﻿ TI介质角度域高斯束逆时偏移方法
 石油地球物理勘探  2019, Vol. 54 Issue (5): 1067-1074  DOI: 10.13810/j.cnki.issn.1000-7210.2019.05.014 0
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### 引用本文

XIAO Jian'en, LI Zhenchun, ZHANG Kai, LIU Qiang. Angle-domain reverse time migration with Gaussian beams for TI media. Oil Geophysical Prospecting, 2019, 54(5): 1067-1074. DOI: 10.13810/j.cnki.issn.1000-7210.2019.05.014.

### 文章历史

TI介质角度域高斯束逆时偏移方法

Angle-domain reverse time migration with Gaussian beams for TI media
XIAO Jian'en , LI Zhenchun , ZHANG Kai , LIU Qiang
School of Geosciences, China University of Petroleum(East China), Qingdao, Shandong 266580, China
Abstract: The reverse time migration with Gaussian beams combines the high efficiency and flexibility of Gaussian beam migration and the high precision of reverse time migration, which can be used for target-oriented imaging. In this paper, an anisotropic ray tracing algorithm based on phase velocity is introduced into the reverse time migration with Gaussian beams, and combined with the propagation angle information of Gaussian beam calculation, a more efficient angle-domain Gaussian beams reverse time migration method for TI media is realized. According to our model trial, compared with the conventional anisotropic algorithm based on elastic parameters, the proposed method has higher computational efficiency and extracted angle-domain common imaging gathers (ADCIGs) can not only provide support for subsequent migration velocity analysis, but also be used for stack imaging to suppress imaging noise and improve image quality.
Keywords: TI media    reverse time migration with Gaussian beams    Green function    angle-domain common imaging gather (ADCIG)
0 引言

1 基本原理

1.1 正、反向波场延拓

 $LU = \left[ {\Delta - \frac{1}{{{C^2}(\mathit{\boldsymbol{x}})}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right]U = 0$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {LG\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right) = {\rm{ \mathsf{ δ} }}\left( {t - {t_0}} \right){\rm{ \mathsf{ δ} }}\left( {\mathit{\boldsymbol{x}} - {\mathit{\boldsymbol{x}}_0}} \right)}\\ {{{\left. G \right|}_{t < {t_0}}} = 0} \end{array}} \right.$ (2)

 $\begin{gathered} U\left( {{\mathit{\boldsymbol{x}}_0},{t_0}} \right) = \int_{{t_0}}^T {\text{d}} t\iint_{\partial \Omega } {\left[ {G\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right)\frac{\partial }{{\partial {n_x}}}{U^0} - } \right.} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{U^0}\frac{\partial }{{\partial {n_x}}}G\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right)} \right]{\text{d}}{S_x} \hfill \\ \end{gathered}$ (3)

 ${\left. {G\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right)} \right|_{z = 0}} = 0$ (4)

 $\begin{gathered} {U_{\text{B}}}\left( {{\mathit{\boldsymbol{x}}_0},{t_0}} \right) = - 2\int_{{t_0}}^T {\text{d}} t\iint_{z = 0} {{U^0}}(\mathit{\boldsymbol{x}},t) \times \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{\partial }{{\partial z}}G\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right){\text{d}}x{\text{d}}y \hfill \\ \end{gathered}$ (5)

 ${U_{\text{F}}}\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_{\text{s}}}} \right) = {\rm{Re}} \int_0^\infty {\frac{1}{{\rm{\pi }}}} {G_{{\text{GB}}}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_{\text{s}}};\omega } \right){f_{\text{F}}}(\omega ){{\text{e}}^{ - {\text{i}}\omega t}}{\text{d}}\omega$ (6)

1.2 格林函数渐近式

Popov[21]给出了高斯束表征的格林函数渐近式为

 $G\left( {\mathit{\boldsymbol{x}},t;{\mathit{\boldsymbol{x}}_0},{t_0}} \right) = {\rm{Re}} \int_0^\infty {\frac{1}{{\rm{\pi }}}} {G_{{\text{GB}}}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_0};\omega } \right){{\text{e}}^{ - {\text{i}}\omega \left( {t - {t_0}} \right)}}{\text{d}}\omega$ (7)

