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Given a CMP gather and a radon transform operator , we want to minimize the least-squares objective function
| |
(3) |

where is the unknown radon domain. The main idea of this
paper is to decompose into its positive and negative parts
by imposing simple bounds on with the L-BFGS-B
algorithm. Therefore, the two problems
| |
(4) |

and
| |
(5) |

need to be solved. Note that we could decompose into more
subdomains as well. It is important to solve both problems of finding
and independently, and not
simultaneously as it can be done with linear programming techniques
Claerbout and Muir (1973).
Here, the main idea is to decrease the null space
and its effects by constraining the model, similar to what is
accomplished with the Cauchy regularization. Once the two models and are estimated with the L-BFGS-B algorithm,
the sparse model is obtained by computing .
In the following section, I illustrate this technique with a synthetic
and real data example using the hyperbolic radon transform.

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** Previous:** The L-BFGS-B algorithm
Stanford Exploration Project

5/3/2005