Revisiting the cross-correlation and SPatial AutoCorrelation (SPAC) of the seismic ambient noise based on the plane wave model
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摘要: 自
Aki（1957） 提出微震的空间自相关（SPatial AutoCorelation, SPAC）技术以来，SPAC技术一直独立发展，并在工程地震领域获得了广泛应用. 近20年来，地震干涉（Seisimic Interferometry, SI）在多个领域引起人们的关注，该技术的核心思想是连续地震背景噪声的互相关函数（Noise Crosscorrelation Function, NCF）可以重建系统的格林函数（Green's Function, GF），对该技术的回溯性研究建立了SPAC和NCF的关系：它们是对同一物理现象的不同描述，SPAC在频率域中描述随机平稳噪声的空间相干，NCF在时间域中描述扩散场的互相关. 理论上SAPC和NCF技术要求背景噪声源均匀分布，这样的噪声场可以用平面波叠加来模拟. 本文基于平面波模型重访地震背景噪声的互相关和空间自相关技术，从单色平面波的互相关表示出发，对地震背景噪声互相关及空间自相关技术进行评述，试图使这些概念更易于理解. 与之前众多研究地震干涉技术的理论相比，本文尤其关注以下几点：（1）基于简单的平面波模型，给出不同维度下，源或台站对方位均匀分布时，平面波互相关对入射波的方位平均和台阵对的方位平均结果，并对格林函数GF和时域互相关函数NCF的关系进行总结. （2）给出声源和（或）交叉台站方位分布不均匀时的互相关表示，指出这种非均匀性对方位的依赖关系，与弱各向异性介质中面波速度的方位依赖关系类似，因此，非均匀源的影响在反演时可能会映射到面波方位各向异性结果中. （3）互相关运算中，哪一个台站是虚拟源. NCF包含因果性和非因果性两部分，NCF的非对称性通常用于研究噪声源的方位分布，但由于源和接收的互易关系，及对互相关运算的不同定义和不同的傅里叶变换习惯，哪一个台站是虚拟源在目前的文献中并不明确. （4）方位平均和时间平均的关系. 在SPAC处理中，需要对不同方位分布的台站对进行方位平均，本文从理论上说明，单个平面波入射时，交叉台站互相关系数对台站对的方位平均，等价于单个台站对互相关系数对入射波的时间平均. （5）几种特定分布非均匀噪声源的SPAC表示. 包括单独的因果性噪声源和非因果性噪声源给出的互相关函数表示，及由此带来的相移问题. （6）利用SPAC、NCF和面波GF之间的关系，给出交叉分量的空间自相关系数表示. （7）衰减介质的空间相干表示. 虽然利用地震干涉技术研究介质衰减在理论上仍然存在一些争议，但人们正试图研究从连续背景噪声记录中提取介质衰减的可能性. 本文基于平面波模型，给出了不同坐标选择下，衰减介质的空间相干表示，这种表达的不同，指示了由地震干涉技术提取介质衰减的困难. 与众多研究地震干涉的理论相比，比如稳相近似理论、互易定理、时间反转声学等，本文主要考虑均匀介质，不涉及非均匀介质的散射，从最简单的平面波模型，理解背景噪声重建系统格林函数这一地震干涉的核心思想和相应的基本概念.Abstract: SinceAki (1957) proposed the spatial autocorrelation (SPAC) technology based on the microtremor, the SPAC technique has been independently developed and widely used to infer the S wave velocity at the shallow structure in the field of civil engineering. In the past two decades, seismic interferometry has attracted people's attention in many fields. The key idea of seismic interferometry (SI) is the Green's function (GF) of the system can be extracted via noise cross-correlation function (NCF), which is calculated by cross correlating the continuous seismic ambient noise. The relation between SPAC and NCF is established by the retrospective study of SI technology: two theories describe the same physics with different language. SPAC of microtremors is mainly conducted in the frequency domain, while the retrieval of Green's function is done in the time domain. In theory, both of them require a uniform distribution of ambient noise sources. Such a noise model can be simulated by plane wave superposition. In this paper, starting from the cross-correlation representation of monochromatic plane waves, we review the SPAC and NCF technique of the seismic ambient noise based on the plane wave model. Compared with previous references on seismic interference technology, special attentions are focused on the following: (1) Under the assumption that the source and station-pair orientation are uniform distributed, the averaged SPAC representation is given over the wave incidence and over the inter-station orientation. The relationship between the GF and NCF are reviewed for 1D, 2D and 3D diffuse field constructed by plane wave superposition. (2) The SPAC representation is given for the uneven distribution of the source or the inter-station orientation. It is pointed out the dependence of the SPAC expression on the azimuth of the source or the inter-station