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Applied to a common-midpoint gather, equation (1) produces an approximate unstacked
zero-offset image of pseudo-primaries from water bottom multiple reflections.
In this section, I introduce a least squares scheme to compute self-consistent images of
primaries and pseudo-primaries which are in turn consistent with the data. First I define
some terms:
With these definitions in hand, we can now write the forward modeling operator for joint
NMO of primaries and multiples of order 1 to *p*.
| |
(6) |

In words, equation (6) takes a collection of psuedo-primary panels, divides
each by the appropriate reflection coefficient, applies inverse (adjoint) NMO to each, and
then sums them together to create something that should resemble ``data''.
We define the data residual as the difference between the input data and the
forward-modeled data:
| |
(7) |

Viewed as a standard least-squares inversion problem, minimization of *L*_{2} norm of the
data residual by solution of the normal equations is underdetermined. Additional regularization
terms, defined in later sections, force the problem to be overdetermined.

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Stanford Exploration Project

6/10/2002