Nonlinear regularization techniques for seismic tomography

The effects of several nonlinear regularization techniques are discussed in the framework of 3D seismic tomography. Traditional, linear, $\ell_2$ penalties are compared to so-called sparsity promoting $\ell_1$ and $\ell_0$ 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 $\ell_2$ 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 $\ell_1$ damping of Daubechies-4 wavelet coefficients. It has the additional advantage of yielding a noiseless reconstruction, contrary to simple $\ell_2$ minimization (`Tikhonov regularization') which should be avoided. In some of our examples, the $\ell_0$ method produced notable artifacts. In addition we show how nonlinear $\ell_1$ methods for finding sparse models can be competitive in speed with the widely used $\ell_2$ methods, certainly under noisy conditions, so that there is no need to shun $\ell_1$ penalizations.