Fundamentals of Magnetotellurics
The basic theory of magnetotellurics (MT) is Maxwell equations.
Faraday's law:
\begin{equation}
\nabla\times\boldsymbol{E}(t,\boldsymbol{r})=-\frac{\partial(\mu(\boldsymbol{r})\boldsymbol{H}(t,\boldsymbol{r}))}{\partial t}, \tag{1}
\end{equation}
Ampère-Maxwell law, or Ampère's law with Maxwell's addition:
\begin{equation}
\nabla\times\boldsymbol{H}(t,\boldsymbol{r})=\sigma(\boldsymbol{r})\boldsymbol{E}(t,\boldsymbol{r})+\frac{\partial(\varepsilon(\boldsymbol{r})\boldsymbol{E}(t,\boldsymbol{r}))}{\partial t}+\boldsymbol{J_{ext}(t,\boldsymbol{r})}, \tag{2}
\end{equation}
where \(\boldsymbol{E}\) and \(\boldsymbol{H}\) are electric and magnetic fields, respectively. \(\sigma\), \(\varepsilon\), and \(\mu\) are electrical conducitivity, electric permittivity, and magnetic permeability, respectively.
\(\boldsymbol{J_{ext}}\) is electric current density, which is usually supposed to be non-ohmic electric current to be given as a source of an external origin. In MT analysis, \(\boldsymbol{J_{ext}}\) is supposed to be a horizontal plane wave or field.
In usual cases of MT measurements, phase differences or delays of propagation of electromagnetic wave are negligible, the second term of RHS of eq. (2), that is, a displacement current density can be omitted.
\begin{equation}
\nabla\times\boldsymbol{H}(t,\boldsymbol{r})=\sigma(\boldsymbol{r})\boldsymbol{E}(t,\boldsymbol{r})+\boldsymbol{J_{ext}}(t,\boldsymbol{r}). \tag{3}
\end{equation}
It is supposed that the speed of light or electromagnetic wave is infinte, and it is Ampère's law without Maxwell's addition. Thus, eqs. (1) and (3) are called pre-Maxwell equations.
Hereafter, \(t\) and \(\boldsymbol{r}\) are omitted for simplicity.
According to eqs. (3) and (4),
\begin{equation}
\nabla\times\frac{\nabla\times\boldsymbol{E}(t,\boldsymbol{r})}{\mu(\boldsymbol{r})}=-\frac{\partial(\sigma(\boldsymbol{r})\boldsymbol{E}(t,\boldsymbol{r}))}{\partial t}-\frac{\partial(\boldsymbol{J_{ext}}(t,\boldsymbol{r}))}{\partial t}, \tag{4}
\end{equation}
\begin{equation}
\nabla\times\frac{\nabla\times\boldsymbol{H}(t,\boldsymbol{r})}{\sigma(\boldsymbol{r})}=-\frac{\partial(\mu(\boldsymbol{r})\boldsymbol{H}(t,\boldsymbol{r}))}{\partial t}+\nabla\times\frac{\boldsymbol{J_{ext}}(t,\boldsymbol{r})}{\sigma(\boldsymbol{r})}. \tag{5}
\end{equation}
Eqs. (4) and (5) show that both \(\boldsymbol{E}(t,\boldsymbol{r})\) and \(\boldsymbol{H}(t,\boldsymbol{r})\) are linearly related to \(\boldsymbol{J_{ext}}(t,\boldsymbol{r})\), but non-linearly related to \(\sigma(\boldsymbol{r})\).
Hereafter, a time variation with only a single frequency is considered. Time dependency is supposed to be \(\exp(i\omega t)\), where \(\omega\) is an angular frequency. In addition, \(\mu(\boldsymbol{r})\) is supposed to be the magnetic constant \(\mu_0\), or the magnetic permeability of free space.
Eqs. (1) and (3) can be rewritten as follows,
\begin{equation}
\nabla\times\boldsymbol{E}(\omega,\boldsymbol{r})=-i\omega\mu_0\boldsymbol{H}(\omega,\boldsymbol{r}),\tag{6}
\end{equation}
\begin{equation}
\nabla\times\boldsymbol{H}(\omega,\boldsymbol{r})=\sigma(\boldsymbol{r})\boldsymbol{E}(\omega,\boldsymbol{r})+\boldsymbol{J_{ext}}(\omega,\boldsymbol{r}).\tag{7}
\end{equation}
Hereafter, an external electric current source \(J_{ext}\) is approximated to a plane wave field, and a Cartesian coordinate is considered: \(x\), \(y\), and \(z\) directions are northward, eastward, and downward, respectively.
Homogeneous (half) space case is considered here, \(\sigma(\boldsymbol{r})=\sigma\).
In the case that an external electric current directs to \(x\) direction, \(\boldsymbol{J_{ext}}={J}\boldsymbol{\hat{x}}\delta(z-z_0)\), then \(\boldsymbol{E}={E_x(\omega,z)}\boldsymbol{\hat{x}}\) and \(\boldsymbol{H}={H_y(\omega,z)}\boldsymbol{\hat{y}}\).
\begin{equation}
\partial_zE_x(\omega,z)=-i\omega\mu_0H_y(\omega,z),\tag{8}
\end{equation}
\begin{equation}
-\partial_zH_y(\omega,z)=\sigma E_x(\omega,z),\tag{9}
\end{equation}
where \(z>z_0\).
\begin{equation}
H_y(\omega,z)=H_0\exp(\pm\sqrt{i\omega\mu_0\sigma}z),\tag{10}
\end{equation}
where \(H_0\) is a constant.
Accroding to the condition that \(H_y(\omega,z)\rightarrow0\) at \(z\rightarrow+\infty\),
\begin{equation}
H_y(\omega,z)=H_0\exp(-\sqrt{i\omega\mu_0\sigma}z).\tag{11}
\end{equation}
\begin{equation}
E_x(\omega,z)=\sqrt{\frac{i\omega\mu_0}{\sigma}}H_0\exp(-\sqrt{i\omega\mu_0\sigma}z).\tag{12}
\end{equation}
A skin depth is \(z=\sqrt{2/{\omega\mu_0\sigma}}\) at which an amplitude of \(E_x\) and \(H_y\) decay to \(1/e\). MT impedance \(Z\) is,
\begin{equation}
Z_{xy}(\omega,z)\equiv\frac{E_x(\omega,z)}{H_y(\omega,z)}=\sqrt{\frac{i\omega\mu_0}{\sigma}}=\sqrt{\frac{\omega\mu_0}{\sigma}}e^{i(\pi/4)}. \tag{13}
\end{equation}
It indicates that, regardless of \(\sigma\), an argument of \(Z_{xy}\) is \(\pi/4\). \(\rho\equiv 1/\sigma=|Z_{xy}(\omega,z)|^2/\omega\mu_0\).
In the case that an external electric current directs to \(y\) direction,
\begin{equation}
Z_{yx}(\omega,z)\equiv\frac{E_y(\omega,z)}{H_x(\omega,z)}=-\sqrt{\frac{i\omega\mu_0}{\sigma}}=\sqrt{\frac{\omega\mu_0}{\sigma}}e^{i(-3\pi/4)}. \tag{14}
\end{equation}
Thus, an argument of \(Z_{yx}\) is \(-3\pi/4\).