How to design a robust solver

The need for a robust solver may be addressed using the l¹ 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 l¹ 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

$$f(\bold{m})= \vert\bold{Hm}-\bold{d}\vert _1+\sigma\vert\bold{m}\vert _1,$$ (3)

where | |₁ is the l¹ norm. Since we wish to utilize the l¹ norm, the minimization of f is a cumbersome problem. The l¹ norm is not differentiable everywhere, which makes its use rather difficult. The next section presents some alternatives to the l¹ norm using hybrid l¹-l² objective functions. These functions are differentiable and allow the use of iterative methods.