Adaptive subtraction

Given a model of the multiple ${\bf M}$ and the data ${\bf d}$, a bank of non-stationary filters ${\bf f}$ is estimated such that  

$$g({\bf f})=\Vert{\bf Mf -d}\Vert^2+\epsilon^2\Vert{\bf Rf}\Vert^2$$ (1)

is minimum Rickett et al. (2001). In equation (1), ${\bf R}$ is the Helix derivative Claerbout (1998) that smooths the filter coefficients across micro-patches Crawley (2000) and ${\epsilon}$ a trade-off parameter between data fitting and smoothing. The filters have two dimensions. In the following results, the filters are 20 $\times$ 3 and the patch size is 44 $\times$ 20 (the first number corresponds to the time axis and the second number corresponds to the offset axis). Once the filters are estimated, the signal becomes

$${\bf \hat{s}} = {\bf Mf -d}.$$ (2)

The subtraction is done one shot gather at a time.