One-way wave propagation in the ray-centred coordinate system for vertical transversely isotropic media

Abstract

Seismic wave imaging in complex media requires a suitable wave field simulation method that can accurately describe real-media wave propagation. Reverse time migration is currently the preferred method; however, it is not optimal for simulating wave field propagation as it is based in solving the wave equation (qP-wave equation). The objective of this study is to develop a wave propagation simulation method to accurately describe the P-wave energy, which is less affected by complex surface conditions, and easily integrate anisotropy and attenuation by absorption media. Herein, qP-wave propagation is simulated in vertical transversely isotropic (VTI) media using the one-way wave equation in the ray-centred coordinate system (15°), which combines the flexibility of ray theory and accuracy of wave theory to describe wave propagation. Based on the qP-wave equation of VTI media, the wave equation in the ray-centred coordinate system and the one-way wave equation (15°) in the ray-centred coordinate system are derived. The 15° one-way wave equation can simulate the wave propagation process in complex media. Numerical experiments verify that the simulation results for the 15° equation in the ray-centred coordinate system exhibit high accuracy.

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Acknowledgements

The authors thank the sponsors of WPI (Wave phenomena and Intelligent Inversion Imaging) group for their financial support and helps. WPI’s research works are also financially supported by National Key R&D Program of China (Grant Number: 2018YFA0702503, 2019YFC0312004), National Natural Science Foundation of China (Grant Number: 41774126, 42074143), Southern Marine Science and Engineering Guangdong Laboratory (Zhanjiang) (ZJW-2019-04).

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Correspondence to Bohan Zhang.

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The authors declare that they have no conflict of interest.

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Communicated by Michal Malinowski (CO-EDITOR-IN-CHIEF)/Sanyi Yuan (ASSOCIATE EDITOR).

Appendices

Appendix 1: Relation between ray-centred coordinate system and Cartesian coordinate system

From Eq. (5), we can know:

$${\text{d}}{\mathbf{r}} = \left[ {\frac{{{\text{d}}{\mathbf{r}}\left( {0,0,s} \right)}}{{{\text{d}}s}} + q_{1} \frac{{{\text{d}}{\mathbf{e}}_{1} }}{{{\text{d}}s}} + q_{2} \frac{{{\text{d}}{\mathbf{e}}_{2} }}{{{\text{d}}s}}} \right]{\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} + {\mathbf{e}}_{2} {\text{d}}q_{2}$$
(26)

The basis vectors of the ray-centred coordinate system satisfies Eqs. (27) and (28):

$$\frac{{{\text{d}}{\mathbf{r}}\left( {0,0,s} \right)}}{{{\text{d}}s}} = \frac{{{\text{d}}s}}{{{\text{d}}s}}{\mathbf{t}}_{{\text{g}}} = {\mathbf{t}}_{{\text{g}}}$$
(27)
$$\frac{{{\text{d}}{\mathbf{e}}_{i} \left( s \right)}}{{{\text{d}}s}} = - \left( {{\mathbf{p}}\left( s \right) \cdot {\mathbf{p}}\left( s \right)} \right)^{ - 1} \left( {{\mathbf{e}}_{i} \left( s \right) \cdot \frac{{{\text{d}}{\mathbf{p}}\left( s \right)}}{{{\text{d}}s}}} \right)\;{\mathbf{p}}\left( s \right),\;\left( {i = 1,2} \right)$$
(28)

Substitute Eqs. (27) and (28) into Eq. (26) and we can get:

$$\begin{array}{*{20}l} {{\text{d}}{\mathbf{r}} = \left[ {{\mathbf{t}}_{{\text{g}}} - q_{1} ({\mathbf{p}} \cdot {\mathbf{p}})^{ - 1} \left( {{\mathbf{e}}_{1} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right){\mathbf{p}} - q_{2} ({\mathbf{p}} \cdot {\mathbf{p}})^{ - 1} \left( {{\mathbf{e}}_{2} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right){\mathbf{p}}} \right]{\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} + {\mathbf{e}}_{2} {\text{d}}q_{2} } \hfill \\ {\quad \, = \left[ {{\mathbf{t}}_{{\text{g}}} - q_{1} |{\mathbf{p}}|^{ - 2} \left( {{\mathbf{e}}_{1} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right)|{\mathbf{p}}|{\mathbf{t}}_{{\text{g}}} - q_{2} |{\mathbf{p}}|^{ - 2} \left( {{\mathbf{e}}_{2} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right)|{\mathbf{p}}|{\mathbf{t}}_{{\text{g}}} } \right]{\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} + {\mathbf{e}}_{2} {\text{d}}q_{2} } \hfill \\ {\quad \, = \left[ {1 - q_{1} |{\mathbf{p}}|^{ - 1} \left( {{\mathbf{e}}_{1} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right) - q_{2} |{\mathbf{p}}|^{ - 1} \left( {{\mathbf{e}}_{2} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right)} \right]{\mathbf{t}}_{{\text{g}}} {\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} + {\mathbf{e}}_{2} {\text{d}}q_{2} } \hfill \\ {\quad \, = h{\mathbf{t}}_{{\text{g}}} {\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} + {\mathbf{e}}_{2} {\text{d}}q_{2} } \hfill \\ \end{array}$$
(29)

