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Estimating a regularly sampled common-azimuth volume from our irregular input data
can be set up as a least squares inversion problem.
The data consists of irregular traces in a 5-D space ().
I want to apply a wave-number domain AMO operator Biondi and Vlad (2001)
that works on regular cube, so the first step is to map the data onto
a regular mesh. The operator performs linear interpolation between
the irregular dataspace and a regular 5-D mesh.
We want to create a common azimuth volume, specifically the zero azimuth
volume, so we want to create a dataset where *hy*=0.
AMO provides a way to translate between different offsets.
We can create an operator that sums over *hy* volumes
that have been translated to *hy*=0 using AMO.
Unlike Clapp (2005a),
we will also sum over a small range of *hx*. This additional
summation allows for additional mixing of information in the
five-dimensional space.
We can define transforming from *hx*_{1},*hy*_{1} to *hx*_{2},*hy*_{2} using
AMO as *Z*_{hx1,hy1,hx2,hy2}.
We can define our *hx* sampling interval as *d*_{hx}, the number of samples
as *n*_{hx}, and the first location as *o*_{hx}. Similarly our sampling
*hy* is defined by *n*_{hy},*o*_{hy}, and *d*_{hy}. Given samples
we wish to sum over we can write an equation relating our domain *m* and
range *d* through

| |
(1) |

Where *m*(*ix*) and *d*(*ix*',*iy*) are 3-D cubes ().
Finally, we need to add in our regularization term.
Generally, after NMO, our data should be smooth
as a function of offset.
By applying a
derivative operator along
the offset axis we can emphasize this smoothness
and help fill in acquisition holes caused by
data irregularity. We can improve this estimate
even further by applying a derivative on cubes
that have been transformed to the same offset using
AMO^{} .We can write our objective function as

| |
(2) |

where controls the importance
of consistency along the offset axis.
We can speed up the convergence of this
problem by preconditioning the model with
the inverse of our regularization operator.
In this case, we replace taking the derivative
of AMO cubes with performing causal integration
of AMO cubes . Our new fitting goals then become
| |
(3) |

where .

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Stanford Exploration Project

4/5/2006