The results of separating noise and signal using amplitude and spectral estimations appear promising. In the cases considered here, the noise and signal can be separated fairly well, especially in the case where both the noise filter and the signal filter were two-dimensional.
I found that if, before entering the conjugate-gradient solver, the desired result is initialized to a good approximation of the desired result, the number of iterations required by the solver is reduced tremendously. The resulting reduction of the cost makes this technique a practical noise removal method.
In removing coherent noise, I found that using filters of different shapes to characterize the signal and noise worked well. These shaped filters, along with the amplitude weighting, did a good job of removing the coherent noise without harming the signal. The reflection data was well preserved both within the zone where the noise was removed and outside the zone.