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A New Second Order Absorbing Boundary Layer Formulation for Anisotropic-Elastic Wavefield Simulation

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Abstract

The hybrid perfectly matched layer (H-PML) approach to boundary absorption, a combination of convolutional and multiaxial perfectly matched layer (C-PML and M-PML) approaches, is extended to simulate second-order displacement-stress elastic wave equations. The displacement components instead of the velocity components can be directly updated, which can further be used in elastic full waveform inversion. A general stability condition for the second-order displacement-stress elastic wave equations is also proposed. The C-PML and H-PML simulation that results in isotropic and anisotropic media are compared. H-PML is capable of absorbing boundary reflections in both isotropic and anisotropic media, but C-PML suffers severely from the boundary reflections in anisotropic media. The H-PML simulation results for both first- and second-order elastic wave equations show its efficiency in boundary reflections suppression. The computational cost comparison between C-PML and H-PML also demonstrated that H-PML needs smaller computational volumes than C-PML for suppressing the same level of boundary reflections, which is more suitably applied in anisotropic full waveform inversion to reduce the computational volume.

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Acknowledgements

The authors thank the sponsors of CREWES for continued support. This work was funded by CREWES industrial sponsors, NSERC (Natural Science and Engineering Research Council of Canada) through the Grant CRDPJ 461179-13, and by the Canada First Research Excellence Fund.

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Authors and Affiliations

  1. Department of Geoscience, CREWES Project, University of Calgary, Calgary, AB, Canada

    Junxiao Li & Kristopher A. Innanen

  2. Department of Geoscience, China university of Petroleum, Beijing, 102249, China

    Bing Wang

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  1. Junxiao Li
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  2. Kristopher A. Innanen
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  3. Bing Wang
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Correspondence to Junxiao Li.

Appendix

Appendix

The 3-D fourth-order displacement stress staggered grid FD scheme can be expressed as:

