3. SQp and SQs attributes

As in the viscoelastic medium, attenuation and phase velocity of plane wave propagation are governed by the Kramers-Kronig relations [6]. The maximum value of quality factor of P-wave and S-wave which represent the degree of attenuation can be estimated from basic elastic properties of compressional modulus (M) and shear modulus (G) at high- and low-frequency conditions:

$$\begin{aligned} 2Q\_p^{-1} &= \frac{M\_{\infty} - M\_0}{\sqrt{M\_0 M\_{\infty}}}\\ 2Q\_\circ^{-1} &= \frac{G\_{\infty} - G\_0}{\sqrt{G\_0 G\_{\infty}}} \end{aligned} \tag{6}$$

where the indexes (∞) and (0) represent relaxed and unrelaxed conditions, which are still difficult to be measured directly from seismic data.

A high- and low-frequency measurement, in rock physics, can be assumed as an effect of crack by the Hudson crack theory [6]. The changes of anisotropy stiffness component are associated with the difference between compressional modulus at high and low frequencies and can be correlated with Lame parameters: λ and μ. The change in bulk modulus is approximated by

$$\begin{split} M\_{\infty} - M\_{o} &= \Delta c\_{\mathbf{1}\mathbf{1}}^{Hudson} \\ &\approx \varepsilon \frac{\lambda^2}{\mu} \frac{\mathsf{4}(\lambda + 2\mu)}{\mathsf{3}(\lambda + \mu)} \end{split} \tag{7}$$

And the change in the shear modulus is approximated by

$$\begin{aligned} G\_{\infty} - G\_{o} &= \Delta c\_{44}^{Hudson} \\ &\approx \varepsilon \mu \frac{\mathbf{1} \mathsf{6}(\lambda + 2\mu)}{\mathbf{3}(3\lambda + 4\mu)} \end{aligned} \tag{8}$$

where ε is the crack density, which is estimated from porosity and aspect ratio <sup>p</sup>ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>p</sup>ffiffiffiffiffiffiffiffiffiffiffiffi (α) as<sup>ε</sup> <sup>¼</sup> <sup>3</sup>ϕ=ð4παÞ. By assuming that <sup>M</sup> <sup>¼</sup> MoM<sup>∞</sup> and <sup>G</sup> <sup>¼</sup> GoG∞, the Qp and Qs are formulated as [6]

$$\begin{aligned} Qp^{-1} &= \frac{2}{3} \varepsilon \frac{\left(M/G-2\right)^2}{\left(M/G-1\right)}\\ Qs^{-1} &= \frac{8}{3} \varepsilon \frac{\left(M/G\right)}{\left(3M/G-2\right)}\end{aligned} \tag{9}$$

Information on crack density is indicated by Qp�<sup>1</sup> and Qs�<sup>1</sup> . If the crack density of the rock increases, the secondary porosity related to the crack will increase, while the bulk density decreases. In other words, an increase in crack density will be followed by a decrease in bulk density. Hence, Eq. (9) can be approximated as [7]

$$\begin{aligned} SQ^{-1} &\equiv \frac{5}{6} \frac{1}{\rho} \frac{\left(M/G-2\right)^2}{\left(M/G-1\right)}\\ SQ^{-1} &\equiv \frac{10}{3} \frac{1}{\rho} \frac{\left(M/G\right)}{\left(3M/G-2\right)}\end{aligned} \tag{10}$$

where SQp�<sup>1</sup> and SQs�<sup>1</sup> are defined as scaled inverse Qp (SQp) and scaled inverse Qs (SQs), which are indicating the attenuation of P and S wave,

respectively. These parameters can be extracted from seismic data through inversion results where M/G is approximated from P- and S-wave velocity ratio or Vp/Vs.
