following is a paper in latex, please derive all equations for me 
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\title{Multipole borehole acoustic field in a transversely isotropic medium induced by stress}

\author{Jinxia Liu, Ying Zhou, and Zhiwen Cui \ Department of Acoustics and Microwave Physics, College of Physics, Jilin University, Changchun 130012, China\
Tribikram Kundu \ Department of Civil and Architectural Engineering and Mechanics, University of Arizona, Tucson, Arizona 85721, USA}

\date{Received 24 December 2018; revised 29 July 2019; accepted 31 July 2019; published online 22 August 2019}

\begin{document}

\maketitle

\begin{abstract}
A borehole multipole acoustic field in a pre-stressed formation is investigated. The pre-stressed formation is modeled as a transversely isotropic medium induced by uniaxial stress. The formation is assumed to be isotropic in absence of any static stress and then becomes anisotropic due to the applied stress parallel to the borehole axis. The approximate equivalent elastic constants of the stress-induced anisotropic medium are derived from the theory of acoustoelasticity. The nonlinear static stress-strain relation is used for both small and large static deformations. This problem can be solved analytically because of uniformity of deformation induced by static stress applied parallel to the borehole axis. The stress effects on the velocity of guided waves and amplitude of waveforms excited by monopole, dipole, and quadrupole sources are investigated. Numerical results show that the velocities of guided waves increase with uniaxial stress. The uniaxial stress affects both amplitude and arrival time of the acoustic waves in the borehole. The integral amplitude of full waveforms varies almost in a parabolic manner with the increasing stress level and thus shows sensitivity to the uniaxial stress. This result may be helpful for remote stress measurements in boreholes. \copyright 2019 Acoustical Society of America. \href{https://doi.org/10.1121/1.5122974}{https://doi.org/10.1121/1.5122974}
\end{abstract}

\section{INTRODUCTION}

The study of elastic wave propagation in cylindrical structures is a topic of great interest for many applications in acoustics, engineering, and geophysics (Aristegui \textit{et al.}, 2001; Tang and Cheng, 2004; Liew and Wang, 2007; Markova \textit{et al.}, 2010; Gao \textit{et al.}, 2013; Wang \textit{et al.}, 2016; Glushkov \textit{et al.}, 2017; Iakovlev \textit{et al.}, 2017; Pérez-Arancibia \textit{et al.}, 2018). Multipole acoustic logging is important for understanding characteristics of elastic wave propagation in cylindrical structures. The desire to know local formation information such as wave velocities in the medium prompted the development of borehole logging devices and acoustic methods (Kitsunezaki, 1980; Schmitt \textit{et al.}, 1991; Tang and Cheng, 2004; Che \textit{et al.}, 2014; He \textit{et al.}, 2013). A laboratory scale model of ultrasonic borehole was designed by Chen (1988, 1989) to prove that both dipole and quadrupole sources can be used in the fast and slow formations.

The acoustic field excited by multipole sources in a transversely isotropic formation is available in the literature (Tongtaow, 1982; Rice, 1987, Ellefsen \textit{et al.}, 1988; Renlie, 1994). Acoustic multipole logging in anisotropic formation through numerical solution and perturbation method was presented by Zhang \textit{et al.} (1994) and Zhang and Wang (1996). The acoustic field generated by an axisymmetric source in a borehole embedded in an anisotropic formation caused by the presence of crack-like inclusions was investigated by Markov \textit{et al.} (2016).

It is well known that the horizontal or vertical stress often exists in underground reservoirs and engineered structures. The velocity change of the acoustic waves in presence of a stress field, i.e., the acoustoelastic effect, has been investigated by many researchers (e.g., Toupin and Bernstein, 1961, Thurston and Brugger, 1964, Pao \textit{et al.}, 1984, Norris \textit{et al.}, 1994). Acoustoelasticity theory supports the intuitive prediction that an isotropic medium subjected to a uniaxial stress becomes approximately transversely isotropic (Rasolofosaon, 1998; Sarkar \textit{et al.}, 2003; Prioul \textit{et al.}, 2004). Strictly speaking, a uniaxially stressed isotropic medium is not identical to an unstressed transversely isotropic crystal because elastic properties for an unstressed orthorhombic crystal and a stressed isotropic crystal have slightly different relations between themselves (Rasolofosaon, 1998). Fortunately, the deviations are so small that such subtleties can be neglected (Rasolofosaon, 1998). The reflection coefficients of a medium subjected to a uniaxial stress field was considered by Liu \textit{et al.} (2009, 2016). A seismic velocity change caused by an overburden stress on the basis of acoustoelasticity was investigated by Korneev and Glubokovsk (2013). They presented the nonlinear stress-strain relationship for applied stress (Korneev and Glubokovsk 2013), which is justified for both small and large static deformations.

