## 3 MT Method
### Dimensionality Analysis  
Dimensionality analysis of MT data is important to identify the Earth's subsurface geometry characteristics by utilizing the information in the impedance tensor \(\mathbf{Z}\)[39]. Some of the dimensionality analysis is performed by using a polar diagram, induction arrows [39], tipper magnitude [36], and phase tensor [44]. A polar diagram is a diagram that depicts the dependency of the impedance tensor on the strike direction [45,46,47]. The off-diagonal components of the impedance tensor given by Equation (3) represent several parameters of MT data, including a polar diagram and phase tensor. By rotating Equation (3) with rotation operator \(\mathbf{R}\) (Equation (7)), Equation (8) can be rewritten:  
\[[\mathbf{R}(\alpha)]= \begin{bmatrix}\cos\alpha&\sin\alpha\\ -\sin\alpha&\cos\alpha\end{bmatrix}  \tag{7}\]  
\[[\mathbf{Z}(\alpha)]=[\mathbf{R}(\alpha)][Z][\mathbf{R}(-\alpha)] \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Line** & **Acquisition Time** & **Recording Time** & **Spacing (km)** & **Direction** & **Site Name** \\ \hline B & August 2009 & 12 h & 9–50 & SW-NE & GBM, MANE, TANG, & KEMA, PIDIB \\ \hline C & July–August 2012 & 12 h & 9–13 & N-N & C1, C2, C3, C4, C5, C6 \\ \hline D & July–August 2012 & 17 h & 9–50 & SE-NW & D1, D2, D3, D4, D5, D6, D7, D8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._where \(\alpha\) is the rotation angle. Then, we obtain each component of the impedance tensor \(\mathbf{Z}\), and its results can be given as Equations (9) and (10) [39]:  
\[\left|Z_{xx}(\alpha)\right|=\sqrt{F(\alpha)}\qquad\qquad\left|Z_{xy}(\alpha) \right|=\sqrt{G(\alpha)} \tag{9}\]  
\[\left|\arg Z_{xy}(\alpha)\right|=\left|H(\alpha)\right| \tag{10}\]  
with \(F(\alpha)\), \(G(\alpha)\), and \(H(\alpha)\) given by Equations (11)-(13):  
\[F(\alpha) =\left|Z_{2}\right|^{2}+\left|Z_{3}\right|^{2}sin^{2}\ 2\alpha+ \left|Z_{4}\right|^{2}cos^{2}\ 2\alpha+2 Re(Z_{2}Z_{3}^{*})\sin 2\alpha+2 Re(Z_{2}Z_{4}^{*}) \cos 2\alpha+Re(Z_{3}Z_{4}^{*})\sin 4\alpha \tag{11}\] \[G(\alpha) =\left|Z_{1}\right|^{2}+\left|Z_{3}\right|^{2}cos^{2}\ 2\alpha+ \left|Z_{4}\right|^{2}sin^{2}\ 2\alpha+2 Re(Z_{1}Z_{3}^{*})\cos 2\alpha-2 Re(Z_{1}Z_{4}^{*}) \sin 2\alpha-Re(Z_{3}Z_{4}^{*})\sin 4\alpha\] (12) \[H(\alpha)=\frac{Im(Z_{1}+Z_{3}\cos 2\alpha-Z_{4}\sin 2\alpha)}{ Re(Z_{1}+Z_{3}\cos 2\alpha-Z_{4}\sin 2\alpha)} \tag{13}\]  
Because the angle \(\alpha\) varies from 0 to \(2\pi\), the tips of the resultant segments show closed curves called impedance polar diagrams. The components of \(\left|Z_{xx}(\alpha)\right|\) and \(\left|Z_{xy}(\alpha)\right|\) are known as the amplitude polar diagrams. At the same time, the component of \(\left|\arg Z_{xy}(\alpha)\right|\) is known as the phase polar diagram. They may form an oval, figure of eight, or a flower with four petals. In the case of a one-dimensional model (1D model), the polar diagram for \(\left|Z_{xx}(\alpha)\right|\) reduces to a point. In contrast, the polar diagrams for \(\left|Z_{xy}(\alpha)\right|\) and \(\left|\arg Z_{xy}(\alpha)\right|\) are circles of radius \(\left|Z\right|\) and \(\left|\arg Z\right|\), respectively. In the two-dimensional Earth model (2D model), the diagram of \(\left|Z_{xx}(\alpha)\right|\) forms a flower with four identical petals. The bisectors between these petals are oriented in the longitudinal and transverse directions. The diagrams of \(\left|Z_{xy}(\alpha)\right|\) and \(\left|\arg Z_{xy}(\alpha)\right|\) are ovals. The major and minor diameters of them, \(2\left|Z^{\parallel}\right|\), \(2\left|Z^{\perp}\right|\), and \(\left|\arg Z^{\parallel}\right|\), \(\left|\arg Z^{\perp}\right|\), are longitudinally and transversally directed. In the three-dimensional Earth model (3D model), the polar diagram is distorted and forms a peanut shape because of the asymmetry of the medium [18]. The next dimensionality analysis is induction arrows, which qualitatively indicate the location of the induction field.