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
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)
Irregular seismic data reconstruction using a percentile-half-thresholding algorithm