In this section we present simple examples of discrete Kirchhoff implementations as matrix-vector multiplication on irregularly sampled data. The experiments were designed to illustrate several approaches for efficient handling of irregular geometry: (a) by normalized imaging, (b) by two step equalization and (c) by Inversion to Common Offset (ICO). The implementations are all done in the domain for comparison with the ICO results.
We use the fold distribution shown in Figure figure1. The fold chart represents a subset from a real 3-D land survey. We extracted the header values of traces whose source-receiver azimuth is between -30o and 30o with an absolute-offset range from 1000 to 1200 m. The reflectivity model consists of a single dipping bed with a strike of 60o from the inline direction. A monochromatic planewave is used to create the synthetic input data, and therefore, only one frequency slice is processed. 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.
In the first experiment, the model is a common-offset section of 50 by 50 CMP's with 35m spacing. This is the nominal CMP spacing of the real survey. The second panel in Figure figure1 shows the ideal result from a synthetic experiment which simulates zero-azimuth acquisition and a constant offset of 1100 m. This is the unknown model that solves the set of equations in equ1. The bottom two panels are the results of processing with true-amplitude AMO Chemingui and Biondi (1995). The effects of varying fold are noticeable on the un-normalized image. In absence of empty bins there are no aliasing artifacts other than amplitude distortions. The result of normalized AMO is nearly perfect on this low frequency model.
The challenging task in handling irregular sampling is dealing with aliasing which is a missing data problem. In the second experiment we used the same subset of traces and created a zone of missing coverage by removing traces over some large area. The size of the uncovered area is about 8 bins in both the inline and crossline directions. We also used a higher frequency to simulate the response of the steeply dipping bed. We chose a model resolution of 17.5 m which corresponds to half the nominal CMP-spacing of the data.
Figure figure2 shows the results of processing with AMO. These are the outputs of applying AMO to reconstruct the data with zero common-azimuth and 1100-m effective offset. The result of un-normalized AMO displays poor quality in the area of missing coverage. As result of the normalization, the interpolated values for the missing data are calibrated on the normalized image and result into better resolution. Some aliasing artifacts are still noticeable and are due to the limited interpolation by AMO. As in the previous experiment, fold variations in form of trace redundancy were properly handled by the diagonal normalization.
The result of the two-step solution (equalization with a data covariance operator + imaging with AMO) is shown in Figure figure3. We compare it to the output of ICO with and without preconditioning. After 5 iterations with a conjugate gradient solver the preconditioned inversion yielded a reasonably good solution where the solution without preconditioning is still far from convergence.
Overall, the preconditioned ICO yielded the best picture in terms of fold equalization and dealiasing. The two step solution, with explicit analytic computation of the data covariance operator, presented a good and cost effective solution to the problem of irregular sampling. This is an important result considering the alternative cost of estimating the equalization filter numerically using iterative solvers. The normalized one step imaging is still an attractive approach given that it can greatly eliminate the effects of fold variations at only twice the cost of conventional imaging.