石油物探  2018, Vol. 57 Issue (4): 505-511, 536  DOI: 10.3969/j.issn.1000-1441.2018.04.002
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引用本文 

刘洪, 黄继伟, 袁雨欣, 等. 横向非均匀弹性介质中波传播的线性预测理论[J]. 石油物探, 2018, 57(4): 505-511, 536. DOI: 10.3969/j.issn.1000-1441.2018.04.002.
LIU Hong, HUANG Jiwei, YUAN Yuxin. Linear prediction theory for wave propagation in laterally inhomogeneous elastic media[J]. Geophysical Prospecting for Petroleum, 2018, 57(4): 505-511, 536. DOI: 10.3969/j.issn.1000-1441.2018.04.002.

基金项目

国家自然科学基金重点项目(41630319)资助

作者简介

刘洪(1959—), 男, 研究员, 主要从事地震波成像和油储地球物理研究。Email:liuhong@mail.igcas.ac.cn

文章历史

收稿日期:2018-04-16
改回日期:2018-05-27
横向非均匀弹性介质中波传播的线性预测理论
刘洪1,2,3 , 黄继伟1,2,3 , 袁雨欣1,2,3     
1. 中国科学院地质与地球物理研究所, 北京 100029;
2. 中国科学院大学, 北京 100049;
3. 中国科学院油气资源研究重点实验室, 北京 100029
摘要:基于层状介质的声波方程, 将自回归算子的逆用于表示透射波, 称为透射波的线性预测表示。将该表示与反射波的透射波自相关表示相结合, 可以得到层状介质的Yule-Walker方程和Levinson递归反演方法。Levinson递归反演方法的缺陷是反演层数一般小于12层, 而当反演层数再增加的时候不够稳定。已有的研究表明, 自回归算子是一个齐次方程的解, 采用李代数方法进行正、反演, 其反演结果比采用Levinson递归反演方法得到的结果更为稳定。在横向非均匀弹性介质情况下, 采用拟微分算子的象征方法, 推导出自回归算子齐次方程, 并利用该齐次方程, 进一步研究强不均匀介质的共振现象, 这对强耦合裂缝介质响应的研究有重要意义, 可合理地解释低频阴影现象。此外, 纠正了以往推导中存在的一个符号错误。
关键词横向非均匀    弹性介质    线性预测算子    递归反演    自回归算子    透射波    
Linear prediction theory for wave propagation in laterally inhomogeneous elastic media
LIU Hong1,2,3, HUANG Jiwei1,2,3, YUAN Yuxin1,2,3     
1. Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Key Laboratory of Petroleum Resources research, Chinese Academy of Sciences, Beijing 100029, China
Foundation item: This research is financially supported by the National Natural Science Foundation of China (Grant No.41630319)
Abstract: In the case of a layered medium, based on the acoustic wave equation, transmission waves can be expressed by the inverse of the autoregressive operator, which is called the expression of linear prediction of the transmission wave.Combining the expression of reflected waves with the autocorrelation representation of transmitted waves can lead to the Yule-Walker equation and Levinson recursive inversion method for layered media.The disadvantage of the Levinson recursive inversion method is that the calculation is not stable enough when the number of inversion layers is too large, such as when there are more than 12 layers.The previous studies indicated that the autoregressive operator satisfies a homogeneous equation and is more stable than the Levinson recursive inversion if the Lie algebra method is used for forward modeling and inversion.By using the symbolic method in pseudo-differential operators, the homogeneous equation of the autoregressive operator can be derived in the case of a laterally inhomogeneous elastic medium.By using the homogeneous equation, it is possible to further study the resonance phenomenon of a strong heterogeneous medium, which is of great significance to the study of the response of a strong coupling crack medium.It is also possible to explain the phenomenon of a low-frequency shadow more reasonably.There is a symbol error in the previous deduction, and it is corrected in the derivation presented.
Key words: laterally inhomogeneous    elastic medium    linear prediction operator    recursive inversion    autoregressive operator    transmission wave    

自20世纪80年代以来, Thom的突变论、Prigogine的耗散结构、孤立子和Radon变换等数学物理研究取得了重大突破[1], 卡兹穆迪代数(Kac-Moody Algebra)圈群、圈代数(Loop Group and loop Algebra)和Virasoro代数将孤立子、黎曼群面、超弦、统一场论和凝聚态物理研究联系了起来[2], 在数学和物理领域取得了大量进展, 如拓扑绝缘体、半金属等。在地震波传播领域, 将亚波长尺度的不均匀性效应在波长尺度上表达出来[3], 在等效速度、等效各向异性等静态等效介质理论基础上, 发展了等效Q[3]和等效慢波理论, 以解释薄互层波动等效、裂缝波动等效和低频阴影[4]等现象。1988—2004年间油储项目成果研究[5-8]将数学和物理学的最新成果迅速引入到勘探地震波传播领域, 并发展出了多维逆散射[9-10]、波场延拓辛几何算法[6]、非对称走时算法[11-13]和横向非均匀弹性介质方向波响应等理论[14]

我们基于横向非均匀弹性介质方向波响应理论[14], 利用本征函数将频率域波动方程展开, 推导出了方向波耦合方程, 利用方向波相互耦合矩阵的对称性质, 推导出自回归算子齐次方程[9]

我们基于层状介质的声波方程, 将自回归算子的逆表示透射波, 称为透射波的线性预测表示[15], 将该表示与反射波的透射波自相关表示(又称为谱乘积表示、谱分解表示)相结合, 可以得到层状介质的Yule-Walker方程和Levinson递归反演方法[15], 该方法已经推广应用于P-SV波[16]。Levinson递归反演方法的缺陷是反演层数一般小于12层, 而反演层数再增加的时候不够稳定。已有的研究表明, 自回归算子满足一个齐次方程[9], 如果采用李代数方法[17]进行正、反演, 反演结果比采用Levinson递归反演方法得到的结果更为稳定。

在非均匀弹性介质情况下, 利用自回归算子齐次方程, 采用Magnus方法归入圈代数和维拉宿代数[1-2], 可以进一步研究强不均匀介质的共振现象, 这对强耦合裂缝介质响应的研究有重要意义, 使合理地解释低频阴影现象变为可能。可分表示、快速多极算法和多尺度算法的出现, 使单程波指数映射的快速计算成为可能[18]。以往的推导中[14], 存在一个符号错误, 本文纠正了符号错误, 并补充了微分散射矩阵的对称性质。

1 传播向量微分方程组

方向波积分表示的基础是方向波微分方程组, 方向波微分方程组可从(1)式垂向一阶弹性波微分方程出发导出, 其形式如下:

$ {\partial _z}{\bf{B}} = {\rm{i}}\omega {\bf{HNB}} $ (1)

式中:B为传播向量; ω为圆频率。其中, H, N分别为:

$ {\bf{H}} = \left( {\begin{array}{*{20}{c}} {{{\bf{H}}_{11}}}&{{{\bf{H}}_{12}}}\\ {{{\bf{H}}_{21}}}&{{{\bf{H}}_{22}}} \end{array}} \right) $ (2)
$ {\bf{N}} = \left( {\begin{array}{*{20}{c}} {\bf{0}}&I\\ I&{\bf{0}} \end{array}} \right) $ (3)
$ {\bf{H}} = {{{\bf{\bar H}}}^{\rm{T}}} $ (4)

