Waveform Tomography of Two-Dimensional Three-Component Seismic Data for HTI Anisotropic Media

Abstract

Reservoirs with vertically aligned fractures can be represented equivalently by horizontal transverse isotropy (HTI) media. But inverting for the anisotropic parameters of HTI media is a challenging inverse problem, because of difficulties inherent in a multiple parameter inversion. In this paper, when we invert for the anisotropic parameters, we consider for the first time the azimuthal rotation of a two-dimensional seismic survey line from the symmetry of HTI. The established wave equations for the HTI media with azimuthal rotation consist of nine elastic coefficients, expressed in terms of five modified Thomsen parameters. The latter are parallel to the Thomsen parameters for describing velocity characteristics of weak vertical transverse isotropy media. We analyze the sensitivity differences of the five modified Thomsen parameters from their radiation patterns, and attempt to balance the magnitude and sensitivity differences between the parameters through normalization and tuning factors which help to update the model parameters properly. We demonstrate an effective inversion strategy by inverting velocity parameters in the first stage and updates the five modified Thomsen parameters simultaneously in the second stage, for generating reliably reconstructed models.

Appendices

Appendix A: The Gradients with Respect to Thomsen Parameters

Following Eq. (10), the gradient calculation is divided into three steps. First, using the differential of the objective function for general anisotropic media (Kamath and Tsvankin 2016), we derive the derivatives of the objective function, with respect to each of the nine elastic coefficients for HTI media in survey coordinate system, as

$$ \begin{aligned} \frac{\partial \varphi }{{\partial c'_{11} }} = - \int\limits_{t} {\frac{{\partial w_{x} }}{\partial x}\frac{{\partial \tilde{u}_{x} }}{\partial x}{\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{13} }} = - \int\limits_{t} {\left( {\frac{{\partial w_{x} }}{\partial x}\frac{{\partial \tilde{u}_{z} }}{\partial z} + \frac{{\partial w_{z} }}{\partial z}\frac{{\partial \tilde{u}_{x} }}{\partial x}} \right){\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{16} }} = - \int\limits_{t} {\left( {\frac{{\partial w_{\text{x}} }}{\partial x}\frac{{\partial \tilde{u}_{y} }}{\partial x} + \frac{{\partial w_{y} }}{\partial x}\frac{{\partial \tilde{u}_{x} }}{\partial x}} \right){\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{33} }} = - \int\limits_{t} {\frac{{\partial w_{z} }}{\partial z}\frac{{\partial \tilde{u}_{z} }}{\partial z}{\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{36} }} = - \int\limits_{t} {\left( {\frac{{\partial w_{\text{z}} }}{\partial z}\frac{{\partial \tilde{u}_{y} }}{\partial x} + \frac{{\partial w_{y} }}{\partial x}\frac{{\partial \tilde{u}_{z} }}{\partial z}} \right){\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{44} }} = - \int\limits_{t} {\frac{{\partial w_{y} }}{\partial z}\frac{{\partial \tilde{u}_{y} }}{\partial z}{\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{45} }} = - \int\limits_{t} {\left( {\frac{{\partial w_{y} }}{\partial z}\left( {\frac{{\partial \tilde{u}_{x} }}{\partial z} + \frac{{\partial \tilde{u}_{z} }}{\partial x}} \right) + \frac{{\partial \tilde{u}_{y} }}{\partial z}\left( {\frac{{\partial w_{x} }}{\partial z} + \frac{{\partial w_{z} }}{\partial x}} \right)} \right){\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{55} }} = - \int\limits_{t} {\left( {\frac{{\partial w_{x} }}{\partial z} + \frac{{\partial w_{z} }}{\partial x}} \right)\left( {\frac{{\partial \tilde{u}_{x} }}{\partial z} + \frac{{\partial \tilde{u}_{z} }}{\partial x}} \right){\text{d}}t} , \hfill \\ \frac{\partial \varphi }{{\partial c'_{66} }} = - \int\limits_{t} {\frac{{\partial w_{y} }}{\partial x}\frac{{\partial \tilde{u}_{y} }}{\partial x}{\text{d}}t} . \hfill \\ \end{aligned} $$

(A1)

where $\tilde{u}$ and $w$ are the encoded forward and backward seismic wavefields, and the subscripts x, y, z represent the x-, y-, and z-components of $\tilde{u}$ and $w$. For simplicity, Eq. (10) only shows the gradient calculated using one supershot. For the case with multiple supershots, it requires a sum of gradients over the supershots.

