Elastic wave-vector decomposition in heterogeneous anisotropic media

Two-layered heterogeneous triclinic model

For our final test, we consider the two-layered French model (French, 1974) where the interface is made of various geometrical shapes. The model $c_{ij}$ parameters of both sublayers are heterogeneous and changing vertically as

$$c_{ij} = c^0_{ij}\bigg(1+\beta(z-z_0)\bigg)~,$$ (29)

where $c^0_{ij}$ denotes the value of stiffness at the reference depth $z_0$ , $\beta$ is the gradient, and $c_{ij}$ represents the value at other depths. For the top layer, $c^0_{ij}$ is 0.75 times the stiffnesses of the tricnic model (equation 20), $\beta = 0.6375$ ,and $z_0 = 0.4~km$ . For the bootm layer, $c^0_{ij}$ is equal to the stiffnesses of the tricnic model (equation 20), $\beta = 0.425$ , and $z_0 = 0.8~km$ . Figure 14 shows the plot of the density normalized $c_{11}$ in this setting. Other stiffnesses have similar appearance but with different values. A time snapshot at time $0.12~s$ of the full elastic wavefield is shown in Figure 15. We use the same oriented source as in the previous cases and put it at the middle of the model. Figures 16 and 17 show the resultant y-component of separated S1 and S2 wavefields. The final results with corrected amplitudes are shown in Figures 16c and 17c. Similar conclusions can be drawn as in the previous cases.

TRIc-11 Figure 14. Density normalized $c_{11}$ for the two-layered heterogeneous triclinic model. The parmeters are subjected to the heterogeneity specified in equation 29.

TRIw-lr-x,TRIw-lr-y,TRIw-lr-z
Figure 15. Original elastic wavefield in ${[x,z]}$ , ${[y,z]}$ , and ${[x,y]}$ planes generated from the stiffness tensor coefficients of the two-layered heterogenous triclinic model (equation 29) a) x-component b) y-component c) z-component. One can observe more complicated S-wave behaviors that those in the homogeneous orthorhombic model (Figure 7) and homogeneous triclinic model (Figure 15).

noTRIw-dlr-S1-y,TRIw-dlr-S1-y,comTRIw-dlr-S1-y
Figure 16. Separated y-component of S1 elastic wavefield in the two-layered heterogenous triclinic model (equation 29) with $\tau$ equal to a) 0 (no smoothing) b) 0.2. The final seprated wavefield with amplitude compensation (equation 28) is shown in c). Notice planar artifacts disappearing when the proposed smoothing filter is applied as shown in b) and with the restored amplitude asshown in c). The clipping has been adjusted to enhance visualization and stay constant in all three plots.

noTRIw-dlr-S2-y,TRIw-dlr-S2-y,comTRIw-dlr-S2-y
Figure 17. Separated y-component of S2 elastic wavefield in the two-layered heterogenous triclinic model (equation 29) with $\tau$ equal to a) 0 (no smoothing) b) 0.2. The final seprated wavefield with amplitude compensation (equation 28) is shown in c). Notice planar artifacts disappearing when the proposed smoothing filter is applied as shown in b) and with the restored amplitude as shown in c). The clipping has been adjusted to enhance visualization and stay constant in all three plots.

Elastic wave-vector decomposition in heterogeneous anisotropic media