orientation is similar to the azimuth dependence of the surface wave velocity in weakly anisotropic media. The influence of anisotropic source and inter-station orientation may therefore be projected into the inversed surface wave azimuthal anisotropy. (3) Which station is the virtual source when calculating the cross correlation using the given definition. Causal and noncausal parts are involved in NCFs. The asymmetry of NCFs is usually used to study the azimuth distribution of noise sources. However, due to the reciprocal relationship between the source and the receiver, and different convention on the cross-correlation and Fourier transform, it is not clear stated in the literature which station is the virtual source. (4) The relationship of the average over the azimuth and over the time. It is usually in SPAC technique to conduct the azimuthal average over the inter-station orientation. For one incident plane wave, it is illustrated in this paper that the averaged SPAC expression over the inter-station orientation is equivalent to that averaged over the time. (5) SPAC representations are given for several noise source model with non-uniform distribution. The phase shift in causal and noncausal part of NCFs is discussed. (6) The SPAC expressions for cross component is derived based on the relations between SPAC, NCF and surface wave GF. (7) The SPAC representation for the attenuation medium is given. Although there is still some controversy in theory on the extracting of attenuation by SI technology, people have been trying to study the possibility on extracting attenuation of the earth from continuous ambient seismic noise. Based on the plane wave model, the SPAC expressions are given for the attenuating medium. The difference of SPAC expressions for different normalization and the selection of the coordinate origin indicates the difficulty of extracting the attenuation of the medium using NCF. Compared with other theories studying SI, such as the stationary phase approximation, reciprocity theorem, time reversal acoustics, etc., homogeneous media is considered in this paper. The key idea and concept on SI is illustrated from the simple plan wave model. -
图 1 单色平面波入射示意图，
$\theta $ 为台站对的方位，$\varphi $ 为入射波的传播方向，a、b为台站位置Figure 1. Illustration on the incidence of a monochromatic plane wave.
$\theta $ is the azimuth of the station pair, and$\varphi $ is the propagation direction of the incident wave. Stations a and b are denoted by triangles
图 2 一维情形下，平面单色波分别从左（a）右（b）单边入射的示意图，黑色实线箭头表示x轴正向，蓝色虚线表示虚拟源到虚拟接收的方向
Figure 2. The plane monochromatic waves are incident from the left (a) or from the right (b). The solid black arrow indicates the positive direction of the x-axis, and the blue dotted line indicates the direction from the virtual source to the virtual receiver
图 3 一维情形下单色平面波从左侧入射，对台站对方位进行平均
Figure 3. A monochromatic plane wave is incident from the left, and average is take over the station pair azimuth
图 4 一维情形下单色平面波双侧入射，单方向台站的互相关结果对入射波的方位平均
Figure 4. A monochromatic plane wave is incident on both sides, the cross-correlation for single station pair is averaged over the incident wave
图 5 二维情况下，单个平面波入射，接收台站沿圆周均匀连续分布
Figure 5. A single monochromatic plane wave is incident and stations are distributed uniformly along the circumference
图 6 二维情况下，一个台站对接收，入射波各向同性入射
