Characteristics of Azimuthal Seismic Reflection Response in Horizontal Transversely Isotropic Media under Horizontal In Situ Stress

Abstract

Anisotropy is ubiquitous in the Earth's crust, which causes the elastic characteristics of seismic waves to change with direction. The study of seismic wave anisotropy is of great significance to seismic exploration, prediction and geodynamics. As one of the sources of seismic anisotropy, in situ stress belongs to secondary anisotropy as common as the intrinsic and fracture-induced anisotropy, but it is often ignored among the sources of seismic anisotropy. Therefore, we focus on the study of seismic anisotropy under the influence of in situ stress using the nonlinear acoustoelasticity theory. Based on a horizontal transversely isotropic (HTI) model and the linear slip theory, the characteristics of azimuthal seismic reflection response in anisotropic media under horizontal in situ stress are discussed in this paper. Firstly, by using the quasi-linear relationship between stress and Tsvankin’s anisotropic parameters and the transformation relationship between anisotropic and fracture parameters in HTI medium, the elastic stiffness matrix of an HTI medium with the effect of horizontal in situ stress is established. Secondly, the reflection coefficient of PP-wave seismic data for a planar weak-contrast interface separating two weak-anisotropy and small-stress HTI half-spaces is derived using both the seismic scattering theory and the stiffness matrix under horizontal in situ stress, building the quantitative relationship between azimuthal seismic reflection characteristics and the model parameters, such as the background elastic parameters, the fracture parameters and the horizontal-stress-induced anisotropic parameters. Finally, the variation rules of azimuthal seismic reflection response characteristics of four elastic interfaces under different in situ stress conditions are analyzed. The results demonstrate that the seismic inversion for fracture parameters and horizontal-stress-induced anisotropic parameters is more favorable under the condition of large incident angle. In addition, the effect of horizontal in situ stress on the reflection coefficient depends on the second- and third-order elastic properties of the rock itself. Also, the established seismic PP-wave reflection coefficient equation has provided an alternative approach to calculate the magnitude of horizontal in situ stress.

Article Highlights

• A novel linearized PP-wave reflection coefficient is presented for HTI media with the effect of horizontal in situ stress

• The response law of azimuthal seismic reflection characteristics induced by horizontal in situ stress is demonstrated

• A simple inversion method is provided to calculate the magnitude of horizontal in situ stress

Appendices

Appendix A: Acoustoelasticity Theory Model in Natural State

As illustrated in Fig. 1, following Tian and Wang (2006) and Song et al. (2020), the particle displacement of static deformation from the natural state to the initial state is given by

$${\mathbf{u}}^{i} \left( {{\varvec{\upxi}}} \right) = {\mathbf{X}} - {{\varvec{\upxi}}}.$$

(57)

The particle displacement of dynamic deformation from the natural state to the final state is given by

$${\mathbf{u}}^{f} \left( {{{\varvec{\upxi}}},t} \right) = {\mathbf{x}} - {{\varvec{\upxi}}},$$

(58)

where the particle displacement of static deformation is far more than that of dynamic deformation, i.e., ${\mathbf{u}}^{i} \left( {{\varvec{\upxi}}} \right) \gg {\mathbf{u}}^{f} \left( {{{\varvec{\upxi}}},t} \right)$. Then, the displacement caused by the seismic wave can be obtained as

$${\mathbf{u}}\left( {{{\varvec{\upxi}}},t} \right) = {\mathbf{x}} - {\mathbf{X}} = {\mathbf{u}}^{f} - {\mathbf{u}}^{i} .$$

(59)

Now, the Lagrangian strains $E$ of the initial and final states, respectively, can be expressed as

$$E_{\alpha \beta }^{i} = \frac{1}{2}\left( {\frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\beta } }} + \frac{{\partial u_{\beta }^{i} }}{{\partial \xi_{\alpha } }} + \frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\alpha } }}\frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\beta } }}} \right) = e_{\alpha \beta }^{i} + \frac{1}{2}\frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\alpha } }}\frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\beta } }},$$

