The offset dimension adds important aspects to reflection seismology. Mainly, it provides robust analysis of the velocity of seismic waves and enables enhancement of signal-to-noise ratio by stacking. During conventional NMO-stacking, traces obtained from independent measurements are treated as redundant and therefore summed together to reduce random noise by destructive interference. In Chapter 5 of the dissertation, I use the offset dimension to formulate an inversion technique that is suitable for data with irregular geometry and takes advantage of the abundance of seismic traces in multichannel recording to interpolate beyond aliasing. Posing partial stacking as an optimization process, the inversion improves the stack when the data are spatially aliased. Furthermore, the solution provides an opportunity for potentially reducing the cost of 3D surveys by acquiring data with sparse offset sampling. Since the modeling operator that relates the data to the reflectivity model is AMO, the inversion is not restricted to zero-offset models or to a particular azimuth. The model, in general, simulates a regular common-offset experiment.
The inversion of multichannel seismic data is generally perceived as an overdetermined problem. In reality, the problem is often locally underdetermined due to inadequate sampling of 3D subsets and gaps in seismic coverage that result into missing data. Moreover an ill-conditioned system of equations behaves numerically as underdetermined. Therefore, based on the least-squares solutions for overdetermined and underdetermined systems, I discuss two formulations for the inversion which I refer to as data-space and model-space inverses. The data-space solution represents a two-step reflectivity inversion where the data is equalized in a first stage with an inverse AMO filter and an imaging operator is then applied to the equalized data to invert for a model. The model-space solution poses the inverse of AMO stacking as a modeling process and iteratively solves for a regularly sampled partial stack from the irregularly sampled data.
The application of a time-variant operator as an optimization process is not practical for 3D data because of its computing costs. I present a cost-effective implementation of the inversion based on a log-stretch transformation Bolondi et al. (1982); Ronen (1987), after which AMO becomes time invariant and the inversion can be split into independent frequencies. To accelerate the convergence of the iterative solution, I propose a new technique for preconditioning based on proper scaling of both rows and columns of the operator. Finally, to regularize the inversion, one needs to limit the variability of the model and guide the iterative optimization to the desired solution. The trick is to augment the problem with a model penalty operator that adds constraints about the model smoothness.
Results of applying the inversion technique to a 3D land survey showed that we can decimate the prestack data to simulate sparse acquisition geometries while still able to reconstruct the high frequency features of the reflectivity function (i.e., buried channels). The cost of the iterative solution is reasonable because of the limited aperture of the modeling operator (AMO), the practical Log-stretch Fourier domain implementation, and the suitable preconditioning of the linear system. The costs incurred by the iterative solution are quite negligeable in comparison to potential savings in acquisition costs.