Figure 6. The monochromatic plane waves incidence isotropic from all direction and single station pair receives
图 7 背景噪声互相关时间叠加平均（左）和SPAC的方位平均（右）（刘庆华等，2015）
Figure 7. Ambient noise cross-correlation is averaged over the long time (left) and SPAC is averaged over the station-pair azimuth (right) (Liu et al., 2015)
图 8 二维情况下，平面波强度随方位变化（a）或交叉台站方位分布不均匀（b）.（a）入射平面波入射强度随方位变化，蓝色实线表示入射波强度随方位的变化. （b）单个平面波入射，台站对沿圆周非均匀分布，红色实线是台站对密度随方位的变化（修改自Lu et al., 2018）
Figure 8. An illustration of uneven distribution of sources and station-pair orientations. (a) A single station pair and incident plane waves with azimuthal varying intensity. The blue solid line denotes the amplitude intensity of incident wave as a function of azimuth. (b) A single plane wave and station-pair orientations with azimuthal varying distribution. The red solid line denotes the distribution of the station number as the function of the azimuth（modified from Lu et al., 2018）
图 9 二维情况下，入射波密度和台站方位分布均不均匀
Figure 9. Both the incident plane wave and the station-pair distribution are not isotropic
图 10 三维情形下单色平面波入射时，波和台站方位坐标示意图
Figure 10. Schematic diagram on the coordinates of the incident wave and station azimuth for 3D case
图 11 源主要分布在稳相区域示意图
Figure 11. Sources are mainly distributed in the stationary phase area
图 12 格林函数因果性（红色五角星）和非因果性（蓝色五角星）部分对应的源所在的位置
Figure 12. The location of the sources corresponding to the causal (red star) and non-causal (blue star) part of Green's function
图 13 源主要分布在非稳相区域
Figure 13. Sources are mainly distributed in nonstationary phase area
图 14 三维情形下，特定方向入射的平面波源
Figure 14. Plane wave source incident from a specific direction in a 3D case
图 15 源沿半径为R的圆周均匀分布和圆周内任意两点a、b的位置坐标关系（图中仅给出了单个源）
Figure 15. The source is uniformly distributed along the circle containing two stations (a and b) with radius R (only a single source is shown in the figure)
图 17 将坐标原点置于a、b两点连线的中点
Figure 17. The midpoint between a and b is taken as the origin of the coordinates
表 1 均匀入射的平面波不同维度下格林函数（GF）、空间自相关（SPAC）系数和时域互相关函数（NCF）的关系
Table 1. The relationship between Green's function (GF), spatial autocorrelation coefficient (SPAC) and cross-correlation function (NCF) for the uniform incident plane waves at 1D, 2D and 3D
空间
维度单色平面波SPAC系数和互相关函数[${C_{{\rm{ab}}}}(r,t)$]的表示${\hat \phi _{{\rm{ab}}}}(r,\omega) = {C_{{\rm{ab}}}}(r,\omega ;\tau = 0)$ ${C_{ {\rm{ab} } } }(r,t) = \displaystyle\int\limits_{ - \infty }^{ + \infty } { { {\hat \phi }_{ {\rm{ab} } } }(r,\omega)} { {\rm{e} }^{i\omega t} }{\rm{d} }t$ 时间域格林函数GF[$G(r,t)$]和NCF[${C_{{\rm{ab}}}}(r,t)$]的关系 频率域GF[$G(r,\omega)$]、SPAC[${\hat \phi _{{\rm{ab}}}}(r,\omega)$]的关系 SPAC
[${\hat \phi _{{\rm{ab}}}}(r,\omega)$]单色波NCF[${C_{{\rm{ab}}}}(r,\omega,\tau)$] GF GF和NCF的关系 GF GF和SPAC的关系 1D $\cos kx$ $\cos \omega \tau \cos kx$ $G(x,t) = \dfrac{c}{2}H\left(t - \dfrac{x}{c}\right)$ ${C_{{\rm{ab}}}}(x,t) = \dfrac{1}{c}\left[ {\dfrac{{{\rm{d}}G(x,t)}}{{{\rm{d}}t}} - \dfrac{{{\rm{d}}G(x, - t)}}{{{\rm{d}}t}}} \right]$ $G(x,\omega) = \dfrac{i}{{2k}}{{\rm{e}}^{ - ikx}}$ $\begin{array}{l} { {\hat \phi }_{ {\rm{ab} } } }(x,\omega) = - ik\left[ {G(x,\omega) - {G^*}(x,\omega)} \right] \\ \qquad\quad\;\;\;\, = 2k{\rm{Im} } \left[ {G(x,\omega)} \right] \end{array}$ 2D ${J_0}(kr)$ $\cos \omega \tau {J_0}(kr)$ $G(r,t) = - \dfrac{1}{{2{ \text{π}}}}\dfrac{{H \left(t - \tfrac{r}{c}\right)}}{{\sqrt {{t^2} - \tfrac{{{r^2}}}{{{c^2}}}} }}$ ${\cal H} [{C_{{\rm{ab}}}}(r,t)] = 2\left[ {G(r,t) - G(r, - t)} \right]$ $G(r,\omega) = \dfrac{i}{4}H_0^{(2)}(kr)$ $\begin{array}{l} { {\hat \phi }_{ {\rm{ab} } } }(r,\omega) = - 2i\left[ {G(r,\omega) - {G^*}(r,\omega)} \right] \\ \qquad\quad\;\;\; = 4{\rm{Im} } \left[ {G(r,\omega)} \right] \end{array}$ 3D ${j_0}(kr) = \dfrac{{\sin kr}}{{kr}}$ $\cos \omega \tau {j_0}(kr)$ $G(r,t) = - \dfrac{1}{ {4{\text{π} }r} }\delta \left(t - {\tfrac{r}{c}}\right)$ $\dfrac{{{\rm{d}}{C_{{\rm{ab}}}}(r,t)}}{{{\rm{d}}t}} = 2{ \text{π}}c\left[ {G(r,t) - G(r, - t)} \right]$ $G(r,\omega) = - \dfrac{1}{{4{ \text{π}}}}\dfrac{{{{\rm{e}}^{ - ikr}}}}{r}$ $\begin{array}{l} { {\hat \phi }_{ {\rm{ab} } } }(r,\omega) = - \dfrac{ {2{\text{π} }i} }{k}\left[ {G(r,\omega) - {G^*}(r,\omega)} \right] \\ \qquad\quad\;\;\; = \dfrac{ {4{\text{π} } } }{k}{\rm{Im} } \left[ {G(r,\omega)} \right] \end{array}$ 注：模型为均匀入射的互不相干的单位幅度平面波，方位平均后的结果进行了幅度归一化. $H(t)$为单位阶跃函数，${\cal{H}}$[ ]为希尔伯特变换符号，*表示复共轭，${J_0}(x)$为第一类零阶贝塞尔函数，${j_0}(x)$为零阶第一类球贝塞尔函数，$H_0^{(2)}$为第二类Hankel函数，$r = |{\boldsymbol{r}} - {\boldsymbol{r}}'|$为场点和源点之间的距离，其中${\boldsymbol{r}}(x,y,z)$和${\boldsymbol{r}}'(x',y',z')$ 分别为场点和源点坐标，常数$c$为均匀介质的相速度. 傅里叶变换习惯为$f(t) = \dfrac{1}{ {2{\text{π} } } }\displaystyle\int\limits_{ - \infty }^\infty {F(\omega){ {\rm{e} }^{i\omega t} }{\rm{d} }\omega }$，函数$f(t)$和$g(t)$的互相关定义为${C_{{\rm{fg}}}}(\tau) = \displaystyle\int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t)g(t + \tau){\rm{d}}t} $. 
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表 2 二维情况下，不同分布的源对应的空间自相关系数表示
Table 2. The SPAC expressions for specific distributed sources for 2D case
源的分布 空间自相关（SPAC）系数 
$\dfrac{ {({\text{π} } - 2\alpha)} }{ {2{\text{π} } } }{J_0}(kr) - \dfrac{ {\rm{1} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha }$ 
$\dfrac{ {({\text{π} } - 2\alpha)} }{ {2\text{π} } }{J_0}(kr) - \dfrac{ {\rm{1} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha }$ 
$\dfrac{ {({\text{π} } - 2\alpha)} }{ {\text{π} } }{J_0}(kr) - \dfrac{ {\rm{2} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha }$ 
$\dfrac{\alpha }{ {\text{π} } }{J_0}(kr) + \dfrac{ {\rm{1} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha + } \dfrac{2}{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- i)}^{2m - 1} } } }{ {2m - 1} }{J_{2m - 1} }(kr)\sin (2m - 1)\alpha }$ 
$\dfrac{\alpha }{ {\text{π} } }{J_0}(kr) + \dfrac{ {\rm{1} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha - } \dfrac{2}{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- i)}^{2m - 1} } } }{ {2m - 1} }{J_{2m - 1} }(kr)\sin (2m - 1)\alpha }$ 
$\dfrac{ { {\rm{2} }\alpha } }{ {\text{π} } }{J_0}(kr) + \dfrac{ {\rm{2} } }{ {\text{π} } }\displaystyle\sum\limits_{m = 1}^\infty {\dfrac{ { { {(- 1)}^m} } }{m}{J_{2m} }(kr)\sin 2m\alpha }$ 
$\dfrac{{\rm{1}}}{2}\left[ {{J_0}(kr) - i{{\rm H}_0}(kr)} \right]$ 
$\dfrac{{\rm{1}}}{2}\left[ {{J_0}(kr){\rm{ + }}i{{\rm H}_0}(kr)} \right]$ 
${J_0}(kr)$ 注：远场平面波源沿圆周在一定范围内均匀分布，并假定源的分布密度为1，${J_0}(x)$为第一类零阶贝塞尔函数，$r$为a、b两点之间的距离，${{\rm{H}}_0}(x)$为第一类零阶Struve函数，傅里叶变换习惯为$f(t) = \dfrac{1}{ {2{\text{π} } } }\int\limits_{ - \infty }^\infty {F(\omega){ {\rm{e} }^{i\omega t} }{\rm{d} }\omega }$，函数$f(t)$和$g(t)$的互相关定义为${C_{{\rm{fg}}}}(\tau) = \int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t)g(t + \tau){\rm{d}}t} $.