式中:HTH的转置; HH的共轭。该微分方程组的积分解为传播矩阵, 故我们称其为传播向量微分方程组。传播向量微分方程组是FRYER[19-20]和宋海斌[21]所研究的方程组的一般形式, 更全面的评述见文献[14]。

2 方向波象征分解 2.1 象征的定义和计算[11, 13, 22, 23]

定义一个算子$\hat A$的象征$\sigma \left( {\hat A} \right)$是一个函数a(x, y, kx, ky), 它是拟微分算子作用于正弦波的响应, 即:

$ \begin{array}{*{20}{c}} {a\left( {x,y,{k_x},{k_y}} \right) \equiv \sigma \left( {\hat A} \right) = \exp \left( { - {\rm{i}}{k_x}x - {\rm{i}}{k_y}y} \right) \cdot }\\ {\left[ {\hat A\left( {x,y,{D_x},{D_y}} \right)\exp \left( {{\rm{i}}{k_x}x + {\rm{i}}{k_y}y} \right)} \right]} \end{array} $ (5)
$ \left\{ \begin{array}{l} {D_x} = \frac{\partial }{{{\rm{i}}\partial x}} = \frac{1}{{\rm{i}}}{\partial _x}\\ {D_y} = \frac{\partial }{{{\rm{i}}\partial y}} = \frac{1}{{\rm{i}}}{\partial _y} \end{array} \right. $ (6)

式中:这里x, y是水平坐标; kx, ky是水平波数。Dxb/i∂x, Dyb/i∂y, b为对b求导。

假设算子${\hat A}$${\hat B}$的象征函数分别为$a = \sigma \left( {\hat A} \right)$$b = \sigma \left( {\hat B} \right)$, 则算子乘法运算的象征[13-15]为:

$ a\# b = \sigma \left( {\hat A\hat B} \right) = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}{{\left( {\frac{{{\partial ^a}}}{{\partial {k_x}}}D_x^b + \frac{{{\partial ^a}}}{{\partial {k_y}}}D_y^b} \right)}^n}ab} $ (7)
$ \left\{ \begin{array}{l} u{,_{{k_x}}} = \frac{\partial }{{\partial {k_x}}}u\\ u{,_{{k_y}}} = \frac{\partial }{{\partial {k_y}}}u \end{array} \right. $ (8)

此运算运用了莱布尼兹法则, 称为Witt积。

2.2 方向波分解方程的象征齐次函数求解法

算子矩阵的方向波分解方程如下:

$ {\bf{HW}} = {\bf{NWQ}} $ (9)
$ {\bf{Q}} = {\bf{diag}}\left( {Q_p^u,Q_{s1}^u,Q_{s2}^u, - Q_p^D, - Q_{s1}^D, - Q_{s2}^D} \right) $ (10)
$ {\bf{Q}} = {\bf{diag}}\left( {{Q_1},{Q_2},{Q_3},{Q_4},{Q_5},{Q_6}} \right) $ (11)
$ {\bf{Q}} = {\bf{diag}}\left( {{{\bf{Q}}^ + }, - {{\bf{Q}}^ - }} \right) $ (12)

式中:Q, W为6×6阶矩阵, 矩阵元素为象征, Q为垂直慢度算子; W的列向量是方向波的运动-应力向量。Q+, Q-为3×3对角阵, 分别为上行波和下行波垂直慢度算子, 一般情况有:

$ {{\bf{Q}}^ + } \ne {{\bf{Q}}^ - } $ (13)

算子方程(9)在象征域求解较为简便, 在象征域有:

$ {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{H}} \right)\# {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{W}} \right) = {\bf{N}}\# \sigma \left( {\bf{W}} \right)\# {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{Q}} \right) $ (14)

即:

$ {\bf{h}}\# {\bf{w}} = {\bf{N}}\# {\bf{w}}\# {\bf{q}} $ (15)

其中:

$ {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{H}} \right) = {\bf{h}} = {{\bf{h}}^{\left[ 0 \right]}} + {{\bf{h}}^{\left[ { - 1} \right]}} + {{\bf{h}}^{\left[ { - 2} \right]}} $ (16)
$ {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{W}} \right) = {\bf{w}} = {{\bf{w}}^{\left[ 0 \right]}} + {{\bf{w}}^{\left[ { - 1} \right]}} + {{\bf{w}}^{\left[ { - 2} \right]}} $ (17)
$ {\bf{ \pmb{\mathsf{ σ}} }}\left( {\bf{Q}} \right) = {\bf{q}} = {{\bf{q}}^{\left[ 0 \right]}} + {{\bf{q}}^{\left[ { - 1} \right]}} + {{\bf{q}}^{\left[ { - 2} \right]}} $ (18)

式中:q[0], q[-1], q[-2]为最高齐次性分别为0次, -1和-2次; w[0]w[-1]w[-2]最高齐次性分别为0次, -1和-2次。按方程的齐次性分级求解, 对比象征方程(15)两边齐次性, 可得:

$ {{\bf{h}}^{\left[ 0 \right]}}{{\bf{w}}^{\left[ 0 \right]}} = {\bf{N}}{{\bf{w}}^{\left[ 0 \right]}}{{\bf{q}}^{\left[ 0 \right]}} $ (19)

方程(19)是传统的方程, 其解法已由FRYER等[19-20]阐明, 本文简述如下。

wj[0]w[0]的列向量, 广义本征向量问题表示为:

$ {{\bf{h}}^{\left[ 0 \right]}}{\bf{w}}_i^{\left[ 0 \right]} = {\bf{Nw}}_i^{\left[ 0 \right]}{\bf{q}}_i^{\left[ 0 \right]} $ (20)

式中:qi[0]是垂直慢度高频近似(或局部慢度, 因为它只在局部有效, 没有考虑横向变速)。

由于N的存在, 该式形式上与本征展开不同。根据方程(4)和方程(20)容易证明:

$ {\bf{\bar w}}_i^{\left[ 0 \right]T}{\bf{Nw}}_j^{\left[ 0 \right]} = \left\{ {\begin{array}{*{20}{c}} 0&{i \ne j}\\ {\varepsilon _i^{\left[ 0 \right]}}&{i = j} \end{array}} \right. $ (21)
$ \varepsilon _i^{\left[ 0 \right]} = {\bf{\bar w}}_i^{\left[ 0 \right]T}{\bf{Nw}}_i^{\left[ 0 \right]} $ (22)

式中:εi[0]是能流密度高频近似, εi[0]≥0时为上行波, εi[0]≤0时为下行波。相对于传统本征矩阵的逆矩阵表达式, 新的表达式多了N, 其它并无太大变化, 这实际上是向量正交方式发生变化, 向量按照内积度量不正交, 按照能流密度度量是正交的; 按照$\sqrt {\left| {\varepsilon _i^{\left[ 0 \right]}} \right|} $归一化并适当排序后, 可以得到:

