## Construction of the block Hankel matrix for 5-D seismic data

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