Joint Inversion of Frequency Components of PP- and PSV-Wave Amplitudes for Attenuation Factors Using Second-Order Derivatives of Anelastic Impedance

Abstract

Attenuation factor (1/Q) is an important indicator that is sensitive to hydrocarbon-bearing reservoirs. In attenuative media, seismic wave velocity is complex and frequency dependent, and 1/Q appears in the imaginary part of wave velocity. Focusing on the real part of complex wave velocity in attenuative media, we present new-parameterized P- and S-wave velocities, in which a maximum P-wave attenuation factor ($1/Q_\mathrm{Pm}$) is involved. Using solutions of Zoeppritz equations, we derive approximate PP- and PSV-wave reflection coefficients and anelastic impedances (AEI) as a function of $1/Q_\mathrm{Pm}$, and we employ an empirical relationship to replace the S-wave attenuation factor with $1/Q_\mathrm{Pm}$ in the derivation of PSV-wave reflection coefficient. Using the derived reflection coefficients and AEI, we establish an inversion approach and workflow of utilizing PP- and PSV-wave seismic amplitudes to estimate elastic parameters (i.e. P- and S-wave moduli, density) and anelastic parameter (i.e. $1/Q_\mathrm{Pm}$). Inputting different frequency components of PP- and PSV-wave seismic wave amplitudes, a model-based least-squares (LS) algorithm is employed to implement the deconvolution for estimating the PP- and PSV-wave AEI of different dominant frequencies and incidence angles, and the joint inversion of PP- and PSV-wave AEI for elastic parameters and attenuation factor is implemented using a Newton method. We employ synthetic seismic data of different signal-to-noise ratios (SNR) to verify the stability and robustness of the established inversion approach, and we finally apply the approach to real datasets acquired over a hydrocarbon reservoir. It illustrates that stable elastic parameters and attenuation factors may be estimated; and combining the inverted elastic parameters and attenuation factor, we may realize the reliable identification and interpretation of hydrocarbon reservoirs.

Article Highlights

• PP- and PSV-wave reflection coefficients and anelastic impedances are derived in terms of P- and S-wave moduli, density, and P-wave attenuation factor at the characterize frequency.

• Approach of joint inversion of PP- and PSV-wave seismic datasets for estimating P and S-wave moduli, density and P-wave attenuation factor based on first- and second-order derivatives of anelastic impedances is established.

• Tests on synthetic and real datasets imply the proposed inversion approach can be used as a valuable tool for generating reliable results of elastic properties and attenuation factor for hydrocarbon reservoir characterization.

Appendices

Appendix A: Elastic Parameters and Incidence Angles Expressed Using Scatterings in Anelastic Media

Following Moradi and Innanen (2016), we rewrite frequency-dependent P- and S-wave velocities, P- and S-wave incidence and transmission angles, and density of top and bottom layers as

$$\begin{aligned} \begin{aligned} {\rho _{1}}&={{\rho }}\left( 1-\frac{1}{2}\frac{\varDelta {\rho }}{{{\rho }}}\right) ,~ {\rho _{2}}={{\rho }}\left( 1+\frac{1}{2}\frac{\varDelta {\rho }}{{{\rho }}}\right) ,\\ {\mathcal {V}_\mathrm{{P1}}}&={\mathcal {V}_\mathrm{{P}}}\left( 1-\frac{1}{2}\frac{\varDelta {{\mathcal {V}_\mathrm{{P}}}}}{{{{\mathcal {V}_\mathrm{{P}}}}}}\right) ,~ {\mathcal {V}_\mathrm{{P2}}}={\mathcal {V}_\mathrm{{P}}}\left( 1+\frac{1}{2}\frac{\varDelta {{\mathcal {V}_\mathrm{{P}}}}}{{{{\mathcal {V}_\mathrm{{P}}}}}}\right) ,\\ {\mathcal {V}_\mathrm{{S1}}}&={\mathcal {V}_\mathrm{{S}}}\left( 1-\frac{1}{2}\frac{\varDelta {{\mathcal {V}_\mathrm{{S}}}}}{{{{\mathcal {V}_\mathrm{{S}}}}}}\right) ,~ {\mathcal {V}_\mathrm{{S2}}}={\mathcal {V}_\mathrm{{S}}}\left( 1+\frac{1}{2}\frac{\varDelta {{\mathcal {V}_\mathrm{{S}}}}}{{{{\mathcal {V}_\mathrm{{S}}}}}}\right) ,\\ {\sin {{\theta }_\mathrm{P1}}}&= {\sin {{{\theta }_\mathrm{P}}}\left( 1-\frac{1}{2}\frac{{\varDelta }\theta _\mathrm{P}}{\tan {\theta _\mathrm{P}}}\right) },~ {\sin {{\theta }_\mathrm{P2}}}={\sin {{{\theta }_\mathrm{P}}}}\left( 1+\frac{1}{2}\frac{{\varDelta {\theta }_\mathrm{P}}}{\tan {{{{\theta }_\mathrm{P}}}}}\right) ,\\ {\sin {{\theta }_\mathrm{S1}}}&={\sin {{{\theta }_\mathrm{S}}}}\left( 1-\frac{1}{2}\frac{{\varDelta {\theta }_\mathrm{S}}}{\tan {{{{\theta }_\mathrm{S}}}}}\right) ,~ {\sin {{\theta }_\mathrm{S2}}}={\sin {{{\theta }_\mathrm{S}}}}\left( 1+\frac{1}{2}\frac{{\varDelta {\theta }_\mathrm{S}}}{\tan {{{{\theta }_\mathrm{S}}}}}\right) . \end{aligned} \end{aligned}$$

