Iterative least-squares inversion can be expressed simply as the conjugate-gradient minimization of this objective function:

(1) |

where **bold**L is a linear modeling operator, **bold**d is the data, and
**bold**m is the model. In this paper, **L** is the adjoint of the
downward continuation migration operator explained by
Prucha et al. (1999b). This
migration algorithm produces a model with the axes depth (z), common
reflection point (CRP), and offset ray parameter (*p*_{h}). Offset
ray parameter is related to the reflection angle (where is half of the opening angle between incident and reflected rays) by:

(2) |

where is the local dip and *V*(*z*,*CRP*) is the local
velocity at the reflection point.

The minimization can be expressed more concisely as a fitting goal:

(3) |

(4) | ||

The first expression in (4) is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second expression is the ``model styling goal,'' meaning that it allows us to impose some idea of what the model should look like using the regularization operator . The strength of the regularization is controlled by the regularization parameter .

Unfortunately, the inversion process described by fitting goals (4) can take many iterations to produce a satisfactory result. We can reduce the necessary number of iterations by making the problem a preconditioned one. We use the preconditioning transformation Fomel et al. (1997); Fomel and Claerbout (2003) to give us these fitting goals:

(5) | ||

is obtained by mapping the multi-dimensional regularization operator to helical space and applying polynomial division Claerbout (1998). This makes our imaging method a Regularized Inversion with model Preconditioning (RIP).

5/23/2004