The dispersion equation under two-dimensional VTI medium conditions:

${\omega }^4-\left(1+2\epsilon \right)v_{qp}^2{\omega }^2{k}_x^2-v_{qp}^2{\omega }^2{k}_z^2+2\left(\epsilon -\delta \right)v_{qp}^4{k}_x^2{k}_z^2=0$. (1)

where $\omega$ is the circular frequency, $v_{qp}$ is the velocity parameter, $\epsilon$ and $\delta$ are the anisotropy parameters, and $k_x$ and $k_z$ are divided into transverse and vertical wavenumbers. After simplification, I obtain:

$k_x^2+k_z^2=\frac{{\omega }^2/v_{qp}^2-\chi k_x^2+\eta k_z^2}{\eta }$, (2)

where $\chi =1+2\epsilon$, $\eta =1+2\left(\epsilon -\delta \right)\frac{v_{qp}^2}{{\omega }^2}{k}_x^2$.

Transform Formula (2) into the frequency space domain and add spatial coordinates:

$\begin{array}{l}\frac{{\partial }^2}{\partial x^2}p\left(x,z;\omega \right)+\frac{{\partial }^2}{\partial z^2}p\left(x,z;\omega \right)\\ =-\frac{\left[\frac{{\omega }^2}{v_{qp}^2\left(x,z\right)}+\chi \left(x,z\right)\frac{{\partial }^2}{\partial x^2}+\eta \left(x,z\right)\frac{{\partial }^2}{\partial x^2}\right]p\left(x,z;\omega \right)}{\eta \left(x,z\right)},\end{array}$ (3)

where $\chi \left(x,z\right)=1+2\epsilon \left(x,z\right)$, $\eta \left(x,z\right)=1+2\left[\epsilon \left(x,z\right)-\delta \left(x,z\right)\right]\frac{v_{qp}^2\left(x,z\right)}{{\omega }^2}\frac{{\partial }^2}{\partial x^2}$, $p\left(x,z;\omega \right)$ is the frequency space domain wavefield.

I set $v_0\left(z\right)$ as the background velocity, ${\epsilon }_0\left(z\right)$ and ${\delta }_0\left(z\right)$ as background anisotropy parameters, which are only functions of depth $z$ and are constant within each layer. Let:

$F_0\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)=\frac{\left[\frac{{\omega }^2}{v_0^2\left(z\right)}+{\chi }_0\left(z\right)\frac{{\partial }^2}{\partial x^2}+{\eta }_0\left(x,z\right)\frac{{\partial }^2}{\partial x^2}\right]}{{\eta }_0\left(x,z\right)}$, (4)

where ${\eta }_0\left(x,z\right)=1+2\left[{\epsilon }_0\left(x,z\right)-{\delta }_0\left(x,z\right)\right]\frac{v_0^2\left(x,z\right)}{{\omega }^2}\frac{{\partial }^2}{\partial x^2}$, I add (4) to Equation (3) and further simplify it to obtain:

$\begin{array}{l}\frac{{\partial }^2}{\partial x^2}p\left(x,z;\omega \right)+\frac{{\partial }^2}{\partial z^2}p\left(x,z;\omega \right)+F_0\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)p\left(x,z;\omega \right)\\ =-\left(F\left(x,z,v_{qp},\epsilon ,\delta \right)-F_0\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)\right)p\left(x,z;\omega \right).\end{array}$ (5)

Equation (5) is the non-homogeneous Helmholtz equation for qP waves in VTI media, which can be rewritten as:

$\left({\nabla }^2+F_0^2\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)\right)p\left(x,z;\omega \right)=-O\left(x,z,v_{qp},\epsilon ,\delta \right)p\left(x,z;\omega \right)$, (6)

where $O\left(x,z,v_{qp},\epsilon ,\delta \right)=F\left(x,z,v_{qp},\epsilon ,\delta \right)-F_0\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)$ is the scattering source term (slowness perturbation and anisotropy parameter perturbation), ${\nabla }^2=\partial /\partial x^2+\partial /\partial z^2$ is the Laplacian operator.

