Let
denote the 5-D seismic data in the time domain, and
be the data in the frequency domain. For notation convenience, we omit in the following context and use
to denote
. The traditional rank-reduction based methods require the construction of a level-four block Hankel matrix for 5-D seismic data to meet the low-rank assumption. A level-four block Hankel matrix means that we treat a series of level-three block Hankel matrix as elements and arrange them into a Hankel matrix. In a similar way, a level-three block Hankel matrix is constructed from a series of level-two block Hankel matrices while a level-two block Hankel matrix is formed from several standard Hankel matrices.
The level-four block Hankel matrix has the following explicit expression:
|
(1) |
where
|
(2) |
and
|
(3) |
and
|
(4) |
In order to make all target matrices (from equation 1 to 4) close to square matrices, parameters are defined as
, , where denotes the size of the th dimension. Here,
denotes the integer part of an input argument.
The process of transforming a four-dimensional hypercube
to the block Hankel matrix
is referred to as the Hankelization process. We can briefly denote this process as:
|
(5) |
Another important step in the rank-reduction based method is the rank reduction process, which can be denoted as
.
Reconstructing the missing data aims at solving the following equation:
|
(6) |
where
is a sampling matrix,
, and
denotes the block Hankel
matrix with missing entries. denotes element-wise product.
Equation 6 is seriously ill-posed and the low-rank assumption is applied to constrain the model,
|
(7) |
denotes the Frobenius norm of an input matrix. The constraint in equation 7 means that we constrain the rank of the block Hankel matrix to be .
The problem expressed in equation 7 can be solved via the following iterative solver:
|
(8) |
is an iteration-dependent scalar that linearly decreases from to
. is used to alleviate the influence of random noise existing in the observed data.
2020-12-06