It will be noted that because the ground roll is very aliased and dispersive in these gathers, the linear radon operator has great difficulty describing these events with a single kernel in the model domain. While the least squares inversion will introduce energy into the model-space to combine these events, as well as cancel acausal energy above the direct arrival in the data-space, the sparse inversions are not capable of stopping events that do not cross the entire data-space. For this reason, only one-sided gathers have been used for this analysis.
These techniques are particularly sensitive to noisy, unbalanced traces. For this reason all gathers have been trace-balanced, gained as a function of time, and noisy traces zeroed. Last, it should be noted that CMP-gathers should be used for this analysis instead of shot-gathers to insure that all of the subsurface hyperbolas do not have an apex shift.
While the bound-constrained, Cauchy, and l1 inversion schemes produce a more pleasing model-space, this investigation shows it is of limited use in this application of separating the linear from hyperbolic events in a CMP gather. While the least-squares inversion does allow the introduction of cross-talk between the two model-spaces, the noise subtraction technique is better implemented within this framework. This conclusion can be evaluated in terms of the sometimes disparate goals of analysis versus synthesis. If analysis is the goal, the sparseness optimized inversion schemes clearly outperform the least-squares model product. Velocity picking would be much better performed with these results. Of the three, this exploration shows the l1 scheme to be more tolerant in terms of the combined ease of parameter selection and quality of the model-space. For noise (linear event) separation, the least-squares solution enjoys both a high quality result and tolerance in terms of parameter selection. Further, the least-squares solution is much faster and requires fewer iterations. Purposefully halting the inversion after less than 30 iterations was important to avoid the inversion making efforts to fit the noise in the data. However, to realize a sparse model domain, at least 100 iterations were required for the other, slower, techniques. Full investigation into the ramifacations of prematurely stopping the sparse inversion schemes was not performed.
For the purpose of removing linear events with a combined HRT-LRT inversion scheme a data-space solution is required. For this problem, the extra expense of fine-tuning the model-space is wasted.