Anelastic acoustic impedance and the correspondence principle

Abstract

A B S T R A C TA general definition of seismic wave impedance is proposed as a matrix differential operator transforming the displacement boundary conditions into traction ones. This impedance is proportional to the standard acoustic impedance at all incidence angles and allows extensions to attenuative media and to the full elastic case. In all cases, reflection amplitudes at the contact of two media are uniquely described by the ratios of their impedances. Here, the anelastic acoustic impedance is studied in detail and attenuation contrasts are shown to produce phase-shifted reflections. Notably, the correspondence principle (i.e., the approach based on complex-valued elastic modules in the frequency domain) leads to incorrect phase shifts of the impedance due to attenuation and consequently to wrong waveforms reflected from attenuation contrasts. Boundary conditions and the Lagrange formulation of elastodynamics suggest that elastic constants should remain real in the presence of attenuation and the various types of energy dissipation should be described by their specific mechanisms. The correspondence principle and complex-valued elastic moduli appear to be applicable only to homogeneous media and therefore they should be used with caution when applied to heterogeneous cases.

Thin-layer effects in glaciological seismic amplitude-versus-angle (AVA) analysis: implications for characterising a subglacial till unit, Russell Glacier, West Greenland

“…Seismic quality factor, Q, also contributes to the observable reflectivity of an interface, although only significantly where the Q-contrast exceeds an order of magnitude (Bourbié and Nur, 1984;Odebeatu et al, 2006). Two proposed mechanisms for Q to contribute to the apparent reflectivity of an interface are (a) frequency dependency of the reflection coefficient, since different frequency components of a wavelet propagate at different velocities in dispersive materials (Xu et al, 2011), and (b) introduction of a phase lag into the recorded waveform (Lines et al, 2008), since the reaction time of a high-to-low-Q interface is frequency-dependant (Quintal et al, 2009;Morozov, 2011). Q-based reflectivity is undoubtedly an important consideration in quantitative seismic interpetatoin but, at this stage of analysis, we assume small Q-contrasts allowing the associated reflectivity contributions to be neglected.…”

Research Note: Experimental measurements of Q‐contrast reflections

“…The pressure reflection coefficient for such media (for a given reference frequency), as presented by White (1965), Morozov (2011), Bourbié and Nur (1984) and Lines et al (2008) is given by:…”

“…The paper 'Reflections on Q' by Lines, Vasheghani and Treitel (2008) derived the reflection coefficients for Q-contrasts and showed its effect on synthetic seismograms using codes due to Carcione (2007) that compute finite-difference solutions to an anelastic wave equation. Other recent papers showing the Q-effects on seismic reflections include papers by Morozov (2011), Innanen (2011 and Odebeatu et al (2006) In our analysis of Q-reflections, we first briefly review the mathematics describing reflection coefficients and then describe the experiments used to validate these reflection amplitude expressions.…”

“…In the case of normal incidence (NI) P-wave reflections in an anelastic acoustic medium, the boundary conditions require continuity of pressure and normal components of particle velocity. The pressure reflection coefficient for such media (for a given reference frequency), as presented by White (1965), Morozov (2011), Bourbié and Nur (1984) and Lines et al (2008) is given by:…”

Temporal variations of coda Q: An attenuation-coefficient view

“…The attenuation-coefficient view on seismic attenuation (Morozov, 2008(Morozov, , 2010a starts from a generalization of the classic scattering-theory approximation (Chernov, 1960;Sato, 1978) and further extends into a critique of the concept of Q and the theory of viscoelasticity (Morozov, 2009c(Morozov, , 2010c. Currently, this approach is undergoing some debate (Morozov, 2009a, b;Xie and Fehler, 2009;Mitchell, 2010;Xie, 2010).…”