Signal and noise separation in prestack seismic data using velocity-dependent seislet transform |

which means that all transforms taken together constitute a

Because of its overcompleteness, a frame representation for a given
signal is not unique. In order to assure that different wavefield
components do not leak into other parts of the frame, it is
advantageous to employ sparsity-promoting inversion
(Fomel and Liu, 2010). We use a nonlinear shaping-regularization scheme
(Fomel, 2008) and define sparse decomposition as an iterative
thresholding process (Daubechies et al., 2004)

where are coefficients of the seislet frame at th iteration, is an auxiliary quantity, is a soft thresholding operator, and are frame construction and deconstruction operators

The iteration in equations 12 and 13
starts with
and
and is related to the
linearized Bregman iteration (Cai et al., 2009), which converges to the
solution of the constrained minimization problem:

Separated wavefield can be calculated by
, where is
iteration number, masking operator is a diagonal matrix
as

Signal and noise separation in prestack seismic data using velocity-dependent seislet transform |

2015-10-24