next up previous [pdf]

Next: Examples Up: Chen et al.: Similarity-weighted Previous: Weighted semblance

Review of local similarity

Fomel (2007) defined local similarity between vectors $ \mathbf{a}$ and $ \mathbf{b}$ as:

$\displaystyle \mathbf{c}=\sqrt{\mathbf{c}_1^T\mathbf{c}_2}$ (4)

where $ \mathbf{c}_1$ and $ \mathbf{c}_2$ come from two least-squares minimization problems:

$\displaystyle \mathbf{c}_1$ $\displaystyle =\arg\min_{\mathbf{c}_1}\Arrowvert \mathbf{a}-\mathbf{B}\mathbf{c}_1 \Arrowvert_2^2,$ (5)
$\displaystyle \mathbf{c}_2$ $\displaystyle =\arg\min_{\mathbf{c}_2}\Arrowvert \mathbf{b}-\mathbf{A}\mathbf{c}_2 \Arrowvert_2^2,$ (6)

where $ \mathbf{A}$ is a diagonal operator composed from the elements of $ \mathbf{a}$ : $ \mathbf{A}=diag(\mathbf{a})$ and $ \mathbf{B}$ is a diagonal operator composed from the elements of $ \mathbf{b}$ : $ \mathbf{B}=diag(\mathbf{b})$ . Least-squares problems 5 and 6 can be solved with the help of shaping regularization with a smoothness constraint:

$\displaystyle \mathbf{c}_1$ $\displaystyle = [\lambda_1^2\mathbf{I} + \mathcal{T}(\mathbf{B}^T\mathbf{B}-\lambda_1^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{B}^T\mathbf{a},$ (7)
$\displaystyle \mathbf{c}_2$ $\displaystyle = [\lambda_2^2\mathbf{I} + \mathcal{T}(\mathbf{A}^T\mathbf{A}-\lambda_2^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{A}^T\mathbf{b},$ (8)

where $ \mathbf{\mathcal{T}}$ is a smoothing operator, and $ \lambda_1$ and $ \lambda_2$ are two parameters controlling the physical dimensionality and enabling fast convergence when inversion is implemented iteratively. These two parameters can be chosen as the least-squares norms of $ \mathbf{B}$ and $ \mathbf{A}$ , respectively.

syn weights
syn,weights
Figure 1.
A demonstration of similarity-weighted semblance. (a) NMO corrected gather. (b) Weights applied to each trace for semblance calculation based on the local similarity between each trace and a reference trace.
[pdf] [pdf] [png] [png] [scons]


next up previous [pdf]

Next: Examples Up: Chen et al.: Similarity-weighted Previous: Weighted semblance

2015-06-25