Decompose the wavefield $p\left(x,z;\omega \right)$ of the qP wave in Formula (6) into two parts: the background velocity and background anisotropic reference medium wavefield $p_0\left(x,z;\omega \right)$, and the anisotropic perturbation and velocity perturbation scattering wavefield $p_F\left(x,z;\omega \right)$. Then, rewrite Equation (6) as:

$\begin{array}{l}\left({\nabla }^2+F_0^2\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)\right)\left(p_0\left(x,z;\omega \right)+p_F\left(x,z;\omega \right)\right)\\ =-O\left(x,z,v_{qp},\epsilon ,\delta \right)p\left(x,z;\omega \right).\end{array}$ (7)

The wavefield in an anisotropic constant velocity background medium satisfies the homogeneous Helmholtz equation:

$\left({\nabla }^2+F_0^2\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)\right)p_0\left(x,z;\omega \right)=0$. (8)

If I substitute Formula (8) into (7), the equation satisfied by the scattered wavefield can be written as:

$\left({\nabla }^2+F_0^2\left(x,z,v_0,{\epsilon }_0,{\delta }_0\right)\right)p_F\left(x,z;\omega \right)=-O\left(x,z,v_{qp},\epsilon ,\delta \right)p\left(x,z;\omega \right)$. (9)

Equation (9) can be solved using the Green’s function method, resulting in:

$p_F\left(x,z;\omega \right)={\displaystyle {\int }_{\Omega }G\left(x,z;x^{\prime },z^{\prime };\omega \right)O\left(x,z,v_{qp},\epsilon ,\delta \right)p\left(x,z;\omega \right)\text{d}{x}^{\prime }\text{d}{z}^{\prime }}$. (10)

Equation (10) is the formula for calculating the scattering field. The total wavefield of qP waves is:

$p\left(x,z;\omega \right)=p_0\left(x,z;\omega \right)+{\displaystyle {\int }_{\Omega }G\left(x,z;x^{\prime },z^{\prime };\omega \right)O\left(x,z,v_{qp},\epsilon ,\delta \right)p\left(x,z;\omega \right)\text{d}{x}^{\prime }\text{d}{z}^{\prime }}$. (11)

Equation (11) is the Lippmann-Schwinger (L-S) equation for VTI media.

If the scattering in VTI is relatively weak, the background wavefield $p_0\left(x,z;\omega \right)$ is generally used instead of the total wavefield $p\left(x,z;\omega \right)$. To improve the adaptability of the Born approximation in complex media, the De Wolf approximation is introduced. Firstly, simplify Equation (11):

$p=p_0+G_{0}Op$, (12)

where O is a diagonal operator in the spatial domain, $G_0$ is a non-diagonal integral operator. If the reference medium is uniform, then $G_0$ is the integral with Green’s function $G\left(x,z;x^{\prime },z^{\prime };\omega \right)$ as the kernel.

The approximate expression for De Wolf (Wu, 1994) [17] is:

$p=p_f+G_f{O}_b{p}_f$, (13)

where $G_f={\displaystyle \underset{m=0}{\overset{M}{\sum }}{\left[G_0{O}_f\right]}^m{G}_0}$, $p_f={\displaystyle \underset{m=0}{\overset{M}{\sum }}{\left[G_0{O}_f\right]}^m{p}_0}$.

The De Wolf approximation is a method based on multiple forward scattering and single backward scattering approximations, which ignores internal reverberation and partitions the model. $z_0$ is the entrance of the first thin plate, $z^{\prime }$ is the exit of the first thin slab and also the entrance of the second thin plate; $z^{″}$ is the outlet of the second thin slab. The position of the outgoing wave field is $\left(x^{*},z^{*}\right)$, and the position of the incident wave is $\left(x^{\prime },z^{\prime }\right)$. For the forward scattering field (positive direction of the Z-axis), the outgoing field is located at $\left(x^{*}=x^{″},z^{*}=z^{″}\right)$.

By using local Born approximations in each layer of the thin slab, the scattered wavefield can be written as:

$p_s\left(x^{*},z^{*};\omega \right)=k_0^2\left(z;\omega \right){\displaystyle {\iint }_V{\tilde{G}}_0\left(x^{*},z^{*};r^{\prime };\omega \right)O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)p_0\left(r^{\prime };\omega \right)\text{d}{r}^{\prime }}$, (14)

where $r^{\prime }=\left(x^{\prime },z^{\prime }\right)$u3002 I use the Fourier transform to transform Equation (14):

$p_s\left(k_x^{*},z^{*};\omega \right)=k_0^2\left(z;\omega \right){\displaystyle {\int }_{z^{\prime }}^{z^{*}}\text{d}z{\displaystyle \int \text{d}x{\tilde{G}}_0\left(k_x^{*},z^{*};r^{\prime };\omega \right)}}\times O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)p_0\left(r^{\prime };\omega \right)$. (15)

The entrance wavefield $p_0\left(r^{\prime };\omega \right)$ of a thin slab can be written as (Wu et al., 2007) [18] :

$p_0\left(r^{\prime };\omega \right)=\frac{1}{2\pi }{\displaystyle \iint \text{d}{k^{\prime }}_x{p}_0\left({k^{\prime }}_x,z^{\prime };\omega \right){\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\left(z-z^{\prime }\right)}{\text{e}}^{\text{i}{k^{\prime }}_x{x}^{\prime }}}$. (16)

