}dV\). Thus, the electric field intensity during the ON-Time can be solved using the following equation [35]:  
\[(2\Delta t\mathbf{M}_{curl}+3\mathbf{M}_{\sigma})\mathbf{e}^{n}=\mathbf{M}_{\sigma}\Big{(}4 \mathbf{e}^{n-1}-\mathbf{e}^{n-2}\Big{)}-\mathbf{M}_{inner}\Big{(}3\mathbf{J}_{s}^{n}-4\mathbf{J}_{ s}^{n-1}+\mathbf{J}_{s}^{n-2}\Big{)}. \tag{5}\]  
After the current is turned off (OFF-Time), \(\mathbf{J}_{s}\) in Equation (3) becomes zero and can be rewritten as follows:  
\[\frac{\partial\mathbf{e}}{\partial t}=-\mathbf{M}_{\sigma}^{-1}\mathbf{M}_{curl}\mathbf{e}, \tag{6}\]  
Using the SAI-Krylov subspace method to solve Equation (6) [36; 37; 38], the approximate solution can be expressed in the following form:  
\[\mathbf{e}(t)\approx\mathbf{e}_{m}(t)=\mathbf{V}_{m}e^{-\frac{t}{\gamma(\mathbf{M}_{m}^{n-1}- \mathbf{t})}}\parallel\mathbf{e}_{0}\parallel_{2}\mathbf{e}_{1}, \tag{7}\]  
where \(m\) is the order of the SAI-Krylov subspace, \(\gamma\) is the shift parameter, \(\mathbf{I}\) is the identity matrix, and \(\mathbf{e}_{0}\) represents the electric field intensity at the time of current shutdown. The vector \(\mathbf{e}_{1}=[1,0,\ldots,0]_{m}^{T}\). \(\mathbf{V}_{m}\) and \(\mathbf{H}_{m}\) are the basis vector matrix and the projection matrix, respectively, which can be solved using the Arnoldi algorithm [39].