An obvious necessary step is an effective method for finding the velocities at offset. In particular, the method must be able to handle the situation where velocity is multi-valued.
Also, in the simple synthetic test described here, both the exact and pull adjoint inversions converge to the same value after a sufficient number of iterations. It remains to be seen how their behaviors will differ when applied to real data (if we can change the objective function by changing the operator, perhaps we will discover noticeably higher or lower minima in some cases).
A large number of wave equation based processes seem likely candidates for further investigation of pull adjoints. Sergey Fomel has done work with Kirchoff migration, and Jon Claerbout Claerbout (1995) discusses the application of pull adjoints to DMO and other forms of migration.
An important future step is the investigation of whether the use of the new moveout and stacking operator pair can reduce the number of iterations required to perform inversions using more complex series of operators. This opens the possibility of accelerating and/or improving various sorts of inversions in velocity space.