Velocity analysis using semblance |

Suppose that the reference sequence has a trend , where is a known function. The trend can be, for example, an expression of the reflection coefficient in Shuey's approximation (Shuey, 1985), where and are the AVO intercept and gradient, , and corresponds to the reflection angle at trace . 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 and from least-square fitting of the trend amounts to the minimization of

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

Substituting the trend with and defined from the least-squares equations (5) and (6) into the correlation coefficient equation (1) and squaring the result leads to equation

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

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

Substituting equations (5) and (6) into (8) is an alternative way of deriving equation (7). This is the

Velocity analysis using semblance |

2013-03-02