I substitute Equation (16) into Equation (15):

$\begin{array}{c}{p}_F\left(k_x^{*},z^{*};\omega \right)={\displaystyle {\iiint }_{\Omega }G\left(k_x^{*},z^{*};r^{\prime };\omega \right)O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)}\\ \times \frac{1}{2\pi }{\displaystyle \iint \text{d}{k^{\prime }}_{xy}p\left({k^{\prime }}_x,z^{\prime };\omega \right){\text{e}}^{i{k}_{z0}\left(z^{\prime }\right)}{\text{e}}^{\text{i}{k^{\prime }}_{x}x}}.\end{array}$ (17)

For convenience in calculation, the Green function of the local background medium is used here instead of the locally renormalized Green function. The expression for the Green function of the locally uniform background medium is as follows:

$G_0\left(k_x^{*},z^{*};r^{\prime };\omega \right)=\frac{\text{i}}{2k_{z0}\left(z^{\prime }\right)}{\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\left(z^{*}-z^{\prime }\right)}{\text{e}}^{-\text{i}{k}_x^{*}{x}^{\prime }}$, (18)

where $k_{z0}\left(z^{\prime }\right)=\sqrt{F_0^2\left(r^{\prime },v_0,{\epsilon }_0,{\delta }_0\right)-{k^{\prime }}_x^2}$. $k_{z0}\left(z^{\prime }\right)$ can be written as:

$\begin{array}{c}{k}_{z0}\left(z^{\prime }\right)=\sqrt{\frac{\left[\frac{{\omega }^2}{v_0^2\left(z^{\prime }\right)}-{\chi }_0\left(z^{\prime }\right){k^{\prime }}_x^2+{\eta }_0\left({k^{\prime }}_x,z^{\prime }\right){k^{\prime }}_x^2\right]}{{\eta }_0\left({k^{\prime }}_x,z^{\prime }\right)}-{k^{\prime }}_x^2}\\ =\frac{\omega }{v_0\left(z^{\prime }\right)}\sqrt{\frac{1-\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right)\frac{v_0^2\left(z^{\prime }\right){k^{\prime }}_x^2}{{\omega }^2}}{1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right)\frac{v_0^2\left(z^{\prime }\right){k^{\prime }}_x^2}{{\omega }^2}}},\end{array}$ (19)

where ${\eta }_0\left({k^{\prime }}_x^2,z\right)=1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right)\frac{v_0^2\left(z^{\prime }\right){k^{\prime }}_x^2}{{\omega }^2}$.

$O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)=F^2\left(r^{\prime },v_{qp},\epsilon ,\delta \right)-F_0^2\left(r^{\prime },v_0,{\epsilon }_0,{\delta }_0\right)$ can be written as:

$\begin{array}{l}O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)=\frac{\left[\frac{{\omega }^2}{v_{qp}^2\left(r^{\prime }\right)}-\chi \left(r^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}+\eta \left(x^{\prime },z^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}\right]}{\eta \left(x^{\prime },y^{\prime },z^{\prime }\right)}\\ -\frac{\left[\frac{{\omega }^2}{v_0^2\left(z^{\prime }\right)}-{\chi }_0\left(z^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}+{\eta }_0\left(z^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}\right]}{{\eta }_0\left(z^{\prime }\right)}.\end{array}$ (20)

I further simplify Equation (20) as follows:

$O\left(r^{\prime },v_{qp},\epsilon ,\delta \right)=\frac{A{\eta }_0\left(z^{\prime }\right)-A_0\eta \left(x^{\prime },z^{\prime }\right)}{\eta \left(x^{\prime },z^{\prime }\right){\eta }_0\left(z^{\prime }\right)}$, (21)

where $A=\frac{{\omega }^2}{v_{qp}^2\left(r^{\prime }\right)}-\chi \left(r^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}+\eta \left(r^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}$ $A_0=\frac{{\omega }^2}{v_0^2\left(z^{\prime }\right)}-{\chi }_0\left(z^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}+{\eta }_0\left(z^{\prime }\right)\frac{{\partial }^2}{\partial {x^{\prime }}^2}$, to simplify the calculation, the denominator term is replaced with ${\eta }_0\left(z^{\prime }\right)$ for $\eta \left(x^{\prime },z^{\prime }\right)$. To avoid the calculation of term $\frac{{\partial }^2}{\partial {x^{\prime }}^2}$, it is transformed into the frequency wavenumber domain, making term $O\left({k^{\prime }}_x^2,z^{\prime },v_{qp},\epsilon ,\delta \right)$ a dual domain calculation.

