Bortfeld's 3 term reflectivity equation

Bortfeld's three term reflectivity equation is a linearized form of the Zoeppritz equation. It predicts the amplitude (R) at various reflection angles ($\theta$) given three reflectivity terms. These three terms are the zero-offset reflectivity (R_O), the P-wave reflectivity (R_P), and a gradient term (R_sh). This is its basic form:

$$R(\theta_{i})\ = \ R_{O}+ R_{sh} \sin^{2}(\theta_{i}) + R_{P}\tan^{2}(\theta_{i})\sin^{2}(\theta_{i})$$ (1)

where

$$\begin{eqnarray} R_{P}\ =\ \frac{\Delta V_{p}}{2 V_{p}} \hspace{1.cm} R_{\rho}\ ... ...V_{s}}) \hspace{1.cm} k\ = \ (\frac{2 V_{s}}{V_{p}})^{2} \nonumber\end{eqnarray}$$ (2)

This form is easily obtained algebraically from the form derived in Aki and Richards (1980). Both forms are well known and frequently used, but may not be accurate in areas where the interval velocities are not well known. Therefore, we choose to look at a different form.