Robust and stable velocity analysis using the Huber function

Antoine Guitton and William W. Symes

antoine@sep.stanford.edu, symes@caam.rice.edu

ABSTRACT

The Huber function is one of several robust error measures which interpolates between smooth (l²) treatment of small residuals and robust (l¹) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient-based optimizer. Tests with a linear inverse problem for velocity analysis, using both synthetic and field data, suggest that (1) the Huber function gives far more robust model estimates than does least squares, (2) its minimization using a standard quasi-Newton method is comparable in computational cost to least squares estimation using conjugate gradient iteration, and (3) the result of Huber data fitting is stable over a wide range of choices for $l^2 \rightarrow l^1$ threshold and total number of quasi-Newton steps.