# Application of Transient Electromagnetic Method for Investigating the Water-Enriched Mined-Out Area
## 2 Materials and Methods
### Theory of Large-Loop TEM  
For the large-fixed loop source given in Figure 4, the four sides of the loop are named A, B, C, and D, respectively, its dimensions are 2a and 2b in the X and Y directions, Receiver (\(x_{R},y_{R}\)) located inside the transmitter loop. The total vertical component contribution \(Hz\) is the sum of each side [27]:  
\[H_{\rm Z}=H_{\rm ZA}+H_{\rm ZB}+H_{\rm ZC}+H_{\rm ZD} \tag{1}\]  
where \(H_{\rm ZA},H_{\rm ZB},H_{\rm ZC}\), and \(H_{\rm ZD}\) represent the electromagnetic field contributions from sides A, B, C, and D, respectively. For a uniform half space, the vertical component of the magnetic field due to a single dipole is [28]  
\[H_{\rm Z\it d}=\frac{\it Idlr}{4\pi r^{2}}\frac{y}{r}\bigg{[}(1-\frac{3}{2u^{2 }})\bigg{]}erf(u)+\frac{3}{u\sqrt{\pi}}e^{-u^{2}} \tag{2}\]  
\[u=\sqrt{\
icefrac{{\mu_{0}r^{2}}}{{4\rho t}}},\ r=\sqrt{\left(x-x_{R}\right) ^{2}+\left(y-y_{R}\right)^{2}} \tag{3}\]  
_erf_ (\(u\)) is the complementary error function, \(I\) is the transmitter current (A), \(dl\) is the dipole length, and \(r\) is the distance between the transmitter and the receiver.  
Therefore, the electromagnetic field contributions from the transmitter loop sides A, B, C, and D are given by:  
\[H_{\rm ZA}=(b+y_{R})\int\limits_{-a}^{a}\frac{\Big{\{}\frac{I}{4\pi r^{2}} \Big{[}(1-\frac{3}{2u^{2}})\Big{]}erf(u)+\frac{3}{u\sqrt{\pi}}e^{-u^{2}}\Big{\}} dx}{\sqrt{\left(x-x_{R}\right)^{2}+\left(b+y_{R}\right)^{2}}} \tag{4}\]  
Figure 3: Topographic map of the survey area.  
\[H_{\text{Z}B} = (a-x_{R})\int\limits_{-a}^{a}\frac{\left\{\frac{I}{4\pi r^{2}} \Big{[}(1-\frac{3}{2u^{2}})\Big{]}erf(u)+\frac{3}{u\sqrt{\pi}}e^{-u^{2}}\right\} dx}{\sqrt{\left(a-x_{R}\right)^{2}+\left(y-y_{R}\right)^{2}}} \tag{5}\] \[H_{\text{ZC}} = (b-y_{R})\int\limits_{-a}^{a}\frac{\left\{\frac{I}{4\pi r^{2}} \Big{[}(1-\frac{3}{2u^{2}})\Big{]}erf(u)+\frac{3}{u\sqrt{\pi}}e^{-u^{2}}\right\} dx}{\sqrt{\left(x-x_{R}\right)^{2}+\left(b-y_{R}\right)^{2}}}\] (6) \[H_{\text{ZD}} = (a+x_{R})\int\limits_{-a}^{a}\frac{\left\{\frac{I}{4\pi r^{2}} \Big{[}(1-\frac{3}{2u^{2}})\Big{]}erf(u)+\frac{3}{u\sqrt{\pi}}e^{-u^{2}}\right\} dx}{\sqrt{\left(a+x_{R}\right)^{2}+\left(y-y_{R}\right)^{2}}} \tag{7}\]  
In general, the data collected from a large loop TEM system consist of vertical voltage measurements at different time intervals after the current is switched off. The relationship between the vertical voltage \(V(t)\) and vertical component of the magnetic field \(H_{Z}\) is given by [29; 30]:  
\[V(t)=-\mu_{0}(\frac{\partial H_{Z}}{\partial t})M \tag{8}\]  
\(M\) is the magnetic moments of received loop, \(\mu_{0}\) is the permeability of free space, and \(t\) is the decay time after the current is switched off.  
The apparent resistivity can be calculated by [27; 31]  
\[\rho_{s}=6.62\times 10^{-3}L^{\frac{5}{3}\left[V(t)/t\right]^{-\frac{2}{3}}t^{- \frac{5}{3}}} \tag{9}\]  
\(L\) is the side length of the transmitter loop, \(L=2a\).  
The relationship between Detecting depth (\(D(t)\)) the decay time \(t\) can be described, as follows [32]:  
\[D(t)=\sqrt{\frac{\rho_{a}\times\ t}{2\mu_{0}}} \tag{10}\]