A primary goal in designing an iterative optimization method for inversion is to minimize the number of iterations necessary for convergence. Often, we tackle this by applying an appropriate filter.
When conjugate gradient (CG) methods are applied to a problem like velocity transform in 2-D, the principal artifact faced is the half-integration waveform Claerbout (1993, 1995). Logically, the best filter to suppress a half-integration waveform is the half-derivative filter. In practice, filters like the half-derivative filter have been reported as causing CG residuals to diverge Lumley (1994). However, when recently revisiting this approach, I encountered no similar obstacle to convergence.
For this work, I applied the half-derivative filter to a CG inversion used for velocity transformations for both a synthetic and a real case. I experimented with the order of the weighting functions in an attempt to improve the rate of convergence. The results in all of the cases showed that the half-derivative filter did not cause divergence in any of the cases and improved the convergence rate beyond that of CG inversion without the half-derivative filter.