Combining acoustoelasticity theory and cross-dipole acoustic logging, the horizontal uniaxial stress was estimated by Sinha and Kostek (1996) and Sinha (1997). Winkler \textit{et al.} (1998) investigated the effects of borehole stress concentrations on dipole anisotropy measurements. Liu and Sinha (2003) simulated acoustic fields of monopole and dipole in a borehole in triaxially stressed formation by a three-dimensional (3D) cylindrical PML/FDTD method. Acoustoelastic effects of guided waves in the presence of borehole pressurization and formation of stress concentration was investigated by Cao \textit{et al.} (2004). Wang and Tang (2005) simulated acoustic waveforms of borehole for monopole, dipole, and quadrupole sources in a pre-stressed formation by an effective medium approach. Lei \textit{et al.} (2012) described a method to estimate the minimum and maximum horizontal stresses by using monopole and cross-dipole sonic data from a vertical well. In those works, the formation is heterogeneous due to the local stress concentrations around the borehole induced by horizontal stress. This heterogeneity prevented those authors from using analytical solutions. Moreover, the static stress-strain relationship was considered linear in the above works, which can be justified for only small static deformations.

Although the triaxial stress case is available in the literature (Liu and Sinha, 2003), there is still a lack of understanding of the multipole acoustic field in a vertical borehole with vertical stress or a horizontal borehole with horizontal stress. In real world applications, one can find many horizontal boreholes for exploration. A schematic diagram showing a uniaxial static stress (S) applied to a formation containing a fluid-filled vertical borehole is shown in Fig. 1. To our knowledge, no efforts have been made to investigate the multipole acoustic field in a vertical borehole with vertical stress or a horizontal borehole with horizontal stress. An isotropic medium becomes approximately transversely isotropic when it is induced by stress where the formation axis of symmetry and borehole axis are parallel. The borehole acoustic wave field in such a stress-induced anisotropic medium is investigated in this paper. It can be solved analytically with the equivalent elastic constants because of uniformity of static deformation. One can use any existing method (e.g., Zhang \textit{et al.}, 1994; Tang and Cheng, 2004) to study it. The second new development of this study is that the nonlinear stress-strain relationship for applied stress (Korneev and Glubokovsk, 2013), which can be justified for both small and large static deformations is used. Dispersion curves of the guided waves, full waveforms, and the integral of full wave amplitudes are analyzed. This work will be helpful for remote stress measurements in a borehole surrounded by uniaxial stress-induced anisotropic medium whose axis of stress is parallel to the well axis.

The organization of this paper is as follows: in Sec. II, equivalent elastic constants of the uniaxial stress-induced anisotropic medium are given. Section III gives a brief review of formulations of the multipole borehole acoustic field. In Sec. IV, numerical results are presented to show the stress effects on the multipole borehole acoustic field. Then the conclusion of the paper is given.

\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{Figure1.png}
\caption{Schematic diagram showing a uniaxial static stress (S) applied to a formation containing a fluid-filled borehole. The uniaxial stress direction is parallel to the borehole axis. An acoustic source (T) emits energy and receivers (R) detect echoes.}
\end{figure}

