Elastic wave-vector decomposition in heterogeneous anisotropic media

Numerical algorithm

We can summarize the steps of the proposed elastic wave-vector decomposition as follows:

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Considering the decomposition operator given in equation 14, we define each component of the decomposed wavefield corresponding to $\alpha$ wave mode as

$$\widetilde{U}^{\alpha}_x = A^{\alpha}_{xx} \widetilde{U}_x + A^{\alpha}_{xy}\widetilde{U}_y + A^{\alpha}_{xz} \widetilde{U}_z~,$$ (26)

$$\widetilde{U}^{\alpha}_y = A^{\alpha}_{yy} \widetilde{U}_y + A^{\alpha}_{xy} \widetilde{U}_x + A^{\alpha}_{yz} \widetilde{U}_z~,$$

$$\widetilde{U}^{\alpha}_{z} = A^{\alpha}_{zz} \widetilde{U}_z + A^{\alpha}_{xz} \widetilde{U}_x + A^{\alpha}_{yz} \widetilde{U}_y~,$$

where each $A^{\alpha}_{ij}$ with $i$ and $j$ denoting different components $x,~y,$ and $z$ is given in equation 16.

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To implement the proposed filtering of singularities, we multiply each component by a weighting factor defined as

$$\hat{A}^{\alpha}_{ij} = A^{\alpha}_{ij}~w(\nu,\tau)~,$$ (27)

where $w(\nu,\tau) =$   min$\left(\frac{\sin(\frac{\nu}{3})}{\tau},1 \right)$ , $\hat{A}^{\alpha}_{ij}$ denotes the modified version of $A^{\alpha}_{ij}$ , which is used in equation 26, and $\tau$ is a thresholding parameter. This filtering process results in small $\hat{A}^{\alpha}_{ij}$ in places where the given phase direction $\mathbf {n}$ is close to the direction of a singularity. Figure 6 shows how the weight $w(\nu ,\tau )$ changes with respect to different values of $\tau$ in the case of the example orthorhombic model.

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Figure 6. Weight $w(\nu ,\tau )$ in equation 27 with a) $\tau =0.2$ , b) $\tau =0.1$ , and c) $\tau =0.05$ for the case of the example orthorhombic model.

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We then implement equation 26 with modified coefficients according to equation 27 applying the low-rank approximation as formulated in equation 18.

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As the last step, we compensate for the lost in amplitude information from the previous smoothing step using local signal-noise orthogonalization (Chen and Fomel, 2015) as described by

$$\mathbf{U}^{\alpha}\mathbf{(x)} = \bigg(1+\Big\langle\frac{\mathb... ...^{\alpha}\mathbf{(x)}}\Big\rangle\bigg)\mathbf{U}^{\alpha}_{\tau}\mathbf{(x)}~,$$ (28)

where $\mathbf{U}^{\alpha}_{0}\mathbf{(x)}$ and $\mathbf{U}^{\alpha}_{\tau}\mathbf{(x)}$ denote the separated wavefield without smoothing ($\tau=0$ ) and with the specified smoothing respectively. $\mathbf{U}^{\alpha}\mathbf{(x)}$ denotes the final separated wavefield after amplitude compensation and $\langle\cdot\rangle$ represents a smooth division operator. The notion behind equation 28 is that the desired signal ( $\mathbf{U}^{\alpha}_{\tau}\mathbf{(x)}$ ) is assumed to be locally orthogonal to the noise ( $\mathbf{U}_{0}^{\alpha}\mathbf{(x)}-\mathbf{U}_{\tau}^{\alpha}\mathbf{(x)}$ ). Therefore, we can extract the remaining part of the signal in the noise--the missing amplitudes from the smoothing process--and simply add it back for the signal reconstruction.

Elastic wave-vector decomposition in heterogeneous anisotropic media