Conclusion

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 the $\ell^1$ and $\ell^2$ norm. It not only improves robustness in the presence of noise and outliers with a $\ell^1$measure, but also keeps smoothness for small residuals with a $\ell^2$ measure.

In this Chapter I have proposed minimizing the Huber function with a quasi-Newton method called limited-memory BFGS. This method has the potential of being faster and more robust than conjugate-gradient for solving non-linear problems. Tests with noisy synthetic and field data examples demonstrate that our method is robust to outliers present in the data space, as expected.

The possible applications of the Huber norm are endless in geophysics. I illustrate in Chapter how the Huber norm helps removing spikes in bathymetry data. In addition, I show in Chapter how the Huber norm helps separating primaries and multiple reflections better.