Diffraction-Transmission

How do the resolution characteristics of the diffraction-transmission wavepaths in Figure 4.4d compare to those for the reflection-transmission wavepaths in Figure 4.4b-c? Figure 4.8 suggests that the diffraction resolution limit will be significantly better because the diffraction propagation distance is effectively halved, leading to a narrower wavepath. This means that, if the waveform residuals are used to update the velocity, then the diffraction updates will have significantly better resolution than the reflection updates.

The resolution limits for diffraction-transmission migration can be quantified according to equation 4.11, which says that the maximum resolution limits perpendicular to the diffraction and reflection central rays should be, respectively, $\Delta r^{\textrm{diff}} \approx \sqrt{\lambda L/4}$ and $\Delta r^{\textrm{refl}} \approx \sqrt{\lambda L}$ . In this case, $L/2$ is the effective length of the central ray between the geophone and the scatterer in Figure 4.8a.

Figure 4.8: Wavepaths for migrating a) diffraction (red ellipse) and b) specular reflection (black ellipse) events along their transmission wavepaths. The diffraction resolution limit $2 \Delta r$ perpendicular to the widest part of the diffraction wavepath is about half smaller than that for the specular reflection. Dashed wavepath in b) is the mirror image of the source-side wavepath, with the mirror source denoted by the white star. These wavepaths were obtained by first generating acoustic data for a a) diffractor model and a b) two-layer reflector model. Windowing about the scattered arrivals, the diffraction and reflection traces were then migrated, respectively, with the kernels $[G({\bf{g}}\vert{\bf{x}})^{\textrm{diff}}G({\bf{x}}\vert{\bf{s}})^{\textrm{dir}}]^*$ and $[G({\bf{g}}\vert{\bf{x}})^{\textrm{dir}}G({\bf{x}}\vert{\bf{s}})^{\textrm{dir}}]*$ .

These limits can be rigorously derived by defining the diffraction Green's function $G({\bf {x}}\vert{\bf {s}})^{\textrm{diff}}$ as

$$G({\bf {x}}\vert{\bf {s}})^{\textrm{diff}}= A_{sx_ox}^{\textrm{diff}} e^{-i\omega (\tau_{sx_o}+\tau_{x_ox})},$$ (4.14)

where the diffractor is located at ${\bf {x}}_o$ , the trial image point is at ${\bf {x}}$ , and $A_{sx_ox}^{\textrm{diff}}$ accounts for the effects of geometrical spreading, reflection amplitude, and phase changes due to scattering. Replacing the migration kernel in equation 4.6 by $[G({\bf{g}}\vert{\bf{x}})^{\textrm{diff}}G({\bf{x}}\vert{\bf{s}})^{\textrm{dir}}]^*$ and the data kernel by $[G({\bf {g}}\vert{\bf {y}})^{\textrm{diff}}G({\bf {y}}\vert{\bf {s}})^{\textrm{dir}}]$ , and using the explicit expression for the Green's functions yields the model resolution function for diffraction imaging:

$$\delta m({\bf {x}})^{\textrm{mig}}=\omega^4 \!\!\int_{ D} [A_{sx_... ...y} \, e^{i\omega(\tau_{gx}-\tau_{gy}+\tau_{x_ox}-\tau_{x_oy})} dy^2 dx_g dx_s .$$ (4.15)

The salient difference between this formula and the one for reflections in equation F.4 is that $\tau_{x_ox}$ and $\tau_{x_oy}$ replace $\tau_{sx}$ and $\tau_{sy}$ . This says that the diffraction wavepath is generated by a ``virtual'' source at the diffractor ${\bf {x}}_o$ rather than at the actual source location ${\bf {s}}$ . Hence, the diffraction wavepath should be thinner than the specular reflection wavepath in Figure 4.8. In addition, every source-geophone pair has a diffraction wavepath that intersects the the diffractor. This means that, similar to diffraction migration, many more diffraction wavenumbers will be available for velocity updates compared to specular reflection-transmission wavepaths.