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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 *l*^{1}
and *l*^{2} norms, not only boasting robustness in the presence of noise and outlier effects like *l*^{1}
measures, but also smoothness for small residuals characteristic of *l*^{2} 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 *l*^{2} 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 *l*^{2} 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 *l*^{2} 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 *l*^{2} norm
in many geophysical applications.

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Stanford Exploration Project

4/20/1999