A Generalized 2.5-D Time-Domain Seismic Wave Equation to Accommodate Various Elastic Media and Boundary Conditions

Abstract

The 2.5-D seismic wave numerical simulation method employs point sources from 2-D geological models, enabling the calculation of point source wavefields at pseudo-2-D computational cost. We present herein a generalized 2.5-D first-order time-domain governing equation to model seismic wave propagation in different (acoustic, elastic isotropic, and anisotropic) media, then derive different formulae that incorporate topographic free-surface and fluid–solid interfaces. Furthermore, by assigning different model parameters from point to point, accommodating different boundary conditions, and applying the finite difference approach, we achieve the numerical simulation of seismic wave propagation with just one computer program. Comparisons with 3-D analytic and numerical solutions obtained using different full-space homogeneous models (acoustic, elastic isotropic, and anisotropic) verify the correctness of the 2.5-D method. Comparison of the results with a 3-D pseudospectral method show that the proposed 2.5-D method can simulate seismic wave propagation in various media with different boundary conditions. In addition, unlike the problems encountered when using 2-D numerical solutions for real 3-D applications, the 2.5-D method can be employed directly as a forward modeling method in seismic reverse-time migration and an efficient wavefield conversion tool between practical point source data and artificial line source data for 2-D seismic full waveform inversion.

Appendices

Appendix 1: Coefficient Matrices for Elastic Anisotropic Medium

Equations (3) can be rewritten in the following form:

$$ \begin{aligned} \left( {\begin{array}{*{20}c} {\dot{\tilde{v}}_{x} } \\ {\dot{\tilde{v}}_{y} } \\ {\dot{\tilde{v}}_{z} } \\ \end{array} } \right) & = \left( {\begin{array}{*{20}c} {\rho^{ - 1} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\rho^{ - 1} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho^{ - 1} } & 0 & 0 \\ \end{array} } \right)\partial_{x} \left( {\begin{array}{*{20}c} {\tilde{\sigma }_{xx} } \\ {\tilde{\sigma }_{xy} } \\ {\tilde{\sigma }_{yy} } \\ {\tilde{\sigma }_{xz} } \\ {\tilde{\sigma }_{yz} } \\ {\tilde{\sigma }_{zz} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} 0 & 0 & 0 & {\rho^{ - 1} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho^{ - 1} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho^{ - 1} } \\ \end{array} } \right)\partial_{z} \left( {\begin{array}{*{20}c} {\tilde{\sigma }_{xx} } \\ {\tilde{\sigma }_{xy} } \\ {\tilde{\sigma }_{yy} } \\ {\tilde{\sigma }_{xz} } \\ {\tilde{\sigma }_{yz} } \\ {\tilde{\sigma }_{zz} } \\ \end{array} } \right) \\ & + \left( {\begin{array}{*{20}c} 0 & {ik_{y} \rho^{ - 1} } & 0 & 0 & 0 & 0 \\ 0 & 0 & {ik_{y} \rho^{ - 1} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {ik_{y} \rho^{ - 1} } & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\tilde{\sigma }_{xx} } \\ {\tilde{\sigma }_{xy} } \\ {\tilde{\sigma }_{yy} } \\ {\tilde{\sigma }_{xz} } \\ {\tilde{\sigma }_{yz} } \\ {\tilde{\sigma }_{zz} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {\rho^{ - 1} s_{x} } \\ {\rho^{ - 1} s_{y} } \\ {\rho^{ - 1} s_{z} } \\ \end{array} } \right). \\ \end{aligned} $$

