Spatial sampling is an important consideration in the design of seismic surveys. It affects acquisition costs and is directly tied to processing and imaging requirements. During the acquisition stage, economic constraints, obstructions, cable feathering, environmental objectives, and many other factors cause seismic data to be sampled irregularly. Therefore, 3D surveys typically have sparse and irregular geometry that often results in spatial aliasing. The possibility of overcoming aliasing in multichannel data was discussed by many authors Bolondi et al. (1982); Claerbout (1985, 1992); Rocca and Ronen (1984); Ronen (1985). In this progress report, we discuss an inversion technique that is suitable for sparse and uneven sampling and takes advantage of the abundance of seismic traces in multifold seismic data to interpolate beyond aliasing. The process is based on least-squares theory and wave-equation interpolation for missing data using the azimuth-moveout operator (AMO).
The application of wave equation techniques for dealiasing
was explored by
Ronen 1985. He applied the adjoint of dip
moveout and stacking with zero traces in place of missing data
to solve an inversion in which the model is a
regularly sampled zero-offset section, and the data are seismic data
after normal moveout.
Ronen 1987 wrote:
``Adding zero traces to aliased data is a linear process,
and so are dip moveout
and stacking. Lump all three linear processes in one,
and describe them by a (huge) matrix 
. is dip moveout
and stacking, the inverse of the transpose, 
, is what you need''.
Biondi and Chemingui 1994 first introduced a new partial prestack imaging operator, named azimuth moveout (AMO), that organizes the data into common-azimuth/common-offset subsets and allows the analysis of partial stacks. The AMO operator can model an input dataset with a given geometry at a new output geometry. This ability makes it possible to apply AMO as an optimization process to equalize the irregular geometry of 3D data and improve the quality of partial stacks Chemingui and Biondi (1996a, 1997). In their formulation of the problem, the inversion is not restricted to zero offset models or to a particular azimuth. The model, in general, simulates a regular common-offset experiment.
If we consider processing as the inverse of modeling irregular data from a regularly sampled constant-offset model, both forward and inverse mappings represent AMO transformations. Chemingui and Biondi 1997, discussed two solutions for the optimization problem which they refer to as data-space and model-space inverses. For the equalization of irregularly sampled data they only explored the effectiveness of the data-space inverse solution. The equalization consists of a two-step inversion where the data is equalized in a first stage with an inverse filter, and an imaging operator is then applied to the preconditioned data to invert for a model. The method yielded very promising results on 2D synthetic data. However, the time-space implementation of a time-variant operator was not practical for 3D data because of its computing costs.
In this paper we present a more cost-effective implementation based on a log-stretch transformation Bolondi et al. (1982), after which AMO becomes time invariant and the inversion can be split into independent frequencies. To accelerate the convergence of the iterative solution, we precondition the inversion by the stacking fold of the data. This is a first approximation to column scaling preconditioning, which is a normalization by the coverage after AMO.
The paper presents the theory and outlines our work in progress, including our preliminary results from processing synthetic 3D data.