# Influence of Source Shape on Semi-Airborne Transient Electromagnetic Surveys
## 2 Methods
### SATEM 3D Modeling  
The 3D numerical simulation of SATEM is implemented through finite volume spatial discretization coupled with model order reduction techniques for efficient time-domain solutions. The governing Maxwell's equations in the time domain applicable to the SATEM configuration can be expressed as [22]  
\[\left\{ \begin{aligned} &\frac{\partial\overset{\rightarrow}{ \leftarrow}}{\partial t}=-\
abla\times\overset{\rightarrow}{E}\\ &\
abla\times\mu^{-1}\overset{\rightarrow}{B}-\sigma\overset{ \rightarrow}{E}=\overset{\rightarrow}{s}\end{aligned} \right. \tag{3}\]  
where \(\overset{\rightarrow}{E}\) denotes the electric field intensity vector (V/m), \(\overset{\rightarrow}{B}\) represents the magnetic flux density vector (T), \(t\) is the time variable (s), \(\mu\) is the magnetic permeability (H/m), \(\sigma\) corresponds to the electrical conductivity (S/m), and \(\overset{\rightarrow}{s}\) designates the external source current density term (A/m\({}^{2}\)). The system is subject to natural boundary conditions as described by [PERSON] [23]:  
\[\overset{\rightarrow}{B}\times\overset{\rightarrow}{n}=0;\overset{ \rightarrow}{n}\times\overset{\rightarrow}{E}=0 \tag{4}\]  
The governing Equation (3) is spatially discretized using the finite volume method implemented on a structured hexahedral mesh. The discretization process yields the following semi-discrete system:  
\[\left\{ \begin{aligned} &\frac{\partial\mathbf{p}}{\partial t}+\text{ CURL}\mathbf{e}=0\\ &\text{CURL}^{\text{T}}\mathbf{M}_{f/\mathbf{h}}\mathbf{b}-\mathbf{M }_{\omega}\mathbf{e}=\mathbf{M}_{\epsilon}\mathbf{s}\end{aligned} \right. \tag{5}\]where \(\mathbf{e}\) and \(\mathbf{b}\) represent the orthogonal projections of the electric field vector \(\overset{\rightarrow}{E}\) and magnetic flux density vector \(\overset{\rightarrow}{B}\) onto the edges and face centers of the computational grid elements, respectively. CURL denotes the discrete curl operator matrix derived from the spatial discretization scheme. The matrices \(\mathbf{M}_{f\mu}\) and \(\mathbf{M}_{ev}\) are the mass matrices incorporating the magnetic permeability \(\mu\) and electrical conductivity \(\sigma\) distributions within the computational domain. For step-function excitation, the electric field can be eliminated from the system, resulting in the following discrete governing equation for the magnetic flux density:  
\[\frac{\partial\mathbf{b}}{\partial t}=\mathbf{A}\mathbf{b} \tag{6}\]  
where \(\mathbf{A}=-\mathrm{CURL}\mathbf{M}_{ev}^{-1}\mathrm{CURL}^{\mathrm{T}} \mathbf{M}_{f\mu}\). The magnetic field at the time \(t\) obtained by Equation (6) is a matrix exponential function:  
\[\mathbf{b}^{t}=\mathrm{exp}(t\mathbf{A})\mathbf{b}^{0} \tag{7}\]  
where \(\mathbf{b}^{0}\) denotes the initial magnetic flux density distribution at \(t=0\). For a grounded long wire source configuration, the initial magnetic flux density corresponds to the solution of a magnetostatic problem, which can be obtained by applying the finite volume method to solve the steady-state Maxwell's equations as described by [PERSON] et al. [22]. Substituting this initial magnetic flux density into Equation (6) enables the computation of \(\partial\mathbf{b}/\partial t\). Equation (7) is subsequently solved using the rational Krylov subspace method. Based on the system matrix \(\mathbf{A}\) and the initial magnetic flux density vector \(\mathbf{b}^{0}\), we construct the rational Krylov subspace \(\kappa\Big{(}\Big{(}\mathbf{A}-\zeta_{i}\mathbf{I})^{-1},\mathbf{b}^{0}\Big{)}\), where \(\zeta_{i}\) represents the shift parameters. The orthonormal basis \(\mathbf{V}_{m+1}\) of this subspace and the projection matrix \(\mathbf{A}_{m+1}=\mathbf{V}_{m+1}^{\mathrm{T}}\mathbf{A}\mathbf{V}_{m+1}\) are then computed.