Noise attenuation using SVD

2D seismic data can be expressed as a data matrix $\mathbf{D}$ ($M\times N$), consisting of $N$ traces and $M$ time samples. The SVD of the data matrix $\mathbf{D}$ can be expressed as:

$$\mathbf{D}=\mathbf{U\Sigma V}^T.$$ (1)

Here, $\mathbf{U}$ is composed of the eigenvectors of $\mathbf{DD}^T$. $\mathbf{V}$ is composed of the eigenvectors of $\mathbf{D}^T\mathbf{D}$. $\mathbf{\Sigma}$ is a diagonal matrix composed of the decreasing singular values. Let us denote $\mathbf{U}$, $\mathbf{\Sigma}$, and $\mathbf{V}$ in the following form:

$$\begin{array}{l} \mathbf{U}=[u_1,u_2,\cdots,u_r],\\ \mathbf{... ...dots,\sigma_r),\\ \mathbf{V}=[v_1,v_2,\cdots,v_r], \end{array}$$ (2)

where $r$ is the rank of $\mathbf{D}$. The vectors $\mathbf{u}_k$ and $\mathbf{v}_k$ are also called the propagation vectors and the eigen-wavelets, respectively (Vrabie et al., 2004). The singular values $\sigma_k$ are sorted such that $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_r$. They can be obtained by calculating the positive square roots of the eigenvalues of the data covariance matrix $\mathbf{D}\mathbf{D}^T$.

Equation 1 can be expressed as:

$$\mathbf{D} = \sum_{k=1}^{r}\lambda_k\mathbf{u}_k\mathbf{v}^T_{k},$$ (3)

where $\mathbf{u}_k\mathbf{v}^T$ is the rank-one matrix called the $k$th eigenimage of $\mathbf{D}$. Thus, from equation 3, the seismic image can be decomposed into $r$ eigenimages, the energy of which corresponds to the value of each element in matrix $\mathbf{\Sigma}$.

We can remove the random noise in seismic data in order to enhance the seismic reflections by only selecting the first several eigenimages (Freire and Ulrych, 1988):

$$\hat{D}_{svd} = \sum_{k=1}^{p} \sigma_k\mathbf{u}_k\mathbf{v}_k^T.$$ (4)

Random noise attenuation approach by SVD utilizes the property that useful seismic signals are horizontally coherent to separate signal and noise. The decomposition is based on the criterion of horizontal coherency. The energy of coherent signals lays in the first several parts of the decomposition, which is represented by the value of bigger values in $\mathbf{\Sigma}$. For seismic data, the horizontal coherent part corresponds to useful signal while random noise or coherent dipping events are not horizontally coherent. Thus, by choosing the first several elements in $\mathbf{\Sigma}$ and making the others equal to zero, we can remove the noise and enhance seismic data quality. SVD acts as a data-driven, low-pass filter by rejecting highly uncorrelated traces (Bekara and van der Baan, 2007). Here, we refer to those denoising approaches that simply apply SVD to the original seismic data in order to remove random noise as global SVD (GSVD).