Revisiting the cross-correlation and SPatial AutoCorrelation (SPAC) of the seismic ambient noise based on the plane wave model
-
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 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 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)
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} $. 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} $. 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}$. 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. -
[1] Abramowitz M, Stegun I A. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[M]. Massachusetts: Courier Corporation [2] Aki K. Space and time spectra of stationary stochastic waves with special reference to microtremors[J]. Bulletin of the Earthquake Research Institute, 1957,35:415-456. [3] Aki K, Richards P G. 2002. Quantitative Seismology[M]. Sausalito: University Science Books. [4] Arfken G B, Weber H J. 2005. Mathematical Methods for Physicists, sixth edition[M]. Amsterdam: Elsevier Academic Press. [5] Asten M. On bias and noise in passive seismic data from finite circular array data processed using SPAC methods[J]. Geophysics, 2006, 71:153-162. [6] Bateman H. 1954. Tables of Integral Transforms[M]. New York: McGraw-Hill. [7] Ben-Menahem A, Singh S J. Multipolar elastic fields in a layered half space[J]. Bulletin of the Seismological Society of America, 1968, 58(5): 1519–1572. [8] Ben-Menahem A, Singh S J. 1981. Seismic Waves and Sources[M]. New York: Seismic Waves and Sources. Springer. [9] Bensen G D, Ritzwoller M H, Barmin M P, et al. Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion mensurements[J]. Geophysical Journal International, 2007, 169: 1239-1260. doi: 10.1111/j.1365-246X.2007.03374.x [10] Berg E M, Lin F -C, Allam A, et al. Tomography of southern California via Bayesian joint inversion of Rayleigh wave ellipticity and phase velocity from ambient noise cross-correlations[J]. Journal of Geophysical Research: Solid Earth, 2018, 123(11): 9933–9949. https://doi.org/10.1029/2018JB016269. [11] Boschi L, Weemstra C, Verbeke J, et al. On measuring surface wave phase velocity from station–station cross-correlation of ambient signal[J]. Geophysical Journal International, 2013, 192: 346-358. doi: 10.1093/gji/ggs023 [12] Boschi L, Weemstra C. Stationary-phase integrals in the cross correlation of ambient noise[J]. Reviews of Geophysics, 2015, 53: 411-451. doi: 10.1002/2014RG000455. [13] Brooksa L A, Gerstoft P. Ocean acoustic interferometry[J]. Journal of the Acoustical Society of America, 2007,121:3377. doi: 10.1121/1.2723650 [14] Buckingham M J. On the two-point cross-correlation function of anisotropic,spatially homogeneous ambient noise in the ocean and its relationship to the Green’s function[J]. Journal of the Acoustical Society of America, 2011,129(6): 3562-3576. doi: 10.1121/1.3573989 [15] Campillo M, Paul A. Long-range correlations in the diffuse seismic coda[J]. Science, 2003, 299:547. doi: 10.1126/science.1078551 [16] Chavez-Garcia F, Luzon F. On the correlation of seismic microtremors[J]. Journal of Geophysical Research, 2005, 110, B11313. [17] Chavez-Garcia F, Rodriguez M, Stephenson W. An alternative approach to the SPAC analysis of microtremors: Exploiting stationarity of noise[J]. Bulletin of the Seismological Society of America, 2005, 95(1): 277–293. doi: 10.1785/0120030179 [18] Chavez-Garcia F, Rodriguez M, Stephenson W. Subsoil structure using SPAC measurements along a line[J]. Bulletin of the Seismological Society of America, 2006, 96(2):729-736. doi: 10.1785/0120050141 [19] Chen X. Seismogram synthesis in multi-layered half-space Part I. Theoretical formulations[J]. Earthquake Research in China, 1999, 13(2):149-174. [20] 陈运泰, 多层弹性半空间中的地震波(一)[J]. 