Figure shows the results of the BP method when addressing the problem of missing data. We can see that the predicted data looks as accurate as the Huber norm result. The velocity model space, however, shows considerable difference. Notice the resolution increase over the same range of velocities and the lack of appreciable chatter away from basis atoms. With this figure, and those to come dealing with the synthetic examples, the predicted data looses the wavelet character and the amplitude seems to diminish with depth.
Figure shows the results of the BP method when a slow plane wave is superimposed on the data. The overcomplete dictionary now shows significantly less chatter about the velocity panel, and very distinguishable differences in the predicted data panel are emerging on the right side of the CMP where the events cross. Combination operators, linear and hyperbolic hybrid operators Trad et al. (2001), may be ideal for this situation, but have not been tried exhaustively yet.
Figure shows the results of the BP method when randomly distributed spikes contaminate the data. BP had significant trouble resolving this model. Unlike the Huber norm implementations of Guitton and Symes (1999), the method has no capacity to utilize the properties of the l1 norm in the data space, and so cannot handle the large spikes. Manually limiting the number of outer loops to seven was the only way to avoid instability. However, this point is easy to find as the duality gap begins increasing and the CG solver fails repeatedly to attain the input tolerance. Regardless, the predicted data looks pretty bad, and while the model space is sparse, the atoms that do have energy are inappropriate.