(60)

$$E_{\alpha \beta }^{f} = \frac{1}{2}\left( {\frac{{\partial u_{\alpha }^{f} }}{{\partial \xi_{\beta } }} + \frac{{\partial u_{\beta }^{f} }}{{\partial \xi_{\alpha } }} + \frac{{\partial u_{\gamma }^{f} }}{{\partial \xi_{\alpha } }}\frac{{\partial u_{\gamma }^{f} }}{{\partial \xi_{\beta } }}} \right) = e_{\alpha \beta }^{f} + \frac{1}{2}\frac{{\partial u_{\gamma }^{f} }}{{\partial \xi_{\alpha } }}\frac{{\partial u_{\gamma }^{f} }}{{\partial \xi_{\beta } }}.$$

(61)

where the prestrain is assumed to be a small deformation, and the strains can be expressed as

$$e_{\alpha \beta }^{i} = \frac{1}{2}\left( {\frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\beta } }} + \frac{{\partial u_{\beta }^{i} }}{{\partial \xi_{\alpha } }}} \right).$$

(62)

$$e_{\alpha \beta }^{f} = \frac{1}{2}\left( {\frac{{\partial u_{\alpha }^{f} }}{{\partial \xi_{\beta } }} + \frac{{\partial u_{\beta }^{f} }}{{\partial \xi_{\alpha } }}} \right).$$

(63)

Then, the strain caused by seismic-wave small disturbance from the initial state to the final state is approximate as

$$E_{\alpha \beta }^{{}} = E_{\alpha \beta }^{f} - E_{\alpha \beta }^{i} \approx e_{\alpha \beta }^{{}} + \frac{1}{2}\frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\alpha } }}\frac{{\partial u_{\gamma }^{{}} }}{{\partial \xi_{\beta } }} + \frac{1}{2}\frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\beta } }}\frac{{\partial u_{\gamma }^{{}} }}{{\partial \xi_{\alpha } }}.$$

(64)

Assuming that the rock is elastic, and the relation between Piola–Kirchhoff stress tensor $T_{ij}$ and strain energy function $W$ can be written as

$$T_{ij} = \frac{\partial W}{{\partial E_{ij} }},$$

(65)

where the strain energy function can be approximated as

$$W = \frac{1}{2!}C_{ijkl} E_{ij} E_{kl} + \frac{1}{3!}C_{ijklmn} E_{ij} E_{kl} E_{mn} + \cdots ,$$

(66)

where $C_{ijkl}$ and $C_{ijklmn}$ represent the second- and third-order elastic stiffness components, respectively. Generally, the independent second- and third-order elastic constants are 21 and 56, respectively, when taking the symmetry into account (Murnaghan, 1951). For the isotropic rock, the independent second- and third-order elastic constants are 2 and 3, respectively.

According to Eqs. 8 and 9, we neglect the higher-order terms, and get

$$T_{\alpha \beta }^{i} = C_{\alpha \beta \gamma \delta } E_{\gamma \delta }^{i} + \frac{1}{2}C_{\alpha \beta \gamma \delta \varepsilon \eta } E_{\gamma \delta }^{i} E_{\varepsilon \eta }^{i} ,$$

(67)

$$T_{\alpha \beta }^{f} = C_{\alpha \beta \gamma \delta } E_{\gamma \delta }^{f} + \frac{1}{2}C_{\alpha \beta \gamma \delta \varepsilon \eta } E_{\gamma \delta }^{f} E_{\varepsilon \eta }^{f} .$$

(68)

Subtracting Eq. 68 from Eq. 67 and ignoring the higher-order term, we can obtain the increment of stress as

$$T_{\alpha \beta }^{{}} = C_{\alpha \beta \gamma \delta } E_{\gamma \delta }^{{}} + C_{\alpha \beta \gamma \delta \varepsilon \eta } e_{\gamma \delta }^{i} e_{\varepsilon \eta }^{{}} .$$

