Under the stationary phase approximation, the equation is in the
standard linear form: . This has a well-known damped least-squares
Gauss-Newton solution (e.g., Gill et al., 1981):
, where
and are the gradient and Hessian of *E* respectively, and
is the adjoint operator to . The adjoint operator is
defined by the scalar product:

(11) |

Writing down the integral equation implied in (11), interchanging the order of integration, and taking the complex conjugate, results in an integral expression for the adjoint operator:

(12) |

In this case, the gradient can be derived as

(13) |

Neglecting second-derivative terms, the Hessian can be evaluated in the usual Gauss-Newton sense as:

(14) |

This representation of the Hessian requires a complete forward modeling
step as well as a complete migration step; i.e., it costs one full
forward modeling of all the recorded data, plus one full prestack migration.
In this form, the Hessian is spatially non-diagonal in the sense that it
tries to incorporate and compensate for all point-to-point correlations
of reflectivity in the subsurface. This non-diagonality should not be
confused with *multi-parameter* crosscorrelations which occur in the
case of acoustic or elastic parameter inversions
(e.g., Beydoun and Mendes, 1989). There is only one single parameter being
inverted for here: , and it is likely to be spatially correlated due
to data bandwidth, and in (14) is trying to deconvolve
that spatial correlation.

In the interest of efficiency, the Hessian can be diagonalized (i.e., ignoring spatial correlations of ), by assuming the ``1'' in the volume integrand of (14) is a spatially uncorrelated delta function (e.g., Clayton and Le Bras, 1988). In this case, the diagonalized Hessian reduces to:

(15) |

Combining (13) and (15), results in an approximate *l _{2}*
reflectivity solution under the stationary phase condition:

(16) |

The approximate diagonal Hessian solution (16) costs little more than a single conventional prestack migration in terms of floating point operations. However, one drawback is that it requires twice the memory, since the and images must be updated separately during the estimation (migration) process. They may only be combined as in (16) after both have been completely imaged. This seems to be the price to pay for accurate amplitude recovery.

It should be noted that the form and weight structure
of the *l _{2}* inverse solution (16) is somewhat similar, but different
in detail, to the trial solution of Bleistein (1987). In
particular, (16) involves two separate but simultaneous images
to be evaluated, the gradient (numerator) being a weighted Kirchhoff
prestack depth migration, and the approximate Hessian (denominator)
being an accumulation
of the squared migration weights. Bleistein's solution involves only
one migration integral since it is not a least-squares estimate, and
his different migration kernel amplitude weights will give a different
amplitude result in general.

11/17/1997