$ {{{\bf{\bar w}}}^{\left[ 0 \right]T}}{\bf{N}}{{\bf{w}}^{\left[ 0 \right]}} = {\bf{M}} $ (23)
$ \begin{array}{l} {\bf{M}} \buildrel \Delta \over = {\bf{diag}}\left( {1,1,1, - 1, - 1, - 1} \right)\\ = \left( {\begin{array}{*{20}{c}} 1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&{ - 1}&0&0\\ 0&0&0&0&{ - 1}&0\\ 0&0&0&0&0&{ - 1} \end{array}} \right) \end{array} $ (24)

文献[14]中少了共轭的符号, q[-1]+q[-2]见文献[14]。

2.3 方向波象征分解的能流归一化

方程(15)可写为:

$ {\bf{h}}\# {{\bf{w}}_i} = {\bf{N}}\# {{\bf{w}}_i}\# {{\bf{q}}_i} $ (25)

记算子转置的象征为:

$ {{\bf{w}}^T} = {\bf{ \pmb{\mathsf{ σ}} }}\left( {{{\bf{W}}^T}} \right) $ (26)

根据齐次性分级计算方法和方程(25)可以证明:

$ \begin{array}{*{20}{c}} {{\bf{\bar w}}_i^T\# {\bf{N}}\# {w_j} = 0}&{i \ne j} \end{array} $ (27)

$ {{\varepsilon '}_i} = \left| {{\bf{\bar w}}_i^T\# {\bf{N}}\# {{\bf{w}}_j}} \right| $ (28)

假定${\left( {\varepsilon _i^{'}} \right)^{\frac{1}{2}}},{\left( {\varepsilon _i^{'}} \right)^{ - \frac{1}{2}}}$为象征方程, 有:

$ {\left( {{{\varepsilon '}_i}} \right)^{\frac{1}{2}}}\# {\left( {{{\varepsilon '}_i}} \right)^{\frac{1}{2}}} = {{\varepsilon '}_i} $ (29)
$ {\left( {{{\varepsilon '}_i}} \right)^{ - \frac{1}{2}}}\# {{\varepsilon '}_i}\# {\left( {{{\varepsilon '}_i}} \right)^{ - \frac{1}{2}}} = 1 $ (30)

引入能流归一化的解:

$ {{{\bf{w'}}}_i} = {{\bf{w}}_i}\# {\left( {{{\varepsilon '}_i}} \right)^{ - \frac{1}{2}}} $ (31)

有:

$ {\left( {{{{\bf{w'}}}_j}} \right)^T}\# {\bf{N}}\# {{{\bf{w'}}}_i} = \left\{ {\begin{array}{*{20}{c}} 0&{i \ne j}\\ { \pm 1}&{i = j} \end{array}} \right. $ (32)

将归一化后的wi′仍记为wi, 可得:

$ {\bf{\bar w}}_i^T\# {\bf{N}}\# {{\bf{w}}_j} = \left\{ {\begin{array}{*{20}{c}} 0&{i \ne j}\\ { \pm 1}&{i = j} \end{array}} \right. $ (33)

有:

$ {{{\bf{\bar w}}}^T}\# {\bf{N}}\# {\bf{w}} = {\bf{M}} $ (34)

据此可以将逆矩阵表示为:

$ {{\bf{w}}^{ - 1}} = {\bf{M}}{{{\bf{\bar w}}}^T}\# {\bf{N}} $ (35)

当垂直慢度为实数时, 所有本征向量象征均为实数, 矩阵元素为实数。

3 微分散射矩阵的对称性 3.1 方向波微分方程组

时间空间域中W是6×6阶矩阵, 其象征为w, 矩阵元素为象征, 即σ(W)=w, 在时间空间域, 设:

$ {\bf{NB}} = {\bf{W}}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) $ (36)

式中:U是上行波波场; D是下行波波场。

由方程(15)可得象征域(频率波数域)算子部分本征分解(区别于完全本征分解):

$ {\bf{h}} = {\bf{N}}\# {\bf{w}}\# {\bf{q}}\# {{\bf{w}}^{ - 1}} $ (37)

其时间空间域形式为:

$ {\bf{H}} = {\bf{NWQ}}{{\bf{W}}^{ - 1}} $ (38)

将其带入(1)式, 可得:

$ \left( {{\partial _z}{\bf{W}}} \right)\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) + {\bf{W}}{\partial _z}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) = {\rm{i}}\omega {\bf{WQ}}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) $ (39)

即:

$ {\partial _z}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) = {\rm{i}}\omega {\bf{Q}}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) - {W^{ - 1}}\left( {{\partial _z}{\bf{W}}} \right)\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) $ (40)

令中间变量S:

$ {\bf{S}} = - {{\bf{W}}^{ - 1}}\left( {{\partial _z}{\bf{W}}} \right) $ (41)

可得:

$ {\partial _z}\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) = \left( {{\rm{i}}\omega {\bf{Q}} + {\bf{S}}} \right)\left( {\begin{array}{*{20}{c}} {\bf{U}}\\ {\bf{D}} \end{array}} \right) $ (42)

该式称为方向波微分方程组。

3.2 微分散射矩阵象征关于转置运算的对称性

方程(41)给出的S称为微分散射矩阵算子, 对应的象征形式(或者频率波数域方程)为:

$ \begin{array}{l} {\bf{s}} = - {{\bf{w}}^{ - 1}}\# {\partial _z}\left( {\bf{w}} \right)\\ \;\; = - {\bf{M}}{{{\bf{\bar w}}}^{\rm{T}}}\# N\left( {{\partial _z}{\bf{w}}} \right) \end{array} $ (43)

式中:s是描述频率波数域单反射、单透射的微分散射矩阵象征, 其元素为微分散射算子的象征。根据方程(33)有:

$ {\partial _z}\left( {{{{\bf{\bar w}}}^T}\# {\bf{N}}\# {\bf{w}}} \right) = 0,\;\;\;\;{\partial _z}\left( {{\bf{w}}\# {{{\bf{\bar w}}}^T}} \right) = 0 $ (44)
$ \left( {{\partial _z}{{{\bf{\bar w}}}^T}} \right)\# {\bf{N}}\# {\bf{w}} + {{\bar w}^T}\# {\bf{N}}\# {\partial _z}\left( {\bf{w}} \right) = 0 $ (45)
$ \left( {{\partial _z}{\bf{w}}} \right)\# {{{\bf{\bar w}}}^T} + {\bf{w}}\# \left( {{\partial _z}{{{\bf{\bar w}}}^T}} \right) = 0 $ (46)
$ {{\bf{w}}^{ - 1}}\# \left( {{\partial _z}{\bf{w}}} \right) + \left( {{\partial _z}{{{\bf{\bar w}}}^T}} \right)\# {\left( {{{\bar w}^T}} \right)^{ - 1}} = 0 $ (47)
$ {\bf{s}} + {{{\bf{\bar s}}}^T} = 0 $ (48)

利用wT#N#w=M可得:

