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$ 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

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

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

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


next up previous [pdf]

Next: Sensitivity analysis of semblance Up: Theory Previous: Semblance as correlation

2013-07-26