Secondly, exploiting relations between the two sets of coefficients $(c'_{11} , \, c'_{13} , \, c'_{16} , \, c'_{33} , c'_{36} , \, c'_{44} , \, c'_{45} , \, c'_{55} , \, c'_{66} )$ and $(c_{11} , \, c_{13} , \, c_{33} , \, c_{44} , \, c_{55} )$, we can obtain the derivatives of the objective function, with respect to $c_{11}$, for example, in the intrinsic coordinate system, by

$$\begin{aligned} \frac{\partial }{{\partial c_{{11}} }} & = \sum\limits_{{i,j}} {\frac{{\partial {c^{\prime}}_{{ij}} }}{{\partial c_{{11}} }}\frac{\partial }{{\partial {c^{\prime}}_{{ij}} }}} \\ & = \cos ^{4} \theta \frac{\partial }{{\partial {c^{\prime}}_{{11}} }} + \frac{1}{2}\cos ^{2} \theta \sin 2\theta \frac{\partial }{{\partial {c^{\prime}}_{{16}} }} + \frac{1}{4}\sin ^{2} 2\theta \frac{\partial}{{\partial {c^{\prime}}_{{66}} }}. \\ \end{aligned} $$

(A2)

The derivatives for the rest of the elastic coefficients, $(\partial /\partial c_{13} , \, \partial /\partial c_{33} , \, \partial /\partial c_{44} , \, \partial /\partial c_{55} )$, can be derived in the same way following $\frac{\partial \varphi }{{\partial c_{ij} }} = \sum\nolimits_{k,\ell } {\frac{{\partial c'_{k\ell } }}{{\partial c_{ij} }}} \frac{\partial \varphi }{{\partial c'_{k\ell } }}$.

Finally, based on Eq. (2), we derive the gradients of the objective function, with respect to the modified Thomsen parameters, as