(69)

Substituting Eq. 64 into Eq. 69, we can get

$$T_{\alpha \beta }^{{}} = C_{\alpha \beta \gamma \delta } \left( {\delta_{\kappa \gamma } + \frac{{\partial u_{\kappa }^{i} }}{{\partial \xi_{\gamma } }}} \right)\frac{{\partial u_{\kappa } }}{{\partial \xi_{\delta } }} + C_{\alpha \beta \gamma \delta \varepsilon \eta } \frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\delta } }}\frac{{\partial u_{\varepsilon } }}{{\partial \xi_{\eta } }},$$

(70)

where $\delta_{\kappa \gamma }$ represents a Kronecker delta. In addition, the motion equation of rocks from natural state to initial state under prestressed condition is

$$\frac{\partial }{{\partial \xi_{\beta } }}\left[ {T_{\beta \gamma }^{i} \left( {\frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\gamma } }} + \delta_{\alpha \gamma } } \right)} \right] = 0.$$

(71)

Also, the motion equation of rocks from the natural state to the final state under the micro-disturbance and stress of seismic wave is

$$\frac{\partial }{{\partial \xi_{\beta } }}\left[ {T_{\beta \gamma }^{i} \left( {\frac{{\partial u_{\alpha }^{f} }}{{\partial \xi_{\gamma } }} + \delta_{\alpha \gamma } } \right)} \right] = \rho_{b} \frac{{\partial^{2} u_{\alpha }^{f} }}{{\partial t^{2} }}.$$

(72)

Subtracting Eq. 72 from Eq. 71, we can get the motion equation of rocks under the micro-disturbance of seismic wave as

$$\frac{\partial }{{\partial \xi_{\beta } }}\left[ {T_{\beta \gamma }^{{}} \left( {\frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\gamma } }} + \delta_{\alpha \gamma } } \right) + T_{\beta \gamma }^{i} \frac{{\partial u_{\alpha }^{{}} }}{{\partial \xi_{\gamma } }}} \right] = \rho_{b} \frac{{\partial^{2} u_{\alpha }^{{}} }}{{\partial t^{2} }}.$$

(73)

Substituting Eq. 70 into Eq. 73, we can get

$$\frac{\partial }{{\partial \xi_{\beta } }}\left[ {\Gamma_{\alpha \beta \gamma \delta } \frac{{\partial u_{\gamma } }}{{\partial \xi_{\delta } }} + T_{\beta \gamma }^{i} \frac{{\partial u_{\alpha }^{{}} }}{{\partial \xi_{\gamma } }}} \right] = \rho_{b} \frac{{\partial^{2} u_{\alpha }^{{}} }}{{\partial t^{2} }},$$

(74)

where $\Gamma_{\alpha \beta \gamma \delta } = C_{\alpha \beta \gamma \delta } + C_{\alpha \beta \kappa \delta } \frac{{\partial u_{\gamma }^{i} }}{{\partial \xi_{\kappa } }} + C_{\kappa \beta \gamma \delta } \frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\kappa } }} + C_{\alpha \beta \gamma \delta \varepsilon \eta } e_{\varepsilon \eta }^{i}$.

Finally, the acoustoelasticity equation in the natural coordinate system can be expressed as

$$\Psi_{\alpha \beta \gamma \delta } \frac{{\partial^{2} u_{\gamma } }}{{\partial x_{\beta } \partial x_{\delta } }} = \rho_{b} \frac{{\partial^{2} u_{\alpha }^{{}} }}{{\partial t^{2} }},$$

(75)

where $\Psi_{\alpha \beta \gamma \delta } = T_{\beta \delta }^{i} \delta_{\alpha \gamma } + \Gamma_{\alpha \beta \gamma \delta }$.

