## Hankel matrix embedding

The rank-reduction based methods discussed in this paper deal with a block Hankel matrix (Inline and Xline) in the frequency-space domain. Let (of size ) represent a 3D seismic dataset. First, we transform in time-space domain to in the frequency-space domain. At a given frequency slice, the 2D data can be expressed as (Oropeza and Sacchi, 2011):

 (1)

From here on, is omitted for notational convenience. A Hankel matrix is then constructed from . We first construct a Hankel matrix as:

 (2)

and then construct the block Hankel matrix as:

 (3)

Parameters and are chosen to make and close to square matrices, e.g., and . The symbol outputs the integer of an input value. The matrix is of size , with , . The block Hankel matrix is considered to be lowrank (Chen et al., 2019a; Trickett, 2008; Oropeza and Sacchi, 2011; Huang et al., 2016), i.e., it can be approximated by a small number of eigen-images.

2020-12-06