To illustrate the performance of the multichannel inversion technique, we use the fold distribution shown in Figure fold. The fold chart represents a subset from a real wide-azimuth land 3D survey. We extracted the header values of 21,400 traces whose source-receiver azimuth is between -30o and 30o with an absolute-offset range from 9000 to 11000 ft. The reflectivity model consists of a single dipping bed whith a strike of 60o from the inline direction. We use a monochromatic planewave to create the synthetic input data and process a single frequency slice. All the results are displayed in the Fourier domain of the log-stretched data. We analyze the effects of fold variations on the imaginary part of the wavefield.
The model is a common-offset section of 100 by 100 CMP's with 80-foot spacing. Figure model shows the ideal result from a synthetic experiment which simulates zero-azimuth acquisition and a constant offset of 10,000 ft. This is the unknown model that solves the set of equations in equ1.
Figure norm-stack is the output of normal moveout and stacking of the irregular subset after fold normalization. We plot the imaginary part of the output normalized by the amplitude of the complex wavefield. This describes the phase of the dipping reflector after NMO. The phase map is shifted compared to the ideal model since the NMO action does not account for the dip of the reflector. In addition, there are phase and amplitude distortions between the CMP bins due to incoherent partial stacking of the traces within local bins.
Figures amo and norm-amo are the output of AMO processing (inverse of modeling). These are the results of applying the adjoint operator, AMO, to reconstruct the data with zero common-azimuth and 10,000-ft effective offset. As result of the coherent partial stacking of the dipping bed, the phase of the output is now reconstructed correctly. The output of the normalized AMO is rougher but displays better resolution than the unnormalized result. Both results do not show a good quality, they were generated using the inversion algorithm (one iteration), which is still in the development stage. No special care was taken for the amplitude weights and the tapering of the operator. In addition, the algorithm doesn't account for the effects of operator aliasing since the goal of the inversion is the dealiasing of the data.
The results of the dealiasing inversion, after 5 conjugate gradient iterations, are shown in Figures (6-8). The results of the model-space inversion are significantly better than the data-space inverse solution. Comparing Figures mod-inv and prec-mod, the fold preconditioning has slightly improved the inversion (sped the convergence). The fold normalization seems to be most effective at the very first few iterations, then, the two solutions converge at comparable rates.
Overall, the model-space dealiasing inversion with fold preconditioning yielded better results than conventional processing. It also provided better results than the data-space inversion and is computationally less expensive.