Summary
Study of linearized reflectivity is very important for amplitude versus offset (AVO) analysis. Linearized reflection coefficients for interfaces of a low contrast, separating two isotropic, lowloss viscoelastic media are derived using the Zeoprittz equation. To calculate the phase and attenuation angles in each layer in terms of the perturbations, we linearized the generalized Snells law in viscoelastic medium and showed that perturbation in attenuation angles is a sum of the perturbations in corresponding velocity and quality factor weighted by phase and attenuation angle averages. The ray parameter and slowness vectors are introduced as a function of attenuation angle and incident angle. For incident inhomogeneous P-wave, we show that reflection coefficient is a complex function, that its imaginary part is due to anelasticity in medium.
Introduction
Reflection coefficients have a complicated dependency upon density, velocities and quality factors. Using the exact form of these functions we can not explicitly determine the dependency of the reflection coefficients upon the physical properties. For a two layer elastic medium with a low contrast in physical properties, the approximate form of the reflectivities is derived by Aki and Richards(2002). There are two main assumption in the procedure of linearization. First of all the differences in the properties along the boundary are small compared to the average of the properties above and below the interface. Second assumption is that the angle of incident should be smaller that any critical angle.
Another approach for linearizing the reflection coefficients is the Born approximation based on perturbation theory. In this method the actual medium in which the wave propagates is decomposed to a reference medium whose properties are known, plus unknown perturbations in the properties.
The approach to calculating the viscoelastic linearized reflectivity is to express the quantities like density, P- and S-wave velocities and corresponding quality factors, in terms of differences between these quantities in upper and lower media. Besides the above physical quantities, we have the incident and transmitted phase and attenuation angles that should be expressed in terms of perturbations in phase and attenuation angles. In this paper we introduce the Zeoperitz equations for viscoelastic medium and linearize them according to the assumptions we mentioned above. In this procedure we analyze Snell’s law in anelastic medium and linearized that. Finally we introduce a map that converts the linearized reflectivities to the scattering potential obtained by Born approximation.
