Synthetic problems

We use the same data panels as Guitton and Symes (1999) to compare the results of BP to Huber norm results as they are both l¹ minimization strategies. Synthetic tests are designed evaluate the method's performance to invert the HRT under circumstances of missing data, a slow plane wave superimposed on hyperbolic events, and spiky data space noise. Two field CMP's are also analyzed and compared to the Huber norm result. The dissimilarities in axes origin and formating of the plots are unimportant. The spikes are at the correct values, and the important thing to note in the plots is the distribution of energy around the model space. The HRT operator used for this implementation has no AVO qualities, although the synthetics were modeled with a wavelet and amplitude variation.

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.

 

miss
Figure 1 Missing data. Left column is velocity model space. Right column is data space. Row 1 is input velocity and modeled data. Row 2 is Huber norm inversion and modeled data. Row 3 and 4 are BP inversion and predicted data results. Row 3 model space has approximately the same number of model variable as data points. Row 4 has four times the number of model variables.

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.

 

surf
Figure 2 Slow plane wave superposition. Same format as explained in the caption of Figure .

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.

 

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Figure 3 Randomly spiked data. Same format as explained in the caption of Figure .