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 .
The Huber solver is fairly stable with respect to two major choices: the number of iterations and . 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 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 , 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 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.