Plane-wave propagation in attenuative transversely isotropic media

Abstract

Directionally dependent attenuation in transversely isotropic (TI) media can influence significantly the body-wave amplitudes and distort the results of the AVO (amplitude variation with offset) analysis. Here, we develop a consistent analytic treatment of plane-wave properties for TI media with attenuation anisotropy. We use the concept of homogeneous wave propagation, assuming that in weakly attenuative media the real and imaginary parts of the wave vector are parallel to one another. The anisotropic quality factor can be described by matrix elements Q ij , defined as the ratios of the real and imaginary parts of the corresponding stiffness coefficients. To characterize TI attenuation, we follow the idea of the Thomsen notation for velocity anisotropy and replace the components Q ij by two reference isotropic quantities and three dimensionless anisotropy parameters Q , δ Q , and γ Q. The parameters Q and γ Q quantify the difference between the horizontal-and vertical-attenuation coefficients of P-and SH-waves, respectively, while δ Q is defined through the second derivative of the P-wave attenuation coefficient in the symmetry direction. Although the definitions of Q , δ Q , and γ Q are similar to those for the corresponding Thomsen parameters, the expression for δ Q reflects the coupling between the attenuation and velocity anisotropy. Assuming weak attenuation as well as weak velocity and attenuation anisotropy allows us to obtain simple attenuation coefficients linearized in the Thomsen-style parameters. The normalized attenuation coefficients for P-and SV-waves have the same form as the corresponding approximate phase-velocity functions, but both δ Q and the effective SV-wave attenuation-anisotropy parameter σ Q depend on the velocity-anisotropy parameters in addition to the elements Q ij. The linearized approximations not only provide valuable analytic insight, but they also remain accurate for the practically important range of small and moderate anisotropy parameters-in particular, for nearvertical and near-horizontal propagation directions.

Ultrasonic velocity and attenuation anisotropy of shales, Whitby, United Kingdom

“…Figure 1a and 1b shows, respectively, a photo and a schematic diagram of the three samples prepared from block WUK47B. Assuming the validity of (V)TI representation for our shales, five independent velocity-attenuation pairs for complete anisotropy characterization were needed (Thomsen, 1986;Zhu and Tsvankin, 2006).…”

“…According to Zhu and Tsvankin (2006), the anisotropic quality factor can be described by matrix elements Q ij and for the case of TI media with TI attenuation can be written as follows:…”

“…The three samples originate approximately 1 m (WUK2), 4 m (WUK70), and 7 m (WUK47B) from the base of the Jet Dogger section of the Whitby Mudstone Formation (Houben et al, 2014). All cores were then used to study velocity and attenuation anisotropy at different stress levels by calculating, respectively, ϵ, γ, δ (Thomsen, 1986) and ϵ Q , γ Q , δ Q (Zhu and Tsvankin, 2006) from ultrasonic measurements (see Appendix A). Figure 1a and 1b shows, respectively, a photo and a schematic diagram of the three samples prepared from block WUK47B.…”

“…Then, the results of measured velocity and attenuation at ultrasonic frequencies (0.3-1 MHz) in mudstone samples as a function of axial stress and with respect to the bedding plane are presented. Following Thomsen (1986) and Zhu and Tsvankin (2006), the definition of velocity and attenuation anisotropy parameters are given and then used to quantify the anisotropy in our shales.…”

An Approximate Method for the Acoustic Attenuating VTI Eikonal Equation

“…In this expanded abstract, we will propose a method to solve the acoustic eikonal equation (Hao and Alkhalifah, 2017) for attenuating transversely isotropic media with a vertical symmetry axis (VTI). Alkhalifah's (2000) notation and Zhu and Tsvankin's (2006) notation are combined to describe the Pwave traveltimes in an attenuating VTI medium, where Alkhalifah's notation and Zhu and Tsvankin's notation correspond to the nonattenuating and attenuating parts of the VTI medium, respectively. Alkhalifah's notation includes the P-wave vertical velocity 0 P  , the P-wave normal moveout velocity n  , and the anellipticity parameter  .…”