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# NUMERIC TEST

For a simple numeric test I choose the stacking DMO operator. The problem is formulated as an iterative least-square inversion of inverse DMO. The input data set is a synthetic three-dimensional common-azimuth common-offset gather containing a reflection response from a point diffractor (Figure 4) The data cube has 64 by 64 traces with the midpoint spacing 20 m. The half-offset is 500 m. Figure 1 compares convergence of the conjugate-gradient inversion with two different types of a DMO operator: adjoint (Hale's) DMO with the weighting function defined by formula (99) and asymptotic pseudounitary DMO with formula (98). Both operators include antialiasing with the method described by Fomel and Biondi 1995. The impulse responses of inverse DMO are plotted in Figure 2. The impulse responses of DMO are plotted in Figure 3. The asymptotic pseudounitary DMO has a noticeably higher amplitudes than the adjoint DMO. We can see that the convergence of the pseudonutary operator is better at the first 5 iterations, though the difference is negligible after 7-th iteration (Figure 1.) The zero-offset data cube obtained after the inversion with 10 conjugate-gradient iterations is shown in Figure 5. Despite some boundary artifacts, caused by the data truncation in the in-line direction, the main kinematic and dynamic features of the solution are correct, and the model of the input data (Figure 6) is accurate. The residual error after 10 iterations is shown in Figure 7. It possesses less than 1% energy of the original signal. To reduce data aliasing artifacts in DMO/inversion, it is desirable to use data with more than one offset Ronen et al. (1991); Ronen (1987) and/or add some model constraints in the inversion Ronen et al. (1995).

 adjcon Figure 1 Comparison of convergence of the iterative inversion with different DMO operators. The relative squared residual error is plotted against the number of iteration.

Figure 2
Impulse responses of inverse DMO operators. Left column: adjoint DMO. Right column: asymptotic pseudo-unitary DMO.

Figure 3
Impulse responses of DMO operators. Left column: adjoint DMO. Right column: asymptotic pseudo-unitary DMO.

Figure 4
Input data for the numeric test: a synthetic common-azimuth common-offset gather recording reflection from a point diffractor (an appropriate NMO correction has been applied).

Figure 5
Zero-offset diffraction from a point diffractor obtained as the result of 10 iterations of iterative least-square DMO/inversion with the asymptotic pseudounitary operator.