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Similarly, Eq. (4) becomes

$$ \left( {\begin{array}{*{20}c} {\dot{\tilde{\sigma }}_{xx} } \\ {\dot{\tilde{\sigma }}_{xy} } \\ {\dot{\tilde{\sigma }}_{yy} } \\ {\dot{\tilde{\sigma }}_{xz} } \\ {\dot{\tilde{\sigma }}_{yz} } \\ {\dot{\tilde{\sigma }}_{zz} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {c_{11} } & {c_{16} } & {c_{15} } \\ {c_{16} } & {c_{66} } & {c_{56} } \\ {c_{12} } & {c_{26} } & {c_{25} } \\ {c_{15} } & {c_{56} } & {c_{55} } \\ {c_{14} } & {c_{46} } & {c_{45} } \\ {c_{13} } & {c_{36} } & {c_{35} } \\ \end{array} } \right)\partial_{x} \left( {\begin{array}{*{20}c} {\tilde{v}_{x} } \\ {\tilde{v}_{y} } \\ {\tilde{v}_{z} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {c_{15} } & {c_{14} } & {c_{13} } \\ {c_{56} } & {c_{46} } & {c_{36} } \\ {c_{25} } & {c_{24} } & {c_{23} } \\ {c_{55} } & {c_{45} } & {c_{35} } \\ {c_{45} } & {c_{44} } & {c_{34} } \\ {c_{35} } & {c_{34} } & {c_{33} } \\ \end{array} } \right)\partial_{z} \left( {\begin{array}{*{20}c} {\tilde{v}_{x} } \\ {\tilde{v}_{y} } \\ {\tilde{v}_{z} } \\ \end{array} } \right) + \mathrm{i}k_{y} \left( {\begin{array}{*{20}c} {c_{16} } & {c_{12} } & {c_{14} } \\ {c_{66} } & {c_{26} } & {c_{46} } \\ {c_{26} } & {c_{22} } & {c_{24} } \\ {c_{56} } & {c_{25} } & {c_{45} } \\ {c_{46} } & {c_{24} } & {c_{44} } \\ {c_{36} } & {c_{23} } & {c_{34} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\tilde{v}_{x} } \\ {\tilde{v}_{y} } \\ {\tilde{v}_{z} } \\ \end{array} } \right). $$

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Accordingly, we obtain the following coefficient matrices for Eq. (6):

$$ {\mathbf{A}}^{(\mathrm{S})} = \left( {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{M}}_{1} } & {{\mathbf{M}}_{2} } \\ {{\mathbf{S}}_{1} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{S}}_{4} } & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right), \, {\mathbf{B}}^{(\mathrm{S})} = \left( {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{3} } \\ {{\mathbf{S}}_{2} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{S}}_{5} } & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right), \, {\mathbf{C}}^{(\mathrm{S})} = \left( {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{M}}_{4} } & {{\mathbf{M}}_{5} } \\ {{\mathbf{S}}_{3} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{S}}_{6} } & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right), $$

