$\nabla ^2 \mathbf{E} + k^2 \mathbf{E} = 0$
$\nabla ^2 \mathbf{H} + k^2 \mathbf{H} = 0$
$ k = \sqrt{\omega ^2 \mu \varepsilon - i \omega \mu \sigma } $
$k_{ground} \simeq (1-i) \sqrt{ \frac{\omega \mu \sigma}{2} }$
$k_{air} \simeq \omega \sqrt{ \mu_0 \varepsilon_0}$
$\begin{split}\left(\begin{matrix} E_{x} \\ E_{y} \end{matrix} \right) = \left(\begin{matrix} \hat{Z}_{xx} & \hat{Z}_{xy} \\ \hat{Z}_{yx} & \hat{Z}_{yy} \end{matrix} \right) \left(\begin{matrix} H_x \\ H_y \end{matrix} \right)\end{split}$
$\begin{split} H_z = \left(\begin{matrix} T_{zx} & T_{zy} \end{matrix} \right) \left(\begin{matrix} H_x \\ H_y \end{matrix} \right)\end{split}$
$\delta = \sqrt{ \frac{2}{\omega \mu \sigma}} \simeq \frac{500}{\sqrt{\sigma f}}$
$ \phi_d = \sum \left(\frac{d^{obs}_i-d^{pred}_i}{\varepsilon} \right) ^2 $
$\phi = \phi_d +\beta \phi_m$
In [3]:
%%latex
\begin{align}
\nabla \times \textbf{E} + i \omega \textbf{B} &= 0 \\
\nabla \times \frac{1}{\mu} \textbf{B} - \sigma \textbf{E} &= \textbf{s}_E
\end{align}
In [4]:
%%latex
\begin{align}
\textbf{Z}(\omega) =
\begin{bmatrix}
Z_{xx}(\omega) & Z_{xy}(\omega)\\
Z_{yx}(\omega) & Z_{yy}(\omega)
\end{bmatrix} =
\begin{bmatrix}
\frac{E_x(\omega)}{H_x(\omega)} & \frac{E_x(\omega)}{H_y(\omega)}\\
\frac{E_y(\omega)}{H_x(\omega)} & \frac{E_y(\omega)}{H_y(\omega)}
\end{bmatrix} \label{eq:ImpFull}
\end{align}
In [6]:
%%latex
\begin{equation}
\rho^{app}(\omega) = \frac{1}{ \omega \mu_0} \left| Z_{n}(\omega) \right|^{2}
\qquad
\Phi(\omega) = \arctan \left(\frac{\Im(Z_{n}(\omega))}{\Re(Z_{n}(\omega))}\right)
\label{eq:AppResPhs}
\end{equation}
In [7]:
%%latex
\begin{align}
\textbf{T}(\omega) =
\begin{bmatrix}
T_{zx}(\omega) \\
T_{zy}(\omega)
\end{bmatrix} =
\begin{bmatrix}
\frac{H_z(\omega)}{H_x(\omega)} \\
\frac{H_z(\omega)}{H_y(\omega)}
\end{bmatrix} \label{eq:TipFull}
\end{align}
$ Z_{xy}, Z_{yx} $
$T_{zx}, T_{zy}$
In [ ]: