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The effects of several nonlinear regularization techniques are discussed inthe framework of 3D seismic tomography. Traditional, linear, $\ell_2$ penaltiesare compared to so-called sparsity promoting $\ell_1$ and $\ell_0$ penalties,and a total variation penalty. Which of these algorithms is judged optimaldepends on the specific requirements of the scientific experiment. If thecorrect reproduction of model amplitudes is important, classical dampingtowards a smooth model using an $\ell_2$ norm works almost as well asminimizing the total variation but is much more efficient. If gradients (edgesof anomalies) should be resolved with a minimum of distortion, we prefer$\ell_1$ damping of Daubechies-4 wavelet coefficients. It has the additionaladvantage of yielding a noiseless reconstruction, contrary to simple $\ell_2$minimization (`Tikhonov regularization') which should be avoided. In some ofour examples, the $\ell_0$ method produced notable artifacts. In addition weshow how nonlinear $\ell_1$ methods for finding sparse models can becompetitive in speed with the widely used $\ell_2$ methods, certainly undernoisy conditions, so that there is no need to shun $\ell_1$ penalizations.

Nonlinear regularization techniques for seismic tomography