where \({\mathbf{p}} = \left| {\mathbf{p}} \right|{\mathbf{t}}_{{\text{g}}}\) (Thomsen 1986), \(h = 1 - q_{1} |{\mathbf{p}}|^{ - 1} \left( {{\mathbf{e}}_{1} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right) - q_{2} |{\mathbf{p}}|^{ - 1} \left( {{\mathbf{e}}_{2} \cdot \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}s}}} \right)\).

Equation (29) is the relation between ray-centred coordinate system and Cartesian coordinate system.

Appendix 2: Wave equations in the ray-centred coordinate system

The dispersion relation of a VTI media is as follows:

$$k_{z}^{2} = \frac{{v^{2} }}{{v_{v}^{2} }}\left( {\frac{{\omega^{2} }}{{v^{2} }} - \frac{{\omega^{2} \left( {k_{x}^{2} + k_{y}^{2} } \right)}}{{\omega^{2} - 2v^{2} \eta \left( {k_{x}^{2} + k_{y}^{2} } \right)}}} \right),$$
(30)

where \(v\) is the NMO velocity,\(V_{{{\text{NMO}}}}\); and \(v_{v}\) is the vertical velocity,\(V_{{p_{0} }}\). Both satisfy the following relationship:

$$V_{{{\text{NMO}}}} (0) = V_{{p_{0} }} \sqrt {1 + 2\delta } ,\eta = \frac{\varepsilon - \delta }{{1 + 2\delta }}.$$

Simplification in two dimensions and assuming that\(\varepsilon = \delta\), resulting in\(\eta = 0\), leads to the following relation:

$$k_{z}^{2} = \frac{{v^{2} }}{{v_{v}^{2} }}\left( {\frac{{\omega^{2} }}{{v^{2} }} - k_{x}^{2} } \right)$$
(31)

The partial derivative of the wave field with respect to space is expressed in the ray-centred coordinate system, as follows:

$$\frac{\partial U}{{\partial x}} = \frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}} + \frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}$$
(32)

and

$$\begin{gathered} \frac{{\partial^{2} U}}{{\partial x^{2} }} = \frac{\partial }{\partial x}\left( {\frac{\partial U}{{\partial x}}} \right) = \frac{\partial }{\partial x}\left( {\frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}} + \frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}} \right) = \frac{\partial }{\partial x}\left( {\frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}}} \right) + \frac{\partial }{\partial x}\left( {\frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\, = \frac{\partial }{\partial s}\left( {\frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}}} \right)\frac{\partial s}{{\partial x}} + \frac{\partial }{\partial q}\left( {\frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}}} \right)\frac{\partial q}{{\partial x}} + \frac{\partial }{\partial s}\left( {\frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}} \right)\frac{\partial s}{{\partial x}} + \frac{\partial }{\partial q}\left( {\frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}} \right)\frac{\partial q}{{\partial x}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, = \frac{{\partial^{2} U}}{{\partial s^{2} }}\left( {\frac{\partial s}{{\partial x}}} \right)^{2} + \frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial x}}} \right) + \frac{{\partial^{2} U}}{\partial q\partial s}\frac{\partial s}{{\partial x}}\frac{\partial q}{{\partial x}} + \frac{\partial U}{{\partial s}}\frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial x}}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{\partial^{2} U}}{\partial s\partial q}\frac{\partial q}{{\partial x}}\frac{\partial s}{{\partial x}} + \frac{\partial U}{{\partial q}}\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial x}}} \right) + \frac{{\partial^{2} U}}{{\partial q^{2} }}\left( {\frac{\partial q}{{\partial x}}} \right)^{2} + \frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial x}}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\, = \frac{{\partial^{2} U}}{{\partial s^{2} }}\left( {\frac{\partial s}{{\partial x}}} \right)^{2} + \frac{{\partial^{2} U}}{{\partial q^{2} }}\left( {\frac{\partial q}{{\partial x}}} \right)^{2} + 2\frac{{\partial^{2} U}}{\partial q\partial s}\frac{\partial q}{{\partial x}}\frac{\partial s}{{\partial x}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{\partial U}{{\partial s}}\left( {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial x}}} \right)} \right) + \frac{\partial U}{{\partial q}}\left( {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial x}}} \right)} \right). \hfill \\ \end{gathered}$$
(33)