地球物理学报, 1974, 17(1):20-43.Chen Y-T. Seismic Waves in Multilayered elastic half-space(I)[J]. Chinese Journal of Geophysics, 1974, 17(1):20-43(in Chinese). [21] Cox H. Spatial correlation in arbitrary noise fields with application to ambient sea noise[J]. Journal of the Acoustical Society of America, 1973, 54(5): 1289-1301. doi: 10.1121/1.1914426 [22] Cron B F, Sherman C H. Spatial-correlation functions for various noise models[J]. Journal of the Acoustical Society of America, 1962, 34(11):1732-1736. doi: 10.1121/1.1909110 [23] Cron B F, Hassell B C, Keltonic F J. Comparison of theoretical and experimental values of spatial correlation[J]. Journal of the Acoustical Society of America, 1965, 37(3):523-529. doi: 10.1121/1.1909361 [24] Debayle E, Sambridge M. Inversion of massive surface wave data sets: Model construction and resolution assessment[J]. Journal of Geophysical Research, 2004, 109: B02316. doi: 10.1029/2003JB002652 [25] Ditmar P G, Yanovskaya T B. A generalization of the Backus-Gilbert method for estimation of lateral variations of surface wave velocity[J]. Izvestiya-Physics of the Solid Earth,1987,23(61):470-477. [26] Ekstrom G, Abers G, Webb S. Determination of surface-wave phase velocities across USArray from noise and Aki’s spectral formulation[J]. Geophysical Research Letters, 2009, 36: L18301. doi: 10.1029/2009GL039131 [27] Fang H, Yao H, Zhang H, et al. Direct inversion of surface wave dispersion for three-dimensional shallow crustal structure based on ray tracing: methodology and application[J]. Geophysical Journal International,2015, 201(3): 1251–1263. https://doi.org/10.1093/gji/ggv080. [28] Friederich W, Wielandt E, Stange S. Multiple forward scattering of surface waves: Comparison with an exact solution and Born single-scattering methods[J]. Geophysical Journal International, 1993, 112: 264-275. doi: 10.1111/j.1365-246X.1993.tb01454.x [29] Froment B, Campillo M, Roux, P,et al. Estimation of the effect of nonisotropically distributed energy on the apparent arrival time in correlations[J]. Geophysics, 2010,75(5): SA85-SA93. doi: 10.1190/1.3483102. [30] Gallot T, Catheline S, Roux P, et al. Passive elastography: Shear-wave tomography from physiological-noise correlation in soft tissues[J]. IEEE Transactions on Ultrasonics, Ferroelectrics, and Freqequency Control, 2011, 58(6):1122-1126. doi: 10.1109/TUFFC.2011.1920 [31] Gerstoft P, Tanimoto T. A year of microseisms in southern California[J]. Geophysical Research Letters,2007,34(20): L20304. doi: 10.1029/2007gl031091 [32] Gradshteyn I S, Ryzhik I M. 2007. Table of Integrals, Series, and Products[M], Elsevier Inc: Academic Press. [33] Haney M M, Mikesell T D,Van Kasper W. Extension of the spatial autocorrelation (SPAC) method to mixed-component correlations of surface waves[J]. Geophysical Journal International, 2012,191:189–206. doi: 10.1111/j.1365-246X.2012.05597.x [34] Haney M M, Nakahara H. Surface-wave Green’s tensors in the near field[J]. Bulletin of the Seismological Society of America, 2014, 104(2):1578-1586. doi: 10.1785/0120130113. [35] Harkrider D G. Surface waves in multilayered elastic media I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space[J]. Bulletin of the Seismological Society of America,1964, 54(2): 627–679. [36] Harmon N, Forsyth, D and Webb, S. Using ambient seismic noise to determine short-period phase velocities and shallow shear velocities in young oceanic lithosphere. The Bulletin of the Seismological Society of America. 