The main motivation in writing this paper is to better understand why the subtraction method attenuates the noise better than the filtering approach Guitton and Claerbout (2003). This is difficult to answer since both methods, although having their roots in the inversion theory framework, follow two different philosophies. The subtraction method models simultaneously the noise and the signal components with a noise and signal modeling operator, respectively, whereas the filtering method approximates the noise covariance operator with filters. It is not clear from these definitions why both approaches should lead to different results. This short note attempts to fill this gap.
In the first section following the introduction, I present the subtraction method as developed by Nemeth. Then I show that the subtraction method approximates the inverse noise covariance operator with a projection filter, as opposed to a single filter, e.g., a prediction-error filter, for the filtering approach. To illustrate the difference between both methods, I present a 3-D land data example and show that the subtraction approach gives the best noise attenuation result.