$$ \begin{aligned} \frac{\partial }{{\partial v_{{P{\text{v}}}} }} &= \sum\limits_{k,\ell } {\frac{{\partial c_{k\ell } }}{{\partial v_{{P{\text{v}}}} }}} \frac{\partial }{{\partial c_{k\ell } }} \\ &= 2\rho v_{{P{\text{v}}}} \times \left( {\tilde{\varepsilon }^{(E)} \frac{\partial }{{\partial c_{11} }} + \frac{\partial }{{\partial c_{33} }}} \right.\left. { + \frac{{\tilde{\delta }^{(E)} v_{{P{\text{v}}}}^{ 2} - \frac{1}{2}\left( {1 + \tilde{\delta }^{(E)} } \right)v_{{S{\text{v}}}}^{ 2} }}{{\sqrt {(\tilde{\delta }^{(E)} v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} )(v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} )} }}\frac{\partial }{{\partial c_{13} }}} \right) \, , \\ \frac{\partial }{{\partial v_{{S{\text{v}}}} }} &= \sum\limits_{k,\ell } {\frac{{\partial c_{k\ell } }}{{\partial v_{{S{\text{v}}}} }}} \frac{\partial }{{\partial c_{k\ell } }} \\ &= 2\rho v_{{S{\text{v}}}} \times \left( {\frac{1}{{\tilde{\gamma }^{(E)} }}\frac{\partial }{{\partial c_{44} }} + \frac{\partial }{{\partial c_{55} }}\left. { - \left( {\frac{{\frac{1}{2}\left( {1 + \tilde{\delta }^{(E)} } \right)v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} }}{{\sqrt {\left( {\tilde{\delta }^{(E)} v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} } \right)\left( {v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} } \right)} }} + 1} \right)\frac{\partial }{{\partial c_{13} }}} \right)} \right., \\ \frac{\partial }{{\partial \tilde{\varepsilon }^{(E)} }} &= \sum\limits_{k,\ell } {\frac{{\partial c_{k\ell } }}{{\partial \tilde{\varepsilon }^{(E)} }}} \frac{\partial }{{\partial c_{k\ell } }} = \rho v_{{P{\text{v}}}}^{ 2} \frac{\partial }{{\partial c_{11} }} \, , \\ \frac{\partial }{{\partial \tilde{\gamma }^{(E)} }} &= \sum\limits_{k,\ell } {\frac{{\partial c_{k\ell } }}{{\partial \tilde{\gamma }^{(E)} }}} \frac{\partial }{{\partial c_{k\ell } }} = - \rho v_{{S{\text{v}}}}^{ 2} \left( {\frac{1}{{\tilde{\gamma }^{(E)} }}} \right)^{2} \frac{\partial }{{\partial c_{44} }} \, , \\ \frac{\partial }{{\partial \tilde{\delta }^{(E)} }} &= \sum\limits_{k,\ell } {\frac{{\partial c_{k\ell } }}{{\partial \tilde{\varepsilon }^{(E)} }}} \frac{\partial }{{\partial c_{k\ell } }} = \frac{1}{2}\rho v_{{P{\text{v}}}}^{ 2} \sqrt {\frac{{v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} }}{{\tilde{\delta }^{(E)} v_{{P{\text{v}}}}^{ 2} - v_{{S{\text{v}}}}^{ 2} }}} \frac{\partial }{{\partial c_{13} }}. \\ \end{aligned} $$

(A3)

Appendix B: Formulas of Radiation Patterns

The radiation pattern due to perturbation of the model parameters is generally defined as (Pan et al. 2016a; Chapman 2004)

$$ R_{{P - \alpha }} (\varphi _{{{\rm{in}}}} ,\phi _{{{\rm{in}}}} ,\varphi _{{{\rm{sc}}}}^{\alpha } ,\phi _{{{\rm{sc}}}}^{\alpha } ) = [{\bf{\hat{g}}}_{{{\rm{sc}}}}^{\alpha } ]^{T} {{\partial {\bf{\hat{T}}}} \over {\partial {\bf{m}}}}{\bf{\hat{p}}}, $$

(B1)

where $\varphi$ is the inclination angle of the wave, departing from the z-axis, and is defined in the x-z plane, $\phi$ is the angle departing from the x-axis, and is defined in the x-y plane, the subscript ‘in’ and ‘sc’ stand for incident and scattered waves, respectively, and $\alpha$ indicates either P or SV mode of the reflection wave. Hence, $R_{P - P}$ is the P-P wave radiation pattern, and $R_{P - SV}$ is the P-SV wave radiation pattern. We focus on the case with a plane P-wave incidence in this paper.

On the right-hand side of Eq. (B1), ${\hat{\mathbf{T}}}$ is the reduced equivalent moment tensor,

$$ {\hat{\mathbf{T}}} = \left[ {\begin{array}{*{20}c} {\hat{\sigma }_{11} } & {\hat{\sigma }_{12} } & {\hat{\sigma }_{13} } \\ {\hat{\sigma }_{12} } & {\hat{\sigma }_{22} } & {\hat{\sigma }_{23} } \\ {\hat{\sigma }_{13} } & {\hat{\sigma }_{23} } & {\hat{\sigma }_{33} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{t}}}_{1} } & {{\hat{\mathbf{t}}}_{2} } & {{\hat{\mathbf{t}}}_{3} } \\ \end{array} } \right]. $$

(B2)