Appendix B: Acoustoelasticity Theory Model in Initial State

The motion equation of the rock in the final state after seismic-wave disturbance superimposed on the static deformation can be expressed in the initial coordinate system as

$$\frac{\partial }{{\partial X_{L} }}\left( {T_{KL}^{f} \frac{{\partial X_{j}^{{}} }}{{\partial X_{K} }}} \right) = \rho^{i} \frac{{\partial^{2} x_{j}^{{}} }}{{\partial t^{2} }},$$

(76)

where $\rho^{i}$ denotes the density term in the initial state, $T_{KL}^{f}$ denotes the Piola–Kirchhoff stress tensor, and can be written as

$$T_{KL}^{f} = T_{KL}^{{}} - T_{KL}^{i} ,$$

(77)

where $T_{KL}^{i}$ denotes the Cauchy stress tensor in the initial state. Similarly, the Piola–Kirchhoff stress tensor and the Cauchy stress tensor in the natural coordinate system can be expressed as Eqs. 67 and 68.

In Eq. 76, the relationship between the Piola–Kirchhoff stress tensor $T_{KL}^{f}$ and the Cauchy stress tensor $T_{ij}^{f}$ in the final state can be expressed as

$$T_{ij}^{f} = \frac{1}{{\left| {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial X}}} \right. \kern-\nulldelimiterspace} {\partial X}}} \right|}}\frac{{\partial x_{i} }}{{\partial X_{K} }}\frac{{\partial x_{j} }}{{\partial X_{L} }}T_{KL}^{f} = \frac{1}{{\left| {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial X}}} \right. \kern-\nulldelimiterspace} {\partial X}}} \right|}}\left( {\delta_{iK} + \frac{{\partial u_{i} }}{{\partial X_{K} }}} \right)\left( {\delta_{jL} + \frac{{\partial u_{j} }}{{\partial X_{L} }}} \right)T_{KL}^{f} .$$

(78)

In the initial coordinate system, the static equilibrium equation of the rock in the initial state under the action of static stress $T_{IJ}^{i}$ can be expressed as

$$\frac{{\partial T_{IJ}^{i} }}{{\partial X_{J} }} = 0.$$

(79)

Subtracting Eq. 76 from Eq. 79 and combining Eq. 78, we can get the motion equation of rocks under the micro-disturbance of seismic wave in the initial coordinate system as

$$T_{JK}^{i} \frac{{\partial^{2} u_{I} }}{{\partial X_{J} \partial X_{K} }} + \frac{{\partial T_{IJ} }}{{\partial X_{J} }} = \rho^{i} \frac{{\partial^{2} u_{I}^{{}} }}{{\partial t^{2} }}.$$

(80)

In Eq. 80, the stress increment $T_{IJ}$ in the initial coordinate system can be expressed by the stress increment $T_{\alpha \beta }$ in the natural coordinate system, that is

$$T_{IJ} = \frac{{\rho^{i} }}{{\rho_{b} }}\frac{{\partial X_{I} }}{{\partial \xi_{\alpha } }}\frac{{\partial X_{J} }}{{\partial \xi_{\beta } }}T_{\alpha \beta } .$$

(81)

Substituting Eq. 78 into Eq. 81, the stress increment $T_{IJ}$ can be further expressed as

$$T_{IJ} = \delta_{I\alpha } \delta_{J\beta } \left( {T_{\alpha \beta } + T_{\alpha \gamma } \frac{{\partial u_{\beta }^{i} }}{{\partial \xi_{\gamma } }} + T_{\beta \gamma } \frac{{\partial u_{\alpha }^{i} }}{{\partial \xi_{\gamma } }}} \right) = C_{IJKL} \frac{{\partial u_{K} }}{{\partial X_{L} }},$$

(82)

where

$$C_{IJKL} = c_{IJKL} + c_{IJKLMN} e_{MN}^{i} + c_{MJKL} \frac{{\partial u_{I}^{i} }}{{\partial X_{M} }} + c_{IMKL} \frac{{\partial u_{J}^{i} }}{{\partial X_{M} }} + c_{IJML} \frac{{\partial u_{K}^{i} }}{{\partial X_{M} }} + c_{IJKM} \frac{{\partial u_{L}^{i} }}{{\partial X_{M} }},$$

