The *velocity domain* representation of seismic data is an alternative to the standard
CMP presentation. Transformation of CMP data into the velocity domain (producing a velocity
*model* or *panel* of the data) exhibits clearly the moveout inherent in the data and therefore,
forms a convenient basis for velocity analysis as a linear inverse problem.
The velocity transform **A** from the model space (velocity domain)
into the data space (CMP gathers) stretches the velocities back
in the offset plane (superposition of hyperbolas) whereas the adjoint operation (**A'**)
squeezes the data (summation over hyperbolas):

with with where is a weighting function, is a filter that we define later. is related to the velocity stack as defined by Taner and Koehler (1969).

The problem is: given a CMP gather can we find a velocity panel which synthesizes it ? In equations, given data , we want to solve for model :

A simple way to solve this problem is to find a model that minimizes the

This optimization problem is equivalent to the linear system (``normal equations'')

This system is easy to solve if , i.e if is close to unitary: then .In general, is far from an unitary operator for many reasons. However, the choice of a weighting function compensates to some extent for geometrical spreading and other effects Claerbout and Black (1997):

The summation in the velocity space boosts low frequencies. Claerbout and Black (1997) suggest that a good choice of filter is a half derivative operator (). These choices for and bring closer to being an unitary operator.

Since the data is noisy, the modeling operator is not unitary and the numbers of equations and unknowns may be large, an iterative data-fitting approach seems reasonable:

where is the model, , the data we want to fit, the modeling operator, and

The next two parts of this paper compare the performance of the CG algorithm to Huber in the velocity analysis application.

4/20/1999