Velocity analysis using $AB$ semblance

Semblance as correlation

The correlation coefficient $\gamma$ between two sequences of numbers $\mathbf{a}=a_1,a_2,\ldots,a_N$ and $\mathbf{b}=b_1,b_2,\ldots,b_N$ is defined as

$$\gamma(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a} \cdot \mathbf{b}... ...\,b_i}{ \displaystyle \sqrt{\sum_{i=1}^{N} a_i^2}\,\sqrt{\sum_{i=1}^{N} b_i^2}}$$ (1)

The correlation coefficient is analogous to the cosine of the angle between two vectors $\mathbf{a}$ and $\mathbf{b}$ . It takes values in the range from $-1$ to $1$ . Taking a correlation of a sequence $\mathbf{a}$ with a constant sequence $\mathbf{c}=C,C,\ldots,C$ produces a measure $\beta$ , defined as

$$\beta(\mathbf{a}) = \gamma(\mathbf{a},\mathbf{c}) = \frac{\displa... ...{\displaystyle \sum_{i=1}^N a_i}{ \displaystyle \sqrt{N\,\sum_{i=1}^{N} a_i^2}}$$ (2)

Squaring the correlation with a constant yields the measure equivalent to semblance

$$\beta^2(\mathbf{a}) = \frac{\displaystyle \left(\sum_{i=1}^N a_i\right)^2}{ \displaystyle N\,\sum_{i=1}^{N} a_i^2}\;.$$ (3)

Semblance is maximized when the sequence $\mathbf{a}$ has a uniform distribution. When seismic amplitude is uniformly distributed along a moveout curve, the semblance of a horizontal slice through the gather will be maximized when the event is flattened. This fact is the basis of the conventional velocity analysis originally developed by Taner and Koehler (1969). The approach fails, however, when the amplitude variation is distinctly non-uniform.

Velocity analysis using $AB$ semblance