## 2 Theoretical Background  
### Model and Field Expressions  
Figure 1 shows the SH-TE logging geometry, \(a=0.12\) m is the radius of the borehole. Outside the borehole, we use a homogeneous saturated porous medium extending to infinity in the radial direction to simulate the typical sandstone formation. In order to excite and receive pure cylindrical shear waves, we take an annular shear source and place the acoustic receivers on the borehole wall. Since the electromagnetic field can radiate into the fluid through the borehole wall, we place the electromagnetic receiver in the borehole and take a radial distance of \(a/2\) from the borehole axis.  
Figure 1: Schematic diagram for SH-TE logging.  
Next we derive the expressions of the borehole electromagnetic fields. According to [PERSON] et al. [68], the borehole radial and tangential electric and magnetic field expressions can be expressed as Equations (1)-(4).  
\[E_{r}^{b}=-\frac{i}{\eta_{e}^{2}}\left(k\frac{\partial E_{z}^{b}}{\partial r}+ \frac{\omega\mu}{r}\frac{\partial H_{z}^{b}}{\partial\theta}\right) \tag{1}\]  
\[H_{r}^{b}=-\frac{i}{\eta_{e}^{2}}\left(k\frac{\partial H_{z}^{b}}{\partial r}+ \frac{\omega e_{b}}{r}\frac{\partial E_{z}^{b}}{\partial\theta}\right) \tag{2}\]  
\[E_{\theta}^{b}=-\frac{i}{\eta_{e}^{2}}\left(k\frac{\partial E_{z}^{b}}{\partial \theta}-\omega\mu\frac{\partial H_{z}^{b}}{\partial r}\right) \tag{3}\]  
\[H_{\theta}^{b}=-\frac{i}{\eta_{e}^{2}}\left(k\frac{\partial H_{z}^{b}}{\partial \theta}-\omega e_{b}\frac{\partial E_{z}^{b}}{\partial r}\right) \tag{4}\]  
where \(E_{r}^{b}\), \(E_{\theta}^{b}\) and \(H_{r}^{b}\), \(H_{\theta}^{b}\) represent the electric field and the magnetic field, respectively. The subscript \"\(b\)\" represents the quantities in the borehole and the subscripts \"\(r\)\" and \"\(\theta\)\" represent the radial component and the tangential component, respectively. \(\eta_{e}\) is the electromagnetic radial wave number in the borehole. \(k\) is the axial wave number. \(\omega\) is the circular frequency and \(\mu=4\pi\times 10^{-7}\) H/m is the vacuum permeability. \(E_{z}^{b}\) cannot be excited due to the shape of the source. Combining Equations (1)-(4), the frequency-wavenumber domain expressions for \(\hat{E}_{\theta}^{b}\) and \(\hat{H}_{r}^{b}\) are given by Equations (5) and (6).  
\[\hat{E}_{\theta}^{b}=\frac{i}{\eta_{e}}\omega\mu A_{H}I_{1}(\eta_{e}r) \tag{5}\]  
\[\hat{H_{r}^{b}}=-\frac{i}{\eta_{e}}kA_{H}I_{1}(\eta_{e}r) \tag{6}\]  
where \(I_{1}\) is the first-order first-kind modified Bessel function. Combining the Helmholtz decomposition method, the field expressions of the formation are given by Equations (7)-(10).  
\[\hat{u}_{\theta}^{o}=\eta_{sh}^{o}K_{1}(\eta_{sh}^{o}r)B_{sh}^{o}+\eta_{em}^ {o}K_{1}(\eta_{em}^{o}r)B_{em}^{o} \tag{7}\]  
\[\hat{\tau}_{r\theta}^{0}=-G^{o}\left[\eta_{sh}^{o}\,{}^{2}K_{2}(\eta_{sh}^{o} r)B_{sh}^{o}+\eta_{em}^{o}\,{}^{2}K_{2}(\eta_{em}^{o}r)B_{em}^{o}\right] \tag{8}\]  
\[\hat{E}_{\theta}^{o}=\beta_{sh}^{o}\eta_{sh}^{o}K_{1}(\eta_{sh}^{o}r)B_{sh}^{ o}+\beta_{em}^{o}\eta_{em}^{o}K_{1}(\eta_{em}^{o}r)B_{em}^{o} \tag{9}\]  
\[\hat{H}_{z}^{o}=\frac{i}{\omega\mu}\left[\beta_{sh}^{o}\eta_{sh}^{o}\,{}^{2}K_ {0}(\eta_{sh}^{o}r)B_{sh}^{o}+\beta_{em}^{o}\eta_{em}^{o}\,{}^{2}K_{0}(\eta_{ em}^{o}r)B_{em}^{o}\right] \tag{10}\]  
\(\hat{u}_{\theta}^{o}\), \(\hat{\tau}_{r\theta}^{0}\), and \(\hat{E}_{\theta}^{o}\) represent the tangential displacement, stress, and electric field, respectively. \(\hat{H}_{z}^{o}\) is the axial magnetic field. Superscript \"\(o\)\" represents the quantities corresponding to the porous formation. \(G^{o}\) represents the formation shear modulus. For the meanings of \(\beta_{sh}^{o}\) and \(\beta_{em}^{o}\) the readers are referred to [PERSON] et al. [10]. \(\eta_{j}^{o}{}^{2}=k^{2}-k_{j}^{o}{}^{2},j=sh,em\) are the formation's SH and electromagnetic radial wave numbers. \(B_{sh}^{o}\) and \(B_{em}^{o}\) are the undetermined coefficients. \(K_{0}\), \(K_{1}\), and \(K_{2}\) are the second kind modified Bessel functions.