The Fourier transform of Eqs. (32) and (33) results in the following:

$$k_{x} = k_{s} \frac{\partial s}{{\partial x}} + k_{q} \frac{\partial q}{{\partial x}}$$
(34)

and

$$\begin{gathered} k_{x}^{2} = k_{s}^{2} \left( {\frac{\partial s}{{\partial x}}} \right)^{2} + k_{q}^{2} \left( {\frac{\partial q}{{\partial x}}} \right)^{2} + 2k_{q} k_{s} \frac{\partial q}{{\partial x}}\frac{\partial s}{{\partial x}} \hfill \\ \,\,\,\,\,{\kern 1pt} \,\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - ik_{s} \left[ {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial x}}} \right)} \right] \hfill \\ \,\,\,\,\,{\kern 1pt} \,\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - ik_{q} \left[ {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial x}}} \right)} \right]. \hfill \\ \end{gathered}$$
(35)

Similarly, we can obtain the following:

$$\frac{\partial U}{{\partial z}} = \frac{\partial U}{{\partial s}}\frac{\partial s}{{\partial z}} + \frac{\partial U}{{\partial q}}\frac{\partial q}{{\partial z}},$$
(36)
$$\begin{gathered} \frac{{\partial^{2} U}}{{\partial z^{2} }} = \frac{{\partial^{2} U}}{{\partial s^{2} }}\left( {\frac{\partial s}{{\partial z}}} \right)^{2} + \frac{{\partial^{2} U}}{{\partial q^{2} }}\left( {\frac{\partial q}{{\partial z}}} \right)^{2} + 2\frac{{\partial^{2} U}}{\partial q\partial s}\frac{\partial q}{{\partial z}}\frac{\partial s}{{\partial z}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{\partial U}{{\partial s}}\left( {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial z}}} \right)} \right) + \frac{\partial U}{{\partial q}}\left( {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial z}}} \right)} \right), \hfill \\ \end{gathered}$$
(37)
$$k_{z} = k_{s} \frac{\partial s}{{\partial z}} + k_{q} \frac{\partial q}{{\partial z}},$$
(38)

and

$$\begin{gathered} k_{z}^{2} = k_{s}^{2} \left( {\frac{\partial s}{{\partial z}}} \right)^{2} + k_{q}^{2} \left( {\frac{\partial q}{{\partial z}}} \right)^{2} + 2k_{q} k_{s} \frac{\partial q}{{\partial z}}\frac{\partial s}{{\partial z}} \hfill \\ \,\,\,\,\,{\kern 1pt} \,\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - ik_{s} \left[ {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial z}}} \right)} \right] \hfill \\ \,\,\,\,\,{\kern 1pt} \,\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - ik_{q} \left[ {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial z}}} \right)} \right]. \hfill \\ \end{gathered}$$
(39)

Combining Eq. (35) with Eq. (39) and the dispersion relation in the two-dimensional ray-centred coordinate system results in the following:

$$\begin{gathered} v_{v}^{2} \left( \begin{gathered} k_{s}^{2} \left( {\frac{\partial s}{{\partial z}}} \right)^{2} + k_{q}^{2} \left( {\frac{\partial q}{{\partial z}}} \right)^{2} + 2k_{s} k_{q} \frac{\partial q}{{\partial z}}\frac{\partial s}{{\partial z}} \hfill \\ {\kern 1pt} - ik_{s} \left[ {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial z}}} \right)} \right] \hfill \\ {\kern 1pt} - ik_{q} \left[ {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial z}}} \right)} \right] \hfill \\ \end{gathered} \right) \hfill \\ = \left( {\omega^{2} - v^{2} \left( \begin{gathered} k_{s}^{2} \left( {\frac{\partial s}{{\partial x}}} \right)^{2} + k_{q}^{2} \left( {\frac{\partial q}{{\partial x}}} \right)^{2} + 2k_{q} k_{s} \frac{\partial q}{{\partial x}}\frac{\partial s}{{\partial x}} \hfill \\ - ik_{s} \left[ {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial x}}} \right)} \right] \hfill \\ - ik_{q} \left[ {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial x}}} \right)} \right] \hfill \\ \end{gathered} \right)} \right) \hfill \\ \end{gathered}$$
(40)

