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.
adjinv
Figure 2 Impulse responses of
inverse DMO operators. Left column: adjoint DMO. Right column:
asymptotic pseudo-unitary DMO.
adjimp
Figure 3 Impulse responses of
DMO operators. Left column: adjoint DMO. Right column: asymptotic
pseudo-unitary DMO.
adjdat
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).
adjmod
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.
adjrdt
Figure 6 Modeled
common-offset common-azimuth data after 10 iterations of iterative
least-square DMO/inversion with the asymptotic pseudounitary
operator.
adjres
Figure 7 Residual error after
10 iterations of iterative least-square DMO/inversion, plotted at the
same scale as the data.