 ${G_{{\text{GB}}}}\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_0};\omega } \right) = \mathit{\Phi }\int u \left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_0},\mathit{\boldsymbol{p}},\omega } \right){\text{d}}\theta$ (8)

 $\left\{ {\begin{array}{*{20}{l}} {u\left( {\mathit{\boldsymbol{x}},{\mathit{\boldsymbol{x}}_0},\mathit{\boldsymbol{p}},\omega } \right) = \sqrt {\frac{V}{Q}} \exp \left[ { - {\rm{i}}\omega \tau (s) + \frac{{{\rm{i}}\omega }}{2}\frac{P}{Q}{n^2}} \right]}\\ {\mathit{\Phi } = \frac{{ - {\rm{i}}}}{{4{\rm{ \mathsf{ π} }}V\left( {{\mathit{\boldsymbol{x}}_0}} \right)}}} \end{array}} \right.$ (9)

1.3 互相关成像

 $I\left( {{\mathit{\boldsymbol{x}}_0},{\mathit{\boldsymbol{x}}_{\rm{s}}}} \right) = \int {{U_{\rm{B}}}} \left( {{\mathit{\boldsymbol{x}}_0},{t_0}} \right){U_{\rm{F}}}\left( {{\mathit{\boldsymbol{x}}_0},t;{\mathit{\boldsymbol{x}}_{\rm{s}}}} \right){\rm{d}}{t_0}$ (10)

2 各向异性射线追踪

2.1 各向异性运动学射线追踪

Červený[22]基于各向异性弹性波波动方程推导了各向异性运动学射线追踪方程为

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}{x_i}}}{{{\rm{d}}\tau }} = {a_{ijkl}}{p_l}{g_j}{g_k}}\\ {\frac{{{\rm{d}}{p_i}}}{{{\rm{d}}\tau }} = - \frac{1}{2}\frac{{\partial {a_{mjkl}}}}{{\partial {x_i}}}{p_m}{p_l}{g_j}{g_k}} \end{array}} \right.$ (11)

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}{x_i}}}{{{\rm{d}}\tau }} = V}\\ {\frac{{{\rm{d}}{p_i}}}{{{\rm{d}}\tau }} = - \frac{1}{v}\frac{{\partial v}}{{\partial {x_i}}}} \end{array}} \right.$ (12)

 $V = \frac{1}{v}\frac{{\partial v}}{{\partial {p_i}}} + {v^2}{p_i}$ (13)

2.2 各向异性动力学射线追踪

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}Q}}{{{\rm{d}}\tau }} = MP + NQ}\\ {\frac{{{\rm{d}}P}}{{{\rm{d}}\tau }} = - NP - HQ} \end{array}} \right.$ (14)

 $\left\{ {\begin{array}{*{20}{l}} {M = \frac{1}{2}\frac{{{\partial ^2}Z}}{{\partial {n^2}}} - \frac{1}{4}{{\left( {\frac{{\partial Z}}{{\partial n}}} \right)}^2}}\\ {N = \frac{1}{2}\frac{{{\partial ^2}Z}}{{\partial {p_n}\partial n}} - \frac{1}{4}\frac{{\partial Z}}{{\partial {p_n}}}\frac{{\partial Z}}{{\partial n}}}\\ {H = \frac{1}{2}\frac{{{\partial ^2}Z}}{{\partial p_n^2}} - \frac{1}{4}{{\left( {\frac{{\partial Z}}{{\partial {p_n}}}} \right)}^2}} \end{array}} \right.$ (15)

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}{P_J}}}{{{\rm{d}}\tau }} = - {A_{JK}}{Q_K} - {B_{JK}}{P_K}}\\ {\frac{{{\rm{d}}{Q_J}}}{{{\rm{d}}\tau }} = {C_{JK}}{Q_K} + {D_{JK}}{P_K}} \end{array}} \right.$ (16)

