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We have presented new developments for accurate
implementations of discrete Kirchhoff operators
as matrix-vector multiplication. The main process is based on
the normalization of the Kirchhoff matrix by a diagonal transformation
using the sum of the rows (summation surfaces) and
columns (impulse responses). The normalization operator
is designed in consistency with the numerical implementation
of the Kirchhoff operator as pull (sum)
or push (spray) operator.
The final image is normalized by a reference
model that is the operator's response to an input vector
with all components equal to one (flat event).
We also presented an explicit formulation of a data covariance matrix
for the solution of two-step inversion of irregularly sampled data.
This data covariance is an AMO matrix that
measures the correlations among data elements and corrects the imaging
operator for the effects of irregular sampling.

Beyond the fold normalization, the diagonal transformations
have proved to be a suitable preconditioner for the Inversion
to Common Offset (ICO).
It accelerates the convergence of the iterative solution and,
therefore, enables a cost effective
technique for 3D dealiasing inversion.

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Stanford Exploration Project

7/5/1998