\section{STRESS-INDUECED ANISOTROPIC MEDIUM}

The effective elastic stiffness tensor as a function of stress can be obtained using acoustoelasticity theory (Pao \textit{et al.}, 1984). In traditional linear elasticity, the potential energy is represented by a quadratic function of strains, while acoustoelasticity theory includes cubic terms and introduces third-order elastic constants to account for the acoustic properties induced by the stress change. This paper analyzes small perturbations superposed on a static bias. A static bias means incremental deviatoric stresses. Three configurations of the body play crucial roles: reference, intermediate (deformed), and current configuration (Toupin and Bernstein, 1961; Thurston and Brugger, 1964). In this paper, the intermediate (deformed) configuration is chosen to describe the wave propagation in the pre-stressed medium. Effective stiffness tensor in intermediate configuration is (Sinha, 1982; Pao \textit{et al.}, 1984)

\begin{equation}
\label{eqn:1}
H_{ijkl}=C_{ijkl}+T_{ik}d_{jl}-C_{ijkl}W_{n,n}+C_{ijklmn}E_{mn}+C_{gjkl}W_{i,g}+C_{igkl}W_{j,g}+C_{ijgl}W_{k,g}+C_{ijkg}W_{l,g}.
\end{equation}

In Eq. (\ref{eqn:1}), $C_{ijkl}$ are second-order elastic constants and $C_{ijklmn}$ are third-order elastic constants. The elastic constants (for instance $C_{11}$, $C_{111}$) follow Voigt's contracted notation for $C_{ijkl}$ and $C_{ijklmn}$, whereby tensorial indices are contracted according to the notation 11 $\rightarrow$ 1, 222 $\rightarrow$ 2, 333 $\rightarrow$ 3, 234 $\rightarrow$ 4, 314 $\rightarrow$ 5, 12 $\rightarrow$ 6. Under isotropic assumption, the second-order elastic constants are defined by two independent constants, the Lame coefficients $\lambda$ and $\mu$, the relations are $C_{11} = \lambda + 2\mu$, $C_{66} = \mu$ and $C_{12} = C_{11} - 2C_{66} = \lambda$. The third-order elastic constants are defined by three independent nonlinear coefficients ($C_{111}$, $C_{112}$, $C_{123}$) with the following two relationships: $C_{112} = 2C_{144} + C_{123}$ and $C_{111} = 4C_{155} + C_{112}$ (Thurston and Brugger, 1964). $\delta_{ji}$ is the Kronecker delta and $T_{ik}$, $E_{mn}$, $W_{i,g}$ are the static stresses, strains, and displacement gradients, respectively. We can see that the effective stiffness tensor for a pre-stressed medium lacks usual symmetry. Rasolofosaon (1998) has discussed the distinction between the stress-induced anisotropy and the intrinsic anisotropy. Prioul \textit{et al.} (2004) deem that $c_{ij}E_{ki}$ is much smaller than the remaining terms of the form $c_{ij}$ and $c_{ijk}E_{mn}$ since, for rocks, $|c_{ij}E_{kl}| \ll |T_{ij}|$. When Tij (as a function of $c_{ij}E_{kl}$) and $c_{ij}E_{kl}$ are neglected (Rasolofosaon, 1998; Prioul \textit{et al.}, 2004) then the effective stiffness tensor $H_{ijkt}$ has the same symmetry as the second-order elastic constants $c_{ij}$. Voigt's notation can be used to contract the indices of $H_{ijkl}$; it may be represented as $H_{pq}$. When the medium is subjected to the uniaxial stress ($T_{33}$, $T_{11} = T_{22} = 0$) effective stiffness tensor can be written as (Prioul \textit{et al.}, 2004)

\begin{align}
\label{eqn:2}
\begin{split}
H_{11} &\approx \lambda + 2\mu + (C_{111} + C_{112})E_{11} + C_{112}E_{33}, \
H_{22} &\approx \lambda + 2\mu + (C_{111} + C_{112})E_{11} + C_{112}E_{33}, \
H_{33} &\approx \lambda + 2\mu + C_{111}E_{33} + 2C_{112}E_{11}, \
H_{12} &\approx \lambda + 2C_{112}E_{11} + C_{123}E_{33}, \
H_{13} &\approx \lambda + (C_{112} + C_{123})E_{11} + C_{112}E_{33}, \
H_{23} &\approx \lambda + (C_{112} + C_{123})E_{11} + C_{112}E_{33}, \
H_{44} &\approx \mu + (C_{144} + C_{155})E_{11} + C_{155}E_{33}, \
H_{55} &\approx \mu + (C_{144} + C_{155})E_{22} + C_{155}E_{11}, \
H_{66} &\approx \mu + C_{144}E_{33} + 2C_{155}E_{11}.
\end{split}
\end{align}