Equation (21) can be simplified as follows:

$O\left({k^{\prime }}_x^2,z^{\prime },v_{qp},\epsilon ,\delta \right)=a_s\frac{\omega }{v_0\left(z^{\prime }\right)}\Delta s+a_{\epsilon }\left[\epsilon \left(r^{\prime }\right)-{\epsilon }_0\left(z^{\prime }\right)\right]+a_{\delta }\left[\delta \left(r^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right]$, (22)

where:

$\{\begin{array}{l}{a}_s=k_0\left(z^{\prime }\right)\frac{\left[1-4\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2+2\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right)\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^4\right]}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\frac{\omega }{v_0\left(z^{\prime }\right)}\\ a_{\epsilon }=-\frac{2k_0^2\left(z^{\prime }\right)\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right){\lambda }^4}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\\ a_{\delta }=-\frac{2k_0^2\left(z^{\prime }\right)\left[1-\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right){\lambda }^2\right]{\lambda }^2}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\end{array}$, (23)

where $\Delta s=v_0^2\left(z^{\prime }\right)/v_{qp}^2\left(r^{\prime }\right)-1$, $\lambda ={k^{\prime }}_x/k_0\left(z^{\prime }\right)$, $k_0\left(z^{\prime }\right)=\omega /v_0\left(z^{\prime }\right)$.

By substituting Equations (18) and (22) into Equation (17), Equation (17) can be simplified as follows:

$\begin{array}{c}{p}_F\left(k_x^{*},z^{*};\omega \right)=\frac{\text{i}}{4\pi k_{z0}\left(z^{\prime }\right)}{\text{e}}^{\text{i}{k}_{z_0}\left(z^{\prime }\right)\left(z^{*}-z^{\prime }\right)}{\displaystyle \int \text{d}{k^{\prime }}_x}\times {\displaystyle {\int }_{z^{\prime }}^{z^{*}}\text{d}z{\displaystyle \int \text{d}{x}^{\prime }}}O\left({k^{\prime }}_x,z^{\prime },v_{qp},\epsilon ,\delta \right)\\ \times p_0\left({k^{\prime }}_x,z^{\prime };\omega \right){\text{e}}^{-\text{i}\left(k_{z_0}\left(z^{*}\right)-k_{z_0}\left(z^{\prime }\right)\right)z^{\prime }}{\text{e}}^{-\text{i}\left(k_x^{*}-{k^{\prime }}_x\right)x^{\prime }},\end{array}$ (24)

where $\Delta z=z^{*}-z^{\prime }$, Equation (24) is an approximate formula for double domain thin slabs.

To improve computational efficiency, the thin slab is compressed into a screen, which compresses the three-dimensional thin slab into a two-dimensional screen to accelerate computation speed. The transverse wave numbers $k_x$ at the inlet and outlet of the thin slab are much smaller than the background wave numbers $k_0\left(z^{\prime }\right)$ inside the thin slab (Wu et al., 2007) [18] .

I use Taylor expansion $k_{z0}\left(z^{\prime }\right)$ term to obtain:

$k_{z0}\left(z^{\prime }\right)=\frac{\omega }{v_0\left(z^{\prime }\right)}\sqrt{1-\frac{\left(1+2{\delta }_0\left(z^{\prime }\right)\right)v_0^2\left(z^{\prime }\right)\frac{{k^{\prime }}_x^2}{{\omega }^2}}{1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right)v_0^2\left(z^{\prime }\right)\frac{{k^{\prime }}_x^2}{{\omega }^2}}}\approx \frac{\omega }{v_0\left(z^{\prime }\right)}\left(1-0.5A^{\prime }\right)$, (25)

where $A^{\prime }=\frac{\left(1+2{\delta }_0\left(z^{\prime }\right)\right)v_0^2\left(z^{\prime }\right){k^{\prime }}_x^2/{\omega }^2}{1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right)v_0^2\left(z^{\prime }\right){k^{\prime }}_x^2/{\omega }^2}$.