For a plane P-wave incidence, three column vectors can be expressed as

$$ {\hat{\mathbf{t}}}_{1} = \left[ {\begin{array}{*{20}c} {c'_{11} \hat{p}_{1}^{2} + c'_{13} \hat{p}_{3}^{2} + 2c'_{16} \hat{p}_{ 1} \hat{p}_{ 2} } \\ {c'_{16} \hat{p}_{1}^{2} + c'_{36} \hat{p}_{3}^{2} + 2c'_{66} \hat{p}_{ 1} \hat{p}_{ 2} } \\ {2c'_{45} \hat{p}_{ 2} \hat{p}_{ 3} + 2c'_{55} \hat{p}_{ 1} \hat{p}_{ 3} } \\ \end{array} } \right], $$

(B3)

$$ {\hat{\mathbf{t}}}_{2} = \left[ {\begin{array}{*{20}c} {c'_{16} \hat{p}_{1}^{2} + c'_{36} \hat{p}_{3}^{2} + 2c'_{66} \hat{p}_{ 1} \hat{p}_{ 2} } \\ 0 \\ {2c'_{44} \hat{p}_{ 2} \hat{p}_{ 3} + 2c'_{45} \hat{p}_{ 1} \hat{p}_{ 3} } \\ \end{array} } \right], $$

(B4)

$$ {\hat{\mathbf{t}}}_{3} = \left[ {\begin{array}{*{20}c} {2c'_{45} \hat{p}_{ 2} \hat{p}_{ 3} + 2c'_{55} \hat{p}_{ 1} \hat{p}_{ 3} } \\ {2c'_{44} \hat{p}_{ 2} \hat{p}_{ 3} + 2c'_{45} \hat{p}_{ 1} \hat{p}_{ 3} } \\ {c'_{13} \hat{p}_{1}^{2} + c'_{33} \hat{p}_{3}^{2} + 2c'_{36} \hat{p}_{ 1} \hat{p}_{ 2} } \\ \end{array} } \right], $$

(B5)

where ${\hat{\mathbf{p}}}$ is the slowness vector, which is in the propagation direction (Chapman 2004),

$$ {\hat{\mathbf{p}}} = \left[ {\begin{array}{*{20}c} {\hat{p}_{1} } \\ {\hat{p}_{2} } \\ {\hat{p}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sin \varphi_{\text{in}} \cos \phi_{\text{in}} } \\ {\sin \varphi_{\text{in}} \sin \phi_{\text{in}} } \\ {\cos \varphi_{\text{in}} } \\ \end{array} } \right]. $$

(B6)

The vectors ${\hat{\mathbf{g}}}_{\text{sc}}^{\text{P}}$ and ${\hat{\mathbf{g}}}_{\text{sc}}^{\text{SV}}$ are polarization vectors for scattered P- and SV-wave in Eq. (B1)

$$ {\hat{\mathbf{g}}}_{\text{sc}}^{\text{P}} = \left[ {\begin{array}{*{20}c} {\hat{g}_{1}^{P} } \\ {\hat{g}_{2}^{P} } \\ {\hat{g}_{3}^{P} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sin \varphi_{\text{sc}}^{P} \cos \phi_{\text{sc}}^{P} } \\ {\sin \varphi_{\text{sc}}^{P} \sin \phi_{\text{sc}}^{P} } \\ {\cos \varphi_{\text{sc}}^{P} } \\ \end{array} } \right], $$

(B7)

$$ {\hat{\mathbf{g}}}_{\text{sc}}^{\text{SV}} = \left[ {\begin{array}{*{20}c} {\hat{g}_{1}^{\text{SV}} } \\ {\hat{g}_{2}^{\text{SV}} } \\ {\hat{g}_{3}^{\text{SV}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \varphi_{\text{sc}}^{\text{SV}} \cos \phi_{\text{sc}}^{\text{SV}} } \\ {\cos \varphi_{\text{sc}}^{\text{SV}} \sin \phi_{\text{sc}}^{\text{SV}} } \\ { - \sin \varphi_{\text{sc}}^{\text{SV}} } \\ \end{array} } \right]. $$

(B8)

