# Influence of Source Shape on Semi-Airborne Transient Electromagnetic Surveys
## 2 Methods
### SATEM 1D Modeling  
We establish a stratified n-layer earth model characterized by resistivity and thickness parameters denoted as \(\rho_{1},h_{1};\rho_{2},h_{2};\ldots;\rho_{n},h_{n}\), where \(h_{n}\rightarrow\infty\) represents the semi-infinite bottom layer. An electric dipole source AB is positioned on the earth's surface and oriented along the positive X-axis direction. The coordinate system is defined such that the origin O coincides with the center of the electric dipole. For any arbitrary measurement point M situated in the air, we define z as its vertical distance from the earth's surface. The horizontal distance between the surface projection point P of measurement point M and the coordinate origin O is denoted as \(r\), while \(\varphi\) represents the azimuthal angle between vector \(r\) and the positive X-axis direction. This geometric configuration is illustrated comprehensively in Figure 1.  
In practical field applications, long wire sources typically extend for several kilometers, rendering the conventional electric dipole approximation inadequate in near-field and transition zones. To address this limitation, we implement a superposition principle wherein the extended source is discretized into \(m\) infinitesimal dipole elements, each with length \(ds\). The electromagnetic field contribution from each elemental dipole is then systematically integrated to accurately reconstruct the composite field generated by the entire source configuration. Employing vector and scalar potential formalism as described  
Figure 1: Schematic representation of the electric dipole coordinate system positioned on a horizontally stratified earth surface. by [PERSON] et al. [25], the frequency-domain magnetic induction intensity generated by an extended wire source can be mathematically expressed as  
\[\left\{\begin{aligned} & B_{x}(\omega)=\sum_{j=1}^{m}\left[ \begin{array}{c}-\frac{P_{E_{j}}}{2\pi}\frac{\mu_{0}sin\varphi_{j}cos\varphi_{ j}}{r_{j}}\int_{0}^{\infty}\frac{2\lambda}{\lambda+\frac{2\pi}{R_{1}}}e^{ \lambda z}I_{1}(\lambda r_{j})d\lambda\\ &+\frac{P_{E_{j}}}{2\pi}\mu_{0}sin\varphi_{j}cos\varphi_{j}\int_{0}^{\infty }\frac{\lambda^{2}}{\lambda+\frac{2\pi}{R_{1}}}e^{\lambda z}I_{0}(\lambda r_{j })d\lambda\\ & B_{y}(\omega)=\sum_{j=1}^{m}\left[ \begin{array}{c}\frac{P_{E_{j}}}{2 \pi}\frac{\mu_{0}(cos^{2}\varphi_{j}-sin^{2}\varphi_{j})}{r_{j}}\int_{0}^{ \infty}\frac{\lambda}{\lambda+\frac{2\pi}{R_{1}}}e^{\lambda z}I_{1}(\lambda r _{j})d\lambda\\ &+\frac{P_{E_{j}}}{2\pi}\mu_{0}sin^{2}\varphi_{j}\int_{0}^{\infty}\frac{ \lambda^{2}}{\lambda+\frac{2\pi}{R_{1}}}e^{\lambda z}I_{0}(\lambda r_{j})d \lambda\\ & B_{z}(\omega)=\sum_{j=1}^{m}\frac{P_{E_{j}}}{2\pi}\mu_{0}sin\varphi_{j}\int_ {0}^{\infty}\frac{\lambda^{2}}{\lambda+\frac{2\pi}{R_{1}}}e^{\lambda z}I_{1}( \lambda r_{j})d\lambda\end{array} \right] \tag{1}\]  
where \(R_{1}=cth\left[u_{1}h_{1}+arcth\frac{u_{1}}{u_{2}}cth\left(u_{2}h_{2}+...+arcth \frac{u_{m-1}}{u_{m}}\right)\right]\), and \(u_{i}=\sqrt{\lambda^{2}-i\omega\mu/\rho_{i}}\). The parameter \(P_{E_{j}}=Ids_{j}\) represents the dipole moment of the \(j\) discretized element, with \(I\) denoting the current intensity and \(ds_{j}\) representing the length of the \(j\) electric dipole segment. The geometric parameters \(\varphi_{j}\) and \(r_{j}\) correspond to the azimuthal angle between the projection point P and the \(j\) electric dipole segment and the horizontal offset distance, respectively. The variable \(m\) denotes the total number of discretized segments comprising the extended electrical source. For step-function excitation, the time-domain SATEM responses \(B(t)\) and \(dB(t)/dt\) can be derived from the frequency-domain response \(B(\omega)\) through the following integral transformation relations as established by [PERSON] et al. [25]:  
\[\left\{ \begin{aligned} & B(t)=\frac{2}{\pi}\int_{0}^{\infty} \frac{\text{Re}[B(\omega)]}{\omega}\text{sin}(\omega t)d\omega\\ &\frac{\partial B(t)}{\partial t}=-\frac{2}{\pi}\int_{0}^{\infty} \text{Re}[B(\omega)]\text{cos}(\omega t)d\omega\end{aligned}