Forward scattering $k_z\left(z;\omega \right)-{k^{\prime }}_z\left(z^{\prime };\omega \right)$ can be simplified as:

$k_z\left(z;\omega \right)-{k^{\prime }}_z\left(z^{\prime };\omega \right)\approx \left(k_0\left(z;\omega \right)-k_0\left(z^{\prime };\omega \right)\right)-\left(A^{\prime }-A\right)\approx 0$. (26)

One backscattering $-k_z\left(z;\omega \right)-{k^{\prime }}_z\left(z^{\prime };\omega \right)$ can be simplified as:

$-k_z\left(z;\omega \right)-{k^{\prime }}_z\left(z^{\prime };\omega \right)\approx -\left(k_0\left(z;\omega \right)+k_0\left(z^{\prime };\omega \right)\right)-\left(A^{\prime }+A\right)\approx 2k_0\left(z;\omega \right)$, (27)

when $\left(x^{*},z^{*}\right)=\left(x^{″},z^{″}\right)$, $\left(x,z\right)=\left(x^{\prime },z^{\prime }\right)$, multiple forward scatterings and one backward scattering can be approximated as:

$\begin{array}{c}{p}_F\left({k^{″}}_x,z^{″};\omega \right)\approx \frac{\text{1}}{2k_{z0}\left(z\right)}{\text{e}}^{\text{i}{k}_z\left(z^{\prime };\omega \right)z^{″}-\text{i}{k}_z\left(z^{\prime };\omega \right)z^{\prime }}\\ \times {\displaystyle \iint {\text{e}}^{-\text{i}{k^{″}}_x{x}^{\prime }}}\left[\text{i}O\left({k^{\prime }}_x,z^{\prime },v_{qp},\epsilon ,\delta \right)\Delta z\right]\text{d}{x}^{\prime }\\ \times \frac{1}{2\pi }{\displaystyle \iint \text{d}{k^{\prime }}_x{p}_0\left({k^{\prime }}_x,z^{\prime };\omega \right){\text{e}}^{\text{i}{k^{\prime }}_x{x}^{\prime }}},\end{array}$ (28)

$\begin{array}{c}{p}_b\left({k^{\prime }}_x,z^{\prime };\omega \right)\approx \frac{1}{2k_z\left(z^{\prime };\omega \right)}\text{sinc}\left(\Delta z\right){\text{e}}^{\text{i}{k}_0\left(z^{\prime }\right)\Delta z}{\text{e}}^{\text{i}{k}_z\left(z^{\prime };\omega \right)z^{″}-\text{i}{k}_z\left(z^{\prime };\omega \right)z^{\prime }}\\ \times {\displaystyle \iint {\text{e}}^{-\text{i}{k^{″}}_x{x}^{\prime }}}\left[\text{i}O\left({k^{\prime }}_x,z^{\prime },v_{qp},\epsilon ,\delta \right)\Delta z\right]\text{d}{x}^{\prime }\\ \times \frac{1}{2\pi }{\displaystyle \iint \text{d}{k^{\prime }}_x{p}_0\left({k^{\prime }}_x,z^{\prime };\omega \right){\text{e}}^{\text{i}{k^{\prime }}_x{x}^{\prime }}},\end{array}$ (29)

where $\text{sinc}\left(\Delta z\right)=\sin \left(\Delta z\right)/\Delta z$.

Performing the small-angle approximation on the multiple forward and one backward operator mentioned above, specifically by using $F_s\left(x^{\prime },z^{\prime }\right)=\frac{1}{2v_0\left(z^{\prime }\right)}\left(\frac{v_0^2\left(z^{\prime }\right)}{v^2\left(x^{\prime },z^{\prime }\right)}-1\right)\approx \frac{1}{v\left(x^{\prime },z^{\prime }\right)}-\frac{1}{v_0\left(z^{\prime }\right)}$ and adding or subtracting the perturbation term $\text{i}\omega F_S\left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)$ in the expression. Therefore, the multiple forward scattering field is:

$\begin{array}{l}{p}_F\left({k^{″}}_x,z^{″};\omega \right)={\text{e}}^{\text{i}{k}_{z_0}\left(z^{\prime }\right)\Delta z}\{FT\left[\text{i}\omega F_s\left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{s1}\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}{F}_s\left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{\epsilon }\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}\Delta \epsilon \left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{\delta }\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}\Delta \delta \left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\}.\end{array}$ (30)

The backscattering field is:

$\begin{array}{l}{p}_b\left({k^{\prime }}_x,z^{\prime };\omega \right)=\text{sinc}\left(\Delta z\right){\text{e}}^{\text{i}{k}_0\left(z^{\prime }\right)\Delta z}{\text{e}}^{\text{i}{k}_{z_0}\left(z^{\prime }\right)\Delta z}\\ \times \{FT\left[\text{i}\omega F_s\left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{s1}\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}{F}_s\left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{\epsilon }\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}\Delta \epsilon \left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\\ +F_{\delta }\left({k^{\prime }}_x,z^{\prime }\right)FT\left[\text{i}\Delta \delta \left(x^{\prime },z^{\prime }\right)\Delta z{p}_0\left(x^{\prime },z^{\prime };\omega \right)\right]\}.\end{array}$ (31)

