where and are the matching filters for the multiples and primaries respectively, is a parameter to balance the relative importance of the two components of the fitting goal (, ), is the data (primaries and multiples), is a regularization operator, (in my implementation a Laplacian operator), and is the usual parameter to control the level of regularization.

Once convergence is achieved, each filter is applied to its corresponding
convolutional matrix, and new estimates for and are computed:

(3) | |||

(4) |

Here represents the index of the outer iteration of the linear problem described by Equations 1 and 2. Notice that I hold constant although it could be changed from iteration to iteration . Also notice that the regularization operator and the regularization parameter in Equation 2 could be different for and . I have chosen to keep them the same to limit the number of adjustable parameters. This choice worked very well in all my tests. The updated versions of the convolutional matrices and are plugged into equations 1 and 2 and the process repeated until the cross-talk has been eliminated or significantly attenuated.

2007-10-24