Robust and stable velocity analysis using the Huber function

The Huber function is an hybrid l1-l2 misfit measure. We demonstrate, on velocity analysis examples, that the Huber function is more robust to outlier effects, and more stable with respect to the number of iterations than the l2 norm. We also show that the Huber threshold which controls the transition between the l1 and the l2 norm may be chosen within a certain range of values without damaging the final result. At this point, the l2 norm with conjugate gradient is twice as fast as the Huber function with a nonlinear optimization method. The comparison may be altered in favor of the nonlinear scheme by algorithm tuning, however. A more natural data dependent expression for the threshold might be also useful. These results encourage use of the Huber function whenever the data are contaminated with noise and, as a robust and stable function, to replacethe l2 norm in othergeophysicalapplications.