Substituting equations (B2)–(B8) into Eq. (B1), we obtain the P-P wave radiation pattern with respect to elastic coefficients $c'_{ij}$ as

$$ \begin{aligned} R_{P - P} \left( {c'_{11} } \right) &= \left( {\hat{g}_{1}^{P} } \right)^{2} \hat{p}_{1}^{2} , \\ R_{P - P} \left( {c'_{13} } \right) &= \left( {\hat{g}_{1}^{P} } \right)^{2} \hat{p}_{3}^{2} + \left( {\hat{g}_{3}^{P} } \right)^{2} \hat{p}_{1}^{2} , \\ R_{P - P} \left( {c'_{16} } \right) &= 2\left( {\hat{g}_{1}^{P} } \right)^{2} \hat{p}_{1} \hat{p}_{2} + 2\hat{g}_{1}^{P} \hat{g}_{2}^{P} \hat{p}_{1}^{2} , \\ R_{P - P} \left( {c'_{33} } \right) &= \left( {\hat{g}_{3}^{P} } \right)^{2} \hat{p}_{3}^{2} , \\ R_{P - P} \left( {c'_{36} } \right) &= 2\hat{g}_{1}^{P} \hat{g}_{2}^{P} \hat{p}_{3}^{2} + 2\left( {\hat{g}_{3}^{P} } \right)^{2} \hat{p}_{1} \hat{p}_{2} , \\ R_{P - P} \left( {c'_{44} } \right) &= 4\hat{g}_{2}^{P} \hat{g}_{3}^{P} \hat{p}_{2} \hat{p}_{3} , \\ R_{P - P} \left( {c'_{45} } \right) &= 4\hat{g}_{1}^{P} \hat{g}_{3}^{P} \hat{p}_{2} \hat{p}_{3} + 4\hat{g}_{2}^{P} \hat{g}_{3}^{P} \hat{p}_{1} \hat{p}_{3} , \\ R_{P - P} \left( {c'_{55} } \right) &= 4\hat{g}_{1}^{P} \hat{g}_{3}^{P} \hat{p}_{1} \hat{p}_{3} , \\ R_{P - P} \left( {c'_{66} } \right) &= 4\hat{g}_{1}^{P} \hat{g}_{2}^{P} \hat{p}_{1} \hat{p}_{2} , \\ \end{aligned} $$

(B9)

and the P-SV radiation pattern as

$$ \begin{aligned} R_{P - SV} \left( {c'_{11} } \right) = \hat{g}_{1}^{P} \hat{g}_{1}^{SV} \hat{p}_{1}^{2} , \\ R_{P - SV} \left( {c'_{13} } \right) = \hat{g}_{1}^{P} \hat{g}_{1}^{SV} \hat{p}_{3}^{2} + \hat{g}_{3}^{P} \hat{g}_{3}^{SV} \hat{p}_{1}^{2} , \\ R_{P - SV} \left( {c'_{16} } \right) = 2\hat{g}_{1}^{P} \hat{g}_{1}^{SV} \hat{p}_{1} \hat{p}_{2} + \hat{g}_{2}^{P} \hat{g}_{1}^{SV} \hat{p}_{1}^{2} + \hat{g}_{1}^{P} \hat{g}_{2}^{SV} \hat{p}_{1}^{2} , \\ R_{P - SV} \left( {c'_{33} } \right) = \hat{g}_{3}^{P} \hat{g}_{3}^{SV} \hat{p}_{3}^{2} , \\ R_{P - SV} \left( {c'_{36} } \right) = \hat{g}_{2}^{P} \hat{g}_{1}^{SV} \hat{p}_{3}^{2} + \hat{g}_{1}^{P} \hat{g}_{2}^{SV} \hat{p}_{3}^{2} + 2\hat{g}_{3}^{P} \hat{g}_{3}^{SV} \hat{p}_{1} \hat{p}_{2} , \\ R_{P - SV} \left( {c'_{44} } \right) = 2\left( {\hat{g}_{3}^{P} \hat{g}_{2}^{SV} \hat{p}_{2} \hat{p}_{3} + \hat{g}_{2}^{P} \hat{g}_{3}^{SV} \hat{p}_{2} \hat{p}_{3} } \right), \\ R_{P - SV} \left( {c'_{45} } \right) = 2(\hat{g}_{3}^{P} \hat{g}_{1}^{SV} \hat{p}_{2} \hat{p}_{3} + \hat{g}_{3}^{P} \hat{g}_{2}^{SV} \hat{p}_{1} \hat{p}_{3} \\ + \hat{g}_{1}^{P} \hat{g}_{3}^{SV} \hat{p}_{2} \hat{p}_{3} + \hat{g}_{2}^{P} \hat{g}_{3}^{SV} \hat{p}_{1} \hat{p}_{3} ), \\ R_{P - SV} \left( {c'_{55} } \right) = 2\left( {\hat{g}_{1}^{P} \hat{g}_{3}^{SV} \hat{p}_{ 1} \hat{p}_{3} + \hat{g}_{3}^{P} \hat{g}_{1}^{SV} \hat{p}_{ 1} \hat{p}_{3} } \right), \\ R_{P - SV} \left( {c'_{66} } \right) = 2\left( {\hat{g}_{2}^{P} \hat{g}_{1}^{SV} \hat{p}_{ 1} \hat{p}_{2} + 2\hat{g}_{1}^{P} \hat{g}_{2}^{SV} \hat{p}_{ 1} \hat{p}_{2} } \right). \hfill \\ \end{aligned} $$

