Conclusions

When a model of the multiples is adaptively subtracted from the data in a least-squares sense, we implicitly assume that the signal (primaries) has minimum energy and is orthogonal to the noise (multiples). This paper demonstrates that the minimum energy assumption might not hold and that another norm, the $\ell^1$ norm, should be used instead. Based on Chapter , I propose using the Huber norm for the filter estimation problem. I demonstrate with 1-D and 2-D data examples that the Huber norm with a small threshold $\alpha$ gives a much improved multiple attenuation results when the signal has not minimum energy: the multiples are well separated and the primaries are preserved. This Chapter exemplifies that the Huber norm can be used for many geophysical applications whenever robustness is sought. Chapter will further illustrate this point with an interpolation problem of noisy data.