 $\left\{ {\begin{array}{*{20}{l}} {{A_{JK}} = \frac{1}{v}\frac{{{\partial ^2}v}}{{\partial {y_J}\partial {y_K}}}}\\ {{B_{JK}} = \frac{{{\partial ^2}\ln v}}{{\partial {y_J}\partial {q_K}}}}\\ {{C_{JK}} = \frac{{{\partial ^2}\ln v}}{{\partial {y_K}\partial {q_J}}}}\\ {{D_{JK}} = \frac{{\partial V}}{{\partial {q_J}}}} \end{array}} \right.$ (17)

 $T(R) = T(X) + \frac{1}{2} {\rm{Re}} \left[ {\frac{{P(X)}}{{Q(X)}}} \right]{n^2}$ (18)
 图 1 二维中心射线坐标系

 $n = \left( {x - {x_X}} \right){t_z} - \left( {z - {z_X}} \right){t_x}$ (19)

 $\frac{{\partial T(R)}}{{\partial x}} = \frac{{\partial T(X)}}{{\partial x}} + n {\rm{Re}} \left[ {\frac{{P(X)}}{{Q(X)}}} \right]\frac{{\partial n}}{{\partial x}}$ (20)

 $\left\{ {\begin{array}{*{20}{l}} {{p_x}(X) = \frac{{\partial T(X)}}{{\partial x}}}\\ {{p_x}(R) = \frac{{\partial T(R)}}{{\partial x}}} \end{array}} \right.$ (21)

 $\begin{array}{l} {p_x}(R) = {p_x}(X) + \left[ {\left( {x - {x_X}} \right)t_z^2 - \left( {z - {z_X}} \right){t_x}{t_z}} \right] \times \\ \;\;\;\;\;\;\;\;\;\;\;{\rm{Re}} \left[ {\frac{{P(X)}}{{Q(X)}}} \right] \end{array}$ (22)

 $\begin{array}{l} {p_z}(R) = {p_z}(X) - \left[ {\left( {x - {x_X}} \right){t_x}{t_z} - \left( {z - {z_X}} \right)t_x^2} \right] \times \\ \;\;\;\;\;\;\;\;\;\;\;\;{\rm{Re}} \left[ {\frac{{P(X)}}{{Q(X)}}} \right] \end{array}$ (23)

 $\phi = \left\{ {\begin{array}{*{20}{l}} {\arctan \frac{{{p_x}}}{{{p_z}}} - \pi }&{{p_x} < 0,{p_z} < 0}\\ {\arctan \frac{{{p_x}}}{{{p_z}}} + \pi }&{{p_x} > 0,{p_z} < 0}\\ {\arctan \frac{{{p_x}}}{{{p_z}}} - \pi }&{其他} \end{array}} \right.$ (24)

4 计算流程

(1) 读入速度场和炮记录，以及相关各向异性参数场；

(2) 通过各向异性运动学(式(12))和动力学(式(16))射线追踪求取中心射线的运动学和动力学信息；

(3) 在震源处利用式(6)计算正向传播波场，同时在接收点处利用式(5)构建反向延拓波场；

(4) 利用式(10)的成像公式，计算震源处正向传播波场和接收点处反向延拓波场的互相关，得到单炮成像值；

5 模型试算 5.1 VTI介质ZY模型

 图 2 VTI介质ZY模型 (a)速度场；(b)ε参数场；(c)δ参数场

 图 3 ZY模型试算结果 (a)各向同性方法；(b)VTI高斯束逆时偏移(基于弹性参数)；(c)本文方法

 图 4 CDP500处不同方法生成的ADCIG (a)各向同性方法；(b)VTI高斯束逆时偏移(基于弹性参数)；(c)本文方法

 图 5 不同角度ADCIG叠加成像 (a)0°~10°叠加结果; (b)11°~20°叠加结果; (c)21°~30°叠加结果

5.2 TTI洼陷模型

 图 6 洼陷模型 (a)速度场；(b)ε参数场；(c)δ参数场；(d)各向异性角度场

 图 7 洼陷模型四种方法的试算结果 (a)各向同性方法；(b)VTI高斯束逆时偏移；(c)TTI高斯束逆时偏移(基于弹性参数)；(d)本文方法

6 结论

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