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 $\mathbf{D}(t,x,y)$ (of size $N_t\times N_x\times N_y$) represent a 3D seismic dataset. First, we transform $\mathbf{D}(t,x,y)$ in time-space domain to $\mathbf{D}(w,x,y)(w=1\cdots N_w)$ in the frequency-space domain. At a given frequency $w_0$ slice, the 2D data can be expressed as (Oropeza and Sacchi, 2011):

$$\mathbf{D}(w_0)=\left(\begin{array}{cccc} D(1,1) & D(1,2) & \cdot... ...ts &\ddots &\vdots \\ D(N_x,1)&D(N_x,2) &\cdots&D(N_x,N_y) \end{array}\right).$$ (1)

From here on, $w_0$ is omitted for notational convenience. A Hankel matrix is then constructed from $\mathbf{D}$. We first construct a Hankel matrix $\mathbf{R}_i$ as:

$$\mathbf{R}_i=\left(\begin{array}{cccc} D(i,1) & D(i,2) & \cdots &... ...dots &\vdots \\ D(i,N_y-m+1)&D(i,N_y-m+2) &\cdots&D(i,N_y) \end{array}\right),$$ (2)

and then construct the block Hankel matrix as:

$$\mathbf{M}=\left(\begin{array}{cccc} \mathbf{R}_1 &\mathbf{R}_2 &... ...{R}_{N_x-n+1}&\mathbf{R}_{N_x-n+2} &\cdots&\mathbf{R}_{N_x} \end{array}\right).$$ (3)

Parameters $m$ and $n$ are chosen to make $\mathbf{R}_i$ and $\mathbf{M}$ close to square matrices, e.g., $m=N_y-\lfloor\frac{N_y}{2}\rfloor$ and $n=N_x-\lfloor\frac{N_x}{2}\rfloor$. The symbol $\lfloor\cdot \rfloor$ outputs the integer of an input value. The matrix $\mathbf{M}$ is of size $I\times J$, with $I=(N_y-m+1)(N_x-n+1)$, $J=mn$. The block Hankel matrix $\mathbf{M}$ 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.