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The need for a robust solver may be addressed using the *l*^{1} norm for the data residual
Claerbout and Muir (1973). Again, robust measures are related to the
long-tailed density function in the same way that the mean square is related to
the (short-tailed) Gaussian Tarantola (1987). The *l*^{1} norm is then less sensitive to
outliers and will give a more probable fitting of the data.

The requirements in the design of a robust inverse method that gives a sparse model for
the velocity estimation problem leads to the minimization of the objective function

| |
(3) |

where | |_{1} is the *l*^{1} norm.
Since we wish to utilize the *l*^{1} norm, the minimization of *f* is a cumbersome
problem. The *l*^{1} norm is not differentiable everywhere, which makes its use
rather difficult. The next section presents some alternatives to the *l*^{1} norm
using hybrid *l*^{1}-*l*^{2} objective functions. These functions are differentiable
and allow the use of iterative methods.

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Stanford Exploration Project

4/27/2000