Inversion, processing and Q compensation of waves reflecting from absorptive-dispersive targets
The main thing I'm working on at the moment, with Jose Lira, is the development of new direct non-linear inverse theory for waves that have propagated through absorptive-dispersive media. Jose and I are looking at seismic exploration problems, but the dissipative wave problem is completely ubiquitous, and in fact the radar work involves precisely the same problems and something like the same solutions.
LINEAR ABSORPTIVE-DISPERSIVE INVERSE SCATTERING: Q COMPENSATION WITH KNOWN Q
The idea is to try to understand and develop the capability of non-linear inverse scattering to perform not just inversion of absorptive wave data, but some of the sub-tasks of inversion that we typically associate with absorption. For instance, Q smooths out and decays the events we measure, compromising the resolving power of the data: a perfectly well-formed processing strategy is to try to estimate a new, equivalent data set, that only differs from the input in that this smoothing effect has been corrected for. This is called Q compensation. Below we do the simple version of Q compensation based on inverse scattering, which we discussed at the 2008 EAGE meeting in Rome. If the Q structure of the medium is known, a linear inversion, followed by a forward Born approximation in a non-absorptive medium, does the trick.
Here's an example: two attenuated reflected primaries at the top in black, compared to their (idealized) un-attenuated counterparts in red. Solving the inverse-forward absorptive inverse scattering equations, and tapering to stabilize, we produce the results in the middle panel. The bottom panel is just there to assure that it is the tapering that causes the differences between the output and the idealized benchmark result.
NONLINEAR ABSORPTIVE-DISPERSIVE INVERSE SCATTERING: Q COMPENSATION WITH UNKNOWN Q
Next comes the hard part. What if you don't know what Q is? The unique aspect of non-linear direct inverse scattering is that you can trade away this requirement, of knowing the medium parameters, and still get results. The cost is, the new algorithm is highly non-linear in the data. Jose and I are going to discuss prototype ways of carrying this out at the SEG in Las Vegas this Fall. The results are synthetic, but pleasing nonetheless. Below (top panel) you can see a shot gather containing two primaries from an absorptive model -- the bottom primary is significantly attenuated. The direct non-linear algorithm automatically constructs an operator from this attenuated data, enacts it upon the data again, and the bottom panel results. Below are a few detailed traces for a few offsets. We're working on extending it beyond a stratified medium.
