Magnetotelluric responses of a vertical inhomogeneous and anisotropic resistivity structure with a transitional layer

Abstract

The theoretical magnetotelluric responses of a vertical inhomogeneous and anisotropic resistivity structure with a transitional layer in which the resistivity is a linear function of depth are investigated. The expressions of the tangential electric and magnetic fields at the surface of the Earth model and the corresponding impedance have been evaluated. The influence of some model parameters such as the anisotropic dipping angles, the anisotropic coefficients and the resistivity contrast as well as the thickness of the transitional layer on the apparent resistivity and impedance phase are treated in detail. The results are graphically illustrated in the form of apparent resistivity and impedance phase curves, and they may be used in the interpretation of magnetotelluric sounding data in some specified geologic situations.

Appendix A: The MT responses in polarization with electric vector oriented along the \(x^{{\prime}}\)-axis.

When the electric field vector of the normally incident plane wave is along the $x^{{\prime}}$ direction (strike), the existing electric and magnetic fields components in first two isotropic layers are $E_{x^{\prime}}$ and $H_{y^{\prime}}$(in $o - x^{{\prime}} y^{{\prime}} z^{{\prime}}$ coordinate system), and in the dipping anisotropic layer are $E_{x}$, $H_{y}$ and $H_{z}$ (in $o{ - }xyz$ coordinate system).

In $o - x^{{\prime}} y^{{\prime}} z^{{\prime}}$ coordinate system, the electric and magnetic fields may be calculated according to the algebra described in Sect. 3, here only the final results are given. The solution may be refer to the Sect. 3 in this study and the corresponding context in (Qin and Yang 2020).

The electric and magnetic fields in region 1:

$$\left\{ \begin{gathered} E_{{x^{{\prime}} 1}} = A^{{\prime}} e^{{ - k_{1} z^{{\prime}} }} + B^{{\prime}} e^{{k_{1} z^{{\prime}} }} \hfill \\ H_{{y^{{\prime}} 1}} = - \frac{{k_{1} }}{i\omega \mu }\left( { - A^{{\prime}} e^{{ - k_{1} z^{{\prime}} }} + B^{{\prime}} e^{{k_{1} z^{{\prime}} }} } \right) \hfill \\ \end{gathered} \right..$$

(29)

The electric and magnetic fields in region 2:

$$\left\{ \begin{gathered} E_{{x^{{\prime}} 2}} = C^{{\prime}} \gamma I_{1} \left( \gamma \right) + D^{{\prime}} \gamma K_{1} \left( \gamma \right) \hfill \\ H_{{y^{{\prime}} 2}} = - \frac{{k_{2} }}{i\omega \mu }\left[ {C^{{\prime}} \gamma I_{0} \left( \gamma \right) - D^{{\prime}} \gamma K_{0} \left( \gamma \right)} \right] \hfill \\ \end{gathered} \right..$$

(30)

The electric and magnetic fields in region 3:

$$\left\{ \begin{aligned} E_{{x^{{\prime}} 3}} &= F^{{\prime}} e^{{ - k_{3}^{\prime } z^{{\prime}} }} \hfill \\ H_{{y^{{\prime}} 3}} &= N^{{\prime}} F^{{\prime}} e^{{ - k_{3}^{\prime } z^{{\prime}} }} \hfill \\ \end{aligned} \right.,$$

(31)

where $N^{{\prime}} = {{k_{3} } \mathord{\left/ {\vphantom {{k_{3} } {i\omega \mu }}} \right. \kern-\nulldelimiterspace} {i\omega \mu }}$.

The impedance at the surface of the model ($z^{\prime} = 0$) may be calculated as follows:

$$Z^{{\prime}} = \frac{{E_{{x^{{\prime}} 1}} }}{{H_{{y^{{\prime}} 1}} }} = - \frac{i\omega \mu }{{k_{1} }}\frac{{1 + {{A^{{\prime}} } \mathord{\left/ {\vphantom {{A^{{\prime}} } {B^{{\prime}} }}} \right. \kern-\nulldelimiterspace} {B^{{\prime}} }}}}{{1 - {{A^{{\prime}} } \mathord{\left/ {\vphantom {{A^{{\prime}} } {B^{{\prime}} }}} \right. \kern-\nulldelimiterspace} {B^{{\prime}} }}}},$$

(32)

where $\frac{{A^{{\prime}} }}{{B^{{\prime}} }} = \frac{{\left[ {U^{{\prime}} K_{0} \left( {\gamma_{2} } \right) + N^{{\prime}} K_{1} \left( {\gamma_{2} } \right)} \right]\left[ {I_{1} \left( {\gamma_{1} } \right) - I_{0} \left( {\gamma_{1} } \right)} \right] + \left[ {U^{{\prime}} I_{0} \left( {\gamma_{2} } \right) - N^{{\prime}} I_{1} \left( {\gamma_{2} } \right)} \right]\left[ {K_{1} \left( {\gamma_{1} } \right) + K_{0} \left( {\gamma_{1} } \right)} \right]}}{{\left[ {U^{{\prime}} K_{0} \left( {\gamma_{2} } \right) + N^{{\prime}} K_{1} \left( {\gamma_{2} } \right)} \right]\left[ {I_{1} \left( {\gamma_{1} } \right) + I_{0} \left( {\gamma_{1} } \right)} \right] + \left[ {U^{{\prime}} I_{0} \left( {\gamma_{2} } \right) - N^{{\prime}} I_{1} \left( {\gamma_{2} } \right)} \right]\left[ {K_{1} \left( {\gamma_{1} } \right) - K_{0} \left( {\gamma_{1} } \right)} \right]}}e^{{2k_{1} z_{1} }} ,$

$$U^{{\prime}} = - \frac{1}{i\omega \mu }\sqrt {\frac{i\omega \mu }{{\rho_{1} + p^{{\prime}} \left( {z_{2} - z_{1} } \right)}}} ,$$

$$p^{{\prime}} = {{\left( {\rho_{l3} - \rho_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{l3} - \rho_{1} } \right)} {\left( {z_{2} - z_{1} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {z_{2} - z_{1} } \right)}}.$$

Finally, the commonly used MT transfer functions (apparent resistivity and impedance phase) can be easily obtained according to the formulas (28) given in Sect. 3.5.