and

$$\begin{gathered} \omega^{2} = \left[ {v_{v}^{2} \left( {\frac{\partial s}{{\partial z}}} \right)^{2} + v^{2} \left( {\frac{\partial s}{{\partial x}}} \right)^{2} } \right]k_{s}^{2} + \left[ {v_{v}^{2} \left( {\frac{\partial q}{{\partial z}}} \right)^{2} + v^{2} \left( {\frac{\partial q}{{\partial x}}} \right)^{2} } \right]k_{q}^{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \,\;\;\; + 2\left( {v_{v}^{2} \frac{\partial q}{{\partial z}}\frac{\partial s}{{\partial z}} + v^{2} \frac{\partial q}{{\partial x}}\frac{\partial s}{{\partial x}}} \right)k_{s} k_{q} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \,\;\;\; - i\left[ {v_{v}^{2} \left( {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial z}}} \right)} \right) + v^{2} \left( {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial q}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial q}{{\partial x}}} \right)} \right)} \right]k_{q} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \,\;\;\; - i\left[ {v_{v}^{2} \left( {\frac{\partial s}{{\partial z}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial z}}} \right) + \frac{\partial q}{{\partial z}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial z}}} \right)} \right) + v^{2} \left( {\frac{\partial s}{{\partial x}}\frac{\partial }{\partial s}\left( {\frac{\partial s}{{\partial x}}} \right) + \frac{\partial q}{{\partial x}}\frac{\partial }{\partial q}\left( {\frac{\partial s}{{\partial x}}} \right)} \right)} \right]k_{s} . \hfill \\ \end{gathered}$$
(41)

In the two-dimensional ray-centred coordinate system, we have the following:

$${\text{d}}{\mathbf{r}} = h{\mathbf{t}}_{{\text{g}}} {\text{d}}s + {\mathbf{e}}_{1} {\text{d}}q_{1} { = }h{\mathbf{t}}_{{\text{g}}} {\text{d}}s + {\mathbf{n}}{\text{d}}q,$$
(42)

where \({\mathbf{t}}_{{\text{g}}} = \left( {t_{1} ,\;t_{2} } \right)^{{\text{T}}}\) and\({\mathbf{n}} = \left( {n_{1} ,\;n_{2} } \right)^{{\text{T}}}\).

This leads to the following coordinate transformation:

$$\left( \begin{gathered} {\text{d}}x \hfill \\ {\text{d}}z \hfill \\ \end{gathered} \right) = \left( {\begin{array}{*{20}c} {ht_{1} } & {n_{1} } \\ {ht_{2} } & {n_{2} } \\ \end{array} } \right)\left( \begin{gathered} {\text{d}}s \hfill \\ {\text{d}}q \hfill \\ \end{gathered} \right)$$
(43)

and

$$\left( \begin{gathered} {\text{d}}s \hfill \\ {\text{d}}q \hfill \\ \end{gathered} \right) = \frac{1}{h}\left( {\begin{array}{*{20}c} { - n_{2} } & {n_{1} } \\ {ht_{2} } & { - ht_{1} } \\ \end{array} } \right)\left( \begin{gathered} {\text{d}}x \hfill \\ {\text{d}}z \hfill \\ \end{gathered} \right).$$
(44)

This results in the following:

$$\frac{\partial s}{{\partial x}} = - \frac{{n_{2} }}{h},\;\frac{\partial s}{{\partial z}} = \frac{{n_{1} }}{h},\;\frac{\partial q}{{\partial x}} = t_{2} ,\;\frac{\partial q}{{\partial z}} = - t_{1} .$$
(45)

