Fourier Coefficients Variation with Angle for Fracture Detection and Fluid Discrimination in Tilted Transversely Isotropic Media

AbstractSection Abstract

Fracture detection and fluid discrimination play significant roles in the field of geothermics, hydrogeology, and exploration geophysics. However, it is still a challenging task how to reliably and robustly characterize fracture and fluid distribution from seismic data. Therefore, this paper focuses on fracture detection and fluid discrimination from reflected seismic data in the saturated rock with the tilted parallel fractures. And we adopt the dry fracture weakness parameters related with the fracture density and the fluid bulk modulus mainly affected by fluid as indicators of fractures and fluid, respectively. Firstly, using the anisotropic fluid substitution equation and the linear-slip model, the analytical approximate expressions of the stiffness parameters are derived for the gas-bearing rock. Numerical examples illustrate that these approximations possess satisfactory accuracies for the case with the small dry weakness parameters and small fluid bulk modulus. Further, following the connection between the reflection coefficient and the scattering function, and coupling rock physical relationships, we derive a novel linearized PP-wave reflectivity equation parameterized only by six unknown parameters, and rewrite it in terms of the Fourier series to decouple isotropic and anisotropic components of seismic data. Based on the analysis of the contribution of each unknown parameter to the reflection coefficient, a two-step Bayesian inversion method is proposed for estimations of the shear modulus, fluid bulk modulus, porosity, dry weakness parameters, and dip angle, in which the Karhunen–Loève transform is introduced to enhance robustness of inversion. Finally, synthetic and field data are utilized to verify feasibility and stability of the proposed method.

AbstractSection Article Highlights

• Reflectivity is parameterized by only six model parameters for the gas-bearing rock with tilted aligned fractures.

• A two-step inversion incorporating the Fourier series and Karhunen–Loève transform can increase the prediction stability.

• This paper may be useful for the exploration and exploitation of gas-bearing fractured tight sandstones.

Appendices

Appendix A

The stiffness coefficients of the saturated TTI medium can be expressed as

$$\begin{gathered} C_{11}^{\text{sat}} = \left[ {4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{4} \zeta + \left[ {4\mu_{\text{b}} \left( {\delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{2\Lambda \left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right]\cos^{2} \zeta \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + M_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\Lambda^{2} }}{\Upsilon } \hfill \\ \end{gathered}$$

(A-1)

$$C_{22}^{\text{sat}} = M_{\text{b}}^{\text{dry}} \left[ {1 - \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} \delta_{\text{N}}^{\text{dry}} } \right] + \frac{{{\rm T}^{2} }}{\Upsilon }$$

(A-2)

$$C_{12}^{\text{sat}} = \left[ {2\mu_{\text{b}} \chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \frac{{{\rm T}\left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right]\cos^{2} \zeta + \lambda_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\Lambda {\rm T}}}{\Upsilon }$$

(A-3)

$$\begin{gathered} C_{13}^{\text{sat}} = \left[ {4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{4} \zeta + \left[ {4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{2} \zeta \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \lambda_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\Lambda {\rm T}}}{\Upsilon } \hfill \\ \end{gathered}$$

(A-4)

$$\begin{gathered} C_{33}^{\text{sat}} = \left[ {4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{4} \zeta - \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \left[ {4\mu_{\text{b}} \left( {\chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} + \delta_{\text{T}} } \right) - \frac{{2{\rm T}\left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right]\cos^{2} \zeta + M_{\text{b}}^{\text{dry}} \left[ {1 - \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} \delta_{\text{N}}^{\text{dry}} } \right] + \frac{{{\rm T}^{2} }}{\Upsilon } \hfill \\ \end{gathered}$$

(A-5)

$$C_{23}^{\text{sat}} = \left[ {\frac{{{\rm T}\left( {\Lambda - {\rm T}} \right)}}{\Upsilon } - 2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right]\cos^{2} \zeta + \lambda_{\text{b}}^{\text{dry}} \left[ {1 - \chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right] + \frac{{{\rm T}^{2} }}{\Upsilon }$$