Among them, $FT$ represents the Fourier transform, $F{T}^{-1}$ represents the inverse Fourier transform, and the expressions for $F_{s1}\left({k^{\prime }}_x,z^{\prime }\right)$, $F_{\epsilon 1}\left({k^{\prime }}_x,z^{\prime }\right)$, and $F_{\delta 1}\left({k^{\prime }}_x,z^{\prime }\right)$ are as follows:

$\{\begin{array}{l}{F}_{s1}\left({k^{\prime }}_x,{k^{\prime }}_y,z^{\prime }\right)=\frac{k_0\left(z^{\prime }\right)}{k_{z0}\left(z^{\prime }\right)}\frac{\left[1-4\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2+2\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right)\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^4\right]}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\omega -\omega \\ F_{\epsilon 1}\left({k^{\prime }}_x,{k^{\prime }}_y,z^{\prime }\right)=-\frac{k_0^2\left(z^{\prime }\right)}{k_{z0}\left(z^{\prime }\right)}\frac{\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right){\lambda }^4}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\\ F_{\delta 1}\left({k^{\prime }}_x,{k^{\prime }}_y,z^{\prime }\right)=-\frac{k_0^2\left(z^{\prime }\right)}{k_{z0}\left(z^{\prime }\right)}\frac{\left[1-\left(1+2{\epsilon }_0\left(z^{\prime }\right)\right){\lambda }^2\right]{\lambda }^2}{{\left[1-2\left({\epsilon }_0\left(z^{\prime }\right)-{\delta }_0\left(z^{\prime }\right)\right){\lambda }^2\right]}^2}\end{array}$. (32)

I set $\left(x^{*},z^{*}\right)=\left(x^{″},z^{″}\right)$, and the entire forward scattering field $p\left(r^{″};\omega \right)$ can be expressed as:

$p\left(r^{″};\omega \right)=p_0\left(r^{″};\omega \right)+p_s\left(r^{″};\omega \right)+p_{\epsilon }\left(r^{″};\omega \right)+p_{\delta }\left(r^{″};\omega \right)$, (33)

where

$\{\begin{array}{l}{p}_0\left(r^{″};\omega \right)=F{T}^{-1}\left\{{\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\Delta z}FT\left[{\text{e}}^{\text{i}\omega F_s\left(r^{\prime }\right)\Delta z}{p}_0\left(r^{\prime };\omega \right)\right]\right\}\\ p_s\left(r^{″};\omega \right)=F{T}^{-1}\left\{{\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\Delta z}{F}_{s1}\left({k^{\prime }}_x,z^{\prime }\right)FT\left[{\text{e}}^{\text{i}\omega F_s\left(r^{\prime }\right)\Delta z}\text{i}\omega \Delta s\left(r^{\prime }\right)\Delta z{p}_0\left(r^{\prime };\omega \right)\right]\right\}\\ p_{\epsilon }\left(r^{″};\omega \right)=F{T}^{-1}\left\{{\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\Delta z}{F}_{\epsilon 1}\left({k^{\prime }}_x,z^{\prime }\right)FT\left[{\text{e}}^{\text{i}\omega F_s\left(r^{\prime }\right)\Delta z}\text{i}\Delta \epsilon \left(r^{\prime }\right)\Delta z{p}_0\left(r^{\prime };\omega \right)\right]\right\}\\ p_{\delta }\left(r^{″};\omega \right)=F{T}^{-1}\left\{{\text{e}}^{\text{i}{k}_{z0}\left(z^{\prime }\right)\Delta z}{F}_{\delta 1}\left({k^{\prime }}_x,z^{\prime }\right)FT\left[{\text{e}}^{\text{i}\omega F_s\left(r^{\prime }\right)\Delta z}\text{i}\Delta \delta \left(r^{\prime }\right)\Delta z{p}_0\left(r^{\prime };\omega \right)\right]\right\}\end{array}$, (34)

where $\Delta z=z^{″}-z^{\prime }$， $z^{\prime }$ is the depth position of the incident wavefield, and $z^{″}$ is the depth position of the outgoing wavefield. Equation (34) is a generalized screen approximation extension operator for VTI media based on the De Wolf approximation.