Substituting the coordinate transformation into the dispersion relation, the following equation is obtained:

$$\begin{gathered} \omega^{2} = \left( {v_{v}^{2} \frac{{n_{1}^{2} }}{{h^{2} }} + v^{2} \frac{{n_{2}^{2} }}{{h^{2} }}} \right)\,k_{s}^{2} + \left( {v_{v}^{2} t_{1}^{2} + v^{2} t_{2}^{2} } \right)\;k_{q}^{2} - \frac{2}{h}\left( {v_{v}^{2} t_{1} n_{1} + v^{2} t_{2} n_{2} } \right)\;k_{s} k_{q} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - i\left[ {v_{v}^{2} \left( {\frac{{ - n_{1} }}{h}\frac{\partial }{\partial s}\left( {t_{1} } \right) + t_{1} \frac{\partial }{\partial q}\left( {t_{1} } \right)} \right) + v^{2} \left( {\frac{{ - n_{2} }}{h}\frac{\partial }{\partial s}\left( {t_{2} } \right) + t_{2} \frac{\partial }{\partial q}\left( {t_{2} } \right)} \right)} \right]k_{q} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - i\left[ {v_{v}^{2} \left( {\frac{{n_{1} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{1} }}{h}} \right) - t_{1} \frac{\partial }{\partial q}\left( {\frac{{n_{1} }}{h}} \right)} \right) + v^{2} \left( {\frac{{n_{2} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{2} }}{h}} \right) - t_{2} \frac{\partial }{\partial q}\left( {\frac{{n_{2} }}{h}} \right)} \right)} \right]k_{s} . \hfill \\ \end{gathered}$$
(46)

Equation (46) can be written as follows:

$$\omega^{2} = c_{ss} k_{s}^{2} + c_{qq} k_{q}^{2} + c_{sq} k_{s} k_{q} - ic_{q} k_{q} - ic_{s} k_{s} ,$$
(47)

where

$$\begin{gathered} c_{ss} = v_{v}^{2} \left( {\frac{{n_{1} }}{h}} \right)^{2} + v^{2} \left( {\frac{{n_{2} }}{h}} \right)^{2} \hfill \\ c_{qq} = v_{v}^{2} \left( {t_{1} } \right)^{2} + v^{2} \left( {t_{2} } \right)^{2} \hfill \\ c_{sq} = - 2\left( {v_{v}^{2} \frac{{t_{1} n_{1} }}{h} + v^{2} \frac{{t_{2} n_{2} }}{h}} \right) \hfill \\ c_{q} = v_{v}^{2} \left( { - \frac{{n_{1} }}{h}\frac{\partial }{\partial s}\left( {t_{1} } \right) + t_{1} \frac{\partial }{\partial q}\left( {t_{1} } \right)} \right) + v^{2} \left( { - \frac{{n_{2} }}{h}\frac{\partial }{\partial s}\left( {t_{2} } \right) + t_{2} \frac{\partial }{\partial q}\left( {t_{2} } \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = - \left[ {v_{v}^{2} \frac{{n_{1} }}{h}\frac{{\partial \left( {t_{1} } \right)}}{\partial s} + v^{2} \frac{{n_{2} }}{h}\frac{{\partial \left( {t_{2} } \right)}}{\partial s}} \right] \hfill \\ c_{s} = v_{v}^{2} \left( {\frac{{n_{1} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{1} }}{h}} \right) - t_{1} \frac{\partial }{\partial q}\left( {\frac{{n_{1} }}{h}} \right)} \right) + v^{2} \left( {\frac{{n_{2} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{2} }}{h}} \right) - t_{2} \frac{\partial }{\partial q}\left( {\frac{{n_{2} }}{h}} \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = v_{v}^{2} \left( {\frac{{n_{1} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{1} }}{h}} \right) - t_{1} n_{1} \frac{\partial }{\partial q}\left( \frac{1}{h} \right)} \right) + v^{2} \left( {\frac{{n_{2} }}{h}\frac{\partial }{\partial s}\left( {\frac{{n_{2} }}{h}} \right) - t_{2} n_{2} \frac{\partial }{\partial q}\left( \frac{1}{h} \right)} \right). \hfill \\ \end{gathered}$$

Equation (47) is the dispersion relation of the two-way wave equation in the ray-centred coordinate system for anisotropic media.

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Zhang, B., Wang, H. One-way wave propagation in the ray-centred coordinate system for vertical transversely isotropic media. Acta Geophys. (2021). https://doi.org/10.1007/s11600-021-00596-4

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Keywords

  • Ray-centred coordinate system
  • Anisotropic media
  • Paraxial one-way wave forward modelling