Here, the assumption is made that the coordinate axes $X$ (i = 1, 2, 3) are aligned with the directions of principal stresses. It means that in these chosen coordinates, $T_{12} = T_{13} = T_{23} = E_{12} = E_{13} = E_{23} = 0$. The second assumption is that the medium is isotropic without any static stress. Under uniaxial stress, Korneev and Glubokovkikh (2013) defined the only nonzero static-strain component $E_{33} = -(λ+2\mu)/2C_{111}(1-V\sqrt{1+(4C_{111}T_{33})/(λ+2\mu)^2})$, and $E_{11} = E_{22} = 0$. For small static deformations, $E_{33} \approx T_{33}/(λ+2\mu)$ when $|4C_{111}T_{33}/(λ+2\mu)^2| \ll 1$. The nonzero static-strain component is justified for both small and large static deformations and is used in this paper. One can see from here that when the medium is subjected to the uniaxial stress, it becomes approximately transversely isotropic.

\begin{equation}
\begin{aligned}
\sigma_{rr} &= H_{11}e_{rr}+(H_{11}-2H_{66})e_{\theta\theta}+H_{13}e_{zz}+0+0+0,\
\sigma_{\theta\theta} &= (H_{11}-2H_{66})e_{rr}+H_{11}e_{\theta\theta}+H_{13}e_{zz}+0+0+0,\
\sigma_{zz} &= H_{13}e_{rr}+H_{13}e_{\theta\theta}+H_{33}e_{zz}+0+0+0,\
\sigma_{r\theta} &= 0+0+0+H_{44}e_{r\theta}+0+0,\
\sigma_{rz} &= 0+0+0+0+H_{44}e_{rz}+0,\
\sigma_{\theta z} &= 0+0+0+0+0+H_{66}e_{\theta z},
\end{aligned}
\end{equation}

where $\sigma_{ij}$ and $e_{ij}$ (i, j = r, $\theta$, z) represent dynamic stress and dynamic strain, respectively. $H_{11}$, $H_{13}$, $H_{33}$, $H_{44}$, and $H_{66}$ are the elastic constants of the stress-induced anisotropic medium, see Eq. (\ref{eqn:2}). When the stress is zero, or $E_{11}$, $E_{22}$, and $E_{33}$ are zero, then Eq. (\ref{eqn:5}) becomes isotropic, and it reduces to isotropic case. Substitute these stress and displacement fields into elastic wave equation of general form

\begin{equation}
\label{eqn:6}
\rho\ddot{U}=\nabla\cdot\sigma,
\end{equation}

where $\rho=\rho_{0}/(1+E_{11}+E_{22}+E_{33})$, $\rho_{0}$ is the density of the unstressed state. One can find the displacement vector $U$ of the formation using the decomposition: $U=\nabla\phi+\nabla\times(\chi e_{z})+\nabla\times\nabla\times(\psi e_{z})$, where $\phi$, $\chi$, and $\psi$ are the potential functions of P wave, SH wave, and SV wave in formation, respectively. The solution for P-SV and SH waves can be given as (e.g., Zhang \textit{et al.}, 1994; Tang and Cheng, 2004)

\begin{align}
\label{eqn:7}
\begin{split}
\phi &= e^{ikz} \frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[B_{n}K_{n}(q_{p}r)+b'F_{n}K_{n}(q_{s\nu}r)]cosn\theta d k,\
\chi &= e^{ikz} \frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[D_{n}K_{n}(q_{p}r)+F_{n}K_{n}(q_{s\nu}r)]cosn\theta dk,\
\psi &= e^{ikz}\frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[B_{n}K_{n}(q_{SH}r)]sinn\theta dk,
\end{split}
\end{align}

where $q_{p}$, $q_{SH}$, $q_{s\nu}$ are the radial wavenumbers of P, SH, and SV waves, respectively, and $a'$, $b'$ are given in the Appendix. Since in Eq. (\ref{eqn:7}) there are three unknown coefficients, it is necessary to use the boundary conditions of borehole wall to solve for the unknowns. Three unknown coefficients (i.e., $B_{n}$, $D_{n}$, $F_{n}$) are determined from the boundary conditions of continuity of the radial displacement, radial normal stress, and vanishing of the tangential shear stresses on the borehole wall (r = R),

\begin{equation}
\label{eqn:8}
\begin{aligned}
U_{r} &= U_{f}\
\sigma_{rr} &= -P_{f}\
\sigma_{\theta\theta} &= 0\
\sigma_{r\theta} &= 0.
\end{aligned}
\end{equation}

