Velocity analysis using similarity-weighted semblance

Traditional semblance

The traditional semblance is defined by Neidell and Taner (1971) as:

$$s(i) = \frac{\displaystyle\sum_{j=i-M}^{i+M}\left(\sum_{k=0}^{N-1}a(j,k)\right)^2}{\displaystyle N\sum_{j=i-M}^{i+M}\sum_{k=0}^{N-1}a^2(j,k)},$$ (1)

where $i$ and $j$ are time sample indices, $s(i)$ denotes the semblance for time index $i$ , $2M+1$ is the length of the smoothing window (in this formulation a boxcar filter is used), along time axis, $a(j,k)$ is the trace amplitude at time index $j$ and trace number $k$ of the NMO-corrected common midpoint (CMP) gather, and $N$ is the number of traces.