$ \begin{array}{l} {\bf{N}}\# {\bf{w}} = {\left( {{{{\bf{\bar w}}}^T}} \right)^{ - 1}}{\bf{M}}\\ {{{\bf{\bar w}}}^T}\# {\bf{N}} = {\bf{M}}{{\bf{w}}^{ - 1}} \end{array} $ (49)

将方程(49)带入方程(45)可得:

$ \left( {{\partial _z}{{{\bf{\bar w}}}^T}} \right)\# {\left( {{{{\bf{\bar w}}}^T}} \right)^{ - 1}}{\bf{M}} + {\bf{M}}{{\bf{w}}^{ - 1}}\# {\partial _z}\left( {\bf{w}} \right) = 0 $ (50)

即:

$ {{{\bf{\bar s}}}^T}{\bf{M}} + {\bf{Ms}} = 0 $ (51)

记:

$ {\bf{s}} = \left( {\begin{array}{*{20}{c}} {{{\bf{s}}^{uu}}}&{{{\bf{s}}^{ud}}}\\ {{{\bf{s}}^{du}}}&{{{\bf{s}}^{dd}}} \end{array}} \right) $ (52)

则根据公式(48)可得:

$ \begin{array}{*{20}{c}} {{{\bf{s}}^{uu}} = - {{\left( {{{\bar s}^{uu}}} \right)}^T}}&{{{\bf{s}}^{dd}} = - {{\left( {{{\bar s}^{dd}}} \right)}^T}}&{{{\bf{s}}^{ud}} = {{\left( {{{\bar s}^{ud}}} \right)}^T}} \end{array} $ (53)

根据公式(51)可得:

$ \begin{array}{*{20}{c}} {{{\bf{s}}^{uu}} = - {{\left( {{{\bar s}^{uu}}} \right)}^T}}&{{{\bf{s}}^{dd}} = - {{\left( {{{\bar s}^{dd}}} \right)}^T}}&{{{\bf{s}}^{du}} = {{\left( {{{\bar s}^{ud}}} \right)}^T}} \end{array} $ (54)

方程(53)和方程(54)称为微分反射透射矩阵象征关于转置运算的对称性。

3.3 微分散射矩阵象征关于变量倒转的对称性

我们研究微分反射、透射象征矩阵$s = \left( {\begin{array}{*{20}{c}} {{{\bf{s}}^{uu}}}&{{{\bf{s}}^{ud}}}\\ {{{\bf{s}}^{du}}}&{{{\bf{s}}^{dd}}} \end{array}} \right)$关于变量(ω, kx, ky)倒转的对称性, 为此设:

$ {\bf{k}} = \left( {{k_x},{k_y}} \right) $ (55)

方程(10), 方程(11), 方程(12)即为:

$ \begin{array}{l} {\bf{q}} = {\bf{diag}}\left( {\begin{array}{*{20}{c}} {{q_1}}&{{q_2}}&{{q_3}}&{{q_4}}&{{q_5}}&{{q_6}} \end{array}} \right)\\ \;\;\; = {\bf{diag}}\left( {\begin{array}{*{20}{c}} {q_p^u}&{q_{s1}^u}&{q_{s2}^u}&{ - q_p^d}&{ - q_{s1}^d}&{ - q_{s2}^d} \end{array}} \right)\\ \;\;\; = {\bf{diag}}\left( {{q^ + }, - {q^ - }} \right) \end{array} $ (56)

方程(15)或者方程(26)可写为:

$ {\bf{h}}\left( {\omega ,{\bf{k}}} \right)\# {{\bf{w}}_i}\left( {\omega ,{\bf{k}}} \right) = {\bf{N}}{{\bf{w}}_i}\left( {\omega ,{\bf{k}}} \right)\# {{\bf{q}}_i}\left( {\omega ,{\bf{k}}} \right) $ (57)

$ {\bf{h}}\left( { - \omega , - {\bf{k}}} \right)\# {{\bf{w}}_i}\left( { - \omega , - {\bf{k}}} \right) = {\bf{N}}{{\bf{w}}_i}\left( { - \omega , - {\bf{k}}} \right)\# {{\bf{q}}_i}\left( { - \omega , - {\bf{k}}} \right) $ (58)

利用Christoffel方程、传播矩阵微分方程[19-20]和前述象征进行齐次运算, 不难证明:

$ {\bf{q}}_i^ \pm \left( { - \omega , - {\bf{k}}} \right) = {\bf{q}}_i^ \pm \left( {\omega , - {\bf{k}}} \right) = - q_i^ \mp \left( {\omega ,{\bf{k}}} \right) $ (59)

即波数变号, 上下行波换方向:

$ {{\bf{q}}^ + }\left( {\omega ,{\bf{k}}} \right) = {{\bf{q}}^ + }\left( { - \omega ,{\bf{k}}} \right) = - {{\bf{q}}^ - }\left( { - \omega , - {\bf{k}}} \right) $ (60)

据此, 方程(58)改写为:

$ \begin{array}{l} {\bf{h}}\left( { - \omega , - {\bf{k}}} \right)\# \left( {{\bf{w}}\left( { - \omega , - {\bf{k}}} \right){\bf{N}}} \right) = {\bf{N}}\left( {{\bf{w}}\left( { - \omega ,} \right.} \right.\\ \;\;\;\;\;\;\left. {\left. { - {\bf{k}}} \right){\bf{N}}} \right)\# {\bf{q}}\left( {\omega ,{\bf{k}}} \right) \end{array} $ (61)

根据传播矩阵微分方程, 可得:

$ {\bf{h}}\left( {\omega ,{\bf{k}}} \right) = {\bf{h}}\left( { - \omega , - {\bf{k}}} \right) $ (62)

将公式(62)代入方程(61)与方程(57)比较可得:

$ {\bf{w}}\left( { - \omega , - {\bf{k}}} \right){\bf{N}} = {\bf{w}}\left( {\omega ,{\bf{k}}} \right) $ (63)

带入方程(43)可得:

$ \begin{array}{l} {\bf{s}}\left( { - \omega , - {\bf{k}}} \right) = - {\bf{w}}{\left( { - \omega , - {\bf{k}}} \right)^{ - 1}}\# {\partial _z}\left( {{\bf{w}}\left( { - \omega , - {\bf{k}}} \right)} \right)\\ = - {\bf{Nw}}{\left( {\omega ,{\bf{k}}} \right)^{ - 1}}\# {\partial _z}\left( {{\bf{w}}\left( {\omega ,{\bf{k}}} \right){\bf{N}}} \right) = {\bf{Ns}}\left( {\omega ,{\bf{k}}} \right){\bf{N}} \end{array} $ (64)

即:

$ {{\bf{s}}^{uu}}\left( { - \omega , - {\bf{k}}} \right) = {{\bf{s}}^{dd}}\left( {\omega ,{\bf{k}}} \right) $ (65)

$ {{\bf{s}}^{du}}\left( { - \omega , - {\bf{k}}} \right) = {{\bf{s}}^{ud}}\left( {\omega ,{\bf{k}}} \right) $ (66)
4 反射算子的因果分解表达 4.1 Lie代数微分方程的解[17, 9, 10]

