Linearized Zoeppritz Equations

Since the Zoeppritz equations are highly nonlinear with respect to velocities and density, many approximations have been made in order to linearize them. Aki and Richards 1980 assumed small layer contrasts, simplifying the equations to the following form:

$$\begin{eqnarray} \rm R(\theta)&=& {1\over{2\: \cos^2(\theta)}}\: \rm I_{\rm p} \... ...} + \rm v_{\rm s1})/2, \nonumber \\ \rho&= &\: (\rho_2 + \rho_1)/2\end{eqnarray}$$ (2) (3) (4) (5) (6)

The relative contrasts in P-impedance, S-impedance and density are given by $\rm I_{\rm p},\rm I_{\rm s}$ and $\rm D$, respectively. $\theta$ is the reflection angle and $\gamma$ is an estimate of the background shear to compressional velocity ratio $\rm v_{\rm s}/\rm v_{\rm p}$.

The reflectivity curves corresponding to either a unit perturbation in P-impedance contrast ($\rm I_{\rm p} = 1,\rm I_{\rm s} = 0, \rm D = 0$), S-impedance contrast or density contrast can be seen in Figure . For a unit perturbation in relative P-impedance contrast, the P-impedance inversion curve dominates at small angles of incidence and increases with increasing offset. For a unit perturbation in relative S-impedance contrast, the S-impedance inversion curve is zero at normal incidence and is increasingly negative with increasing offset. Over the conventional range of surface reflection data acquisition geometry illumination, which is typically $0^{^{\circ}}$ to $35^{^{\circ}}$, the density inversion curve is not significant, as most of the density contrast contributes to the reflection AVO through the impedance contrasts alone. As the reflection amplitudes are mostly a combination of the P- and S-impedance contrast inversion curves, reflectors with P- and S-impedance contrasts of the same polarity and magnitude are expected to show approximately constant amplitude versus offset. On the other hand, reflectors with P- and S-impedance contrasts of opposite polarities, indicating a transition zone of changing rock pore fluid properties, should show increasing amplitudes versus offset.

 

zoepr-ann
Figure 1 Theoretical P, S, and Density impedance inversion curves.