(A-6)

$$C_{55}^{\text{sat}} = \left[ {4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{4} \zeta - \left[ {4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{2} \zeta + \mu_{\text{b}} \left( {1 - \delta_{\text{T}} } \right)$$

(A-7)

$$C_{15}^{\text{sat}} = \left\{ {\left[ {4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right) + \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{2} \zeta + 2\mu_{\text{b}} \left( {\delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{\Lambda \left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right\}\sin \zeta \cos \zeta$$

(A-8)

$$C_{25}^{\text{sat}} = \left[ {2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \frac{{{\rm T}\left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right]\sin \zeta \cos \zeta$$

(A-9)

$$C_{35}^{\text{sat}} = \left\{ {\left[ {4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right) - \frac{{\left( {\Lambda - {\rm T}} \right)^{2} }}{\Upsilon }} \right]\cos^{2} \zeta + 2\mu_{\text{b}} \left( {\chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} + \delta_{\text{T}} } \right) - \frac{{{\rm T}\left( {\Lambda - {\rm T}} \right)}}{\Upsilon }} \right\}\sin \zeta \cos \zeta$$

(A-10)

$$C_{44}^{\text{sat}} = - \mu_{\text{b}} \delta_{\text{T}} \cos^{2} \zeta + \mu_{\text{b}}$$

(A-11)

$$C_{46}^{\text{sat}} = \mu_{\text{b}} \delta_{\text{T}} \sin \zeta \cos \zeta$$

(A-12)

$$C_{66}^{\text{sat}} = \mu_{\text{b}} \delta_{\text{T}} \cos^{2} \zeta + \mu_{\text{b}} \left( {1 - \delta_{\text{T}} } \right)$$

(A-13)

In addition, $C_{21}^{\text{sat}} = C_{12}^{\text{sat}}$, $C_{31}^{\text{sat}} = C_{13}^{\text{sat}}$, $C_{32}^{\text{sat}} = C_{23}^{\text{sat}}$, $C_{51}^{\text{sat}} = C_{15}^{\text{sat}}$, $C_{52}^{\text{sat}} = C_{25}^{\text{sat}}$, $C_{53}^{\text{sat}} = C_{35}^{\text{sat}}$, $C_{64}^{\text{sat}} = C_{46}^{\text{sat}}$ , and other components equal zero.

The final approximations of the stiffness parameters of the saturated TTI rock can be written as

$$C_{11}^{\text{sat}} \approx 4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right)\cos^{4} \zeta + 4\mu_{\text{b}} \left( {\delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)\cos^{2} \zeta + M_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + G\left( {\phi_{\text{p}} } \right)K_{\text{f}}$$

(A-14)

$$C_{22}^{\text{sat}} \approx M_{\text{b}}^{\text{dry}} \left[ {1 - \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} \delta_{\text{N}}^{\text{dry}} } \right] + G\left( {\phi_{\text{p}} } \right)K_{\text{f}}$$

(A-15)

$$C_{12}^{\text{sat}} \approx 2\mu_{\text{b}} \chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta + \lambda_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + G\left( {\phi_{\text{p}} } \right)K_{\text{f}}$$

(A-16)

$$C_{13}^{\text{sat}} \approx 4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)\cos^{4} \zeta + 4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right)\cos^{2} \zeta + \lambda_{\text{b}}^{\text{dry}} \left( {1 - \delta_{\text{N}}^{\text{dry}} } \right) + G\left( {\phi_{\text{p}} } \right)K_{\text{f}}$$

(A-17)

$$\begin{gathered} C_{33}^{\text{sat}} \approx 4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right)\cos^{4} \zeta - 4\mu_{\text{b}} \left( {\chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} + \delta_{\text{T}} } \right)\cos^{2} \zeta \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + M_{\text{b}}^{\text{dry}} \left[ {1 - \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} \delta_{\text{N}}^{\text{dry}} } \right] + G\left( {\phi_{\text{p}} } \right)K_{\text{f}} \hfill \\ \end{gathered}$$