However, due to the existence of term $1/k_{z0}$, the operator exhibits spatial aliasing effects, making terms $v_0^2\left(z\right)k_x^2/{\omega }^2={\sin }^2\theta$ and $k_{z0}$ equivalent to:

$\begin{array}{c}{k}_{z0}=\frac{\omega }{v_0\left(z\right)}\sqrt{\frac{1-\left(1+2{\epsilon }_0\left(z\right)\right){\sin }^2\theta }{1-2\left({\epsilon }_0(z)-{\delta }_0\left(z\right)\right){\sin }^2\theta }}\\ =\frac{\omega }{v_0\left(z\right)}\sqrt{1-\frac{\left(1+2{\delta }_0\left(z\right)\right){\sin }^2\theta }{1-2\left({\epsilon }_0(z)-{\delta }_0\left(z\right)\right){\sin }^2\theta }}\\ =\frac{\omega }{v_0\left(z\right)}\sqrt{1-\frac{1+2{\delta }_0\left(z\right)}{1/{\sin }^2\theta -2\left({\epsilon }_0(z)-{\delta }_0\left(z\right)\right)}}.\end{array}$ (35)

As the angle approaches 90 degrees, $\sin \theta$ gradually approaches 1, then:

$\frac{1+2{\delta }_0\left(z\right)}{\frac{1}{{\sin }^2\theta }-2\left({\epsilon }_0\left(z\right)-{\delta }_0\left(z\right)\right)}=\frac{1+2{\delta }_0\left(z\right)}{\left(1+2{\delta }_0\left(z\right)\right)-2{\epsilon }_0\left(z\right)}$, (36)

when ${\epsilon }_0\left(z\right)\ge 0$, $\frac{1+2{\delta }_0\left(z\right)}{\left(1+2{\delta }_0\left(z\right)\right)-2{\epsilon }_0\left(z\right)}\ge 1$. Therefore, when the propagation angle approaches 90 degrees, the $1/k_{z0}$ term will still exhibit an unstable phenomenon, namely the spatial aliasing phenomenon. Referring to the isotropic medium processing method, $M^{\prime }$ is introduced here to change the filtering characteristics of the operator on the dip angle of the formation. The improved $1/k_{z0}$ is replaced by $M$:

$\begin{array}{c}M=\frac{1}{k_{z0}}\cdot M^{\prime }=\frac{1}{k_{z0}}\frac{k_{z0}}{\sqrt{b^2{k}_x^2+k_{z0}^2}}\\ =\frac{1}{\sqrt{b^2{k}_x^2+k_{z0}^2}}=\frac{1}{\sqrt{{\omega }^2/v_0^2\left(z\right)+\left(b^2-c\right)k_x^2}},\end{array}$ (37)

where $c=\sqrt{\frac{{\omega }^2}{v_0^2\left(z\right)}\frac{1+2{\delta }_0\left(z\right)}{1/{\sin }^2\theta -2\left({\epsilon }_0\left(z\right)-{\delta }_0\left(z\right)\right)}}$. When $b=c$ occurs, spatial aliasing disappears, but it is often not processed cleanly during actual calculations. When $b>c$ occurs, it can further suppress the spatial aliasing on both wings, but also remove some effective signals. When $b<c$ occurs, the spatial aliasing effect is more severe. The longitudinal amplitude attenuation factor is:

$l\left(z\right)=1-\left[1-s_z\right]\sin \left(\frac{\pi }{2}\frac{j}{N-1}\right)$. (38)

Among them, $l\left(z\right)$ is the correction factor, which takes different values at different depths; N is the grid point in the vertical direction, with specific values of $j=0,1,\cdots ,N-1$; $s_z$ is the attenuation coefficient, with slightly different values at different depths. Assume the top layer is 1.0 and the bottom layer is a very small value.

Figure 1(a) is a two-dimensional velocity model. The grid points of the model in the x and z directions are 401 and 301, respectively; the grid spacing is 5.0 m; the VTI (vertically transversely isotropic) medium is a transversely isotropic model characterized by a vertical axis of symmetry. It approximates the sedimentary stratigraphic characteristics of subsurface layers. Compared to other anisotropic models, the propagation characteristics of seismic waves in the VTI medium are more easily described. The anisotropic parameters $\epsilon \left(x,z\right)$ and $\delta \left(x,z\right)$ have the following linear relationship with velocity $v\left(x,z\right)$ (Huang et al., 2017) [27] :

$\{\begin{array}{l}\epsilon \left(x,z\right)=\frac{0.606\left[v\left(x,z\right)-v_{\min }\right]}{v_{\max }}\\ \delta \left(x,z\right)=\frac{0.485\left[v\left(x,z\right)-v_{\min }\right]}{v_{\max }}\end{array}$. (39)

Among them, $v_{\max }$ and $v_{\min }$ are the maximum and minimum velocities, respectively, and the calculation results are shown in Figure 1(b) and Figure 1(c) . Based on the above model, the parameter settings of the observation system are as follows: there are a total of 200 detectors, with a track spacing of 5.0 meters, arranged in the center of the detectors, and the first detector is located at (0.5 km, 0.0). The seismic source wavelet adopts Rayleigh waves with a dominant frequency of 30 Hz, a sampling interval of 1 s, and a sampling length of 1600.

