Conclusion

Since geophysical inverse problems are often ill-posed due to the presence of inconsistent data, high amplitude anomalies and outliers, relative insensitivity to noise is a desirable characteristic of an inversion method. The Huber function is a compromise misfit measure between l¹ and l² norms, not only boasting robustness in the presence of noise and outlier effects like l¹ measures, but also smoothness for small residuals characteristic of l² measures. The transition between the two norms is governed by a free parameter, the Huber threshold $\epsilon$.

The Huber solver is fairly stable with respect to two major choices: the number of iterations and $\epsilon$. The most striking result arises when we increase the number of iterations: while the l² result explodes, the Huber result looks stable. In addition, we may choose a threshold within a large range without degrading the estimated velocity model (once $\epsilon$ is small enough). We did not apply any regularization on the least squares method: it would make l² less noise-sensitive but requires either a regularization weight or a noise level estimate and results are rather sensitive to these. The Huber function also requires an estimate for the parameter $\epsilon$, but the results seem not to depend strongly on its choice. Furthermore, the Huber function gives better results than the l² when applied to velocity analysis showing its robustness to outlier effects. A data-dependent criterion for choosing the Huber threshold may prove fruitful, i.e., ``treat $X\%$ of the data as Gaussian in the small residuals'', where X is specified interactively by the end-user.

These results encourage the use of the Huber function whenever the data are contaminated with noise and, as a robust and stable measure, to replace the l² norm in many geophysical applications.