Weighted stacking of seismic AVO data using hybrid AB semblance and local similarity

Stacking CMP gathers using local-similarity weights

After the optimal NMO velocity with high fidelity is obtained that addresses the AVO anomaly, we propose to use the following local-similarity-weighted scheme to stack the NMO-corrected gathers using the optimal NMO velocity. The local-similarity-weighted stacking was initially proposed by Liu et al. (2009). Equations 3 and 4 show the calculation of conventional stacking (Mayne, 1962) and weighted stacking with an arbitrary weighting function $v(i,j)$ , respectively:

$$\textrm{stk}(i) = \frac{1}{N}\displaystyle\sum_{j=0}^{N-1}d(i,j)\;,$$ (3)

$$\textrm{stk}_{v}(i) = \frac{1}{\displaystyle\sum_{j=0}^{N-1}v(i,j)}\displaystyle\sum_{j=0}^{N-1}v(i,j)d(i,j)\;,$$ (4)

where $N$ is the number of traces in one CMP gather, $d(i,j)$ is the $i$ th sample amplitude of the $j$ th trace in the NMO-corrected CMP gather. The local-similarity-weighted stacking substitutes the weighting function $v(i,j)$ in equation 4 with the local correlation of each trace and the reference trace in the same CMP gather, where the local correlation should be implemented after the soft thresholding (Donoho, 1995) and the final weighted stack is averaged by the total number of the non-zero weighted samples in this CMP gather. Appendix B gives a brief review of the calculation of local similarity. The algorithm of local similarity can be used for the calculation of signals in any dimension. For 1D signals, the meanings of equations B-4 and B-5 are intuitive. For 2D or higher-dimensional signals, each signal is first reshaped into a 1D signal and then follows equations B-4 and B-5 to calculate the local-similarity vector. The smoothing operator is applied to the 2D or multi-dimensional form of the original signal to enforce the smoothness in any dimension.

Weighted stacking of seismic AVO data using hybrid AB semblance and local similarity