Noniterative f-x-y streaming prediction filtering for random noise attenuation on seismic data

Sherman-Morrison formula in the complex space

To get the inverse of $( \lambda^{2} \mathbf{I} + \mathbf{S^{*}}_{m,n,l} \mathbf{S^{T}}_{m,n,l})$ , one can implement Sherman-Morrison formula to transform the inverse matrix as (13). Whereas $\mathbf{S^{*}}_{m,n,l}$ and $\mathbf{S^{T}}_{m,n,l}$ are complex vectors. Here, we drop the subscript of vectors for a concise proof:

$$\begin{aligned} & (\lambda^{2} \mathbf{I} + \mathbf{S^{*}S^{T}})... ... } { \lambda^{2} + \mathbf{ S^{T}S^{*} }} \\ & = \mathbf{ I }, \end{aligned}$$

where $\mathbf{I}$ denotes the identity matrix, $\mathbf{S}$ is the complex column vector, $\mathbf{S^{T}}$ is the transpose of $\mathbf{S}$ , $\mathbf{S^{*}}$ is the conjugation of $\mathbf{S}$ . Therefore, $\mathbf{S^{*}S^{T}}$ is a complex matrix, and $\mathbf{S^{T}S^{*}}$ is constant. Therefore, we prove that Sherman-Morrison formula can be applied in the complex space.