Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition

$f-x$ SSA

Matrix completion via $f-x$ domain singular spectrum analysis (SSA) can handle complex dipping events well by extracting the first several eigen components after SVD for each frequency slice. Considering 2D seismic data acquired on a regular grid $D(m,w)$ , $m=1,\cdots, M$ , where $M$ denotes the number of traces in the spatial dimension. The Hankel matrix for each frequency slice $D(m,w)$ is constructed as:

$$\mathbf{H}=\left(\begin{array}{cccc} D(1,w) & D(2,w) & \cdots &D(... ...& \vdots &\ddots &\vdots \\ D(L,w)&D(L+1,w) &\cdots&D(M,w) \end{array} \right),$$ (3)

where $L=\lfloor\frac{M}{2}\rfloor+1$ and $K=M-L+1$ , $\lfloor\rfloor$ denotes the integer part of its argument, $w$ denotes frequency. It can be proved that if the processing window contains $k$ plane waves of independent dips, then the Hankel matrix $\mathbf{H}$ is rank k. The added random noise will increase the rank of matrix $\mathbf{H}$ . It's natural that the random noise can be removed by rank reduction for $\mathbf{H}$ (Oropeza and Sacchi, 2011). A singular value decomposition (SVD) is applied to the Hankel matrix in order to select the first several eigen components that correspond to the first several singular values. The $f-x$ SSA approach is based on a pre-known rank of the seismic data. However, for complex seismic data, the rank is hard to select, and for curved events, the rank tends to be high and thus will involve a serious rank-mixing problem. The simpler the seismic profile is, the easier it is to choose the predefined matrix rank because the rank-mixing problem is less serious. Yuan and Wang (2011) came up up a way to simplify $f-x$ SSA. They applied $f-x$ SSA in a local version to reduce the complexity of seismic events in order to make the rank of the aforementioned Hankel matrix lower.

Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition