## Iterative shrinkage thresholding

In order to reconstruct the missing traces in the seismic data, one can use a sparsity-promoting transform to precondition in equation 1, that is:

 (4)

where is a tight frame such that and , and denotes adjoint. A common selection for is the Fourier transform. Inserting equation 4 into equation 1, and let , we obtain

 (5)

Correspondingly, equation 3 turns to:

 (6)

The well-known IST algorithm is used for solving equation 6 with a sparsity constraint:

 (7)

Here corresponds to a thresholding operator performed element-wise with threshold (Yang et al., 2013a). When , , , where is a regularization parameter which controls the weight between misfit and constraint in the minimization problem, corresponds to a soft-thresholding operator:

 (8)

where denotes the amplitude of each position-coordinate vector . When , , , corresponds to a hard-thresholding operator:

 (9)

2020-02-28