(A-18)

$$C_{23}^{\text{sat}} \approx - 2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta + \lambda_{\text{b}}^{\text{dry}} \left( {1 - \chi_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right) + G\left( {\phi_{\text{p}} } \right)K_{\text{f}}$$

(A-19)

$$C_{55}^{\text{sat}} \approx 4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)\cos^{4} \zeta - 4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)\cos^{2} \zeta + \mu_{\text{b}} \left( {1 - \delta_{\text{T}} } \right)$$

(A-20)

$$C_{15}^{\text{sat}} \approx \left[ {4\mu_{\text{b}} \left( {\delta_{\text{T}} - g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} } \right)\cos^{2} \zeta + 2\mu_{\text{b}} \left( {\delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)} \right]\sin \zeta \cos \zeta$$

(A-21)

$$C_{25}^{\text{sat}} \approx 2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} \sin \zeta \cos \zeta$$

(A-22)

$$C_{35}^{\text{sat}} \approx \left[ {4\mu_{\text{b}} \left( {g_{\text{b}}^{\text{dry}} \delta_{\text{N}}^{\text{dry}} - \delta_{\text{T}} } \right)\cos^{2} \zeta + 2\mu_{\text{b}} \left( {\chi \delta_{\text{N}}^{\text{dry}} + \delta_{\text{T}} } \right)} \right]\sin \zeta \cos \zeta$$

(A-23)

$$C_{44}^{\text{sat}} = - \mu_{\text{b}} \delta_{\text{T}} \cos^{2} \zeta + \mu_{\text{b}}$$

(A-24)

$$C_{46}^{\text{sat}} = \mu_{\text{b}} \delta_{\text{T}} \sin \zeta \cos \zeta$$

(A-25)

$$C_{66}^{\text{sat}} = \mu_{\text{b}} \delta_{\text{T}} \cos^{2} \zeta + \mu_{\text{b}} \left( {1 - \delta_{\text{T}} } \right)$$

(A-26)

After the symmetry of the stiffness matrix is satisfied, other components are all zero.

Differentiating the both sides of Formulas (A14)–(A26) and neglecting ${\delta }_{\text{N}}^{\text{dry}}\Delta {M}_{I}$ and ${\delta }_{\text{T}}\Delta {M}_{I}$, we can obtain the perturbations in the stiffness parameters across the interface separating two saturated TTI rocks,

$$\begin{gathered} \Delta C_{11}^{\text{sat}} \approx 4\mu_{\text{b}} \left[ {\Delta \left( {\delta_{\text{T}} \cos^{4} \zeta } \right) - g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{4} \zeta } \right)} \right] + 4\mu_{\text{b}} \left[ {\Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right) - \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right)} \right] \\ + \Delta M_{\text{b}}^{\text{dry}} - M_{\text{b}}^{\text{dry}} \Delta \delta_{\text{N}}^{\text{dry}} + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right) \\ \end{gathered}$$

(A-27)

$$\Delta C_{22}^{\text{sat}} \approx \Delta M_{\text{b}}^{\text{dry}} - \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} M_{\text{b}}^{\text{dry}} \Delta \delta_{\text{N}}^{\text{dry}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right) + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}}$$

(A-28)

$$\Delta C_{12}^{\text{sat}} \approx 2\mu_{\text{b}} \chi_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right) + \Delta \lambda_{\text{b}}^{\text{dry}} - \lambda_{\text{b}}^{\text{dry}} \Delta \delta_{\text{N}}^{\text{dry}} + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right)$$

(A-29)

$$\begin{gathered} \Delta C_{13}^{\text{sat}} \approx 4\mu_{\text{b}} \left[ {g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{4} \zeta } \right) - \Delta \left( {\delta_{\text{T}} \cos^{4} \zeta } \right) + \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right) - g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right)} \right] \\ + \Delta \lambda_{\text{b}}^{\text{dry}} - \lambda_{\text{b}}^{\text{dry}} \Delta \delta_{\text{N}}^{\text{dry}} + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right) \\ \end{gathered}$$