The boundary conditions give rise to a matrix equation for unknown coefficients in the following form:

\begin{equation}
\label{eqn:9}
\begin{bmatrix}
M_{11} & M_{12} & M_{13} & M_{14}\
M_{21} & M_{22} & M_{23} & M_{24}\
M_{31} & M_{32} & M_{33} & M_{34}\
M_{41} & M_{42} & M_{43} & M_{44}
\end{bmatrix}
\begin{bmatrix}
A_{n}\
B_{n}\
F_{n}\
D_{n}
\end{bmatrix}=
\begin{bmatrix}
b_{1}\
b_{2}\
0\
0
\end{bmatrix},
\end{equation}

where $b_{1}$,$b_{2}$ and $M_{ij}$ (i, j = 1,2,3,4) are given in the Appendix. When $n = 0$, the SH wave cannot be excited by a monopole source, and $\sigma_{r\theta} = 0$; then Eq. (\ref{eqn:9}) becomes a 3 × 3 matrix equation. The determinant $D(k, \omega)$ of the matrix in Eq. (\ref{eqn:9}) leads to a dispersion equation for borehole guided waves. The poles can be located by solving

\begin{equation}
\label{eqn:10}
D(k, \omega)=0.
\end{equation}

This is the dispersion equation of guided waves whose excited amplitude is given by the residue of the poles. Let the receivers be placed at the borehole axis to record the pressure, displacement, and spatial derivative of the displacement when the source is monopole, dipole and quadrupole, respectively.

\section{MULTIPOLE ACOUSTIC FIELD IN A BOREHOLE}

In this section, we consider a borehole surrounded by an effective transversely isotropic medium where the anisotropy is induced by the uniaxial stress, and the axis of uniaxial stress is assumed to be parallel to the axis of the borehole (see Fig. 1). It is mechanically similar to a vertical borehole surrounded by a vertical transversely isotropic medium (Zhang \textit{et al.}, 1994). For this reason, the formulation of borehole multipole acoustic field in a transversely isotropic formation is reviewed here.

In the cylindrical coordinate system (r, $\theta$, z), where r, z, and $\theta$ are the radial distance, axial distance and the azimuthal angle, respectively, the borehole acoustic field potential function in the frequency-wavenumber domain can be expressed as (Kurkjian and Chang, 1986)

\begin{equation}
\label{eqn:3}
\phi(k,r,\theta)=\frac{(k r r_{0})^{n}}{2^{n}n!}[A_{n}I_{n}(k_{r}r)+E_{n}K_{n}(k_{r}r)]cosn\theta e^{ikz},
\end{equation}

where $k$ is the axial wave number, $k_{r}=\sqrt{k^{2}-k_{f}^{2}}$ is the radial wavenumber of the fluid, $k_{f}=\omega/V_{f}$ is the wavenumber of the fluid in the borehole, and $V_{f}$ is the wave velocity in the borehole fluid. $\omega$ is the angular frequency, and $r_{0}$ is the multipole source separation. $A_{n}$ represents the reflection coefficient as a function of k, $\theta$ determined by the boundary conditions of the borehole wall. $I_{n}$ and $K_{n}$ represent the nth-order modified Bessel functions of the first and second kind, respectively. $E_{n}$ is 1 for n = 0 and 2 for n > 0. From this potential, the radial displacement and pressure can be obtained as

\begin{equation}
\label{eqn:4}
\begin{aligned}
U_{r}&=\frac{\partial \phi}{\partial r},\
P_{f}&=\rho_{f}\omega^{2}\phi.
\end{aligned}
\end{equation}