为了简化表达, 我们用${\hat \eta }$表示Lie代数微分方程的解。

设以下Lie代数微分方程的解${\hat \eta }$代表一个映射$\hat f \to \hat \eta \left( {\hat f} \right)$, 有:

$ \begin{array}{l} {\partial _\zeta }\hat \eta = \frac{{a{d_{\hat \eta }}}}{{\exp a{d_{\hat \eta }} - 1}}\hat f = \hat f + {B_1}a{d_{\hat \eta }}\hat f + \frac{{{B_2}}}{2}a{d_{\hat \eta }}a{d_{\hat \eta }}\hat f + \\ \;\;\;\;\;\frac{{{B_4}}}{{24}}{\left( {a{d_{\hat \eta }}} \right)^4}\hat f + \frac{{{B_6}}}{{720}}{\left( {a{d_{\hat \eta }}} \right)^6}\hat f + \cdots = \hat f + {B_1}\left[ {\hat \eta ,} \right.\\ \;\;\;\;\;\left. {\hat f} \right] + \frac{{{B_2}}}{2}\left[ {\hat \eta ,\hat \eta ,\hat f} \right] + \frac{{{B_4}}}{{24}}\left[ {\hat \eta ,\hat \eta ,\hat \eta ,\hat \eta ,\hat f} \right] + \frac{{{B_6}}}{{720}} \cdot \\ \;\;\;\;\;\left[ {\overbrace {\hat \eta , \cdots ,\hat \eta }^6,\hat f} \right] + \cdots \end{array} $ (67)

式中:Bn为Bernoulli数。对于方程:

$ {\partial _\zeta }d\left( \zeta \right) = \hat f\left( \zeta \right)d\left( \zeta \right) $ (68)

的解可写为:

$ d\left( \zeta \right) = \exp \left( {\eta \left( {\hat f} \right)} \right)d\left( 0 \right) $ (69)

即:

$ \sum\limits_{n = 0}^\infty {{B_n}\frac{{{t^n}}}{{n!}} = \frac{t}{{{e_t} - 1}}} $ (70)
$ \begin{array}{l} \begin{array}{*{20}{c}} {{B_0} = 1}&{{B_1} = - \frac{1}{2}}&{{B_2} = \frac{1}{6}} \end{array}\\ \begin{array}{*{20}{c}} {{B_4} = -\frac{1}{30}}&{{B_6} = \frac{1}{42}}&{{B_5} = 0} \end{array} \end{array} $ (71)

利用方程(67)式不难证明:

$ \exp {\left( {\hat \eta \left( {\hat f} \right)} \right)^T} = \exp \left( { - \hat \eta \left( { - {{\hat f}^{\rm{T}}}} \right)} \right) $ (72)
4.2 弹性波单程波算子和广义WRW模型

设:

$ \left( {\begin{array}{*{20}{c}} {{\bf{u}}\left( {w,k} \right)}\\ {{\bf{d}}\left( {w,k} \right)} \end{array}} \right) = \\ \exp \left( {\eta \left( {\begin{array}{*{20}{c}} {{\rm{i}}\omega {{\bf{q}}^ + }\left( {\omega ,k} \right) + {\mathit{\boldsymbol{\rm{s}}}^{uu}}\left( {\omega ,k} \right)}&0\\ 0&{ - {\rm{i}}\omega {{\bf{q}}^ - }\left( {\omega ,k} \right) + {\mathit{\boldsymbol{\rm{s}}}^{dd}}\left( {\omega ,k} \right)} \end{array}} \right)} \right)\left( {\begin{array}{*{20}{c}} {{\bf{u_1}}\left( {w,k} \right)}\\ {{\bf{d_1}}\left( {w,k} \right)} \end{array}} \right) $ (73)

则方程(42)可以变为:

$ \begin{array}{l} {\partial _z}\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( \omega \right) = {\rm{diag}}\left( {\exp \left( { - \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + } + {{\bf{s}}^{uu}}} \right)} \right),} \right.\\ \left. {\exp \left( { - \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + {{\bf{s}}^{dd}}} \right)} \right)} \right)\\ \left( {\begin{array}{*{20}{c}} 0&{{{\bf{s}}^{ud}}}\\ {{{{\bf{\bar s}}}^{udT}}}&0 \end{array}} \right){\rm{diag}}\left( {\exp \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + } + {{\bf{s}}^{uu}}} \right),} \right.\\ \left. {\exp \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + {{\bf{s}}^{dd}}} \right)} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {\bf{ \pmb{\mathsf{ ω}} }} \right)\\ = \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega } \right)}\\ {{{\bf{s}}^{ud\left[ 2 \right]}}\left( {z,\omega } \right)}&{\bf{0}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( \omega \right) \end{array} $ (74)

方程(74)为关于振幅$\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( \omega \right)$的辐射传递方程; sud(z, ω)有9个独立元素, 决定u1d1之间的振幅转换; suu, sdd各有3个独立元素在透射中改变u1d1振幅。其中:

$ \begin{array}{l} {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega } \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{x}},{\bf{k}}} \right) = \exp \left( { - \eta \cdot } \right.\\ \left. {\left( {{\rm{i}}\omega {{\bf{q}}^ + } + {{\bf{s}}^{uu}}} \right)} \right)\# {{\bf{s}}^{ud}}\# \exp \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + {{\bf{s}}^{dd}}} \right) \end{array} $ (75)
$ \begin{array}{l} {{\bf{s}}^{ud\left[ 2 \right]}}\left( {z,\omega } \right) = {{\bf{s}}^{ud\left[ 2 \right]}}\left( {z,\omega ,{\bf{x}},{\bf{k}}} \right) = \exp \left( { - \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + } \right.} \right.\\ \left. {\left. {{{\bf{s}}^{dd}}} \right)} \right)\# {\left( {{{{\bf{\bar s}}}^{ud}}} \right)^T}\# \exp \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + } + {{\bf{s}}^{uu}}} \right) \end{array} $ (76)

式中:exp(-η(iωq++suu))和expη(-iωq-+sdd)均为弹性波单程波算子, 是标量情况下真振幅偏移[3]的推广; sud[1](z, ω, k)为广义WRW模型(sud充当R, 弹性波单程波算子充当W), 它只含一次波, 是准确的, 与玻恩近似相比它是一种忽略了多次波的近似。

4.3 上下行波的能流守恒

利用方程(54)和方程(72), 可得:

$ \begin{array}{l} {{\bf{s}}^{ud\left[ 1 \right]}}{\left( {z,\omega ,{\bf{k}}} \right)^T} = {\left( {\exp \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + {{\bf{s}}^{dd}}} \right)} \right)^T}\# \\ {{\bf{s}}^{udT}}\# {\left( {\exp \left( { - \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + } + {{\bf{s}}^{uu}}} \right)} \right)} \right)^T}\\ = \left( {\exp - \eta \left( {{\rm{i}}\omega \left( {{{\bf{q}}^ - }} \right) + {{\bf{s}}^{dd}}} \right)} \right)\# {{\bf{s}}^{udT}}\# \\ \left( {\exp \left( {\eta \left( { - {\rm{i}}\omega \left( {{{\bf{q}}^ + }} \right) + {{\bf{s}}^{uu}}} \right)} \right)} \right)\\ = {{{\bf{\bar s}}}^{ud\left[ 2 \right]}}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (77)