(A-30)

$$\begin{gathered} \Delta C_{33}^{\text{sat}} \approx 4\mu_{\text{b}} \left[ {\Delta \left( {\delta_{\text{T}} \cos^{4} \zeta } \right) - g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{4} \zeta } \right) - \chi_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right) - \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right)} \right] \\ + \Delta M_{\text{b}}^{\text{dry}} - M_{\text{b}}^{\text{dry}} \left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} \Delta \delta_{\text{N}}^{\text{dry}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right) + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}} \\ \end{gathered}$$

(A-31)

$$\Delta C_{23}^{\text{sat}} \approx - 2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right) + \Delta \lambda_{\text{b}}^{\text{dry}} - \lambda_{\text{b}}^{\text{dry}} \chi_{\text{b}}^{\text{dry}} \Delta \delta_{\text{N}}^{\text{dry}} + K_{\text{f}} \Delta G\left( {\phi_{\text{p}} } \right) + G\left( {\phi_{\text{p}} } \right)\Delta K_{\text{f}}$$

(A-32)

$$\Delta C_{55}^{\text{sat}} \approx 4\mu_{\text{b}} \left[ \begin{gathered} g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{4} \zeta } \right) - \Delta \left( {\delta_{\text{T}} \cos^{4} \zeta } \right) - \hfill \\ g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{2} \zeta } \right) + \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right) \hfill \\ \end{gathered} \right] + \Delta \mu_{\text{b}} - \mu_{\text{b}} \Delta \delta_{\text{T}}$$

(A-33)

$$\Delta C_{15}^{\text{sat}} \approx 2\mu_{\text{b}} \left[ \begin{gathered} 2\Delta \left( {\delta_{\text{T}} \cos^{3} \zeta \sin \zeta } \right) - 2g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{3} \zeta \sin \zeta } \right) \hfill \\ + \Delta \left( {\delta_{\text{N}}^{\text{dry}} \sin \zeta \cos \zeta } \right) - \Delta \left( {\delta_{\text{T}} \sin \zeta \cos \zeta } \right) \hfill \\ \end{gathered} \right]$$

(A-34)

$$\Delta C_{25}^{\text{sat}} \approx 2\lambda_{\text{b}}^{\text{dry}} g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \sin \zeta \cos \zeta } \right)$$

(A-35)

$$\Delta C_{35}^{\text{sat}} \approx 2\mu_{\text{b}} \left[ \begin{gathered} 2g_{\text{b}}^{\text{dry}} \Delta \left( {\delta_{\text{N}}^{\text{dry}} \cos^{3} \zeta \sin \zeta } \right) - 2\Delta \left( {\delta_{\text{T}} \cos^{3} \zeta \sin \zeta } \right) \hfill \\ + \chi \Delta \left( {\delta_{\text{N}}^{\text{dry}} \sin \zeta \cos \zeta } \right) + \Delta \left( {\delta_{\text{T}} \sin \zeta \cos \zeta } \right) \hfill \\ \end{gathered} \right]$$

(A-36)

$$\Delta C_{44}^{\text{sat}} \approx - \mu_{\text{b}} \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right) + \Delta \mu_{\text{b}}$$

(A-37)

$$\Delta C_{46}^{\text{sat}} \approx \mu_{\text{b}} \Delta \left( {\delta_{\text{T}} \sin \zeta \cos \zeta } \right)$$

(A-38)

$$\Delta C_{66}^{\text{sat}} \approx \mu_{\text{b}} \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right) + \Delta \mu_{\text{b}} - \mu_{\text{b}} \Delta \delta_{\text{T}}$$

(A-39)

