where is the (migrated) data in the angle domain, is the model 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, I minimize the objective function:

where the second term is a Cauchy regularization (, ) that enforces sparseness in the model space. Here is the size of the model space, and and are two constants chosen a-priori: which controls the amount of sparseness in the model space and which controls the minimum value below which everything in the Radon domain should be zeroed. The least-squares inverse of is given by

(32) |

Because the model space can be large, I estimate iteratively. Notice that the objective function in equation (31) is non-linear because the model appears in the definition of the regularization term. Therefore, I use a limited-memory quasi-Newton method (, ) to find the minimum of .

2007-10-24