Under the assumption of small incident angle, there is a well-known linearized Zoeppritz equation Aki and Richards (1980). Because we only consider the incident angle less than 35 degree, we have omitted the C term in the original form. For the acoustic and elastic media, the expressions for the reflection coefficients are different.
Acoustic AVO approximation
(17) |
where
(18) |
Elastic AVO approximation
(19) |
where
(20) |
Using the reflection coefficient R and specular incident angle , we find the solution for intercept and slope is a least-squares problem.
(21) |
The resulting estimates of A and B are given by
(22) |
Getting AVO intercept and slope is not our final goal. The purpose of AVO analysis is to display the V_{p}/V_{s} anomaly in the subsurface. This anomaly is a very important hydrocarbon indication, especially for gas-charged reservoirs. Here we use the fluid-line technique to highlight this anomaly. Assume there is a linear relation
A X + B = 0 | (23) |
between intercept A and slope B. We specify a window with reasonable size and use least squares algorithm to estimate the coefficient X. Similar to , we get an expression for X
(24) |
The A X + B section is called the fluid-line section, which highlights the V_{p}/V_{s} anomaly.