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.