Appendix B

Assuming that an incident P-wave produces a reflected P-wave at the reflected interface as in Fig. 1 and choosing the average P-wave velocity $\overline{{\alpha }_{\text{b}}^{\text{sat}}}$ and average mass density $\overline{{\rho }_{\text{b}}^{\text{sat}}}$ of the saturated background media as those of the reference medium, the polarization and slowness vectors of the incident and reflected P-waves are

$${\mathbf{t}} = \left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right],\;{\mathbf{t^{\prime}}} = \left[ { - \sin \theta \cos \varphi , - \sin \theta \sin \varphi ,\cos \theta } \right],$$

$$s = {{\left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right]} \mathord{\left/ {\vphantom {{\left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right]} {\overline{{\alpha_{\text{b}}^{\text{sat}} }} }}} \right. \kern-\nulldelimiterspace} {\overline{{\alpha_{\text{b}}^{\text{sat}} }} }},\;s^{\prime} = {{\left[ { - \sin \theta \cos \varphi , - \sin \theta \sin \varphi ,\cos \theta } \right]} \mathord{\left/ {\vphantom {{\left[ { - \sin \theta \cos \varphi , - \sin \theta \sin \varphi ,\cos \theta } \right]} {\overline{{\alpha_{\text{b}}^{\text{sat}} }} }}} \right. \kern-\nulldelimiterspace} {\overline{{\alpha_{\text{b}}^{\text{sat}} }} }},$$

(B-1)

respectively. Thus, the expressions of ${\eta }_{ij}$ are deduced as

$$\eta_{11} = {{\sin^{4} \theta \cos^{4} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \cos^{4} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{12} = {{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{13} = {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{14} = {{2\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{2\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{15} = {{2\sin^{3} \theta \cos \theta \cos^{3} \varphi } \mathord{\left/ {\vphantom {{2\sin^{3} \theta \cos \theta \cos^{3} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{16} = {{2\sin^{4} \theta \sin \varphi \cos^{3} \varphi } \mathord{\left/ {\vphantom {{2\sin^{4} \theta \sin \varphi \cos^{3} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{22} = {{\sin^{4} \theta \sin^{4} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{4} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{23} = {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{24} = {{2\sin^{3} \theta \cos \theta \sin^{3} \varphi } \mathord{\left/ {\vphantom {{2\sin^{3} \theta \cos \theta \sin^{3} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{25} = {{2\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } \mathord{\left/ {\vphantom {{2\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{26} = {{2\sin^{4} \theta \sin^{3} \varphi \cos \varphi } \mathord{\left/ {\vphantom {{2\sin^{4} \theta \sin^{3} \varphi \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{33} = {{\cos^{4} \theta } \mathord{\left/ {\vphantom {{\cos^{4} \theta } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{34} = {{2\sin \theta \cos^{3} \theta \sin \varphi } \mathord{\left/ {\vphantom {{2\sin \theta \cos^{3} \theta \sin \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{35} = {{2\sin \theta \cos^{3} \theta \cos \varphi } \mathord{\left/ {\vphantom {{2\sin \theta \cos^{3} \theta \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{36} = {{2\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } \mathord{\left/ {\vphantom {{2\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{44} = {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{45} = {{ - 4\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{46} = {{ - 4\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{55} = {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},$$

$$\eta_{56} = {{ - 4\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }},\;\eta_{66} = {{4\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{4\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\overline{{\alpha_{\text{b}}^{\text{sat}} }} } \right)^{2} }}.$$

(B-2)

In addition, $\eta_{21} = \eta_{12}$, $\eta_{31} = \eta_{13}$, $\eta_{32} = \eta_{23}$, $\eta_{41} = - \eta_{14}$, $\eta_{42} = - \eta_{24}$, $\eta_{43} = - \eta_{34}$, $\eta_{51} = - \eta_{15}$, $\eta_{52} = - \eta_{25}$, $\eta_{53} = - \eta_{35}$, $\eta_{54} = \eta_{45}$, $\eta_{61} = \eta_{16}$, $\eta_{62} = \eta_{26}$, $\eta_{63} = \eta_{36}$, $\eta_{64} = - \eta_{46}$, $\eta_{65} = - \eta_{56}$. Substituting Formulas (A-27)–(A-39) and (B-1) and (B-2) into Formula (11) and then executing series of mathematical operations, such as the trigonometric transformation, combining like terms and so forth, we can obtain the reflection coefficient of the saturated TTI medium,