$$\begin{aligned} u_{I,J+\frac{1}{2},K+\frac{1}{2}}^{m+1} & = 2u_{I,J+\frac{1}{2},K+\frac{1}{2}}^{m}-u_{I,J+\frac{1}{2},K+\frac{1}{2}}^{m-1} +\frac{\varDelta ^2t}{h}\frac{1}{\rho _{I,J+\frac{1}{2},K+\frac{1}{2}}} \\ & \left[ a\left( \sigma ^{m}_{xx|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}-\sigma ^{m}_{xx|I-\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}\right) \right. \\ & \left. +\,b\left( \sigma ^{m}_{xx|I+\frac{3}{2},J+\frac{1}{2},K+\frac{1}{2}}-\sigma ^{m}_{xx|I-\frac{3}{2},J+\frac{1}{2},K+\frac{1}{2}}\right) \right. \\ & \left. +\,a\left( \sigma ^{m}_{xy|I,J+1,K+\frac{1}{2}}-\sigma ^{m}_{xy|I,J,K+\frac{1}{2}}\right) \right. \\ & \left. +\,b\left( \sigma ^{m}_{xy|I,J+2,K+\frac{1}{2}}-\sigma ^{m}_{xy|I,J-1,K+\frac{1}{2}}\right) \right. \\ & \left. +\,a\left( \sigma ^{m}_{xz|I,J+\frac{1}{2},K+1}-\sigma ^{m}_{xz|I,J+\frac{1}{2},K}\right) \right. \\ & \left. +\,b\left( \sigma ^{m}_{xz|I,J+\frac{1}{2},K+2}-\sigma ^{m}_{xz|I,J+\frac{1}{2},K-1}\right) \right], \\ \end{aligned}$$
(28)
$$\begin{aligned} v_{I+\frac{1}{2},J,K+\frac{1}{2}}^{m+1}=\,&2v_{I+\frac{1}{2},J,K+\frac{1}{2}}^{m}-v_{I+\frac{1}{2},J,K+\frac{1}{2}}^{m-1} +\frac{\varDelta ^2t}{h}\frac{1}{\rho _{I+\frac{1}{2},J,K+\frac{1}{2}}}\\&\left[ a\left( \sigma ^{m}_{yy|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}-\sigma ^{m}_{yy|I+\frac{1}{2},J-\frac{1}{2},K+\frac{1}{2}}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{yy|I+\frac{1}{2},J+\frac{3}{2},K+\frac{1}{2}}-\sigma ^{m}_{yy|I+\frac{1}{2},J-\frac{3}{2},K+\frac{1}{2}}\right) \right. \\&\left. +\,a\left( \sigma ^{m}_{xy|I+1,J,K+\frac{1}{2}}-\sigma ^{m}_{xy|I,J,K+\frac{1}{2}}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{xy|I+2,J,K+\frac{1}{2}}-\sigma ^{m}_{xy|I-1,J,K+\frac{1}{2}}\right) \right. \\&\left. +\,a\left( \sigma ^{m}_{yz|I+\frac{1}{2},J,K+1}-\sigma ^{m}_{yz|I+\frac{1}{2},J,K}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{yz|I+\frac{1}{2},J,K+2}-\sigma ^{m}_{yz|I+\frac{1}{2},J,K-1}\right) \right], \\ \end{aligned}$$
(29)
$$\begin{aligned} w_{I+\frac{1}{2},J+\frac{1}{2},K}^{m+1}=\,&2w_{I+\frac{1}{2},J+\frac{1}{2},K}^{m}-w_{I+\frac{1}{2},J+\frac{1}{2},K}^{m-1} +\frac{\varDelta ^2t}{h}\frac{1}{\rho _{I+\frac{1}{2},J+\frac{1}{2},K}}\\&\left[ a\left( \sigma ^{m}_{zz|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}-\sigma ^{m}_{zz|I+\frac{1}{2},J+\frac{1}{2},K-\frac{1}{2}}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{zz|I+\frac{1}{2},J+\frac{1}{2},K+\frac{3}{2}}-\sigma ^{m}_{zz|I+\frac{1}{2},J+\frac{1}{2},K-\frac{3}{2}}\right) \right. \\&\left. +\,a\left( \sigma ^{m}_{xz|I+1,J+\frac{1}{2},K}-\sigma ^{m}_{xz|I,J+\frac{1}{2},K}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{xz|I+2,J+\frac{1}{2},K}-\sigma ^{m}_{xz|I-1,J+\frac{1}{2},K}\right) \right. \\&\left. +\,a\left( \sigma ^{m}_{yz|I+\frac{1}{2},J+1,K}-\sigma ^{m}_{yz|I+\frac{1}{2},J,K}\right) \right. \\&\left. +\,b\left( \sigma ^{m}_{yz|I+\frac{1}{2},J+2,K}-\sigma ^{m}_{yz|I+\frac{1}{2},J-1,K}\right) \right], \\ \end{aligned}$$
(30)

