AVO inversion

The physical relation between the variation of reflection/transmission coefficients with incident angle (and offset) and rock parameters has been widely investigated. This relation is established in the Zoeppritz equations, which relate reflection and transmission coefficients for plane waves and elastic properties of the medium. Because of the nonlinearity of the Zoeppritz equations, several approximations have been generated, such as those presented by Aki and Richards (1997) and Shuey (1985). The simplified versions of Zoeppritz equations allow the computation of AVO inversion to estimate elastic parameters from the observed reflection amplitude variation with angle. Equation (2) from Castagna and Smith (1994) is a version of Shuey's approximation for the P-wave reflection coefficient as a function of angle of incidence, which is linear in $\sin^2\theta$.This equation characterizes the reflection coefficient, at normal incidence, and at intermediate angles ($0 < \theta < 30$ degrees),  

$$R(\theta) \approx A + B \sin^2\theta$$ (2)

$$A= \left ( {\Delta V_p \over V_p} + {\Delta \rho \over \rho} \right )/2$$ (3)

$$B= \left( -2 {V_s^2 \over V_p^2} {\Delta \rho \over \rho} + ... ...r 2 V_p} - 4{V_s^2 \over V_p^2} {\Delta V_s \over V_s} \right )$$ (4)

where

V_p=(V_p2 + V_p1)/2,

V_s= (V_s2 + V_s1)/2,

$\rho=(\rho_2+\rho_1)/2$,

$\Delta V_p=V_{p2} - V_{p1}$,

$\Delta V_s= V_{s2} - V_{s1}$,

$\Delta \rho=\rho_2-\rho_1$.

The normal incident term, A, is commonly referred to as the AVO intercept attribute, the intermediate angles term, B, is referred to as the AVO gradient attribute. We use this approximation to invert for the intercept and gradient AVO attributes from the observed reflection amplitude variation with angle in the angle-domain common image gathers (CIG). In this domain, we pick the amplitude values at the reflector of interest and fit the amplitude versus $\sin^2\theta$ to a best straight-line approximation using a least-squares curve fitting method.

Providing a reference for the expected AVO response for the shale/brine, shale/oil, and shale/tuff interface, Figure 5 shows the P-wave reflection coefficient from the exact Zoeppritz equations.

 

zoeppritz Figure 5 P-wave reflection coefficient from Zoeppritz equations

At the near offset we expect a similar reflection coefficient (similar intercept attribute) for the shale/oil and the shale/tuff interfaces because of similar acoustic impedance; however this ambiguity can be resolved by the different radio between V_s and V_p (different gradient attribute). Although this calculation is valid only for a 2-layer model, it will be a reference for the expected tendency in the modeled data. Deviation from this tendency should be due to modeling effect, overburden effect, migration operator effect, velocity anomalies effect, and migration-velocity errors. We examined the modeling effect using a 1-D, 2-layer synthetic model, the overburden and migration operator effects using model 1 (overburden with flat interface), and then we use model 2 (overburden with sinusoidal interface) to understand the effect of velocity anomalies and migration-velocity errors.

 

2laypick Figure 6 Picked amplitudes for a 2-layer modeling case