What will I do?

My goal in this paper is to compare three different inversion schemes for the multiples attenuation problem. They all aim to produce a velocity model where primaries are muted out and the predicted multiples are subtracted from the original data. I successively solve

1.

$f(\bold{m})= \Vert\bold{Hm}-\bold{d}\Vert _2$,

2.

$f(\bold{m})= \vert\bold{Hm}-\bold{d}\vert _1$,

3.

$f(\bold{m})= \vert\bold{Hm}-\bold{d}\vert _1+\sigma\vert\bold{m}\vert _1$,

and compare the results. I call arbitrarily ``l¹ norm'' any Huber function with a small threshold. Let us assume now that to the l¹ norm, for the data residual, corresponds a threshold

$$\epsilon = \frac{max\vert\bold{d}\vert}{100}.$$

In addition, for the regularization term, let us say that to the l¹ norm corresponds a threshold

$$\epsilon = \frac{max\vert\bold{d}\vert}{10000}.$$

$\epsilon$ is chosen smaller than before leading to a larger l¹ treatment of the model. I show later on that the convergence is greatly reduced by the addition of this regularization term.