Discussion and Conclusions

Formulas are derived for the resolution limits of the migration-data kernels in Figure 4.4, as well as those for multiple reflections. They are applicable to images formed by RTM, LSM, and FWI. Their salient implications are the following.

1. Low- and intermediate-wavenumber information about the velocity distribution is estimated primarily by transmission migration of primaries and multiples. The intermediate wavenumbers can be supplied by interbed multiples, while the lower wavenumbers are contained in deep primaries and free-surface related multiples. Inverting multiples can be an opportunity for estimating subsurface velocity information not available in the primary reflections.
2. Inverting diffractions can provide twice or more the resolution compared to imaging primaries. Smearing residuals along the transmission wavepath can achieve a resolution of $\lambda$ near the diffractor. This high resolution is observed in MVA tomograms. On the other hand, diffraction energy can be more than an order-of-magnitude weaker than primary energy, so the diffraction data will be noisier.
3. Diving waves that bottom out at a certain depth will have a better vertical resolution than horizontal resolution. Therefore it is important to also invert deep reflections to increase both the vertical and horizontal resolution. Since reflections can be an order-of-magnitude weaker than diving waves, it is recommended that diving waves be filtered from the data after a sufficient number of iterations. This might constitute an iterative multi-physics approach to FWI, where inverting a different type of wavefield should be emphasized at different depths and iteration numbers.
4. The transmission migration kernels in Figure 4.4a-d are of the same type as their data kernels. This leads to velocity updates along the transmission wavepaths. In contrast, the traditional migration kernel $[G({\bf {g}}\vert{\bf {x}})^{\textrm{dir}}G({\bf {x}}\vert{\bf {s}})^{\textrm{dir}}]^*$ in Figure 4.4e is a product of two Green's functions for direct waves, while the data kernel is a product of a reflection and a direct-wave Green's function. This mismatch in the type of kernel does not lead to the traditional wavepath where seismic energy propagates, but gives the untraditional wavepath of a fat ellipse. This is the zone where reflection energy could have originated, i.e., the interface.

The limitation of this study is that it does not take into account the non-linear effects of evanescent energy (Fleming, 2008) in determining resolution. Utilizing evanescent energy with FWI could provide, in theory, resolution much better than $\lambda$ . It is expected that multiple scattering arrivals between neighboring subwavelength scatterers might provide the extra resolution needed, but not accounted for in this current study.