Magnetotellurics

Remarks

Example: a homogeneous half-space of resistivity $\rho = 1/\sigma$. With time dependence $e^{i \omega t}$, the field $E_x = E_0 e^{-kz}$ with $k = \sqrt{i \omega \mu_0 \sigma}$ solves the diffusion equation $\partial_z^2 E_x = i \omega \mu_0 \sigma E_x$, and Faraday's law gives $\partial_z E_x = -i \omega \mu_0 H_y$. So $Z_{xy} = E_x / H_y = \sqrt{i \omega \mu_0 \rho}$, hence $\rho_a = \rho$ at every frequency and the phase is $45^\circ$. The field decays by $1/e$ over the skin depth $\delta = \sqrt{2\rho/(\omega \mu_0)} \approx 503 \sqrt{\rho T}$ metres, with $T = 2\pi/\omega$ in seconds and $\rho$ in ohm metres. Non-example: controlled-source electromagnetic methods use a man-made transmitter near the receivers, so the plane-wave assumption fails and the impedance is not the magnetotelluric one. For a layered Earth $Z$ has the form $\begin{pmatrix} 0 & Z_{xy} \\ -Z_{xy} & 0 \end{pmatrix}$; in two- and three-dimensional structures all four entries carry information.