2007, 97(6): 2009-2023. 10.1785/0120070050. doi: 10.1785/0120070050 [37] Harmon N, Gerstoft P, Rychert C, et al. Phase velocities from seismic noise using beamforming and cross correlation in Costa Rica and Nicaragua[J]. Geophysical Research Letters, 2008, 35:L19303. doi: 10.1029/2008GL035387 [38] Harmon N, Rychert C, Gerstoft P. Distribution of noise sources for seismic interferometry[J]. Geophysical Journal International, 2010, 183: 1470-1484. doi: 10.1111/j.1365-246X.2010.04802.x [39] Hasselmann K. A statistical analysis of the generation of microseisms[J]. Reviews of Geophysics,1963,1(2): 177–210. doi: 10.1029/RG001i002p00177 [40] Hu S, Luo S, Yao H. The Frequency-Bessel Spectrograms of multicomponent cross‐correlation functions from seismic ambient noise[J]. Journal of Geophysical Research: Solid Earth, 2020,125: e2020JB019630. https://doi.org/10.1029/2020JB019630. [41] Jacobsen F, Roisin T. The coherence of reverberant sound fields[J]. Journal of the Acoustical Society of America, 2000, 108(1):204-210. doi: 10.1121/1.429457 [42] Jacobson M J. Space-Time correlation in spherical and circular noise fields[J]. Journal of the Acoustical Society of America,1962, 34(7):971-978. doi: 10.1121/1.1918232 [43] Kastle E, Soomro R, Weemstra C, et al. Two-receiver measurements of phase velocity: cross-validation of ambient-noise and earthquake-based observations[J]. Geophysical Journal International, 2016, 207: 1493–1512. doi: 10.1093/gji/ggw341 [44] Lawrence J F, Prieto G A. Attenuation tomography of the western United States from ambient seismic noise[J]. Journal of Geophysical Research, 2011, 116(B6): B06302. doi: 10.1029/2010JB007836. [45] Lawrence J F, Denolle M, Seats K J, et al. A numeric evaluation of attenuation from ambient noise correlation functions[J]. Journal of Geophysical Research, 2013,118(12): 6134-6145 [46] Lin F C, Moschetti M P, Ritzwoller M H. Surface wave tomography of the western United States from ambient seismic noise: Rayleigh and Love wave phase velocity maps[J]. Geophysical Journal International,2008,173(1): 281-298. doi: 10.1111/j.1365-246X.2008.03720.x [47] Lin F C,Ritzwoller M H,Shen W. On the reliability of attenuation measurements from ambient noise cross-correlations[J]. Geophysical Research Letters, 2011,38(11): L11303. doi:10. 1029/2011GL047366. [48] Lin F C, Schmandt B. Upper crustal azimuthal anisotropy across the contiguous U.S. determined by Rayleigh wave ellipticity[J]. Geophysical Research Letters, 2014, 41(23), 8301–8307. https://doi.org/10.1002/2014gl062362. [49] 刘庆华, 鲁来玉, 王凯明. 2015. 主动源和被动源面波浅勘方法综述[J].