(22)

Combining Eqs. (22) and (11), we express perturbation in incidence angles as

$$\begin{aligned} \begin{aligned}&{{\varDelta {\theta }_\mathrm{P}}}{\approx }\frac{\varDelta \mathcal {V}_\mathrm{P}}{\mathcal {V}_\mathrm{P}}\tan {{\theta }_\mathrm{P}},\\&{{\varDelta {\theta }_\mathrm{S}}}{\approx }\frac{\varDelta \mathcal {V}_\mathrm{S}}{\mathcal {V}_\mathrm{S}}\tan {{\theta }_\mathrm{S}}. \end{aligned} \end{aligned}$$

(23)

Appendix B: Elastic Parameters and Incidence Angles Expressed Using Scatterings in Anelastic Media

Substituting expressions of P- and S-wave velocities and moduli of upper and lower layers to solutions of Zoeppritz equations for PP and PSV waves, we first obtain reflection coefficients of PP and PSV waves after some algebra, which are similar to reflection coefficients that are derived for PP and PSV waves in elastic media, as

$$\begin{aligned} \begin{aligned} R_\mathrm{PP}&{\approx }\frac{1}{4\cos ^2\theta _\mathrm{P}}\frac{{\varDelta }{\mathcal {M}}}{{\mathcal {M}}} -2{{{{g}}}}{\sin ^2{\theta _\mathrm{P}}}\frac{{\varDelta }{\mathcal {U}}}{{\mathcal {U}}}+\frac{\cos {2\theta _\mathrm{P}}}{4\cos ^2{\theta _\mathrm{P}}}\frac{{\varDelta }\rho }{\rho },\\ R_\mathrm{PS}&{\approx }{\sqrt{{{g}}}}{\sin {\theta _\mathrm{P}}}\left( \frac{\sqrt{{{g}}}\sin ^2{\theta _\mathrm{P}}}{\sqrt{1-{{{g}}}{\sin ^2\theta _\mathrm{P}}}}-{\cos {\theta _\mathrm{P}}}\right) \frac{{\varDelta }{\mathcal {U}}}{{\mathcal {U}}}-\frac{{{{g}}}\sin {\theta _\mathrm{P}}}{2\sqrt{1-{{{g}}}{\sin ^2\theta _\mathrm{P}}}}\frac{{\varDelta }\rho }{\rho }, \end{aligned} \end{aligned}$$

(24)

where ${{{g}}}=\frac{{\mathcal {U}}}{{\mathcal {M}}}$. We observe that effects of elastic properties and attenuation factor are hidden in the reflectivities $\frac{{\varDelta }{\mathcal {M}}}{{\mathcal {M}}}$ and $\frac{{\varDelta }{\mathcal {U}}}{{\mathcal {U}}}$, and we next approximately express $\frac{{\varDelta }{\mathcal {M}}}{{\mathcal {M}}}$ and $\frac{{\varDelta }{\mathcal {U}}}{{\mathcal {U}}}$ as

