Weighted stacking based on PCA

After the low-rank approximation of the data matrix $\hat{\mathbf{S}}$ is obtained, a better zero-offset reference trace can be obtained via calculating the arithmetic mean of the data matrix along the spatial direction:

$\displaystyle \hat{s}_{lr}(t)= \hat{S}\mathbf{a},$ (7)

where $\mathbf{a}$ is an averaging column vector $[\frac{1}{N},\frac{1}{N},\cdots,\frac{1}{N}]_{1\times N}^T$. Here $[\cdot]$ denotes the transpose of the input matrix/vector. Substituting $\hat{s}(t)$ in (4) with $\hat{s}_{lr}(t)$, we can obtain a new weighting criteria:

$\displaystyle \omega_i(t)=\left\{\begin{array}{cl}
\beta_{i}(t)-\epsilon, & \quad \beta_i>\epsilon \\
0 & \quad \beta_i \le \epsilon
\end{array},\right.$ (8)

where $\beta_i(t)$ is the local similarity between $i$th prestack trace and the low-rank approximated reference trace:

$\displaystyle \beta_i(t) = \mathcal{S}(s_i(t),\hat{s}_{lr}(t)).$ (9)

Inserting the new weighting criteria, as shown in (8), into (2), we obtain the new PCA-based weighted stacking approach. The detailed algorithm workflow of the proposed weighted stacking approach can be expressed as:

  1. Calculate the SVD of data matrix $\mathbf{D}$:

    $\displaystyle [\mathbf{U},\Sigma,\mathbf{V}]=$SVD$\displaystyle (\mathbf{D})$ (10)

  2. Calculate the low-rank approximated singular value matrix by selecting the $k$ largest diagonal elements and setting others zero:

    $\displaystyle \hat{\Sigma} = \Sigma(1:k,1:k)$ (11)

  3. Calculate the low-rank approximated data matrix

    $\displaystyle \hat{\mathbf{S}} = \mathbf{U}\hat{\Sigma}\mathbf{V}^T.$ (12)

  4. Calculate the arithmetic mean of the low-rank approximated data matrix according to (7).
  5. Calculate the local similarity between each trace and the low-rank approximated zero-offset reference trace.
  6. Calculate the PCA-based weighting function $w_i(t)$ according to (8).
  7. Stack the CMP gather using the calculated weighting function according to (2).
For more complicated cases, the nonlinear equivalent of standard PCA (NPCA) can be used to potentially obtain even better performance. The NPCA reduces the observed variables to a number of uncorrelated principal components. The most important advantages of nonlinear over linear PCA are that it incorporates nominal and ordinal variables, and that it can handle and discover nonlinear relationships between variables, which may indicates the the traces may not need to be exactly flattened before stacking.