地球物理学进展, 30(6): 2906-2922.Liu Q H, Lu L Y, Wang K M. 2015a. Review on the active and passive surface wave exploration method for the near surface structure[J].Progress in Geophysics, 30(6):2906-2922 (in Chinese). [50] Liu X, Ben-Zion Y. Theoretical and numerical results on effects of attenuation on correlation functions of ambient seismic noise[J]. Geophysical Journal International,2013,194(3):1966-1983. doi: 10.1093/gji/ggt215 [51] Liu X, Ben-Zion Y, Zigone D. Extracting seismic attenuation coefficients from cross-correlations of ambient noise at linear triplets of stations[J]. Geophysical Journal International, 2015, 203(2): 1149-1163. Doi: 10.1093/gji/ggv357. [52] Lobkis O, Weaver R. On the emergence of the Green’s function in the correlations of a diffuse field[J]. Journal of the Acoustical Society of America, 2001,110: 3011–3017. doi: 10.1121/1.1417528 [53] Löer K, Riahi N, Saenger E H. Three-component ambient noise beamforming in the Parkfield area[J]. Geophysical Journal International, 2018, 213:1478-1491. doi: 10.1093/gji/ggy058 [54] Longuet-Higgins M S, Jeffreys H. A theory of the origin of microseisms[J]. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1950, 243(857): 1–35. https://doi.org/10.1098/rsta.1950.0012. [55] Lu L, Zhang B. The analysis of dispersion curves of Rayleigh waves in frequency-wavenumber domain[J]. Canadian Geotechnical Journal, 2004, 41:583-598. doi: 10.1139/t04-005 [56] Lu L, Maupin V, Zeng R, Ding Z. Scattering of surface waves modelled by the integral equation method[J]. Geophysical Journal International, 2008, 174: 857–872. doi: 10.1111/j.1365-246X.2008.03787.x [57] Lu L, Ding Z, Zeng R, He Z. Retrieval of Green’s function and generalized optical theorem for the scattering of complete dyadic fields[J]. Journal of the Acoustical Society of America, 2011,129(4):1935-1944. https://doi.org/10.1121/1.3553224. [58] Lu L, Wang K, Ding Z. The effect of uneven noise source and/or station distribution on the estimation of azimuth anisotropy of surface waves[J]. Earthquake Science, 2018, 31(4): 175-186. doi: 10.29382/eqs-2018-0175-1. [59] Luo Y, Yang Y, Xu Y, et al. On the limitations of interstation distances in ambient noise tomography[J].Geophysical Journal International, 2015, 201:652–661. doi: 10.1093/gji/ggv043 [60] Maupin V. A multiple-scattering scheme for modelling surface wave propagation in isotropic and anisotropic three-dimensional structures[J]. Geophysical Journal International, 2001, 146: 332-348. [61] Maupin V, Park J. 2007. Theory and Observations—Wave Propagation in Anisotropic Media. In: Dziewonski A M, Romanowicz B(ed) Treatise on Geophysics, Vol 1: Seismology and Structure of the Earth[M]. Elsevier, 289-321. [62] Meier T, Dietrich K, Stöckhert B, Harjes H-P. One-dimensional models of shear wave velocity for the eastern Mediterranean obtained from the inversion of Rayleigh-wave phase velocities and tectonic implications[J]. Geophysical Journal International, 2004, 156(1): 45-58. 10.1111/j.1365-246X.2004.02121.x. doi: 10.1111/j.1365-246X.2004.02121.x [63] Mohsen A A, Hashish E A. The fast hankel transform[J]. Geophysical Prospecting, 1994, 42:131-139. doi: 10.1111/j.1365-2478.1994.tb00202.x [64] Montagner J P, Nataf H C. A simple method for inversion the azimuth anisotropy of surface waves[J]. Journal of Geophysical Research, 1986, 91(B1):511-520. doi: 10.1029/JB091iB01p00511 [65] Morse P M, Feshbach H. 1953. Methods of Theoretical Physics[M]. McGraw Hill. [66] Nakahara H. A systematic study of theoretical relations between spatial correlation and Green’s function in one-,two- and three-dimensional random scalar wavefields[J]. Geophysical Journal International, 2006,167:1097-1105. doi: 10.1111/j.1365-246X.2006.03170.x [67] Nakahara H. Formulation of the spatial autocorrelation (SPAC) method in dissipative media[J]. Geophysical Journal International, 2012,190(3): 1777-1783. doi: 10.1111/j.1365-246X.2012.05591.x [68] Nakata N, Gualtieri L, Fichtner