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.Comment: 23 pages, 7 figures. Typographical error corrected in accelerated algorithms (14) and (20

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One of the primary causes of poor seismic data quality is the masking of desired reflection signals by seismic waves scattered from topography or near-surface heterogeneities. For successful seismic acquisition and processing, one must be able to identify this scattered noise and quantify its strength relative to the desired signal so that appropriate measures can be taken to suppress it. An economical procedure has been developed that allows analysis of scattered noise. A field test consisting of a series of shots into a 3-D grid of receivers, whose dimensions depend on the range of noise velocities to be analyzed and the choice of temporal frequencies over which to make the analysis, makes it possible to identify events by measuring the direction and magnitude of their horizontal velocity. This 3-D grid also allows the formation of a series of nested areal arrays with graduated attenuation levels that can be used to quantify the signal-to-noise ratio .by attenuating the scattered noise by known amounts and observing the level at which desired signal appears. For any given shot, the seismic traces recorded by the grid can be analyzed and a graphic workstation can be used to produce radar-like displays of seismic events showing their arrival time, horizontal velocity, azimuth, temporal frequency bandwidth, and amplitude. With this information, one can design field arrays of the appropriate size and attenuation level to suppress the scattered noise. Often the required arrays are so large that they must be recorded in segments to prevent loss of desired signal due to necessary static and cross-line dip corrections. This array approach is most suitable for 2-D and wide-line profile acquisition schemes. Noise suppression levels of 20-30 dB have often been achieved, which has successfully revealed signal in many areas, such as Edwards Plateau in West Texas, karstic limestone regions of the former Yugoslavia, and Black Warrior Basin of Mississippi.