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表 3 先叠加后相关情况下，不同坐标系下的（不同归一化条件）空间自相关系数
${\hat R_{{\rm{ab}}}}$ Table 3. The SPAC expressions
${\hat R_{{\rm{ab}}}} $ at different coordinate systems for attenuating media$\begin{array}{l} \hat R_{ {\rm{ab} } }^{ {\rm{(1)} } } \\ \hat R_{ {\rm{ab} } }^{ {\rm{(1)} } }(+) \\ \hat R_{ {\rm{ab} } }^{ {\rm{(1)} } }(-) \\ \end{array}$ $\begin{array}{l} \hat R_{ {\rm{ab} } }^{ {\rm{(2)} } } \\ \hat R_{ {\rm{ab} } }^{ {\rm{(2)} } }(+) \\ \hat R_{ {\rm{ab} } }^{ {\rm{(2)} } }(-) \\ \end{array}$ $\begin{array}{l} \hat R_{ {\rm{ab} } }^{ {\rm{(3)} } } \\ \hat R_{ {\rm{ab} } }^{ {\rm{(3)} } }(+) \\ \hat R_{ {\rm{ab} } }^{ {\rm{(3)} } }(-) \\ \end{array}$ 
$\dfrac{{{J_0}(kr)}}{{\sqrt {{I_0}(2\alpha r)} }}$ ${J_0}(kr)$ $\dfrac{{{J_0}(kr)}}{{{I_0}(2\alpha r)}}$ $\dfrac{{{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)}}{{\sqrt {{I_0}(2\alpha r) - {{\rm L}_0}(2\alpha r)} }}$ ${J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)$ $\dfrac{{{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) - {{\rm L}_0}(2\alpha r)}}$ $\dfrac{{{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)}}{{\sqrt {{I_0}(2\alpha r) + {{\rm L}_0}(2\alpha r)} }}$ ${J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)$ $\dfrac{{{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) + {{\rm L}_0}(2\alpha r)}}$ 
$\dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{\sqrt {{I_0}^2(\alpha r) - {{\rm L}_0}^2(\alpha r)} }}$ $\dfrac{{{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) + {{\rm L}_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) - {{\rm L}_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{\sqrt {{I_0}^2(\alpha r) - {{\rm L}_0}^2(\alpha r)} }}$ $\dfrac{{{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) - {{\rm L}_0}(\alpha r)}}$ $\dfrac{{{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) + {{\rm L}_0}(\alpha r)}}$ 注：${J_0}(x)$为第一类零阶贝塞尔函数，${I_0}(x)$为第一类零阶修正的贝塞尔函数，${{\rm{H}}_0}(x)$为零阶Struve函数，${{\rm L}_0}(x)$为0阶修正的Struve函数，复波数$k = {k_0} - i\alpha $，$\alpha $为介质衰减因子，${\hat R_{{\rm{ab}}}}$中的上标（1）（2）（3）表示不同的归一化因子选择，（+）和（−）表示对应单边正向（因果性格林函数部分）和单边反向（非因果性格林函数部分）分布的源. 傅里叶变换习惯为$f(t) = {(2{ \text{π}})^{ - 1}}\int\limits_{ - \infty }^\infty {F(\omega){{\rm{e}}^{i\omega t}}{\rm{d}}\omega } $，函数$f(t)$和$g(t)$的互相关定义为${C_{{\rm{fg} } } }(\tau) = \int\limits_{ - \infty }^{ + \infty } { {f^{\rm{*} } }(t)g(t + \tau){\rm{d} }t}$.
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表 4 先相关后叠加中不同坐标系下的
${\hat \gamma _{{\rm{ab}}}}$ Table 4. The SPAC expressions
${\hat \gamma _{{\rm{ab}}}} $ at different coordinate systems for attenuating media$\begin{array}{l} \hat \gamma _{ {\rm{ab} } }^{ {\rm{(1)} } } \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(1)} } }(+) \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(1)} } }(-) \\ \end{array}$ $\begin{array}{l} \hat \gamma _{ {\rm{ab} } }^{ {\rm{(2)} } } \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(2)} } }(+) \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(2)} } }(-) \\ \end{array}$ $\begin{array}{l} \hat \gamma _{ {\rm{ab} } }^{ {\rm{(3)} } } \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(3)} } }(+) \\ \hat \gamma _{ {\rm{ab} } }^{ {\rm{(3)} } }(-) \\ \end{array}$ 
${J_0}({k_0}r)$ ${J_0}(kr)$ ${J_0}({k_0}r + i\alpha r)$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) - i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) + i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right]$ 
${J_0}({k_0}r)$ ${J_0}(kr)$ ${J_0}({k_0}r + i\alpha r)$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}(kr) - i{{\rm{H}}_0}(kr)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) - i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$ $\dfrac{1}{2}\left[ {{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)} \right]$ $\dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) + i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right]$ 注：表中各变量说明同表3.
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