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

Pengliang Yang, Romain Brossier, Ludovic Métivier, Jean Virieux

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Pengliang Yang, Romain Brossier, Ludovic Métivier, Jean Virieux. Erratum: A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium. Geophysical Journal International, 2018, 212 (3), pp.1694-1695. 10.1093/gji/ggx493. hal-02009744

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Submitted on 13 Oct 2021

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

by Pengliang Yang,<fnref n="1" /> Romain Brossier,<fnref n="1" /> Ludovic Métivier<fnref n="1" />,<fnref n="2" /> and Jean Virieux<fnref n="1" />

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:

$$ \frac{\partial \mathbf{L}}{\partial \mathbf{m}} = \left\langle \mathbf{w}, \frac{\partial F(\mathbf{m}, \mathbf{w})}{\partial \mathbf{m}} \right\rangle_T \Leftrightarrow \frac{\partial \chi}{\partial \mathbf{m}} = \left\langle \mathbf{w}, \frac{\partial F(\mathbf{m}, \mathbf{w})}{\partial \mathbf{m}} \right\rangle_T \quad (61) $$

$$ \left\langle \frac{\partial \chi}{\partial \mathbf{m}}, \delta \mathbf{m} \right\rangle_{\Omega} = \int_{\Omega} dx \delta \rho \left( \int_{0}^{T} dt \bar{\mathbf{v}}^{\dagger} \partial_{t} \mathbf{v} \right) + \sum_{I=1}^{6} \sum_{J=I}^{6} \int_{\Omega} dx \delta C_{I J} \left( \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} \frac{\partial C^{-1}}{\partial C_{I J}} \left( \partial_{t} \boldsymbol{\sigma} - \mathbf{f}_{\sigma} + (C :: \Gamma) \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell} \right) \right) + \sum_{I=1}^{6} \sum_{J=I}^{6} \int_{\Omega} dx \delta C_{I J} \left( \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \frac{\partial (C :: \Gamma)}{\partial C_{I J}} \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell} \right) + \sum_{\ell=1}^{L} \sum_{I=1}^{6} \sum_{J=I}^{6} y_{\ell} \int_{\Omega} dx \delta Q_{I J}^{-1} \left( \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \left( C :: \frac{\partial \Gamma}{\partial Q_{I J}^{-1}} \right) \boldsymbol{\xi}_{\ell} \right), \quad (66) $$

$$ \frac{\partial \chi}{\partial \rho} = \int_{0}^{T} dt \bar{\mathbf{v}}^{\dagger} \partial_{t} \mathbf{v}, $$

$$ \frac{\partial \chi}{\partial C_{I J}} = \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} \frac{\partial C^{-1}}{\partial C_{I J}} \left( \partial_{t} \boldsymbol{\sigma} - \mathbf{f}_{\sigma} + (C :: \Gamma) \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell} \right) + \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \frac{\partial (C :: \Gamma)}{\partial C_{I J}} \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell}, $$

$$ \frac{\partial \chi}{\partial Q_{I J}^{-1}} = \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \left( C :: \frac{\partial \Gamma}{\partial Q_{I J}^{-1}} \right) \left( \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell} \right), \quad \text{with} \quad \left( \frac{\partial C :: \Gamma}{\partial C_{I J}} \right)_{I J} = \begin{cases} (Q_{I J}^{-1})_{I J}, & \text{if } I J = I J, J I \\ 0, & \text{otherwise.} \end{cases} \quad (69) $$

$$ \frac{\partial \chi}{\partial \rho} = \int_{0}^{T} dt \bar{\mathbf{v}}^{\dagger} \partial_{t} \mathbf{v}, $$

$$ \frac{\partial \chi}{\partial C_{I J}} = - \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \frac{\partial C}{\partial C_{I J}} C^{-1} D^{T} \mathbf{v} + \int_{0}^{T} dt \bar{\mathbf{\sigma}}^{\dagger} C^{-1} \frac{\partial (C :: \Gamma)}{\partial C_{I J}} \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell}, $$

$$ \frac{\partial \chi}{\partial Q_{I J}^{-1}} = \int_{0}^{T} dt (D^{T} \bar{\mathbf{u}})^{\dagger} \left( C :: \frac{\partial \Gamma}{\partial Q_{I J}^{-1}} \right) \left( \sum_{\ell=1}^{L} y_{\ell} \boldsymbol{\xi}_{\ell} \right), \quad (70) $$

<footnote n="1">¹Univ. Grenoble Alpes, ISTerre, F-38000 Grenoble, France. E-mail: pengliang.yang@univ-grenoble-alpes.fr</footnote>

<footnote n="2">²Univ. Grenoble Alpes, CNRS, Laboratoire Jean Kuntzmann (LJK), F-38000 Grenoble, France</footnote>
$$ \Gamma = \begin{bmatrix} \frac{\rho\alpha^2 Q_{\alpha}^{-1}}{\rho\alpha^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & 0 & 0 & 0 \\ \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1}}{\rho\alpha^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & 0 & 0 & 0 \\ \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1} - 2\rho\beta^2 Q_{\beta}^{-1}}{\rho\alpha^2 - 2\rho\beta^2} & \frac{\rho\alpha^2 Q_{\alpha}^{-1}}{\rho\alpha^2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\rho\beta^2 Q_{\beta}^{-1}}{\rho\beta^2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\rho\beta^2 Q_{\beta}^{-1}}{\rho\beta^2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{\rho\beta^2 Q_{\beta}^{-1}}{\rho\beta^2} \end{bmatrix} \quad (86) $$

$$ \frac{\partial \Gamma}{\partial Q_{\alpha}^{-1}} = \begin{bmatrix} 1 & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 \\ \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 1 & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 \\ \frac{\alpha^2}{\alpha^2 - 2\beta^2} & \frac{\alpha^2}{\alpha^2 - 2\beta^2} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad \frac{\partial \Gamma}{\partial Q_{\beta}^{-1}} = \begin{bmatrix} 0 & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 \\ \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 \\ \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & \frac{-2\beta^2}{\alpha^2 - 2\beta^2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \quad (87) $$

$$ \partial_t \xi_{\ell}^{ij} + \omega_{\ell} \xi_{\ell}^{ij} = \omega_{\ell} \dot{\epsilon}_{ij} \Rightarrow \xi_{\ell}^{ij} = -\frac{1}{\omega_{\ell}} \partial_t \xi_{\ell}^{ij} + \dot{\epsilon}_{ij}, \quad (B8) $$

The online version of this paper has been corrected. The publisher apologise for these errors.