$$R_{\text{TTI}} \left( {\theta ,\varphi } \right) = R_{\text{iso}} \left( \theta \right) + A_{4} \Delta \left( {\delta_{\text{N}} \cos^{4} \zeta } \right) + A_{2} \Delta \left( {\delta_{\text{N}} \cos^{2} \zeta } \right) + A_{0} \Delta \delta_{\text{N}} + B_{4} \Delta \left( {\delta_{\text{T}} \cos^{4} \zeta } \right) + B_{2} \Delta \left( {\delta_{\text{T}} \cos^{2} \zeta } \right) + B_{0} \Delta \delta_{\text{T}} ,$$

(B-3)

where

$$R_{\text{iso}} \left( \theta \right) = \frac{{\sec^{2} \theta }}{4}\frac{{g_{\text{b}}^{\text{sat}} }}{{g_{\text{b}}^{\text{dry}} }}\frac{{\Delta M_{\text{b}}^{\text{dry}} }}{{M_{\text{b}}^{\text{dry}} }} - 2g_{\text{b}}^{\text{sat}} \sin^{2} \theta \frac{{\Delta \mu_{\text{b}} }}{{\mu_{\text{b}} }} + \left( {\frac{1}{2} - \frac{{\sec^{2} \theta }}{4}} \right)\frac{{\Delta \rho^{\text{sat}} }}{{\rho_{\text{b}}^{\text{sat}} }} + \frac{{\sec^{2} \theta }}{4}\left( {1 - \frac{{g_{\text{b}}^{\text{sat}} }}{{g_{\text{b}}^{\text{dry}} }}} \right)\left( {\frac{{\Delta G\left( {\phi_{\text{p}} } \right)}}{{G\left( {\phi_{\text{p}} } \right)}} + \frac{{\Delta K_{\text{f}} }}{{K_{\text{f}} }}} \right),$$

(B-4)

$$A_{4} \left( {\theta ,\varphi } \right) = - g_{\text{b}}^{\text{sat}} g_{\text{b}}^{\text{dry}} \left( {\sin^{2} \theta \tan^{2} \theta \cos^{4} \varphi + 2\sin^{2} \theta \cos^{2} \varphi + \cos^{2} \theta } \right),$$

(B-5)

$$A_{2} \left( {\theta ,\varphi } \right) = 2g_{\text{b}}^{\text{sat}} g_{\text{b}}^{\text{dry}} \sin^{2} \theta \cos^{2} \varphi \left( {1 + \tan^{2} \theta \cos^{2} \varphi } \right) - g_{\text{b}}^{\text{sat}} \chi_{\text{b}}^{\text{dry}} \left( {1 - \tan^{2} \theta \cos^{2} \varphi } \right),$$

(B-6)

$$A_{0} \left( {\theta ,\varphi } \right) = g_{\text{b}}^{\text{sat}} \tan^{2} \theta \left[ { - g_{\text{b}}^{\text{dry}} \sin^{2} \theta \cos^{4} \varphi - \chi_{\text{b}}^{\text{dry}} \cos^{2} \varphi - {{\left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {\chi_{\text{b}}^{\text{dry}} } \right)^{2} } {\left( {4g_{\text{b}}^{\text{dry}} \sin^{2} \theta } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {4g_{\text{b}}^{\text{dry}} \sin^{2} \theta } \right)}}} \right],$$

(B-7)

$$B_{4} \left( {\theta ,\varphi } \right) = g_{\text{b}}^{\text{sat}} \left( {\sin^{2} \theta \tan^{2} \theta \cos^{4} \varphi + 2\sin^{2} \theta \cos^{2} \varphi + \cos^{2} \theta } \right),$$

(B-8)

$$B_{2} \left( {\theta ,\varphi } \right) = g_{\text{b}}^{\text{sat}} \left[ { - 2\sin^{2} \theta \tan^{2} \theta \cos^{4} \varphi + \left( {\tan^{2} \theta - 4\sin^{2} \theta } \right)\cos^{2} \varphi + 1 - 2\cos^{2} \theta } \right],$$

