Hybrid l ¹ /l ² minimization with applications to tomography

Least squares or l ² solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors. The l ^p minimization for 1< or =p<2 gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions to these l ^p problems. We apply IRLS to a hybrid l ¹ /l ² minimization problem that behaves like an l ² fit for small residuals and like an l ¹ fit for large residuals. The smooth transition from l ² to l ¹ behavior is controlled by a parameter that we choose using an estimate of the standard deviation of the data error. For linear problems of full rank, the hybrid objective function has a unique minimum, and IRLS can be proven to converge to it. We obtain a robust efficient method. For nonlinear problems, a version of the Gauss-Newton algorithm can be applied. Synthetic crosswell tomography examples and a field-data VSP tomography example demonstrate the improvement of the hybrid method over least squares when there are outliers in the data.