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where

$$ \begin{aligned} {\mathbf{M}}_{1} & = \left( {\begin{array}{*{20}c} {\rho^{ - 1} } & 0 & 0 \\ 0 & {\rho^{ - 1} } & 0 \\ 0 & 0 & 0 \\ \end{array} } \right), \, {\mathbf{M}}_{2} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\rho^{ - 1} } & 0 & 0 \\ \end{array} } \right), \, {\mathbf{M}}_{3} = \left( {\begin{array}{*{20}c} {\rho^{ - 1} } & 0 & 0 \\ 0 & {\rho^{ - 1} } & 0 \\ 0 & 0 & {\rho^{ - 1} } \\ \end{array} } \right), \, \\ {\mathbf{M}}_{4} & = \left( {\begin{array}{*{20}c} 0 & {ik_{y} \rho^{ - 1} } & 0 \\ 0 & 0 & {ik_{y} \rho^{ - 1} } \\ 0 & 0 & 0 \\ \end{array} } \right), \, {\mathbf{M}}_{5} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & {ik_{y} \rho^{ - 1} } & 0 \\ \end{array} } \right), \, {\mathbf{S}}_{1} = \left( {\begin{array}{*{20}c} {c_{11} } & {c_{16} } & {c_{15} } \\ {c_{16} } & {c_{66} } & {c_{56} } \\ {c_{12} } & {c_{26} } & {c_{25} } \\ \end{array} } \right), \, \\ {\mathbf{S}}_{2} & = \left( {\begin{array}{*{20}c} {c_{15} } & {c_{14} } & {c_{13} } \\ {c_{56} } & {c_{46} } & {c_{36} } \\ {c_{25} } & {c_{24} } & {c_{23} } \\ \end{array} } \right), \, {\mathbf{S}}_{3} = \mathrm{i}k_{y} \left( {\begin{array}{*{20}c} {c_{16} } & {c_{12} } & {c_{14} } \\ {c_{66} } & {c_{26} } & {c_{46} } \\ {c_{26} } & {c_{22} } & {c_{24} } \\ \end{array} } \right), \, {\mathbf{S}}_{4} = \left( {\begin{array}{*{20}c} {c_{15} } & {c_{56} } & {c_{55} } \\ {c_{14} } & {c_{46} } & {c_{45} } \\ {c_{13} } & {c_{36} } & {c_{35} } \\ \end{array} } \right), \, \\ {\mathbf{S}}_{5} & = \left( {\begin{array}{*{20}c} {c_{55} } & {c_{45} } & {c_{35} } \\ {c_{45} } & {c_{44} } & {c_{34} } \\ {c_{35} } & {c_{34} } & {c_{33} } \\ \end{array} } \right), \, {\mathbf{S}}_{6} = \mathrm{i}k_{y} \left( {\begin{array}{*{20}c} {c_{56} } & {c_{25} } & {c_{45} } \\ {c_{46} } & {c_{24} } & {c_{44} } \\ {c_{36} } & {c_{23} } & {c_{34} } \\ \end{array} } \right){.} \\ \end{aligned} $$

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Appendix 2: Coefficient Matrices for Acoustic Medium

We combine Eqs. (9) and (10) into the following form:

$$ \begin{aligned} \left( \begin{gathered} \dot{\tilde{v}}_{{\text{x}}} \hfill \\ \dot{\tilde{v}}_{y} \hfill \\ \dot{\tilde{v}}_{z} \hfill \\ {\dot{\tilde{P}}} \hfill \\ \end{gathered} \right) & = \left( \begin{gathered} { 0 0 0 } - \rho^{ - 1} \hfill \\ { 0 0 0 0} \hfill \\ { 0 0 0 0} \hfill \\ - K{ 0 0 0} \hfill \\ \end{gathered} \right)\partial_{x} \left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ {\tilde{P}} \hfill \\ \end{gathered} \right) + \left( \begin{gathered} {0 0 0 0} \hfill \\ {0 0 0 0} \hfill \\ {0 0 0 } - \rho^{ - 1} \hfill \\ {0 0 } - K { 0} \hfill \\ \end{gathered} \right)\partial_{z} \left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ {\tilde{P}} \hfill \\ \end{gathered} \right) \\ & + \left( \begin{gathered} {0 0 0 0} \hfill \\ {0 0 0 } - \mathrm{i}k_{y} \rho^{ - 1} \hfill \\ {0 0 0 }0 \hfill \\ {0 } - \mathrm{i}k_{y} K \, 0{ 0} \hfill \\ \end{gathered} \right)\left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ {\tilde{P}} \hfill \\ \end{gathered} \right) + \left( \begin{gathered} { 0} \hfill \\ { 0} \hfill \\ { 0} \hfill \\ \dot{s}(t)\delta ({\mathbf{x}} - {\mathbf{x}}_{s} ) \hfill \\ \end{gathered} \right). \\ \end{aligned} $$

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Consequently, we obtain the following coefficient matrices for Eq. (12):

$$ {\mathbf{A}}^{(\mathrm{W})} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & { - \rho^{ - 1} } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ { - K} & 0 & 0 & 0 \\ \end{array} } \right),{\mathbf{B}}^{(\mathrm{W})} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - \rho^{ - 1} } \\ 0 & 0 & { - K} & 0 \\ \end{array} } \right),{\mathbf{C}}^{(\mathrm{W})} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - \mathrm{i}k_{y} \rho^{ - 1} } \\ 0 & 0 & 0 & 0 \\ 0 & { - \mathrm{i}k_{y} K} & 0 & 0 \\ \end{array} } \right). $$