(B10)

Once we obtain the radiation patterns for coefficients $c'_{ij}$, we can derive the radiation patterns for coefficients $c_{ij}$, using the chain rule, as

$$ \left[ {\begin{array}{*{20}c} {R_{P - \alpha } \left( {c_{11} } \right)} \\ {R_{P - \alpha } \left( {c_{13} } \right)} \\ {R_{P - \alpha } \left( {c_{33} } \right)} \\ {R_{P - \alpha } \left( {c_{44} } \right)} \\ {R_{P - \alpha } \left( {c_{55} } \right)} \\ \end{array} } \right] = {\varvec{\Theta}} \, \left[ {\begin{array}{*{20}c} {R_{P - \alpha } \left( {c'_{11} } \right)} \\ {R_{P - \alpha } \left( {c'_{13} } \right)} \\ {R_{P - \alpha } \left( {c'_{16} } \right)} \\ {R_{P - \alpha } \left( {c'_{33} } \right)} \\ {R_{P - \alpha } \left( {c'_{36} } \right)} \\ {R_{P - \alpha } \left( {c'_{44} } \right)} \\ {R_{P - \alpha } \left( {c'_{45} } \right)} \\ {R_{P - \alpha } \left( {c'_{55} } \right)} \\ {R_{P - \alpha } \left( {c'_{66} } \right)} \\ \end{array} } \right] \, , $$

(B11)

where ${\varvec{\Theta}}$ is a $5 \times 9$ matrix, ${\varvec{\Theta}} = [\begin{array}{*{20}c} {{\varvec{\uptheta}}_{1} } & {{\varvec{\uptheta}}_{2} } & \cdots & {{\varvec{\uptheta}}_{9} } \\ \end{array} ]$. Each column vectors are

