## Appendix A Nomenclature of Variables in Equations  
\begin{tabular}{l l l} \(\mathbf{\dot{H}}\) & Equation (1) & The radiated magnetic field of the transmitting coil \\ \(N\) & Equation (1) & The number of turns of the transmitting coil \\ \(S\) & Equation (1) & The cross section area of the transmitting coil \\ \(I\) & Equation (1) & The excitation current in the transmitting coil \\ \(k\) & Equation (1) & Wavenumber \\ \((x,y,z)\) & Equation (1) & The coordinate of the field point \\ \(\mathbf{e_{x}}\,\mathbf{e_{y}}\), \(\mathbf{e_{z}}\) & Equation (1) & Unit vectors in Cartesian coordinate system \\ \(H_{0}\) & Equation (2) & The amplitude of the magnetic field intensity before attenuation \\ \(H\) & Equation (2) & The amplitude of the magnetic field intensity after attenuation \\ \(\alpha\) & Equation (2) & The loss factor \\ \(\beta\) & Equation (2) & The phase constant \\ \(h\) & Equation (2) & The thickness of the pipe wall \\ \(n\) & Equation (3) & The number of turns of the receiving coil \\ \(S_{c}\) & Equation (3) & The cross section area of the receiving coil \\ \(\mu_{e}\) & Equation (3) & The effective magnetic permeability of the magnetic core \\ \(r\) & Equation (4) & The distance between dual receivers \\ \(d_{1}\) & Equation (4) & The distance between Transmitter and Receiver 1 in \(y\)-axis direction \\ \(d_{2}\) & Equation (4) & The distance between Transmitter and Receiver 2 in \(y\)-axis direction \\ \(H_{\text{i}}(z)\) & Equation (6) & The MFI with respect to pipe axis direction \\ \(H_{\text{i}max}(y)\) & Equation (7) & The maximum of MFI \(H_{\text{i}}(z)\) \\ \(H_{\text{i}max}(d_{1})\) & Equation (8) & The maximum of MFI \(H_{\text{i}}(z)\) of Tracking Receiver 1 \\ \(H_{\text{i}max}(d_{2})\) & Equation (8) & The maximum of MFI \(H_{\text{i}}(z)\) of Tracking Receiver 2 \\ \(H_{\text{d}}(z)\) & Equation (10) & The difference of the magnetic field intensity \\ \(z_{1}\) & Equation (12) & The extreme point of magnetic field intensity under the distance \(d_{1}\) \\ \(z_{2}\) & Equation (12) & The extreme point of magnetic field intensity under the distance \(d_{2}\) \\ \(w_{\text{0}}\) & Equation (12) & The initial calculated lateral offset \\ \(H_{\text{rd}}\) & Equation (13) & The radial component of the magnetic field \\ \(H_{\text{pd}}\) & Equation (13) & The circumferential component of the magnetic field \\ \(\varphi\) & Equation (13) & The circumferential angle \\ \(r_{\text{new}}\) & Equation (14) & The radial distance between the dual receivers in the lateral offset \\ \(d_{10}\) & Equation (14) & The initial calculated value of \(d_{1}\) \\ \(d_{20}\) & Equation (14) & The initial calculated value of \(d_{2}\) \\ \(w\) & Equation (15) & The lateral offset \\ \(d_{r1}\) & Equation (15) & The distance between Transmitter and Receiver 1 in lateral offset case \\ \(d_{2}\) & Equation (15) & The distance between Transmitter and Receiver 2 in lateral offset case \\ \(\alpha\) & Equation (16) & The pitch angle between pipe axis and the ground plane \\ \(H_{y}(z)\) & Equation (16) & The \(y\)-axis component of the original magnetic field \\ \(H_{x}(z)\) & Equation (16) & The \(z\)-axis component of the original magnetic field \\ \(\Delta z\) & Equation (17) & The difference of positions corresponding to magnetic maximum \\ \(H_{\text{hgn}}(y)\) & Equation (19) & MSI of the measured magnetic field \\ \end{tabular}