Velocity analysis using similarity-weighted semblance

Appendix: Review of AB semblance

Suppose that the weight $w(j,k)$ in equation 2 has a trend of trace amplitude $a(j,k)$ ,

$$w(j,k)=A(j)+B(j)\phi(j,k),$$ (9)

where $\phi(j,k)$ is a known function, and $A(j)$ and $B(j)$ are two coefficients. In the simplest form, $\phi(j,k)$ can be chosen as the offset at trace $k$ . In order to estimate $A(j)$ and $B(j)$ , we can turn to minimize the following objection function of misfit between the trend and trace amplitude:

$$F_j=\sum_{k=0}^{N-1}\left(a(j,k)-A(j)-B(j)\phi(j,k)\right)^2.$$ (10)

Taking derivatives with respect to $A(j)$ and $B(j)$ in equation A-2, setting them to zero, and solving the two linear equations, we can obtain the the following two least-squares fitting coefficients:

$$A(j)=\frac{\displaystyle\sum_{k=0}^{N-1}\phi(j,k)\sum_{k=0}^{N-1}... ...laystyle\left(\sum_{k=0}^{N-1}\phi(j,k)\right)^2-N\sum_{k=0}^{N-1}\phi^2(j,k)},$$ (11)

$$B(j)=\frac{\displaystyle\sum_{k=0}^{N-1}\phi(j,k)\sum_{k=0}^{N-1}... ...laystyle\left(\sum_{k=0}^{N-1}\phi(j,k)\right)^2-N\sum_{k=0}^{N-1}\phi^2(j,k)}.$$ (12)

Substituting $w(j,k)= A(j) + B(j)\phi(j,k)$ into equation 2 leads to the AB semblance.

Velocity analysis using similarity-weighted semblance