式中:sudsud的复数共轭。我们利用如下方程:

$ {{\bf{s}}^{du}}\left( { - \omega , - {\bf{k}}} \right) = {{\bf{s}}^{ud}}\left( {\omega ,{\bf{k}}} \right) = {{{\bf{\bar s}}}^{du}}{\left( {\omega ,{\bf{k}}} \right)^T} $ (78)
$ \begin{array}{l} {q^ + }\left( {z, - \omega , - {\bf{k}}} \right) = - {{\bf{q}}^ - }\left( {z,\omega ,{\bf{k}}} \right)\\ {{\bf{s}}^{uu}}\left( {z, - \omega , - {\bf{k}}} \right) = {{\bf{s}}^{dd}}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (79)

可得:

$ \begin{array}{l} {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right) = \exp \left( { - \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + }\left( {z, - \omega ,} \right.} \right.} \right.\\ \left. {\left. {\left. { - {\bf{k}}} \right) + {{\bf{s}}^{uu}}\left( {z, - \omega , - {\bf{k}}} \right)} \right)} \right)\# {{\bf{s}}^{ud}}\left( {z, - \omega , - {\bf{k}}} \right)\# \\ \exp \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - }\left( {z, - \omega , - {\bf{k}}} \right) + {{\bf{s}}^{dd}}\left( {z, - \omega , - {\bf{k}}} \right)} \right)\\ = \exp \left( { - \eta \left( { - {\rm{i}}\omega {{\bf{q}}^ - } + {{\bf{s}}^{dd}}} \right)} \right)\# {\left( {{{{\bf{\bar s}}}^{ud}}} \right)^T}\# \exp \eta \left( {{\rm{i}}\omega {{\bf{q}}^ + } + } \right.\\ \left. {{{\bf{s}}^{uu}}} \right) = {{\bf{s}}^{ud\left[ 2 \right]}}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (80)

由方程(79)、方程(80)可得:

$ {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 2 \right]}}\left( {z, - \omega , - {\bf{k}}} \right) $ (81)
$ {{{\bf{\bar s}}}^{ud\left[ 2 \right]}}{\left( {z,\omega ,{\bf{k}}} \right)^T} = {{\bf{s}}^{ud\left[ 2 \right]}}\left( {z, - \omega , - {\bf{k}}} \right) $ (82)
$ {{{\bf{\bar s}}}^{ud\left[ 1 \right]}}{\left( {z,\omega ,{\bf{k}}} \right)^T} = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right) $ (83)

利用方程(74)、方程(81)可得:

$ \begin{array}{l} {\partial _z}\left( {{{\left( {\begin{array}{*{20}{c}} {{{{\bf{\bar u}}}_1}}\\ {{{{\bf{\bar d}}}_1}} \end{array}} \right)}^T}\left( {z,\omega ,{\bf{k}}} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ { - {{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)} \right) - {\left( {\begin{array}{*{20}{c}} {{{{\bf{\bar u}}}_1}}\\ {{{{\bf{\bar d}}}_1}} \end{array}} \right)^T} \cdot \\ \left( {z,\omega ,{\bf{k}}} \right)\left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{{\bf{\bar s}}}^{ud\left[ 1 \right]}}{{\left( {z,\omega ,{\bf{k}}} \right)}^T}}&{\bf{0}} \end{array}} \right) \cdot \\ \;\;\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ { - {{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) + {\left( {\begin{array}{*{20}{c}} {{{{\bf{\bar u}}}_1}}\\ {{{{\bf{\bar d}}}_1}} \end{array}} \right)^T}\left( {z,\omega ,{\bf{k}}} \right) \cdot \\ \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{{\bf{\bar s}}}^{ud\left[ 1 \right]}}{{\left( {z,\omega ,{\bf{k}}} \right)}^T}}&{\bf{0}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ { - {d_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)\\ = 0 \end{array} $ (84)

这是能流守恒方程在归一化上下行波系数上的反映。

4.4 振幅辐射传递方程的解

$\left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)$满足以下方程:

$ \begin{array}{l} \left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \exp \eta \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{{\bf{\bar s}}}^{ud\left[ 1 \right]}}{{\left( {z,\omega ,{\bf{k}}} \right)}^T}}&{\bf{0}} \end{array}} \right) \cdot \\ \left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (85)

方程(85)中含转置运算, 算子运算很难跨越。为了便于求取时间域倒转解或象征域共轭解, 我们研究了$\left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)$关于(z, -ω, -k)的方程。将公式(85)带入方程(74), 可得:

$ \begin{array}{l} {\partial _z}\left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)\\ = \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right)}&{\bf{0}} \end{array}} \right) \cdot \\ \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)\\ = \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{{\bf{\bar s}}}^{ud\left[ 1 \right]}}{{\left( {z,\omega ,{\bf{k}}} \right)}^T}}&{\bf{0}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right) \cdot \\ \left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (86)
$ \begin{array}{l} {\partial _z}\left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{22}}}&{{{\bf{f}}_{21}}}\\ {{{\bf{f}}_{12}}}&{{{\bf{f}}_{11}}} \end{array}} \right)\left( {z, - \omega , - {\bf{k}}} \right)\\ = \left( {\begin{array}{*{20}{c}} {\bf{0}}&{{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)}\\ {{{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right)}&{\bf{0}} \end{array}} \right) \cdot \\ \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{22}}}&{{{\bf{f}}_{21}}}\\ {{{\bf{f}}_{12}}}&{{{\bf{f}}_{11}}} \end{array}} \right)\left( {z, - \omega , - {\bf{k}}} \right) \end{array} $ (87)

由方程(86)、方程(87)可得:

$ \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{22}}}&{{{\bf{f}}_{21}}}\\ {{{\bf{f}}_{12}}}&{{{\bf{f}}_{11}}} \end{array}} \right)\left( {z, - \omega , - {\bf{k}}} \right) $ (88)
$ \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{\bf{\bar f}}_{22}^T}&{{\bf{\bar f}}_{21}^T}\\ {{\bf{\bar f}}_{12}^T}&{{\bf{\bar f}}_{11}^T} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) $ (89)

方程(86)、方程(87)可写为:

$ \begin{array}{l} {\partial _z}{{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{21}}\left( {z,\omega ,{\bf{k}}} \right)\\ {\partial _z}{{\bf{f}}_{21}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right){{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right)\\ = {{{\bf{\bar s}}}^{ud\left[ 1 \right]T}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (90)
$ \begin{array}{l} {\partial _z}{{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{22}}\left( {z,\omega ,{\bf{k}}} \right)\\ {\partial _z}{{\bf{f}}_{22}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{s}}^{ud\left[ 1 \right]}}\left( {z, - \omega , - {\bf{k}}} \right){{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right)\\ = {{{\bf{\bar s}}}^{ud\left[ 1 \right]T}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (91)
4.5 反射响应的预测算子分解