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Appendix 3: Coefficient Matrices for the Free Surface of Elastic Medium

Combing Eqs. 26 and 27, we obtain

$$ \begin{aligned} \left( \begin{gathered} \dot{\tilde{v}}_{{\text{x}}} \hfill \\ \dot{\tilde{v}}_{y} \hfill \\ \dot{\tilde{v}}_{z} \hfill \\ \dot{\tilde{\sigma }}_{xx} \hfill \\ \dot{\tilde{\sigma }}_{xy} \hfill \\ \dot{\tilde{\sigma }}_{yy} \hfill \\ \end{gathered} \right) & = \left( \begin{gathered} {0 0 0 } \rho^{ - 1} { 0 } 0 \hfill \\ {0 0 0 0 }\rho^{ - 1} \, 0 \hfill \\ {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} { 0 } 0 \hfill \\ c_{11} \, c_{16} \, c_{15} { 0 0 } 0 \hfill \\ c_{16} \, c_{66} \, c_{56} { 0 0 } 0 \hfill \\ c_{12} \, c_{26} \, c_{25} { 0 0 } 0 \hfill \\ \end{gathered} \right)\partial_{x} \left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ \tilde{\sigma }_{xx} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yy} \hfill \\ \end{gathered} \right) + \left( \begin{gathered} {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} { 0 } 0 \hfill \\ {0 0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} \, 0 \hfill \\ {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )^{2} \rho^{ - 1} { 0 } 0 \hfill \\ c_{15} \, c_{14} \, c_{13} { 0 0 } 0 \hfill \\ c_{56} \, c_{46} \, c_{36} { 0 0 } 0 \hfill \\ c_{25} \, c_{24} \, c_{23} { 0 0 } 0 \hfill \\ \end{gathered} \right)\partial_{z} \left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ \tilde{\sigma }_{xx} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yy} \hfill \\ \end{gathered} \right) \\ & + \left( \begin{gathered} {0 0 0 } 0 \, \mathrm{i}k_{y} \rho^{ - 1} \, 0 \hfill \\ {0 0 0 0 0 } \mathrm{i}k_{y} \rho^{ - 1} \hfill \\ {0 0 0 (}\partial_{xx} {\text{z}}_{{0}} )\rho^{ - 1} { (}\partial_{x} {\text{z}}_{{0}} )ik_{y} \rho^{ - 1} \, 0 \hfill \\ \mathrm{i}k_{y} c_{16} \, \mathrm{i}k_{y} c_{12} \, \mathrm{i}k_{y} c_{14} { 0 0 }0 \hfill \\ \mathrm{i}k_{y} c_{66} \, \mathrm{i}k_{y} c_{26} \, \mathrm{i}k_{y} c_{46} { 0 0 }0 \hfill \\ \mathrm{i}k_{y} c_{26} \, \mathrm{i}k_{y} c_{22} \, \mathrm{i}k_{y} c_{24} { 0 0 }0 \hfill \\ \end{gathered} \right)\left( \begin{gathered} \tilde{v}_{x} \hfill \\ \tilde{v}_{y} \hfill \\ \tilde{v}_{z} \hfill \\ \tilde{\sigma }_{xx} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yy} \hfill \\ \end{gathered} \right) + \left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right). \\ \end{aligned} $$

