Image-domain least-squares migration (Tang, 2009; Valenciano, 2008)
optimizes the reflectivity model by minimizing an objective function defined in the image domain as follows:
where is the reflectivity model, and
is the migrated image
where denotes taking the adjoint,
is the vector of observed primaries, and
is the Born modeling operator, which models only singly scattered waves.
In equation 1,
is the Hessian operator, which
contains all necessary information, including information of acquisition geometry, velocity model and frequency content of seismic waves,
for correcting the effects of distorted illumination.
The second term
in equation 1
is a regularization term that incorporates user-defined model covariance into the inversion,
and determines the strength of the regularization. Objective function can be minimized
with any iterative solver, such as the conjugate gradient method. The most important components in minimizing
are the explicit calculation of the Hessian operator and the definition of the regularization term
In the subsequent subsections, we first demonstrate how to calculate the Hessian efficiently in 3-D.
Then we discuss how to incorporate dip constraints into the inversion and solve it as
a preconditioning problem.
Subsalt imaging by target-oriented wavefield least-squares migration: A 3-D field-data example