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As a linear transformation, the apex-shifted Radon transform can be
represented simply as
| |
(1) |

where is the image in the angle domain,
is the image in the Radon domain and
is the forward apex-shifted Radon transform operator.
To find the model that best fits the data in a least-squares
sense, we minimize the objective function:
| |
(2) |

where the second term is a Cauchy regularization that enforces sparseness
in the model
space. Here *n* is the size of the model space, and and *b* are
two constants chosen a-priori: which controls the
amount of sparseness in the model
space and *b* related to the minimum value below which
everything in the Radon domain should be zeroed Sava and Guitton (2003).
The least-squares inverse of is
| |
(3) |

where defines a diagonal operator.
Because the model space can be large, we estimate iteratively.
Notice that the objective function in Equation (2) is non-linear
because the model appears in the definition of the regularization term.
Therefore, we use a limited-memory quasi-Newton method
Guitton and Symes (2003) to find the minimum of .

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Stanford Exploration Project

5/23/2004