Error analysis

In this section, we compare the accuracy of the approximated analytical stationary path approximation for VTI media with the one obtained by numerically solving the full quartic equation for different $\eta$ values (top panel of Figures 1-3). From the plots, we observe that the approximated analytical equation has similar accuracy as the one obtained by solving the quartic equation. Even when the $\eta$ value exceeds the bound that was given in the previous section, the relative error is still sufficiently small. This indicates that our approximation should hold for a wide range of $\eta$ values. The middle panel of Figures 1-3 shows the relative error between the isotropic and the analytically derived anisotropic stationary path. The bottom panels of figures 1-3 show the vertical wavenumbers computed for isotropic and anisotropic media for different $\eta$ values. It can be seen that there are significant differences between the anisotropic and isotropic k_z values which, if ignored, will lead to significant mispositioning of migrated events.

 

eta08
Figure 1 Relative Error plots for $\eta$ = 0.08, relative error in k_hy between analytical and quartic solutions (top panel), relative error between analytical anisotropic and isotropic k_hy (middle panel), relative error between anisotropic (solid) and isotropic (dashed) k_z (bottom panel)

 

eta17
Figure 2 Relative error plots for $\eta$ = 17, relative error in k_hy between analytical and quartic solutions (top panel), relative error between analytical anisotropic and isotropic k_hy (middle panel), relative error between anisotropic (solid) and isotropic (dashed) k_z (bottom panel)

 

eta33
Figure 3 Relative error plots for $\eta$ = 0.33, relative error in k_hy between analytical and quartic solutions (top panel), relative error between analytical anisotropic and isotropic k_hy (middle panel), relative error between anisotropic (solid) and isotropic (dashed) k_z (bottom panel)