The relationship between the stress and strain fields in the approximately transversely isotropic medium is given by

\begin{equation}
\label{eqn:5}
\begin{aligned}
\sigma_{rr} &= H_{11}e_{rr}+(H_{11}-2H_{66})e_{\theta\theta}+H_{13}e_{zz}+0+0+0,\
\sigma_{\theta\theta} &= (H_{11}-2H_{66})e_{rr}+H_{11}e_{\theta\theta}+H_{13}e_{zz}+0+0+0,\
\sigma_{zz} &= H_{13}e_{rr}+H_{13}e_{\theta\theta}+H_{33}e_{zz}+0+0+0,\
\sigma_{r\theta} &= 0+0+0+H_{44}e_{r\theta}+0+0,\
\sigma_{rz} &= 0+0+0+0+H_{44}e_{rz}+0,\
\sigma_{\theta z} &= 0+0+0+0+0+H_{66}e_{\theta z},
\end{aligned}
\end{equation}

where $\sigma_{ij}$ and $e_{ij}$ (i, j = r, $\theta$, z) represent dynamic stress and dynamic strain, respectively. $H_{11}$, $H_{13}$, $H_{33}$, $H_{44}$, and $H_{66}$ are the elastic constants of the stress-induced anisotropic medium, see Eq. (\ref{eqn:2}). When the stress is zero, or $E_{11}$, $E_{22}$, and $E_{33}$ are zero, then Eq. (\ref{eqn:5}) becomes isotropic, and it reduces to isotropic case. Substitute these stress and displacement fields into elastic wave equation of general form

\begin{equation}
\label{eqn:6}
\rho\ddot{U}=\nabla\cdot\sigma,
\end{equation}

where $\rho=\rho_{0}/(1+E_{11}+E_{22}+E_{33})$, $\rho_{0}$ is the density of the unstressed state. One can find the displacement vector $U$ of the formation using the decomposition: $U=\nabla\phi+\nabla\times(\chi e_{z})+\nabla\times\nabla\times(\psi e_{z})$, where $\phi$, $\chi$, and $\psi$ are the potential functions of P wave, SH wave, and SV wave in formation, respectively. The solution for P-SV and SH waves can be given as (e.g., Zhang \textit{et al.}, 1994; Tang and Cheng, 2004)

\begin{align}
\label{eqn:7}
\begin{split}
\phi &= e^{ikz} \frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[B_{n}K_{n}(q_{p}r)+b'F_{n}K_{n}(q_{s\nu}r)]cosn\theta d k,\
\chi &= e^{ikz} \frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[D_{n}K_{n}(q_{p}r)+F_{n}K_{n}(q_{s\nu}r)]cosn\theta dk,\
\psi &= e^{ikz}\frac{1}{2\pi} \int_{0}^{\infty} (k r r_{0})^{n}\frac{1}{n!}[B_{n}K_{n}(q_{SH}r)]sinn\theta dk,
\end{split}
\end{align}

where $q_{p}$, $q_{SH}$, $q_{s\nu}$ are the radial wavenumbers of P, SH, and SV waves, respectively, and $a'$, $b'$ are given in the Appendix. Since in Eq. (\ref{eqn:7}) there are three unknown coefficients, it is necessary to use the boundary conditions of borehole wall to solve for the unknowns. Three unknown coefficients (i.e., $B_{n}$, $D_{n}$, $F_{n}$) are determined from the boundary conditions of continuity of the radial displacement, radial normal stress, and vanishing of the tangential shear stresses on the borehole wall (r = R),

\begin{equation}
\label{eqn:8}
\begin{aligned}
U_{r} &= U_{f}\
\sigma_{rr} &= -P_{f}\
\sigma_{\theta\theta} &= 0\
\sigma_{r\theta} &= 0.
\end{aligned}
\end{equation}

