Erratum: A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

\begin{eqnarray} \frac{\partial \mathbb {L}}{\partial \mathbf {m}} = \left\langle \bar{\mathbf {w}},\frac{\partial F(\mathbf {m},\mathbf {w})}{\partial \mathbf {m}}\right\rangle _T \Leftrightarrow \frac{\partial \chi }{\partial \mathbf {m}} = \left\langle \bar{\mathbf {w}},\frac{\partial F(\mathbf {m},\mathbf {w})}{\partial \mathbf {m}}\right\rangle _T \end{eqnarray}

\begin{eqnarray} && {\left\langle \frac{\partial \chi }{\partial \mathbf {m}},\delta \mathbf {m} \right\rangle _\Omega =\int _\Omega \mathrm{d}\mathbf {x} \delta \rho \left(\int _0^T \mathrm{d}t \bar{\mathbf {v}}^\dagger \partial _t\mathbf {v} \right)}\nonumber\\ && {\qquad + \sum _{I=1}^6 \sum _{J=I}^6\int _\Omega \mathrm{d}\mathbf {x} \delta C_{IJ} \left(\int _0^T \mathrm{d}t \bar{\boldsymbol{\sigma }}^\dagger \frac{\partial C^{-1}}{\partial C_{IJ}} \left(\partial _t\boldsymbol{\sigma }-\mathbf {f}_\sigma + (C:: \Gamma)\sum^L_{\ell=1}y_\ell{\boldsymbol \xi_\ell}\right)\right) + \sum^6_{I=1}\sum^6_{J=I}\int_\Omega {\rm d}{\bf x}\delta C_{IJ}\left(\int^T_0{\rm d}t\bar{\boldsymbol\sigma}^\dagger C^{-1}\frac{\partial(C::\Gamma)}{\partial C_{IJ}}\sum^L_{\ell=1}y_\ell{\boldsymbol\xi}_\ell\right)}\nonumber\\ && {\qquad +\sum _{\ell =1}^L\sum _{I=1}^6\sum _{J=I}^6 y_\ell \int _{\Omega }\mathrm{d}\mathbf {x}\delta Q_{IJ}^{-1} \left(\int _0^T\mathrm{d}t \bar{\boldsymbol{\sigma }}^\dagger C^{-1} \left(C::\frac{\partial \Gamma }{\partial Q_{IJ}^{-1}}\right) \boldsymbol{\xi }_\ell \right),} \end{eqnarray}

\begin{eqnarray} \frac{\partial \chi }{\partial \rho } &=& \int _0^T \mathrm{d}t \bar{\mathbf {v}}^\dagger \partial _t\mathbf {v}, \nonumber \\ \frac{\partial \chi }{\partial C_{IJ}}&=& \int _0^T \mathrm{d}t \bar{\boldsymbol{\sigma }}^\dagger \frac{\partial C^{-1}}{\partial C_{IJ}} \left(\partial _t\boldsymbol{\sigma } -\mathbf {f}_\sigma + (C::\Gamma)\sum^L_{\ell=1}y_\ell{\boldsymbol\xi}_\ell\right) + \int^T_0{\rm d}t\bar{\boldsymbol\sigma}^\dagger C^{-1}\frac{\partial(C::\Gamma)}{\partial C_{IJ}}\sum^L_{\ell=1}y_\ell{\boldsymbol\xi}_\ell, \nonumber \\ \frac{\partial \chi }{\partial Q_{IJ}^{-1}} &=& \int _0^T\mathrm{d}t \bar{\boldsymbol{\sigma }}^\dagger C^{-1}\left(C::\frac{\partial \Gamma }{\partial Q_{IJ}^{-1}}\right) \left(\sum _{\ell =1}^L y_\ell \boldsymbol{\xi}_\ell \right), \quad {\rm with}\ \left(\frac{\partial C::\Gamma}{\partial C_{IJ}}\right)_{ij} =\left\{\begin{array}{l@{\quad}l} \left(Q^{-1}_{IJ}\right)_{ij}, & {\rm if}\ \ ij=IJ, JI\\ 0, & {\rm otherwise}. \end{array}\right. \end{eqnarray}

\begin{eqnarray} \frac{\partial \chi }{\partial \rho } &=& \int _0^T \mathrm{d}t \bar{\mathbf {v}}^\dagger \partial _t\mathbf {v}, \nonumber \\ \frac{\partial \chi }{\partial C_{IJ}} &=&-\int _0^T \mathrm{d}t\bar{\boldsymbol\sigma}^\dagger C^{-1}\frac{\partial C}{\partial C_{IJ}} C^{-1} D^T {\mathbf {v}} + \int^T_0 {\rm d}t\bar{\boldsymbol\sigma}^\dagger C^{-1} \frac{\partial(C::\Gamma)}{\partial C_{IJ}}\sum^L_{\ell=1}y_\ell{\boldsymbol\xi}_\ell,\nonumber\\ \frac{\partial \chi }{\partial Q_{IJ}^{-1}} &=& \int _0^T\mathrm{d}t \left( D^T \bar{\mathbf {u}} \right)^\dagger \left(C ::\frac{\partial \Gamma }{\partial Q_{IJ}^{-1}}\right) \left(\sum _{\ell =1}^L y_\ell \boldsymbol{\xi}_\ell \right), \end{eqnarray}

\begin{eqnarray} \Gamma= \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \frac{\rho\alpha^2Q^{-1}_\alpha}{\rho\alpha^2} & \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q_\beta^{-1}}{\rho\alpha^2 - 2\rho \beta^2} & \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q^{-1}_\beta}{\rho\alpha^2 - 2\rho\beta^2} & 0 & 0 &0\\ \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q^{-1}_\beta}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2Q^{-1}_\alpha}{\rho\alpha^2} & \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q_\beta^{-1}}{\rho\alpha^2 - 2\rho \beta^2} & 0 & 0 & 0\\ \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q^{-1}_\beta}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2 Q^{-1}_\alpha - 2\rho\beta^2 Q_\beta^{-1}}{\rho\alpha^2 - 2\rho \beta^2} & \frac{\rho\alpha^2Q^{-1}_\alpha}{\rho\alpha^2} & 0& 0 & 0\\ 0& 0 & 0 & \frac{\rho\beta^2 Q^{-1}_\beta}{\rho\beta^2} & 0 &0\\ 0& 0 & 0 & 0 &\frac{\rho\beta^2 Q^{-1}_\beta}{\rho\beta^2} &0\\ 0& 0 & 0 & 0 & 0 &\frac{\rho\beta^2 Q^{-1}_\beta}{\rho\beta^2} \end{array}\right] \end{eqnarray}

\begin{eqnarray} \frac{\partial\Gamma}{\partial Q^{-1}_\alpha}= \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 0 & 0 &0\\ \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 1 & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 0 & 0 &0\\ \frac{\alpha^2}{\alpha^2 - 2\beta^2} & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right],\ \frac{\partial \Gamma}{\partial Q^{-1}_\beta} = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0\\ \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0\\ \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right] \end{eqnarray}

\begin{equation} \partial _t\xi _\ell ^{ij}+\omega _\ell \xi _\ell ^{ij}=\omega _\ell \dot{\epsilon }_{ij}\Rightarrow \xi _\ell ^{ij}=-\frac{1}{\omega _\ell }\partial _t \xi _\ell ^{ij} + \dot{\epsilon }_{ij}, \end{equation}