Abstract: Mechanics of Seismic Q

Seismic-wave attenuation and dispersion are studied in most geophysical curricular and broadly used in both academic and applied research. Applications of these concepts include identification of gas reservoirs and chimneys from frequency-dependent seismic amplitudes, analysis of frequency dependent P-and S-wave velocities, interpretation of the effects of porosity and fluid saturation using laboratory experiments with rock samples, amplitude- and Q-variation with offset studies, time-lapse data analysis and reservoir monitoring, and Q-compensation increasing the resolution of reflection seismic sections. Attenuation effects are also included in many algorithms for waveform modeling, imaging, and full-waveform inversion.

Despite such long-established and often routine use, interpretations of seismic attenuation still contains several popular misconceptions about its nature. The key misconception consists in the core assumption that the energy dissipation rate in a wave-propagating medium can be described regardless of the physical mechanism, by a phenomenological property called the “Q-factor”. This material property was suggested from observations of rock creep in materials science (Lomnitz, 1957) and almost simultaneously introduced in seismology (e.g., Knopoff, 1956). This model describing the wave attenuation and dispersion by spatially- and (often) frequency-dependent Q-factors of the medium is usually called the viscoelastic (VE) model. However, already at that time, at the end of his famous paper entitled “Q”, Knopoff (1964) pointed out an important difficulty of the VE model for the Earth’s mantle. Unfortunately, the analysis of the physical consistency of this model was not pursued further at that time, and we need to continue it today.

The controversial subtlety of the concept of Q can be seen from the fact that it is used both as an observed quantity (measured in various experiments) and an in situ medium property (such as determining the frequency dependences of wave speeds). For example, by assigning Q values to thin rock layers, Qadrouh et al. (2018) recently compared several averaging formulas for layered media, similarly to averaging the density or elastic moduli. Measurements of the strain-stress ratios and phase lags for rock samples in the laboratory are often called “direct observations” of seismic attenuation and velocity dispersion (Batzle et al., 2014). In a study further considered in this paper, Pimienta et al. (2015) examine the empirical frequency dependencies of Q^-1(f) for sandstone samples and identify the characteristic frequencies (“transitions”), which presumably also occur within these rocks at field conditions. However, note that if a material property denoted Q(f) could indeed be measured so directly, this would be the only such case in physics! In physics, in order to determine some property of the medium (for example, wave speed), we typically have to measure some other quantities (such as distance and time in this case) and utilize the appropriate physical theories.

With regard to the seismic Q factor, the understanding of its meaning and implications for modeling or interpreting seismic attenuation is still far from being sufficient. In the following section, I consider the physical meaning of the Q-factor by focusing on three general points:

1. The Q is usually an empirical, or “apparent” quantity, which is specific to practically every type of deformation and experimental set-up an not simply related to material properties. This variability of Q is much broader than the usual differentiation of the Q-factors attributed to the bulk and shear moduli, or the differences between the “intrinsic” and “scattering” Qs. For example, Morozov and Deng (2018a) showed that the bulk-modulus attenuation (Q^-1) measured in a 8-cm long sandstone specimen by Pimienta et al. (2015) is about ten times greater than the Q^-1 measured in a traveling wave at the same frequency.
2. Although the VE model often leads to plausible predictions of decaying amplitudes and dispersive waveforms, such predictions are achieved phenomenologically, by using the “correspondence principle” to insert the expected effective Q values into the “VE moduli” and wave equations. However, the correspondence principle is physically unjustified for waves, and particularly for porous rock (White, 1986).
3. Furthermore, the VE model is incomplete for heterogeneous media, which include practically all cases of interest in seismology. This model lacks boundary- condition relations for the internal variables on material-property contrasts. However, similarly to fluid-saturated porous rock, boundary conditions should often dominate the attenuation and dispersion effects.

The above arguments show that the Q model alone is often inaccurate and/or insufficient for describing seismic-wave attenuation, particularly when links to quantitative interpretation and geomechanics are required. In section “Mechanical approach”, I summarize an alternate, rigorous-mechanics based approach not relying on a Q.