(83)

$$c_{IJKL} = \delta_{I\alpha } \delta_{J\beta } \delta_{K\gamma } \delta_{L\delta } C_{\alpha \beta \gamma \delta } ,$$

(84)

$$c_{{IJKLMN}} = \delta _{{I\alpha }} \delta _{{J\beta }} \delta _{{K\gamma }} \delta _{{L\delta }} \delta _{{M\varepsilon }} \delta _{{N\eta }} C_{{\alpha \beta \gamma \delta \varepsilon \eta }} .$$

(85)

The difference between Eqs. 11 and 83 is the terms containing $T_{IJ}$, and $c_{IJKL} e_{NN}^{i}$. This is because that $\left| {c_{IJKLMN} } \right| \gg \left| {c_{IJKL} } \right| \gg \left| {T_{IJ} } \right|$ and $\left| {c_{IJKL} } \right| \gg \left| {c_{IJKL} e_{NN}^{i} } \right|$ for rocks. The exact effective elastic stiffness tensor $C_{IJKL}$ in initial coordinate system can be found in Sinha (1982), Pao et al. (1984) and Liu et al. (2009).

Then, substituting Eq. 82 into Eq. 80, we can get the acoustoelasticity equation in initial coordinate system as

$$\frac{\partial }{{\partial X_{J} }}\left[ {\left( {\delta_{IK} T_{JL}^{i} + C_{IJKL} } \right)\frac{{\partial u_{K} }}{{\partial X_{L} }}} \right] = \rho^{i} \frac{{\partial^{2} u_{I}^{{}} }}{{\partial t^{2} }}.$$

(86)

Finally, the acoustoelasticity equation in the initial coordinate system can be expressed as

$$\Psi_{IJKL} \frac{{\partial^{2} u_{K} }}{{\partial X_{J} \partial X_{L} }} = \rho^{i} \frac{{\partial^{2} u_{I}^{{}} }}{{\partial t^{2} }},$$

(87)

where $\Psi_{IJKL} = \delta_{IK} T_{JL}^{i} + C_{IJKL}$.

Appendix C: Derivation of PP-Wave Reflection Coefficient in a Stressed HTI Medium

For the incident and reflected P waves, the polarization and slowness vectors are given by (Shaw and Sens 2006)

$$t = \left[ {\sin \theta \cos \phi ,\sin \theta \sin \phi ,cos\theta } \right],$$

(88)

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

(89)

$$p = {1 \mathord{\left/ {\vphantom {1 {\alpha_{b}^{s} }}} \right. \kern-\nulldelimiterspace} {\alpha_{b}^{s} }}\left[ {\sin \theta \cos \phi ,\sin \theta \sin \phi ,cos\theta } \right],$$

(90)

$$p^{\prime} = {1 \mathord{\left/ {\vphantom {1 {\alpha_{b}^{s} }}} \right. \kern-\nulldelimiterspace} {\alpha_{b}^{s} }}\left[ { - \sin \theta \cos \phi , - \sin \theta \sin \phi ,cos\theta } \right],$$

(91)

where $\alpha_{b}^{s}$ represents the P-wave velocity of the background stressed medium, and the expressions for $\xi$ and $\eta_{IJ}$ are also given by (Shaw and Sen 2006)

$$\xi = \cos^{2} \theta - \sin^{2} \theta = cos2\theta ,$$

(92)

and

$$\eta_{11} = {{\sin^{4} \theta \cos^{4} \theta } \mathord{\left/ {\vphantom {{\sin^{4} \theta \cos^{4} \theta } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{12} = {{\sin^{4} \theta \sin^{2} \phi \cos^{2} \theta } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{2} \phi \cos^{2} \theta } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{13} = {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } {\left( {\alpha_{b}^{s} } \right)^{2} ,}}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} ,}}$$