and

$$\begin{aligned} \sigma ^{m}_{xx|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}=\,&\frac{1}{h}\left\{ c_{11|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( u^{m}_{I+1,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I,J+\frac{1}{2},K+\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\,b\left( u^{m}_{I+2,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I-1,J+\frac{1}{2},K+\frac{1}{2}}\right) \right] \right. \\&\left. +\,c_{12|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( v^{m}_{I+\frac{1}{2},J+1,K+\frac{1}{2}}-v^{m}_{I+\frac{1}{2},J,K+\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\,b\left( v^{m}_{I+\frac{1}{2},J+2,K+\frac{1}{2}}-v^{m}_{I+\frac{1}{2},J-1,K+\frac{1}{2}}\right) \right] \right. \\&\left. +\,c_{13|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+1}-w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K}\right) \right. \right. \\&\left. \left. +\,b\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+2}-v^{m}_{I+\frac{1}{2},J+\frac{1}{2},K-1}\right) \right] \right\}, \\ \end{aligned}$$
(31)
$$\begin{aligned} \sigma ^{m}_{yy|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}=\,&\frac{1}{h}\left\{ c_{12|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( u^{m}_{I+1,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I,J+\frac{1}{2},K+\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\,b\left( u^{m}_{I+2,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I-1,J+\frac{1}{2},K+\frac{1}{2}}\right) \right] \right. \\&\left. +\,c_{22|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( v^{m}_{I+\frac{1}{2},J+1,K+\frac{1}{2}}-u^{m}_{I+\frac{1}{2},J,K+\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\,b\left( v^{m}_{I+\frac{1}{2},J+2,K+\frac{1}{2}}-v^{m}_{I+\frac{1}{2},J-1,K+\frac{1}{2}}\right) \right] \right. \\&\left. \left. +\,c_{23|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}}\left[ a\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+1}-w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K}\right) \right. \right. \right. \\&\left. \left. +\,b\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+2}-v^{m}_{I+\frac{1}{2},J+\frac{1}{2},K-1}\right) \right] \right\}, \\ \end{aligned}$$
(32)
$$\begin{aligned} \sigma ^{m}_{zz|I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}=\,&\frac{1}{h}\left\{ c_{13|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}} \left[ a\left( u^{m}_{I+1,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I,J+\frac{1}{2},K+\,\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\, b\left( u^{m}_{I+2,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I-1,J+\frac{1}{2},K+\frac{1}{2}}\right) \right] \right. \\&\left. +\, c_{23|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}} \left[ a\left( v^{m}_{I+\frac{1}{2},J+1,K+\frac{1}{2}}-u^{m}_{I+\frac{1}{2},J,K+\frac{1}{2}}\right) \right. \right. \\&\left. \left. +\, b\left( v^{m}_{I+\frac{1}{2},J+2,K+\frac{1}{2}}-v^{m}_{I+\frac{1}{2},J-1,K+\frac{1}{2}}\right) \right] \right. \\&\left. +\, c_{33|{I+\frac{1}{2},J+\frac{1}{2},K+\frac{1}{2}}} \left[ a\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+1}-w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K}\right) \right. \right. \\&\left. \left. +\, b\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K+2}-v^{m}_{I+\frac{1}{2},J+\frac{1}{2},K-1}\right) \right] \right\}, \\ \end{aligned}$$
(33)
$$\begin{aligned} \sigma ^{m}_{xy|I,J,K+\frac{1}{2}}=\,&\frac{1}{h}c_{66|{I,J,K+\frac{1}{2}}} \left[ a\left( u^{m}_{I,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I,J-\frac{1}{2},K+\frac{1}{2}}\right) \right. \\&\left. +\,b\left( u^{m}_{I,J+\frac{3}{2},K+\frac{1}{2}}-u^{m}_{I,J-\frac{3}{2},K+\frac{1}{2}}\right) \right. \\&\left. +\,a\left( v^{m}_{I+\frac{1}{2},J,K+\frac{1}{2}}-v^{m}_{I-\frac{1}{2},J,K+\frac{1}{2}}\right) \right. \\&\left. +\,b\left( v^{m}_{I+\frac{3}{2},J,K+\frac{1}{2}}-v^{m}_{I-\frac{3}{2},J,K+\frac{1}{2}}\right) \right], \\ \end{aligned}$$
(34)
$$\begin{aligned} \sigma ^{m}_{xz|I,J+\frac{1}{2},K}=&\frac{1}{h}c_{44|{I,J+\frac{1}{2},K}} \left[ a\left( u^{m}_{I,J+\frac{1}{2},K+\frac{1}{2}}-u^{m}_{I,J+\frac{1}{2},K-\frac{1}{2}}\right) \right. \\&\left. +\,b\left( u^{m}_{I,J+\frac{1}{2},K+\frac{3}{2}}-u^{m}_{I,J+\frac{1}{2},K-\frac{3}{2}}\right) \right. \\&\left. +\,a\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K}-w^{m}_{I-\frac{1}{2},J+\frac{1}{2},K}\right) \right. \\&\left. +\,b\left( w^{m}_{I+\frac{3}{2},J+\frac{1}{2},K}-w^{m}_{I-\frac{3}{2},J+\frac{1}{2},K}\right) \right], \\ \end{aligned}$$
(35)
$$\begin{aligned} \sigma ^{m}_{yz|I,J+\frac{1}{2},K}=&\frac{1}{h}c_{44|{I+\frac{1}{2},J,K}} \left[ a\left( v^{m}_{I+\frac{1}{2},J,K+\frac{1}{2}}-v^{m}_{I+\frac{1}{2},J,K-\frac{1}{2}}\right) \right. \\&\left. +\,b\left( v^{m}_{I+\frac{1}{2},J,K+\frac{3}{2}}-v^{m}_{I+\frac{1}{2},J,K-\frac{3}{2}}\right) \right. \\&\left. +\,a\left( w^{m}_{I+\frac{1}{2},J+\frac{1}{2},K}-w^{m}_{I+\frac{1}{2},J-\frac{1}{2},K}\right) \right. \\&\left. +\,b\left( w^{m}_{I+\frac{1}{2},J+\frac{3}{2},K}-w^{m}_{I+\frac{1}{2},J-\frac{3}{2},K}\right) \right], \\ \end{aligned}$$
(36)

in which, a and b (\(a=9/8,b=-\,\frac{1}{2}4\)) are the fourth-order staggered grid parameters.

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Li, J., Innanen, K.A. & Wang, B. A New Second Order Absorbing Boundary Layer Formulation for Anisotropic-Elastic Wavefield Simulation. Pure Appl. Geophys. 176, 1717–1730 (2019). https://doi.org/10.1007/s00024-018-2046-z

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