(B-9)

$$B_{0} \left( {\theta ,\varphi } \right) = g_{\text{b}}^{\text{sat}} \left[ {\sin^{2} \theta \tan^{2} \theta \cos^{4} \varphi + \sin^{2} \theta \left( {1 - \tan^{2} \theta } \right)\cos^{2} \varphi } \right].$$

(B-10)

Note that ${\rho }^{\text{sat}}$ and ${\rho }_{\text{b}}^{\text{sat}}$ are easily confused and they represent the mass densities of the saturated TTI medium and the saturated background medium, respectively. The number and correlations of the parameter reflectivities in formula (B-4) ($\frac{\Delta {M}_{\text{sat}}^{\text{dry}}}{{M}_{\text{sat}}^{\text{dry}}}, \frac{\Delta {\mu }_{\text{b}}}{{\mu }_{\text{b}}}, \frac{\Delta {\rho }^{\text{sat}}}{{\rho }_{\text{b}}^{\text{sat}}}, \frac{\Delta G\left({\phi }_{\text{p}}\right){K}_{\text{f}}}{G\left({\phi }_{\text{p}}\right){K}_{\text{f}}}$) will decrease the solvability of the parameters from wide azimuth seismic data, therefore we will further simplify formula (B-4) in order to obtain the stable inversion results.

In formula (B-4), $M_{{\text{b}}}^{{{\text{dry}}}} = \mu _{{\text{b}}} /{\text{g}}_{{\text{b}}}^{{{\text{dry}}}}$ so that

$${{\Delta \mu_{\text{b}} } \mathord{\left/ {\vphantom {{\Delta \mu_{\text{b}} } {\mu_{\text{b}} }}} \right. \kern-\nulldelimiterspace} {\mu_{\text{b}} }} = {{\Delta M_{\text{b}}^{\text{dry}} } \mathord{\left/ {\vphantom {{\Delta M_{\text{b}}^{\text{dry}} } {M_{\text{b}}^{\text{dry}} }}} \right. \kern-\nulldelimiterspace} {M_{\text{b}}^{\text{dry}} }}$$

(B-11)

due to $g_{{\text{b}}}^{{{\text{dry}}}}$ being considered a constant (Pan and Zhang 2018; Pan et al. 2018a; Russell et al. 2011). The relationship between the mass density and P-wave velocity of the isotropic saturated background medium can be assumed to obey the Gardner equation, that is, ${\rho }_{\text{b}}^{\text{sat}}=k{\left({\alpha }_{\text{b}}^{\text{sat}}\right)}^{q}$, where k and q are constants depending on the lithology, porosity, depth and so forth. With the assumption of the weak contrasts of properties above and below the interface, we can derive

$$\frac{{\Delta \rho_{\text{b}}^{\text{sat}} }}{{\rho_{\text{b}}^{\text{sat}} }} \approx \Delta \ln \rho_{\text{b}}^{\text{sat}} = \Delta \ln \left( {\alpha_{\text{b}}^{\text{sat}} } \right)^{q} = \Delta \ln \left( {\frac{{\left( {\beta_{\text{b}}^{\text{sat}} } \right)^{2} }}{{g_{\text{b}}^{\text{sat}} }}} \right)^{\frac{q}{2}} = \frac{q}{2}\Delta \ln \frac{{\mu_{\text{b}} }}{{\rho_{\text{b}}^{\text{sat}} }} \approx \frac{q}{2}\left( {\frac{{\Delta \mu_{\text{b}} }}{{\mu_{\text{b}} }} - \frac{{\Delta \rho_{\text{b}}^{\text{sat}} }}{{\rho_{\text{b}}^{\text{sat}} }}} \right).$$

(B-12)

Adopting the hypotheses of $\phi \approx {\phi }_{\text{p}}$ and the small mass density of the gas–liquid mixture, the Voigt average of the mass density of the saturated TTI medium can be deduced,

$$\rho^{\text{sat}} = \rho_{\text{b}}^{\text{sat}} \left( {1 - \phi + \phi_{\text{p}} } \right) + \rho_{\text{f}} \left( {\phi - \phi_{\text{p}} } \right) \approx \rho_{\text{b}}^{\text{sat}} ,$$

