Robust inversion of seismic data using the Huber norm

The “Huber function” (or “Huber norm”) is one of several robust error measures which interpolates between smooth (l 2) treatment of small residuals and robust (l 1) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient-based optimizer. We propose to minimize the Huber function with a quasi-Newton method that has the potential of being faster and more robust than conjugate-gradient methods when solving nonlinear problems. Tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the Huber function gives far more robust model estimates than does a least-squares fit with or without damping.