$$\begin{aligned} \begin{aligned} \frac{{\varDelta }\mathcal {M}}{\mathcal {M}}&=\frac{{\varDelta }\left\{ {M}\left[ 1+\frac{2}{\pi {Q_\mathrm{Pm}}}{{\psi \left( \omega \right) }}\right] \right\} }{{M}\left[ 1+\frac{2}{\pi {Q_\mathrm{Pm}}}{{\psi \left( \omega \right) }}\right] }{\approx }\frac{{\varDelta }{M}}{{M}} +\frac{2{{\psi \left( \omega \right) }}}{\pi }{{\varDelta }\left( \frac{1}{{Q_\mathrm{Pm}}}\right) },\\ \frac{{\varDelta }\mathcal {U}}{\mathcal {U}}&=\frac{{\varDelta }\left\{ {\mu }\left[ 1+\frac{2}{\pi {Q_\mathrm{Pm}}}\frac{{{\psi \left( \omega \right) }}}{h\left( \omega \right) }\right] \right\} }{{\mu }\left[ 1+\frac{2}{\pi {Q_\mathrm{Pm}}}\frac{{{\psi \left( \omega \right) }}}{h\left( \omega \right) }\right] }{\approx }\frac{{\varDelta }{\mu }}{{\mu }} +\frac{2}{\pi }\frac{{{\psi \left( \omega \right) }}}{{h\left( \omega \right) }}{{\varDelta }\left( \frac{1}{{Q_\mathrm{Pm}}}\right) }. \end{aligned} \end{aligned}$$

(25)

Substituting Eq. (25) into Eq.(24), we obtain final expressions of frequency-dependent PP- and PSV-wave reflection coefficients as

$$\begin{aligned} \begin{aligned} R_\mathrm{PP}\left( \theta _\mathrm{P},\omega \right) {\approx }&a_{M}\left( \theta _\mathrm{P}\right) \frac{{\varDelta }{M}}{{M}}+a_{\mu }\left( \theta _\mathrm{P}\right) \frac{{\varDelta }{\mu }}{{\mu }}+a_{\rho }\left( \theta _\mathrm{P}\right) \frac{{\varDelta }\rho }{\rho }+a_{Q}\left( \theta _\mathrm{P},\omega \right) {{\varDelta }\left( \frac{1}{{Q_\mathrm{Pm}}}\right) },\\ R_\mathrm{PS}\left( \theta _\mathrm{P},\omega \right) {\approx }&b_{\mu }\left( \theta _\mathrm{P}\right) \frac{{\varDelta }{\mu }}{{\mu }}+b_{\rho }\left( \theta _\mathrm{P}\right) \frac{{\varDelta }\rho }{\rho }+b_{Q}\left( \theta _\mathrm{P},\omega \right) {{\varDelta }\left( \frac{1}{{Q_\mathrm{Pm}}}\right) }, \end{aligned} \end{aligned}$$

(26)

where

$$\begin{aligned} \begin{aligned} a_{M}\left( \theta _\mathrm{P}\right)&=\frac{1}{4}{\sec ^2\theta _\mathrm{P}},~a_{\mu }\left( \theta _\mathrm{P}\right) =-2{{{\gamma }}}{\sin ^2{\theta _\mathrm{P}}},~ a_{\rho }\left( \theta _\mathrm{P}\right) =\frac{\cos {2\theta _\mathrm{P}}}{4\cos ^2{\theta _\mathrm{P}}},\\ a_{Q}\left( \theta _\mathrm{P},\omega \right)&=\frac{1}{2}{\sec ^2\theta _\mathrm{P}}\frac{{{\psi \left( \omega \right) }}}{\pi }-4{{{\gamma }}}{\sin ^2{\theta _\mathrm{P}}}\frac{{{\psi \left( \omega \right) }}}{{\pi }}{h},\\ b_{\mu }\left( \theta _\mathrm{P}\right)&={\sqrt{{\gamma }}}{\sin {\theta _\mathrm{P}}}\left( \frac{\sqrt{{\gamma }}\sin ^2{\theta _\mathrm{P}}}{\sqrt{1-{{\gamma }}{\sin ^2\theta _\mathrm{P}}}}-{\cos {\theta _\mathrm{P}}}\right) ,\\ b_{\rho }\left( \theta _\mathrm{P}\right)&=-\frac{{{\gamma }}\sin {\theta _\mathrm{P}}}{2\sqrt{1-{{\gamma }}{\sin ^2\theta _\mathrm{P}}}},\\ b_{Q}\left( \theta _\mathrm{P},\omega \right)&=\frac{2}{\pi }{\sqrt{{\gamma }}}{\sin {\theta _\mathrm{P}}}\left( \frac{\sqrt{{\gamma }}\sin ^2{\theta _\mathrm{P}}}{\sqrt{1-{{\gamma }}{\sin ^2\theta _\mathrm{P}}}}-{\cos {\theta _\mathrm{P}}}\right) {{{\psi \left( \omega \right) }}}{{h}}, \end{aligned} \end{aligned}$$

(27)

and $\gamma =\frac{\mu }{M}$. We conclude that in Eq.26 the effects of elastic parameters and attenuation factor on PP- and PSV-wave reflection coefficients may be separated explicitly.