$$\eta_{22} = {{\sin^{4} \theta \sin^{4} \phi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{4} \phi } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{23} = {{\sin^{2} \theta \sin^{2} \phi \cos^{2} \theta } \mathord{\left/ {\vphantom {{\sin^{2} \theta \sin^{2} \phi \cos^{2} \theta } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{33} = {{\cos^{4} \theta } \mathord{\left/ {\vphantom {{\cos^{4} \theta } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},$$

$$\eta_{44} = - 4{{\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{55} = - 4{\kern 1pt} {{\sin^{2} \theta \cos^{2} \phi \cos^{2} \theta } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \phi \cos^{2} \theta } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},$$

$$\eta_{66} = 4{{\sin^{4} \theta \cos^{2} \phi \sin^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \cos^{2} \phi \sin^{2} \phi } {\left( {\alpha_{b}^{s} } \right)^{2} }}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{b}^{s} } \right)^{2} }},\eta_{21} = \eta_{12} ,\eta_{31} = \eta_{13} ,\eta_{32} = \eta_{23} .$$

(93)

Combining Eqs. 92 and 93 and Eq. 37, and substituting into Eq. 36, we can get

$$\begin{gathered} R_{PP} (\theta ,\phi ) = \frac{{\Delta \rho_{b}^{s} }}{{4\rho_{b}^{s} }}\xi \sec^{2} \theta + \sum\limits_{m = 1}^{6} {\sum\limits_{n = 1}^{6} {\frac{{\Delta C_{mn} }}{{4\rho_{b}^{s} }}} } \hfill \\ = \frac{{\sec^{2} \theta }}{{4\rho_{b}^{s} }}\left\{ \begin{gathered} \hfill \Delta \rho_{b}^{s} \cos 2\theta + \frac{{\sin^{4} \theta \cos^{4} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta M_{b}^{s} - M_{b}^{s} \Delta \delta_{N}^{{}} + \frac{{M_{b}^{s} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill + \frac{{2\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta \lambda_{b}^{s} - \lambda_{b}^{s} \Delta \delta_{N}^{{}} + \frac{{\lambda_{b}^{s} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill + \frac{{2\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta \lambda_{b}^{s} - \lambda_{b}^{s} \Delta \delta_{N}^{{}} + \frac{{\lambda_{b}^{s} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill + \frac{{\sin^{4} \theta sin^{4} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta M_{b}^{s} - M_{b}^{s} \left( {\chi_{b}^{s} } \right)^{2} \Delta \delta_{N}^{{}} + \frac{{M_{b}^{s} \left( {\chi_{b}^{s} } \right)^{2} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill + \frac{{2\sin^{2} \theta \cos^{2} \theta sin^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta M_{b}^{s} - M_{b}^{s} \left( {\chi_{b}^{s} } \right)^{2} \Delta \delta_{N}^{{}} + \frac{{M_{b}^{s} \left( {\chi_{b}^{s} } \right)^{2} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill + \frac{{\cos^{4} \theta }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta \lambda_{b}^{s} - \lambda_{b}^{s} \chi_{b}^{s} \Delta \delta_{N}^{{}} + \frac{{\lambda_{b}^{s} \chi_{b}^{s} }}{{4g_{b} (1 - g_{b} )}}\Delta \Gamma_{p} } \right] \\ \hfill \_\frac{{4\sin^{2} \theta \cos^{2} \theta sin^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\Delta \mu_{b} \\ \hfill \_\frac{{4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta \mu_{b}^{s} - \mu_{b}^{s} \Delta \delta_{T}^{{}} + \mu_{b}^{s} \Delta \Gamma_{s} } \right] \\ \hfill + \frac{{4\sin^{2} \theta \sin^{2} \phi \cos^{2} \phi }}{{\left( {\alpha_{b}^{s} } \right)^{2} }}\left[ {\Delta \mu_{b}^{s} - \mu_{b}^{s} \Delta \delta_{T}^{{}} + \mu_{b}^{s} \Delta \Gamma_{s} } \right] \\ \end{gathered} \right\}. \hfill \\ \end{gathered}$$