A. 2019. Sismic Ambient Noise[M]. Cambridge, UK: Cambridge University Press. [69] Piessens R. 2000. The Hankel Transform[M]//Poularikas A D. The Transforms and Applications Handbook: Second Edition. Boca Raton: CRC Press LLC. [70] Poli P, Campillo M, Pedersen H, and LAPNET Working Group. Body-wave imaging of Earth’s mantle discontinuities from ambient seismic noise[J]. Science, 2012, 38: 1063–1065. [71] Prieto G A, Lawrence J F, Beroza G C. Anelastic Earth structure from the coherency of the ambient seismic field[J]. Journal of Geophysical Research, 2009,114(B7): B07303, doi: 10.1029/2008JB006067 [72] Prieto G A, Denolle M, Lawrence J F, et al. On amplitude information carried by the ambient seismic field[J]. Comptes Rendus Geoscience, 2011, 343(8): 600-614 [73] Rafaely B, Spatial-temporal correlation of a diffuse sound field[J]. Journal of the Acoustical Society of America, 2000,107(6):3254-3258. doi: 10.1121/1.429397. [74] Roux P, Sabra K G, Kuperman W A, et al. Ambient noise cross correlation in free space: Theoretical approach[J]. Journal of the Acoustical Society of America, 2005, 117(1):79–84. doi: 10.1121/1.1830673 [75] Roux P, Ben-Zion Y. Rayleigh phase velocities in southern California from beamforming short-duration ambient noise[J]. Geophysical Journal International, 2017,211(1):450-454. doi: 10.1093/gji/ggx316 [76] Sabra K, Winkel E, Bourgoyne D, et al. Using cross correlations of turbulent flow-induced ambient vibrations to estimate the structural impulse response. Application to structural health monitoring[J]. Journal of the Acoustical Society of America, 2007, 121(4):1987-1995. doi: 10.1121/1.2710463 [77] Sadeghisorkhani H, Gudmundsson Ó, Roberts R, Tryggvason A.Velocity-measurement bias of the ambient noise method due to source directivity: a case study for the Swedish National Seismic Network[J]. Geophysical Journal International, 2017,209(3):1648–1659. doi: 10.1093/gji/ggx115 [78] Sánchez-Sesma F, Campillo M. Retrieval of the Green’s function from cross-correlation: The canonical elastic problem[J]. Bulletin of the Seismological Society of America,2006, 96: 1182–1191. doi: 10.1785/0120050181 [79] Savage M K, Lin F-C, Townend J. Ambient noise cross-correlation observations of fundamental and higher-mode Rayleigh wave propagation governed by basement resonance[J]. Geophysical Research Letters, 2013, 40(14): 3556–3561. https://doi.org/10.1002/grl.50678. [80] Schuster G. 2009. Seismic Interferometry[M]. Cambridge, UK: Cambridge University Press. [81] Slob E, Wapenaar K. Electromagnetic Green’s function retrieval by cross-correlation and cross-convolution in media with losses[J]. Geophysical Research Letters, 2007, 34: L05307. [82] Smith M L, Dahlen F A. The azimuth dependence of love and rayeigh wave propagation in a slightly anisotropic medium[J]. Journal of Geophysical Research,1973,78(17):3321-3333. doi: 10.1029/JB078i017p03321 [83] Snieder R. Extracting the green's function from the correlation of coda waves: a derivation based on stationary phase[J]. Physical Review E, 2004, 69:046610. doi: 10.1103/PhysRevE.69.046610 [84] Snieder R, Șafak E. Extracting the building response using seismic interferometry: Theory and application to the Millikan library in Pasadena, California[J]. Bulletin of the Seismological Society of America, 2006, 96: 586-598. doi: 10.1785/0120050109 [85] Snieder R, Wapenaar K, Wegler U. Unified Green's function retrieval by cross-correlation; connection with energy principles[J]. Physical Review E, 2007,75: 036103. doi: 10.1103/PhysRevE.75.036103 [86] Snieder R, van Wijk K . 