The boundary conditions give rise to a matrix equation for unknown coefficients in the following form:

\begin{equation}
\label{eqn:9}
\begin{bmatrix}
M_{11} & M_{12} & M_{13} & M_{14}\
M_{21} & M_{22} & M_{23} & M_{24}\
M_{31} & M_{32} & M_{33} & M_{34}\
M_{41} & M_{42} & M_{43} & M_{44}
\end{bmatrix}
\begin{bmatrix}
A_{n}\
B_{n}\
F_{n}\
D_{n}
\end{bmatrix}=
\begin{bmatrix}
b_{1}\
b_{2}\
0\
0
\end{bmatrix},
\end{equation}

where $b_{1}$,$b_{2}$ and $M_{ij}$ (i, j = 1,2,3,4) are given in the Appendix. When $n = 0$, the SH wave cannot be excited by a monopole source, and $\sigma_{r\theta} = 0$; then Eq. (\ref{eqn:9}) becomes a 3 × 3 matrix equation. The determinant $D(k, \omega)$ of the matrix in Eq. (\ref{eqn:9}) leads to a dispersion equation for borehole guided waves. The poles can be located by solving

\begin{equation}
\label{eqn:10}
D(k, \omega)=0.
\end{equation}

This is the dispersion equation of guided waves whose excited amplitude is given by the residue of the poles. Let the receivers be placed at the borehole axis to record the pressure, displacement, and spatial derivative of the displacement when the source is monopole, dipole and quadrupole, respectively.

\section{NUMERICAL RESULTS}

\subsection{Effective stiffness coefficients}

The above theoretical formulation is an extension of the work for simulation of multipole borehole acoustic logging in anisotropic formation (e.g., Zhang \textit{et al.}, 1994). The elastic parameters of the formation are listed in Table I. Formation A is Hanson sandstone, a fast formation where shear wave velocity exceeds the sound velocity of the borehole fluid, and its compressive strength is 64 MPa (Winkler and Liu, 1996). Formation B is dry Berea sandstone (Winkler \textit{et al.}, 1998), a slow formation where shear wave velocity does not exceed the sound velocity of the borehole fluid. Its compressive strength is about 36MPa (Winkler and Liu, 1996). The third-order elastic constants in Table I are obtained with the assumption that the background formation is isotropic (Winkler and Liu, 1996; Winkler \textit{et al.}, 1998). A borehole radius is 0.1 m, and multipole source separation is 0.01 m, the density $p_{f}$ of the borehole fluid is 1000kg/m³and the acoustic velocity of borehole fluid $V_{f}$ = 1500m/s. A cosine envelope pulse is used as in reference Zhang \textit{et al.} (1994).

\begin{table}[H]
\caption{Properties of the formation.}
\centering
\begin{tabularx}{0.9\textwidth}{l*{7}{c}}
\toprule
Rock & $\rho$ ( kg/m ³) & $V_{p}$ (m/s) & $V_{s}$ (m/s) & $C_{111}$ (GPa) & $C_{112}$ (GPa) & $C_{123}$ (GPa)\
\midrule
A & 2270 & 3230 & 2100 & -22076 & -6850 & -2608 \
B & 2062 & 2127 & 1418 & -9550 & -1370 & 1062\
\bottomrule
\end{tabularx}
\end{table}

Effective stiffness coefficients for formations A and B in presence of the uniaxial stress are shown in Fig. 2. One can see that the effective stiffness coefficients increase with increasing uniaxial compressive stress.

\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure2a.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure2b.png}
\caption{}
\end{subfigure}
\caption{Effective stiffness coefficients of the formation in presence of the uniaxial stress for (a) formation A and (b) formation B.}
\end{figure}

\subsection{Dispersion and excitation of guide waves}

It is well known that the monopole source excites two kinds of modes for transversely isotropic formation, Stoneley waves and Pseudo-Rayleigh waves, which are similar to the case for isotropic formation. The dipole sources and quadrupole sources excite only one kind of guide waves, the flexural waves, and the screw waves, respectively. The dispersion and excitation curves of the guided waves for monopole, dipole and quadrupole sources are shown in Figs. 3–6. The excitation of the guided wave is the excited amplitude given by the residues of the poles of Eq. (\ref{eqn:10}). Here, we mainly investigate the dispersion characteristics of first and second order axisymmetric modes (i.e., Stoneley wave and first pseudo-Rayleigh wave), and first order non-axisymmetric mode-flexural wave and screw wave because these modes are most important