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 ep minimization for 1 less than or equal to 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.