AVO THEORY

AVO is a common tool for extracting shear wave information from P-wave seismic data Ostrander (1984). For plane waves, the PP reflection coefficient is given by Zoeppritz equations which give a relation between velocity and density layer contrasts in dependence of the angle of incidence. A linearized form of $R_p(\theta)$ was derived by Aki and Richards Aki and Richards (1980):

$R(\theta) \: = \ c_1(\theta)\: I_p \: + \: c_2(\theta)\: I_s + \: c_3(\theta) \: D$

with

$c_1(\theta) \: = \: {1 \over {2 \cos^2(\theta)}}$,

$c_2(\theta) \: = \: - 4 \: \gamma^2 \: \sin(\theta)^2$, and

$c_3(\theta) \: = 2 \gamma^2 \: \sin^2(\theta)\:- {1\over 2} \: \tan^2(\theta)$

The P-impedance contrast is denoted by I_p, the S-impedance contrast by I_S and the density contrast by D. P-impedance and S-impedance contrast are functions of either P- or S-wave velocity contrast and density contrast. $\theta$ is the angle of incidence and $\gamma$ denotes the background v_s/v_p ratio. For small angles, only $c_1(\theta)$ contributes to the reflection coefficient, meaning that at near zero-offset it is dominated by the P-impedance contrast. For larger angles, the coefficient $c_2(\theta)$ also starts to affect the amplitudes, thus introducing shear wave velocity information into the data. The density contrast will only be included in the amplitude information for very large angles, thus will be neglected in this study.

Based on the analytic form of the reflection coefficient, it can be interpreted as having all the acoustic information in the near-zero offsets and additional shear impedance information in the far offsets. The longer the offsets, that is the higher the angles, the more $c_2(\theta)$ will contribute to the amplitudes and the more the amplitude will be sensitive to changes in the shear properties across layers. If the angle coverage is only medium (i.e. up to 30-35 degrees), the amplitude will be significantly more sensitive to acoustic changes. This means that in case of large amplitude variations i.e. due to noise, the shear impedance contrast will be less well constrained than the acoustic impedance contrast.