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

Erratum: A review on the systematic formulation of 3-D multiparameter full waveform inversion in... Erratum of the paper ‘A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium’, by Yang et al., published in Geophys. J. Int. (2016) 207, 129–149. The following equations were erroneously displayed in this paper and should be read as follow. The gradient of the misfit function with respect to model parameters m is the same as the gradient of the Lagrangian at the saddle points considering w and m are independent variables when performing derivatives:   \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} (61)  \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} (66)  \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} (69)  \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} (70)   \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} (86)  \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} (87)  \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} (B8) The online version of this paper has been corrected. The publisher apologise for these errors. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

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

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Publisher
The Royal Astronomical Society
Copyright
© The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.
ISSN
0956-540X
eISSN
1365-246X
D.O.I.
10.1093/gji/ggx493
Publisher site
See Article on Publisher Site

Abstract

Erratum of the paper ‘A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium’, by Yang et al., published in Geophys. J. Int. (2016) 207, 129–149. The following equations were erroneously displayed in this paper and should be read as follow. The gradient of the misfit function with respect to model parameters m is the same as the gradient of the Lagrangian at the saddle points considering w and m are independent variables when performing derivatives:   \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} (61)  \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} (66)  \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} (69)  \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} (70)   \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} (86)  \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} (87)  \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} (B8) The online version of this paper has been corrected. The publisher apologise for these errors. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

Journal

Geophysical Journal InternationalOxford University Press

Published: Mar 1, 2018

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