$$ \begin{aligned} {\varvec{\uptheta}}_{1} = \left[ {\begin{array}{*{20}c} {\cos^{4} \theta } \\ {2\sin^{2} \theta \cos^{2} \theta } \\ {\sin^{4} \theta } \\ 0 \\ {\sin^{2} (2\theta )} \\ \end{array} } \right],\,{\varvec{\uptheta}}_{2} = \left[ {\begin{array}{*{20}c} 0 \\ {\cos^{2} \theta } \\ {\sin^{2} \theta } \\ { - 2\sin^{2} \theta } \\ 0 \\ \end{array} } \right],\,{\varvec{\uptheta}}_{3} = \left[ {\begin{array}{*{20}c} {\tfrac{1}{2}\cos^{2} \theta \sin \left( {2\theta } \right)} \\ { - \tfrac{1}{4}\sin \left( {4\theta } \right)} \\ { - \tfrac{1}{2}\sin^{2} \theta \sin \left( {2\theta } \right)} \\ 0 \\ { - \tfrac{1}{2}\sin \left( {4\theta } \right)} \\ \end{array} } \right],\, \hfill \\ {\varvec{\uptheta}}_{4} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ \end{array} } \right],\,\,{\varvec{\uptheta}}_{5} = \left[ {\begin{array}{*{20}c} 0 \\ {\tfrac{1}{2}\sin (2\theta )} \\ { - \tfrac{1}{2}\sin (2\theta )} \\ {\sin (2\theta )} \\ 0 \\ \end{array} } \right],\,\,{\varvec{\uptheta}}_{6} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {\cos^{2} \theta } \\ {\sin^{2} \theta } \\ \end{array} } \right],{\varvec{\uptheta}}_{7} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ { - \tfrac{1}{2}\sin (2\theta )} \\ {\tfrac{1}{2}\sin (2\theta )} \\ \end{array} } \right],\,{\varvec{\uptheta}}_{8} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {\sin^{2} \theta } \\ {\cos^{2} \theta } \\ \end{array} } \right], \hfill \\ {\varvec{\uptheta}}_{9} = \left[ {\begin{array}{*{20}c} {\tfrac{1}{4}\sin^{2} (2\theta )} \\ { - \tfrac{1}{2}\sin^{2} (2\theta )} \\ {\tfrac{1}{4}\sin^{2} (2\theta )} \\ 0 \\ {\cos^{2} (2\theta )} \\ \end{array} } \right]. \hfill \\ \end{aligned} $$

(B12)

Then we can derive the radiation patterns for modified Thomsen parameters. Assuming the background is an isotropic media with $\tilde{\varepsilon }^{\left( E \right)} = 1,$ $\tilde{\gamma }^{\left( E \right)} = 1,$ $\tilde{\delta }^{\left( E \right)} = 1$ (Kamath and Tsvankin 2016), we obtain the P-P and P-SV radiation patterns for the modified Thomsen parameters as

$$ \left[ {\begin{array}{*{20}c} {R_{P - \alpha } \left( {v_{{P{\text{v}}}} } \right)} \\ {R_{P - \alpha } \left( {v_{{S{\text{v}}}} } \right)} \\ {R_{P - \alpha } \left( {\tilde{\varepsilon }^{(E)} } \right)} \\ {R_{P - \alpha } (\tilde{\gamma }^{(E)} )} \\ {R_{P - \alpha } (\tilde{\delta }^{(E)} )} \\ \end{array} } \right] = D\left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 0 & 0 \\ 0 & { - 2} & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - 1} & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {R_{P - \alpha } (c_{11} )} \\ {R_{P - \alpha } (c_{13} )} \\ {R_{P - \alpha } (c_{33} )} \\ {R_{P - \alpha } \left( {c_{44} } \right)} \\ {R_{P - \alpha } \left( {c_{55} } \right)} \\ \end{array} } \right] \, , $$

(B13)

where $D = \rho \cdot {\text{diag\{ }}2\rho v_{{P{\text{v}}}} , { }2\rho v_{{S{\text{v}}}} , { }\rho v_{{P{\text{v}}}}^{2} , { }\rho v_{{S{\text{v}}}}^{2} , { }\tfrac{1}{2}\rho v_{{P{\text{v}}}}^{2} {\text{\} }}$. The subscript ‘$P{ - }\alpha$’ in Eqs. (B11) and (B13) represents either the P-P mode or the P-SV mode.

Focusing on the effect of the azimuth angles (shown in Fig. 4), the radiation patterns in Eq. (B13) are normalized by $D$. Therefore, only the S-wave to P-wave velocity ratio is needed for the calculation of radiation patterns which depends on the incident/reflection angles. The radiation patterns shown in Fig. 4 is the 2D case with $\phi_{\text{in}} = \phi_{\text{sc}} = 0$.