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We thus obtain the following coefficient matrices for Eq. (29):

$$ \begin{aligned} {\mathbf{A}}^{{({\text{AS}})}} & = \left( \begin{gathered} {0 0 0 } \, \rho^{ - 1} { 0 } 0 \hfill \\ {0 0 0 0 }\rho^{ - 1} \, 0 \hfill \\ {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} { 0 } 0 \hfill \\ c_{11} \, c_{16} \, c_{15} { 0 0 } 0 \hfill \\ c_{16} \, c_{66} \, c_{56} { 0 0 } 0 \hfill \\ c_{12} \, c_{26} \, c_{25} { 0 0 } 0 \hfill \\ \end{gathered} \right),\begin{array}{*{20}c} {} & {} \\ \end{array} {\mathbf{B}}^{{({\text{AS}})}} = \left( \begin{gathered} {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} { 0 } 0 \hfill \\ {0 0 0 0 (}\partial_{x} {\text{z}}_{{0}} )\rho^{ - 1} \, 0 \hfill \\ {0 0 0 (}\partial_{x} {\text{z}}_{{0}} )^{2} \rho^{ - 1} { 0 } 0 \hfill \\ c_{15} \, c_{14} \, c_{13} { 0 0 } 0 \hfill \\ c_{56} \, c_{46} \, c_{36} { 0 0 } 0 \hfill \\ c_{25} \, c_{24} \, c_{23} { 0 0 } 0 \hfill \\ \end{gathered} \right), \\ {\mathbf{C}}^{{({\text{AS}})}} & = \left( \begin{gathered} {0 0 0 } 0 \, \mathrm{i}k_{y} \rho^{ - 1} \, 0 \hfill \\ {0 0 0 0 0 } \mathrm{i}k_{y} \rho^{ - 1} \hfill \\ {0 0 0 (}\partial_{xx} {\text{z}}_{{0}} )\rho^{ - 1} { (}\partial_{x} {\text{z}}_{{0}} )ik_{y} \rho^{ - 1} \, 0 \hfill \\ \mathrm{i}k_{y} c_{16} \, \mathrm{i}k_{y} c_{12} \, \mathrm{i}k_{y} c_{14} { 0 0 } 0 \hfill \\ \mathrm{i}k_{y} c_{66} \, \mathrm{i}k_{y} c_{26} \, \mathrm{i}k_{y} c_{46} { 0 0 } 0 \hfill \\ \mathrm{i}k_{y} c_{26} \, \mathrm{i}k_{y} c_{22} \, \mathrm{i}k_{y} c_{24} { 0 0 } 0 \hfill \\ \end{gathered} \right), \\ \end{aligned} $$

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Appendix 4: Coefficient Matrices for a Fluid–Solid Interface

Combining Eqs. 40, 41, and 42, we obtain the following form:

$$ \begin{gathered} \left( \begin{gathered} \dot{\tilde{v}}_{x}^{(\mathrm{W})} \hfill \\ \dot{\tilde{v}}_{x}^{(\mathrm{S})} \hfill \\ \dot{\tilde{v}}_{y}^{(\mathrm{S})} \hfill \\ \dot{\tilde{v}}_{z}^{(\mathrm{S})} \hfill \\ {\dot{\tilde{P}}} \hfill \\ \dot{\tilde{\sigma }}_{xy} \hfill \\ \dot{\tilde{\sigma }}_{yz} \hfill \\ \end{gathered} \right) = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & { - \rho_{w}^{ - 1} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{x} z_{0}^{2} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{x} z_{0} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ { - K} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\partial_{x} \left( \begin{gathered} \tilde{v}_{x}^{(\mathrm{W})} \hfill \\ \tilde{v}_{x}^{(\mathrm{S})} \hfill \\ \tilde{v}_{y}^{(\mathrm{S})} \hfill \\ \tilde{v}_{z}^{(\mathrm{S})} \hfill \\ {\tilde{P}} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yz} \hfill \\ \end{gathered} \right) + \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\rho_{s}^{ - 1} \partial_{x} z_{0} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \rho_{s}^{ - 1} \partial_{x} z_{0} } & {\rho_{s}^{ - 1} } \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ { - K\partial_{x} z_{0} } & {K\partial_{x} z_{0} } & 0 & { - K} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\partial_{z} \left( \begin{gathered} \tilde{v}_{x}^{(\mathrm{W})} \hfill \\ \tilde{v}_{x}^{(\mathrm{S})} \hfill \\ \tilde{v}_{y}^{(\mathrm{S})} \hfill \\ \tilde{v}_{z}^{(\mathrm{S})} \hfill \\ {\tilde{P}} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yz} \hfill \\ \end{gathered} \right) \hfill \\ + \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - 2\rho_{s}^{ - 1} \partial_{x} z_{0} \partial_{xx} z_{0} }}{{(\partial_{x} z_{0}^{2} + 1)^{2} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{xx} z_{0} (1 - \partial_{x} z_{0}^{2} )}}{{(\partial_{x} z_{0}^{2} + 1)^{2} }}} & 0 & 0 \\ 0 & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{36} } & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{23} } & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{34} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\left( \begin{gathered} \tilde{v}_{x}^{(\mathrm{W})} \hfill \\ \tilde{v}_{x}^{(\mathrm{S})} \hfill \\ \tilde{v}_{y}^{(\mathrm{S})} \hfill \\ \tilde{v}_{z}^{(\mathrm{S})} \hfill \\ {\tilde{P}} \hfill \\ \tilde{\sigma }_{xy} \hfill \\ \tilde{\sigma }_{yz} \hfill \\ \end{gathered} \right) + \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right). \hfill \\ \end{gathered} $$