(94)

After simplification, the linearized PP-wave reflection coefficient equation can be obtained

$$\begin{gathered} R_{PP} (\theta ,\phi ) = \frac{{\sec^{2} \theta }}{{4M_{b}^{s} }}\Delta M_{b}^{s} - 2\frac{{\sin^{2} \theta }}{{M_{b}^{s} }}\Delta \mu_{b}^{s} + \frac{1}{{2\rho_{b}^{s} }}\left( {1 - \frac{{\sec^{2} \theta }}{2}} \right)\Delta \rho_{b}^{s} \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} - \frac{{\sec^{2} \theta }}{4}\left[ {2g_{b} \left( {\sin^{2} \theta \sin^{2} \phi + cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{N}^{{}} \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + g_{b} \sin^{2} \theta \cos^{2} \phi \left( {1 - \tan^{2} \theta \sin^{2} \phi } \right)\Delta \delta_{T}^{{}} \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + \frac{{\sec^{2} \theta }}{{16g_{b} \left( {1 - g_{b} } \right)}}\left[ {2g_{b} \left( {\sin^{2} \theta \sin^{2} \phi + cos^{2} \theta } \right) - 1} \right]^{2} \Delta \Gamma_{p} \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} - g_{b} \sin^{2} \theta \cos^{2} \phi \left( {1 - \tan^{2} \theta \sin^{2} \phi } \right)\Delta \Gamma_{s} . \hfill \\ \end{gathered}$$

(95)

When the horizontal in situ stress disappears, that is, $T_{11} = 0$. According to Eqs. 25 and 26, $\Gamma_{p} = \Gamma_{s} = 0$. In that way, their perturbations $\Delta \Gamma_{p}$ and $\Delta \Gamma_{s}$ will also disappear.

Therefore, terms with $\Delta \Gamma_{p}$ and $\Delta \Gamma_{s}$ in Eq. 44 will become zero, and Eq. 44 will be reduced to the following form

$$\begin{gathered} R_{PP} (\theta ,\phi ) = \frac{{\sec^{2} \theta }}{{4M_{b}^{s} }}\Delta M_{b}^{s} - 2\frac{{\sin^{2} \theta }}{{M_{b}^{s} }}\Delta \mu_{b}^{s} + \frac{1}{{2\rho_{b}^{s} }}\left( {1 - \frac{{\sec^{2} \theta }}{2}} \right)\Delta \rho_{b}^{s} \\ - \frac{{\sec^{2} \theta }}{4}\left[ {2g_{b} \left( {\sin^{2} \theta \sin^{2} \phi + cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{N}^{{}} \\ + g_{b} \sin^{2} \theta \cos^{2} \phi \left( {1 - \tan^{2} \theta \sin^{2} \phi } \right)\Delta \delta_{T}^{{}} . \\ \end{gathered}$$

(96)

The form is the same as the reflection coefficient equation of unstressed HTI media derived by Pan and Zhang (2018a, b) without the effect of horizontal in situ stress. Compared with the PP-wave reflection coefficient Eq. 16 in Pan and Zhang (2018a, b), the elastic parameter in Eq. 44 is related to the horizontal in situ stress and has two additional terms $\frac{{\sec^{2} \theta }}{{16g_{b} (1 - g_{b} )}}\left[ {2g_{b} (\sin^{2} \theta \sin^{2} \phi + cos^{2} \theta ) - 1} \right]^{2} \Delta \Gamma_{p}$ and $g_{b} \sin^{2} \theta \cos^{2} \phi (1 - \tan^{2} \theta \sin^{2} \phi )\Delta \Gamma_{s}$ caused by the horizontal in situ stress.