diff --git a/docs/api_TDEM.rst b/docs/api_TDEM.rst index 98468280..b835a24f 100644 --- a/docs/api_TDEM.rst +++ b/docs/api_TDEM.rst @@ -48,22 +48,6 @@ \newcommand{\I}{\vec{I}} -.. raw:: html - - - Sensitivity Calculation *********************** @@ -156,7 +140,7 @@ where \right] \end{align} -For the fields $\u$, the measured data is given by +For the fields \\(\\u\\), the measured data is given by .. math:: @@ -173,7 +157,7 @@ The sensitivity matrix **J** is then defined as \end{align} -Defining the function $\\c(m,\\u)$ to be +Defining the function \\(\\c(m,\\u)\\) to be .. math:: @@ -244,9 +228,9 @@ Implementing **J** times a vector Multiplying **J** onto a vector can be broken into three steps -* Compute $\\vec{p} = \\mathbf{G}m$ -* Solve $\\hat{\\mathbf{A}} \\vec{y} = \\vec{p}$ -* Compute $\\vec{w} = -\\mathbf{Q} \\vec{y}$ +* Compute \\(\\vec{p} = \\mathbf{G}m\\) +* Solve \\(\\hat{\\mathbf{A}} \\vec{y} = \\vec{p}\\) +* Compute \\(\\vec{w} = -\\mathbf{Q} \\vec{y}\\) .. math:: @@ -285,7 +269,7 @@ Remaining time steps: \begin{align} \dcurl \vec{y}_{e}^{(t+1)} + \frac{1}{\delta t} \vec{y}_{b}^{(t+1)} - {\color{red}- \frac{1}{\delta t} \vec{y}_{b}^{(t)} } + - \frac{1}{\delta t} \vec{y}_{b}^{(t)} = \vec{p}_b^{(t+1)} \\ \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)} \end{align} @@ -296,7 +280,7 @@ and \begin{align} \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} = - {\color{red} \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} } + \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \MfMui \vec{p}_b^{(t+1)} \\ \vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)} \end{align}