(B-13)

where ${\rho }_{\text{f}}$ is the mass density of the fluid mixture in the background pores and fractures. Integrating formulas (B-12) and (B-13), we can obtain

$$\frac{{\Delta \rho^{\text{sat}} }}{{\rho_{\text{b}}^{\text{sat}} }} \approx \frac{{\Delta \rho_{\text{b}}^{\text{sat}} }}{{\rho_{\text{b}}^{\text{sat}} }} \approx \frac{1}{r}\frac{{\Delta \mu_{\text{b}} }}{{\mu_{\text{b}} }}.$$

(B-14)

For this case, $r=\left(q+2\right)/q$ and can be estimated with

$$r = {{\Xi_{{R_{{\mu_{\text{b}} }} }}^{2} } \mathord{\left/ {\vphantom {{\Xi_{{R_{{\mu_{\text{b}} }} }}^{2} } {\Xi_{{R_{{\mu_{\text{b}} }} R_{\rho } }} }}} \right. \kern-\nulldelimiterspace} {\Xi_{{R_{{\mu_{\text{b}} }} R_{\rho } }} }},$$

(B-15)

where $\Xi_{{R_{{\mu_{\text{b}} }} R_{\rho } }}$ and $\Xi_{{R_{{\mu_{\text{b}} }} }}^{2}$ represent the covariance between $\frac{\Delta {\mu }_{\text{b}}}{{\mu }_{\text{b}}}$ and $\frac{\Delta {\rho }^{\text{sat}}}{{\rho }_{\text{b}}^{\text{sat}}}$ and the variance of$\frac{\Delta {\mu }_{\text{b}}}{{\mu }_{\text{b}}}$, respectively. $\Xi_{{R_{{\mu_{\text{b}} }} R_{\rho } }}$ and $\Xi_{{R_{{\mu_{\text{b}} }} }}^{2}$ can be calculated by well-log interpretations (volumes of minerals, water saturation, porosity, and aspect ratio, dip, strike, and density of fractures) and the seismic rock physical model (Chen et al. 2015). Formula (B-14) is consistent with the conclusions from Zhang et al. (2012) and Russell and Hedlin (2019). By using relationship${\beta }_{0}\approx \varpi {\phi }_{\text{p}}$, the reflectivity of $G\left({\phi }_{\text{p}}\right)$ can be simplified as

$$\frac{{\Delta G\left( {\phi_{\text{p}} } \right)}}{{G\left( {\phi_{\text{p}} } \right)}} = \frac{{\Delta \left( {\varpi^{2} \phi_{\text{p}} } \right)}}{{\varpi^{2} \phi_{\text{p}} }} = \frac{{\Delta \phi_{\text{p}} }}{{\phi_{\text{p}} }}.$$

(B-16)

With substitutions of formulas (B-11), (B-14), and (B-16) into formula (B-4), formula (B-4) can be rewritten as

$$R_{\text{iso}} \left( \theta \right) = \left[ {\frac{{\sec^{2} \theta }}{4}\frac{{g_{\text{b}}^{\text{sat}} }}{{g_{\text{b}}^{\text{dry}} }} - 2g_{\text{b}}^{\text{sat}} \sin^{2} \theta + \frac{1}{r}\left( {\frac{1}{2} - \frac{{\sec^{2} \theta }}{4}} \right)} \right]\frac{{\Delta \mu_{\text{b}} }}{{\mu_{\text{b}} }} + \frac{{\sec^{2} \theta }}{4}\left( {1 - \frac{{g_{\text{b}}^{\text{sat}} }}{{g_{\text{b}}^{\text{dry}} }}} \right)\left( {\frac{{\Delta \phi_{\text{p}} }}{{\phi_{\text{p}} }} + \frac{{\Delta K_{\text{f}} }}{{K_{\text{f}} }}} \right).$$

(B-17)