Why two methods?

The two methods achieve the same goal, but each has its own pros and cons. I will show that the filtering method is easier to implement: the PEF estimation can be done iteratively directly from the residual as the iterations go on. In contrast, for the subtraction method, the coherent noise operator (a PEF) should be pre-estimated and kept constant as the iterations go on. This a priori information can be sometimes rather difficult to have. The convergence of the subtraction scheme is far better than the convergence of the filtering method, however. Ideally, the nature of the coherent noise should guide us in this choice.