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We then obtain the following coefficient matrices for Eq. 44:

$$ {\mathbf{A}}^{(WS)} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & { - \rho_{w}^{ - 1} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{x} z_{0}^{2} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{x} z_{0} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ { - K} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right), \, {\mathbf{B}}^{(WS)} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\rho_{s}^{ - 1} \partial_{x} z_{0} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \rho_{s}^{ - 1} \partial_{x} z_{0} } & {\rho_{s}^{ - 1} } \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} }}{{\partial_{x} z_{0}^{2} + 1}}} & 0 & 0 \\ { - K\partial_{x} z_{0} } & {K\partial_{x} z_{0} } & 0 & { - K} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right), $$

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$$ {\mathbf{C}}^{(WS)} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - 2\rho_{s}^{ - 1} \partial_{x} z_{0} \partial_{xx} z_{0} }}{{(\partial_{x} z_{0}^{2} + 1)^{2} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \rho_{s}^{ - 1} \partial_{xx} z_{0} (1 - \partial_{x} z_{0}^{2} )}}{{(\partial_{x} z_{0}^{2} + 1)^{2} }}} & 0 & 0 \\ 0 & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{36} } & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{23} } & { - \mathrm{i}k_{y} (\partial_{x} z_{0}^{2} + 1)c_{34} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right). $$

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Appendix 5: Analytical Solutions in 3-D Homogeneous Medium

The analytical solutions in homogeneous acoustic media are given by

$$ F(r,t) = {\text{IFFT}}\left( {\mathrm{i} \cdot w \cdot R(w) \cdot G(w)} \right), $$

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where $R(w)$ is the Fourier transform of the Ricker wavelet function (Wang, 2015),

$$ R(w) = \frac{{2w^{2} }}{{\sqrt \pi w_{{\text{p}}}^{3} }}\exp \left( {\frac{{ - w^{2} }}{{w_{{\text{p}}}^{2} }} + \mathrm{i}wt_{0} } \right). $$

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$G(w)$ is the Green’s function in acoustic media (Aki & Richard, 1980).

The analytical solutions in homogeneous isotropic and VTI media are given by

$$ v_{kl} (x,y,z,t) = {\text{IFFT}}\left( {\mathrm{i} \cdot w \cdot R(w) \cdot G_{kl} ({\mathbf{x}},w)} \right), $$

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where $w_{{\text{p}}}$ is the domain frequency corresponding to the maximum amplitude and t₀ is the delay time. IFFT the is fast inverse Fourier transform. $G_{kl} ({\mathbf{x}},w)$ is an asymptotic Green’s function in homogeneous anisotropic medium (Eq. 72, Vavryčuk, 2007).