The Stack-Constrained form

The basic form just described depends on a $\frac{V_{s}}{V_{p}}$ratio, meaning that to get the information on the rock properties, you first need a fairly accurate interval velocity model. Fred Herkenhoff of Chevron realized that by substituting the value of a stacked trace into the problem, he could constrain the result to have values similar to those of the stack and remove the dependence on the velocity ratio. The stack amplitude can be calculated from the basic form:

 

$$S\ =\ R_{O}+ R_{sh} \sin^{2}(\theta_{S1}) + R_{P}\tan^{2}(\theta_{S2})\sin^{2}(\theta_{S1})$$ (3)

Here, $\sin^{2}\theta_{S1}$ and $\tan^{2}\theta_{S2}$ are the averages of $\sin^{2}\theta$ and $\tan^{2}\theta$ over the range of input angles. This stack equation is then used as a substitute for the $\frac{V_{s}}{V_{p}}$ratio:

$$R(\theta_{i})-S \frac{\sin^{2}(\theta_{i})}{\sin^{2}(\theta_... ...an^{2}(\theta_{i})\sin^{2}(\theta_{i})}{\sin^{2}(\theta_{S1})})$$ (4)

This form can now be inverted for the zero-offset reflectivity (R_O) and the P-wave reflectivity (R_P) without needing the interval velocities. Once those reflectivities are obtained, the stack equation (Eqn. 3) can be used to find the gradient term. This inversion is demonstrated in the next section.