$ \begin{array}{l} a\left( {z,\omega ,{\bf{k}}} \right) = - {{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) + {\bf{\bar f}}_{22}^T\left( {z,\omega ,{\bf{k}}} \right)\\ a\left( {z,\omega ,{\bf{k}}} \right) = - {{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) + {{\bf{f}}_{21}}\left( {z, - \omega , - {\bf{k}}} \right) \end{array} $ (92)

利用公式(90)可得:

$ \begin{array}{l} {\partial _z}{{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{21}}\left( {z,\omega ,{\bf{k}}} \right)\\ {\partial _z}{{\bf{f}}_{21}}\left( {z, - \omega , - {\bf{k}}} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right){{\bf{f}}_{11}}\left( {z, - \omega , - {\bf{k}}} \right)\\ {\partial _z}\left( {{{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right) - {{\bf{f}}_{21}}\left( {z, - \omega , - {\bf{k}}} \right)} \right) = {{\bf{s}}^{ud\left[ 1 \right]}}\left( {z,\omega ,{\bf{k}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\left( {{{\bf{f}}_{21}}\left( {z,\omega ,{\bf{k}}} \right) - {{\bf{f}}_{11}}\left( {z, - \omega , - {\bf{k}}} \right)} \right)\\ {\partial _z}a\left( {z,\omega ,{\bf{k}}} \right) = - {{\bf{s}}^{ud\left[ 1 \right]}}a\left( {z, - \omega , - {\bf{k}}} \right) \end{array} $ (93)

(93)式称为预测算子象征的齐次方程。将反射波响应、透射波响应的边界条件:

$ \begin{array}{l} \left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {0,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{{\bf{I}}_3} + {\bf{r}}}\\ { - {\bf{r}}} \end{array}} \right)\left( {0,\omega ,{\bf{k}}} \right)\\ \left( {\begin{array}{*{20}{c}} {{{\bf{u}}_1}}\\ {{{\bf{d}}_1}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ {\bf{e}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (94)

带入方程(85)可得:

$ \begin{array}{l} \left( {\begin{array}{*{20}{c}} 0\\ {\bf{e}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{11}}}&{{{\bf{f}}_{12}}}\\ {{{\bf{f}}_{21}}}&{{{\bf{f}}_{22}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)\# \left( {\begin{array}{*{20}{c}} {{{\bf{I}}_3} + {\bf{r}}}\\ { - {\bf{r}}} \end{array}} \right) \cdot \\ \;\;\;\;\;\;\left( {0,\omega ,{\bf{k}}} \right)\\ \left( {\begin{array}{*{20}{c}} 0\\ {{{{\bf{\bar e}}}^T}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right) = \left( {\begin{array}{*{20}{c}} {{{\bf{f}}_{22}}}&{{{\bf{f}}_{21}}}\\ {{{\bf{f}}_{12}}}&{{{\bf{f}}_{11}}} \end{array}} \right)\left( {z,\omega ,{\bf{k}}} \right)\# \\ \;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {{{\bf{I}}_3} + {{{\bf{\bar r}}}^T}}\\ { - {{{\bf{\bar r}}}^T}} \end{array}} \right)\left( {0,\omega ,{\bf{k}}} \right) \end{array} $ (95)

(95)式的第1和第4行可写为:

$ \begin{array}{l} 0 = {{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right)\left( {{{\bf{I}}_3} + {\bf{r}}\left( {0,\omega ,{\bf{k}}} \right)} \right) - \\ \;\;\;\;\;\;\;{{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right){\bf{r}}\left( {0,\omega ,{\bf{k}}} \right) \end{array} $ (96)
$ \begin{array}{l} {{{\bf{\bar e}}}^T}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right)\left( {{{\bf{I}}_3} + {{{\bf{\bar r}}}^T}\left( {0,\omega ,{\bf{k}}} \right)} \right) - \\ \;\;\;\;\;\;\;{{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right){{{\bf{\bar r}}}^T}\left( {0,\omega ,{\bf{k}}} \right) \end{array} $ (97)

(96)式和(97)式相加, 利用方程(88)、(92)可得:

$ \begin{array}{l} {{{\bf{\bar e}}}^T}\left( {z,\omega ,{\bf{k}}} \right) = \left( {{{\bf{f}}_{12}}\left( {z,\omega ,{\bf{k}}} \right) - {{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right)} \right)\left( {{{\bf{I}}_3} + } \right.\\ \;\;\;\left. {{\bf{r}}\left( {0,\omega ,{\bf{k}}} \right) + {{{\bf{\bar r}}}^T}\left( {0,\omega ,{\bf{k}}} \right)} \right)\\ \;\;\;\; = \left( {{{{\bf{\bar f}}}_{21}}\left( {z,\omega ,{\bf{k}}} \right) - {{\bf{f}}_{11}}\left( {z,\omega ,{\bf{k}}} \right)} \right)\left( {{{\bf{I}}_3} + {\bf{r}}\left( {0,\omega ,{\bf{k}}} \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {{{{\bf{\bar r}}}^T}\left( {0,\omega ,{\bf{k}}} \right)} \right)\\ \;\;\;\; = {\bf{a}}\left( {z,\omega ,{\bf{k}}} \right)\left( {{{\bf{I}}_3} + {\bf{r}}\left( {0,\omega ,{\bf{k}}} \right) + {\bf{\bar r}}\left( {0,\omega ,{\bf{k}}} \right)} \right)\\ \;\;\;\; = {\bf{a}}\left( {z,\omega ,{\bf{k}}} \right){\bf{e}}\left( {z,\omega ,{\bf{k}}} \right){{{\bf{\bar e}}}^T}\left( {z,\omega ,{\bf{k}}} \right) \end{array} $ (98)

可得:

$ {\bf{e}}\left( {z,\omega ,{\bf{k}}} \right)\# {\bf{a}}\left( {z,\omega ,{\bf{k}}} \right) = {{\bf{I}}_3} $ (99)
$ {\bf{a}}\left( {z,\omega ,{\bf{k}}} \right)\# = {\left( {{\bf{e}}\left( {z,\omega ,{\bf{k}}} \right)\# } \right)^{ - 1}} $ (100)

式中:a(z, ω, k)称为预测算子的象征, 它是点源透射波响应的逆算子响应。利用(99)式, (100)式可得:

$ \begin{array}{l} {{\bf{I}}_3} + {\bf{r}}\left( {z,\omega ,{\bf{k}}} \right) + {{{\bf{\bar r}}}^T}\left( {z,\omega ,{\bf{k}}} \right)\\ \;\;\;\;\;\;\; = {\bf{a}}{\left( {z,\omega ,{\bf{k}}} \right)^{ - 1}}\# {{{\bf{\bar a}}}^T}{\left( {z,\omega ,{\bf{k}}} \right)^{ - 1}} \end{array} $ (101)