Figure 2 shows the forward modeling results. Figure 2(a) is the generalized screen seismic record of isotropic media (Acoustic-thin slabs, without false frequencies), and Figure 2(b) and Figure 2(c) are the approximate seismic records of VTI media thin slabs (VTI-thin slabs) without and with spatial aliasing, respectively. Figure 2(d) and Figure 2(e) show the approximate seismic records of VTI media screens (VTI-screens) without and with false frequencies, respectively. As shown in the figure, seismic records with spatial aliasing have relatively weak amplitudes of deep reflection waves. The use of amplitude attenuation factors can effectively eliminate the phenomenon of spatial aliasing, and seismic records with spatial aliasing effects have a higher signal-to-noise ratio.

To verify the effectiveness of the algorithm, I used the traditional visco-acoustic medium finite difference method for comparison. Figure 3(a) shows the finite difference forward simulation results. Compared with the thin slab approximation and screen approximation methods, the seismic records of the three methods are similar. The finite difference method has richer wavefield information, but the phase axis is relatively coarser. Extract one of the waveforms for waveform comparison (as shown in Figure 3(b) ). The phase difference between the acoustic thin slab approximation result and the seismic records of the other three methods is relatively large. Ignoring the anisotropy of the medium directly leads to a significant lag in the seismic records of the acoustic thin slab approximation compared to the other three methods, and the wavefield components are relatively simple. The thin slab approximation and screen approximation results are closer to each other. Due to the relatively simple model, the two algorithms’ small-angle situations yield similar results. The above research results verify the effectiveness of the algorithms. Table 1 shows the calculation time of several methods. The calculation time for the VTI medium is higher than that for the acoustic medium, due to the addition of anisotropic parameter calculation. The efficiency of the thin slab approximation calculation under the same conditions is lower than that of the screen approximation method. The reason is that screen approximation is an approximation method for thin slab approximation at small angles, and the calculation efficiency of both methods is higher than that of the traditional finite difference method.

To further validate the effectiveness of the algorithm, a velocity model is set up as shown in Figure 4(a) , with 401 and 301 grid points in the x and z directions, respectively; the grid spacing is 5.0 m; the anisotropy parameters are shown in Figure 4(b) and Figure 4(c) . Based on the above model, the parameter settings of the observation system are as follows: there are a total of 200 detectors, with a track spacing of 5.0 meters, arranged in the center of the detectors, and the first detector is located at (0.5 km, 0.0); the seismic source wavelet adopts Rayleigh waves with a dominant frequency of 30 Hz, a sampling interval of 1 s, and a sampling length of 1600.

I used the thin slab approximation method and the screen approximation method for forward simulation, and Figure 5 shows the seismic forward simulation records. Among them, Figure 5(a) shows the generalized screen seismic record of isotropic media (Acoustic-thin slabs, without spatial aliasing), and Figure 5(b) shows the seismic record results of the global Born approximation of VTI media (VTI-Born, without spatial aliasing); Figure 5(c) and Figure 5(d) show the approximate seismic records of VTI medium thin slabs (VTI-thin slabs) without and without spatial aliasing removal, respectively. Figure 5(e) and Figure 5(f) show the approximate seismic records of VTI media screens (VTI-screens) without and with spatial aliasing, respectively. As shown in the figure, when the cumulative effect of forward scattering is strong, the calculation error of the global Born approximation is relatively large. The thin slab approximation and screen approximation based on the De Wolf approximation can effectively solve this problem. When the amplitude of deep reflection waves in seismic records with spatial aliasing is relatively weak, the use of an amplitude attenuation factor can effectively eliminate the phenomenon of spatial false frequency, and the signal-to-noise ratio of seismic records without the spatial false frequency effect is higher.

To verify the effectiveness of the algorithm, I used the traditional viscous acoustic medium finite difference method for comparison. Figure 6(a) shows the finite difference forward simulation results. Compared with the thin slab approximation and screen approximation methods, the seismic records of the three methods are similar. The finite difference method has richer wavefield information, but the phase axis is relatively coarser. One of the waveforms is extracted for waveform comparison (as shown in Figure 6(b) ). The phase difference between the acoustic thin slab approximation results and the seismic records of the other three methods is relatively large. Ignoring the anisotropy of the medium directly leads to a significant lag in the seismic records of the acoustic thin slab approximation compared to the other three methods, and the wavefield components are relatively simple. The results of the thin slab approximation and screen approximation are closer to each other. Due to the relatively simple model, the two algorithms have similar results in small-angle situations. The above research results validate the effectiveness of the algorithm.