Nonlinear regularization techniques for seismic tomography

The effects of several nonlinear regularization techniques are discussed in the framework of 3D seismic tomography. Traditional, linear, ℓ2ℓ2 penalties are compared to so-called sparsity promoting ℓ1ℓ1 and ℓ0ℓ0 penalties, and a total variation penalty. Which of these algorithms is judged optimal depends on the specific requirements of the scientific experiment. If the correct reproduction of model amplitudes is important, classical damping towards a smooth model using an ℓ2ℓ2 norm works almost as well as minimizing the total variation but is much more efficient. If gradients (edges of anomalies) should be resolved with a minimum of distortion, we prefer ℓ1ℓ1 damping of Daubechies-4 wavelet coefficients. It has the additional advantage of yielding a noiseless reconstruction, contrary to simple ℓ2ℓ2 minimization (‘Tikhonov regularization’) which should be avoided. In some of our examples, the ℓ0ℓ0 method produced notable artifacts. In addition we show how nonlinear ℓ1ℓ1 methods for finding sparse models can be competitive in speed with the widely used ℓ2ℓ2 methods, certainly under noisy conditions, so that there is no need to shun ℓ1ℓ1 penalizations.