(101)式称为反射响应预测算子分解的象征形式, 由于预测算子a(z, ω, k)具有因果算子象征的性质, 故(101)式又可以称为因果分解表达式。

5 小结

本文提出了横向非均匀弹性介质方向波的积分表示, 正演计算的过程是:①利用象征齐次函数分级求解本征值问题; ②利用本征函数沿深度变化求微分散射矩阵象征; ③求取单程波散射算子象征; ④求解自回归算子齐次方程; ⑤利用自回归算子求取反射算子象征。这种积分表示与一维谱分解、P-SV波谱分解、横向变速谱分解兼容。自回归算子齐次方程对研究谐振散射有重要意义。

致谢: 感谢李幼铭教授领导的“八五”, “九五”油储项目提供了理论和应用相结合的平台, 使得引入李群和拟微分算子改进频率域散射理论成为可能。
参考文献
[1]
YANG K Q. A Periodic solution of Kdv equation in virasoro algebra[J]. Chinese Physics Letters, 1995, 12(2): 65-67. DOI:10.1088/0256-307X/12/2/001
[2]
FUCHS J, SCHWEIGERT C. Symmetries, Lie algebras and representations:a graduate course for physicists[M]. Cambridge: Cambridge University Press, 2003: 108-128.
[3]
CHANG S K. A high-order quasi-static theory for P-SV waves in finely layered media[J]. Expanded Abstracts of 63rd Annul Internat SEG Mtg, 1993, 903-904.
[4]
TANER M T, KOEHLER F, SHERIFF R. Complex seismic trace analysis[J]. Geophysics, 1979, 44(6): 1041-1063. DOI:10.1190/1.1440994
[5]
李幼铭, 刘洪, 吴永刚, 等. 我国陆相油储地球物理研究历程回顾与认识[J]. 地球物理学进展, 2007, 22(4): 1280-1284.
LI Y M, LIU H, WU Y G, et al. Review and understanding on China's terrestrial reservoir geophysical research process[J]. Progress in Geophysics, 2007, 22(4): 1280-1284.
[6]
李幼铭. 面向油气勘探开发提升地震偏移及属性刻划水平[J]. 地球物理学进展, 2002, 17(2): 198-210.
LI Y M. To enhance seismic migration and attribute characterization level face to oil and gas exploration and development[J]. Progress in Geophysics, 2002, 17(2): 198-210.
[7]
李幼铭. 知识创新工程重大项目在大庆油田的技术成果要览[J]. 地球物理学进展, 2003, 18(1): 5-18.
LI Y M. Major achievement of knowledge innovation project in Daqing oilfield[J]. Progress in Geophysics, 2003, 18(1): 5-18.
[8]
李幼铭. 对"油储"项目在大庆实施获得成果的概述及认识[J]. 地球物理学进展, 2006, 21(1): 135-142.
LI Y M. Overview and understanding of the achievement of the reservoir geophysics project in Daqing[J]. Progress in Geophysics, 2006, 21(1): 135-142.
[9]
刘洪, 袁江华, 勾永峰, 等. 地震逆散射波场和算子的谱分解[J]. 地球物理学报, 2007, 50(1): 240-247.
LIU H, YUAN J H, GOU Y F, et al. Spectral factorizationof wavefield and operator in seismic inverse scattering[J]. Chinese Journal of Geophysics, 2007, 50(1): 240-247.
[10]
刘洪, 刘国峰, 武威, 等. 三维波动方程逆散射的基础理论研究[J]. 石油物探, 2007, 46(6): 569-581.
LIU H, LIU G F, WU W, et al. Study on the Fundamental Theory of Inverse Scattering Of Multi-Dimensional Wave Equation[J]. Geophysical Prospecting for Petroleum, 2007, 46(6): 569-581.
[11]
刘洪, 王秀闽, 曾锐, 等. 单程波算子积分解的象征表示[J]. 地球物理学进展, 2007, 22(2): 463-471.
LIU H, WANG X M, ZENG R, et al. Symbol description to integral solution of one-way wave operator[J]. Progress in Geophysics, 2007, 22(2): 463-471.
[12]
刘洪, 袁江华, 陈景波, 等. 大步长波场深度延拓的理论[J]. 地球物理学报, 2006, 49(6): 1779-1793.
LIU H, YUAN J H, CHEN J B, et al. Theory and implementing scheme of large-step wavefield depth extrapolation[J]. Chinese Journal of Geophysics, 2006, 49(6): 1779-1793.
[13]
刘洪, 刘国峰, 李博, 等. 基于横向导数的走时计算方法及其在叠前时间偏移中的应用[J]. 石油物探, 2009, 48(1): 3-10.
LIU H, LIU G F, LI B, et al. The travel time calculation method via laterally derivative to velocity and its application in Prestack time migration[J]. Geophysical Prospecting for Petroleum, 2009, 48(1): 3-10.
[14]
刘洪. 横向非均匀弹性介质方向波响应的积分表示[C]//勘探地球物理学进展暨庆祝贺振华教授七十寿辰学术研讨会论文集. 北京: 石油工业出版社, 2008: 13-28
LIU H. Integral representation of directional wave responses in laterally inhomogeneous elastic media[C]//Progress in Exploration Geophysics—Celebration of 70th anniversary of Professor He Zhenhua. Beijing: Petroleum Industrial Press, 2008: 13-28
[15]
CLAERBOUT J F. Fundamentals of geophysical data processing with applications to petroleum prospecting[M]. NewYork: McGraw Hill, 1976: 1-246.
[16]
FRASIER C W. Discrete time solution of plane P-SV waves in a plane layered medium[J]. Geophysics, 1970, 35(2): 197-219. DOI:10.1190/1.1440085
[17]
ISERLES A, MUNTHE—KAAS H, NFRSTT S, et al. Lie-group methods[J]. Acta Numerica, 2000, 9(1): 215-365.
[18]
LUO S, QIAN J, BURRIDGE R.. Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime[J]. Journal of Computational Physics, 2014, 270: 378-401. DOI:10.1016/j.jcp.2014.03.066
[19]
FRYER G J, FRAZER L N. Seismic waves in stratified anisotropic media[J]. Geophysical Journal International, 1984, 78(3): 691-710. DOI:10.1111/j.1365-246X.1984.tb05065.x
[20]
FRYER G J, FRAZER L N. Seismic waves in stratified anisotropic media-Ⅱ.Elastodynamic eigensolutions for some anisotropic systems[J]. Geophysical Journal International, 1987, 91(1): 73-101. DOI:10.1111/gji.1987.91.issue-1
[21]
宋海斌. 层状介质弹性参数反演研究[D]. 上海: 同济大学, 1998
SONG H B. Inversion of elastic parameters in layered media[D]. Shanghai: Tongji University, 1998
[22]
齐民友, 徐超江, 王维克. 现代偏微分方程引论[M]. 武汉: 武汉大学出版社, 2005: 20-32.
QI M Y, XU C J, WANG W K. Introduction to modern partial differential equations[M]. Wu Han: Wuhan University Press, 2005: 20-32.
[23]
CORDES H O. The technique of pseudodifferential operators[M]. Cambridge: Cambridge University Press, 1995: 69-73.