Velocity analysis using $AB$ semblance

$AB$ semblance: correlation with a trend

Suppose that the reference sequence has a trend $b_i = A + B\,\phi_i$ , where $\phi_i$ is a known function. The trend can be, for example, an expression of the $PP$ reflection coefficient in Shuey's approximation (Shuey, 1985), where $A$ and $B$ are the AVO intercept and gradient, $\phi_i=\sin^2{\theta_i}$ , and $\theta_i$ corresponds to the reflection angle at trace $i$ . In examples of this paper, I use offset instead of angle. Relating offset and reflection angle can be done either by using approximate equations of by ray tracing once the velocity model is established.

Estimating $A$ and $B$ from least-square fitting of the trend amounts to the minimization of

$$F(A,B) = \sum_{i=1}^N \left(a_i - A - B\,\phi_i\right)^2$$ (4)

Differentiating equation (4) with respect to $A$ and $B$ , setting the derivatives to zero, and solving the system of two linear equations produces the well-known linear fit equations

$$A = \frac{\displaystyle \sum_{i=1}^N \phi_i\,\sum_{i=1}^N a_i\,\phi_i... ...displaystyle \left(\sum_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2}\;,$$ (5)

$$B = \frac{\displaystyle \sum_{i=1}^N \phi_i\,\sum_{i=1}^N a_i - N\,\s... ...displaystyle \left(\sum_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2}\;.$$ (6)

Substituting the trend $b_i = A + B\,\phi_i$ with $A$ and $B$ defined from the least-squares equations (5) and (6) into the correlation coefficient equation (1) and squaring the result leads to equation

$$\boxed{ \alpha^2(\mathbf{a}) = \frac{\displaystyle 2\,\sum_{i=1}^... ...\left[\left(\sum_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2\right]}\;.}$$ (7)

Equation (7) generalizes the semblance measure $\beta$ defined in equation (3) to a new measure $\alpha$ . In the absence of a trend (when the numerator in equation (6) is zero), $\alpha$ is equivalent to $\beta$ .

Sarkar et al. (2001) defined semblance using a normalized least-squares objective

$$\alpha^2(\mathbf{a}) = 1 - \frac{F(A,B)}{\displaystyle \sum_{i=1}^{N} a_i^2}\;.$$ (8)

Substituting equations (5) and (6) into (8) is an alternative way of deriving equation (7). This is the $AB$ semblance in terminology of Sarkar et al. (2002,2001).

Velocity analysis using $AB$ semblance