Hybrid ℓ1/ℓ2 minimization with applications to tomography

Abstract

Least squares or [Formula: see text] solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors. The [Formula: see text] minimization for 1 ≤ p < 2 gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions to these [Formula: see text] problems. We apply IRLS to a hybrid [Formula: see text] minimization problem that behaves like an [Formula: see text] fit for small residuals and like an [Formula: see text] fit for large residuals. The smooth transition from [Formula: see text] to [Formula: see text] 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.