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ACKNOWLEDGMENTS

We would like to thank Sergey Fomel for sharing his experience in designing low-rank approximate algorithms for wave propagation. The first author appreciates Tengfei Wang and Junzhe Sun for useful discussion in this study. We acknowledge supports from the National Natural Science Foundation of China (No.41474099) and Shanghai Natural Science Foundation (No.14ZR1442900). This publication is also based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. 2230. We thank SEG, BP and HESS Corporation for making the 2D VTI and TTI models available.

Appendix A

Components of the Christoffel matrix

For a general anisotropic medium, the components of the density normalized Christoffel matrix $\mathbf{\Gamma}$ are given as follows,

$\begin{displaymath}\begin{split} \Gamma_{11} &= [C_{11}k^2_x + C_{66}k^2_y + C_... ...36}+C_{45})k_xk_z +(C_{25}+C_{46})k_xk_y]/{\rho}. \end{split}\end{displaymath}$

(25)

Appendix B

Extended formulations of the pseudo-spectral operators

According to equations 6 and 8, we express the pseudo-spectral operator that can be used to extrapolate the coupled elastic wavefields in its extended formation:

$\begin{displaymath}\begin{array}{lcl} u_{x}(\mathbf{x},t+\Delta{t})&=&-u_{x}(... ...)\tilde{u}_z(\mathbf{k},t)}\,\mathrm{d}\mathbf{k}, \end{array}\end{displaymath}$

(26)

in which $\tilde{u}_x(\mathbf{k},t)$ , $\tilde{u}_y(\mathbf{k},t)$ and $\tilde{u}_z(\mathbf{k},t)$ represent the three components of the elastic wavefields in wavenumber-domain at the time of

For a VTI or orthorhombic medium, we express the stiffness tensor as a Voigt matrix:

$\displaystyle \mathbf{C} = \begin{pmatrix}C_{11} &C_{12} &C_{13} &0 &0 &0 \cr... ...{44} &0 &0 \cr 0& 0& 0 &0 & C_{55} &0 \cr 0& 0& 0 &0 &0 &C_{66}\end{pmatrix},$

(27)

in which there are only five independent coefficient with $C_{12}=C_{11}-2C_{66}$ , $C_{22}=C_{11}$ , $C_{23}=C_{13}$ and $C_{55}=C_{44}$ , for a VTI medium. Therefore, the propagation matrix has the following extended formulation,

$\begin{displaymath}\begin{array}{lcl} w_{xx}(\mathbf{k})=2-\Delta{t}^2[C_{11}k... ...\mathbf{k})=-\Delta{t}^2[C_{23}+C_{44}]{k_y}{k_z}. \end{array}\end{displaymath}$

(28)

If the principal axes of the medium are not aligned with the Cartesian axes, e.g., for the tilted TI and orthorhombic media, we should apply the Bond transformation (Carcione, 2007; Winterstein, 1990) to get the stiffness matrix under the Cartesian system. This will introduce more mixed partial derivative terms in the wave equation, which demands lots of computational effort if a finite-difference algorithm is used to extrapolate the wavefields. Fortunately, for the pseudo-spectral solution, it only introduces negligible computation to prepare the propagation matrix and no extra computation for the wavefield extrapolation.

Similarly, we can write the propagation matrix $\overline{w}_{ij}^{(m)}$ (in equation 20) for the decoupled elastic waves in its extended formulation:

$\begin{displaymath}\begin{array}{lcl} \overline{w}_{xx}^{(m)}(\mathbf{x},\mat... ...athbf{x},\mathbf{k})w_{zz}(\mathbf{x},\mathbf{k}). \end{array}\end{displaymath}$

(29)

Appendix C

K-space adjustment to the pseudo-spectral solution

According to the eigen-decomposition of the Christoffel matrix (see Equations 9 to 12), we can obtain the scalar wavefields for homogeneous anisotropic media using the theory of mode separation (Dellinger and Etgen, 1990),

$\displaystyle \hat{\overline{u}}_i = Q_{ij}\hat{u}_j,$

(30)

in which $\hat{\overline{u}}_i$ with

represents the scalar qP-, qS

- and qS

-wave fields. So these wavefields satisfy the same scalar wave equation

$\displaystyle \partial^2_{tt}\hat{\overline{u}}_i + (v_i{k})^2\hat{\overline{u}}_i = 0.$

(31)

The standard leapfrog scheme for this equation can be expressed as

$\displaystyle \frac{\hat{\overline{u}}^{(n+1)}_i - 2\hat{\overline{u}}^{(n)}_i ... ...\overline{u}}^{(n-1)}_i}{\Delta{t}^2} = -\lambda^2_i\hat{\overline{u}}^{(n)}_i.$

(32)

It is well known that this solution is limited to small time-steps for stable wave propagation.

Fortunately, there is an exact time-steping solution for the second-order time derivatives allowing for any size of time-steps for homogeneous medium (Cox et al., 2007; Etgen and Brandsberg-Dahl, 2009), namely:

$\displaystyle \frac{\hat{\overline{u}}^{(n+1)}_i - 2\hat{\overline{u}}^{(n)}_i ... ...frac{-\sin^2(\lambda_i\Delta{t}/2)}{(\Delta{t}/2)^2}\hat{\overline{u}}^{(n)}_i.$

(33)

Comparing equations C-3 and C-4 shows that, it is possible to extend the length of time-step without reducing the accuracy by replacing $(\lambda_i\Delta{t}/2)^2$ with $\sin^2{(\lambda_i\Delta{t}/2})$ . This opens up a possibility by replacing

with $k^2{sinc^2(\lambda_i\Delta{t}/2)}$ as a

-space adjustment to the spatial derivatives, which may convert the time-stepping pseudo-spectral solution into an exact one for homogeneous media, and stable for larger time-steps (for a given level of accuracy) in heterogeneous media (Bojarski, 1982).

Nowadays, the -space scheme is widely used to improve the approximation of the temporal derivative in acoustic and ultrasound (Fang et al., 2014; Cox et al., 2007; Tabei et al., 2002). As far as we know, Liu (1995) was the first to apply -space ideas to elastic wave problems. He derived a -space form of the dyadic Green's function for the second-order wave equation and used it to calculate the scattered field iteratively in a Born series. Firouzi et al. (2012) proposed a -space scheme on the base of the first-order elastic wave equation for isotropic media.

Accordingly, we apply the -space adjustment to improve the performance of our two-step time-marching pseudo-spectral solution of the anisotropic elastic wave equation. To propagate the elastic waves on the base of equations 6 and 8, we need modify the eigenvalues of Christoffel matrix as in Equation 14.

Next: Bibliography Up: Cheng et al.: Propagate Previous: conclusions

2016-11-21