2015. A Guided Tour of Mathematical Methods for the Physical Sciences[M].Cambridge, UK: Cambridge University Press. [87] Stehly L, Campillo M, Shapiro N M. A study of the seismic noise from its long-range correlation properties[J]. Journal of Geophysical Research- Solid Earth, 2006, 111(B10): B10306. doi: 10.1029/2005JB004237 [88] Takagi R, Nakahara H, Kono T, et al. Separating body and Rayleigh waves with cross terms of the cross-correlation tensor of ambient noise[J]. Journal of Geophysical Research:Solid Earth,2014,119: 2005–2018. doi: 10.1002/2013JB010824. [89] Tanimoto T. Excitation of normal modes by nonlinear interaction of ocean waves[J]. Geophysical Journal International, 2006,168: 571- 582. doi:10.1111/j.1365- 1246X.2006.03240.x. [90] Tsai V C. On establishing the accuracy of noise tomography travel-time measurements in a realistic medium[J]. Geophysical Journal International, 2009, 178: 1555-1564. doi: 10.1111/j.1365-246X.2009.04239.x [91] Tsai V C. The relationship between noise correlation and the Green's function in the presence of degeneracy and the absence of equipartition[J]. Geophysical Journal International, 2010, 182: 1509-1514. doi: 10.1111/j.1365-246X.2010.04693.x. [92] Tsai V C, Moschetti M P, An explicit relationship between time-domain noise correlation and spatial autocorrelation (SPAC) results[J]. Geophysical Journal International, 2010,182(1): 454-460. [93] Tsai V C. Understanding the amplitudes of noise correlation measurements[J]. Journal of Geophysical Research, 2011, 116(B9): B09311. doi: 10.1029/2011JB008483 [94] van Wijk K , Mikesell T D, Schulte-Pelkum V, et al. Estimating the Rayleigh-wave impulse response between seismic stations with the cross terms of the Green tensor[J]. Geophysical Research Letters, 2011,38: L18301. doi: 10.1029/2011GL047442. [95] Walker S C. A model for spatial coherence from directive ambient noise in attenuating, dispersive media[J]. Journal of the Acoustical Society of America,2012, 132(1): EL15-EL21.doi: 10. 1121/1. 4726195. [96] Wang, J., Wu, G., & Chen, X., Frequency-Bessel transform method for effective imaging of higher-mode Rayleigh dispersion curves from ambient seismic noise data. Journal of Geophysical Research: Solid Earth, 2019, 124: 3708–3723. https://doi.org/10.1029/2018JB016595. [97] 王凯明, 鲁来玉, 刘庆华. 由地震背景噪声提取介质衰减: 衰减介质中的空间相干表示[J]. 地球物理学报, 2016, 59(9):3237-3247. doi: 10.6038/cjg20160909Wang K M, Lu L Y, Liu Q H. On the spatial correlation of seismic noise in an attenuating medium[J]. Chinese Journal of Geophysics, 2016, 59(9): 3237-3247(in Chinese). doi: 10. 6038/cjg20160909. [98] 王凯明, 鲁来玉, 刘庆华, 等. 基于地震背景噪声互相关函数研究介质衰减综述[J]. 地球物理学进展, 2018, 33(1): 0112-0124. doi: 10.6038/pg2018AA0552.Wang K M, Lu L Y, Liu Q H,et al. Review on the research of earth’s attenuation based on the ambient seismic noise crosscorrelation function[J]. Progress in Geophysics, 2018, 33(1): 0112-0124(in Chinese). doi: 10. 6038/pg2018AA0552. [99] Wang K, Lu L, Maupin V,et al. Surface wave tomography of northeastern tibetan plateau using beamforming of seismic noise at a dense array[J]. Journal of Geophysical Research: Solid Earth, 2020, 125, e2019JB018416. [100] Wapenaar K, Fokkema J. Green’s function representations for seismic interferometry[J]. Geophysics, 2006,71: SI33. doi: 10.1190/1.2213955 [101] Wapenaar K, Slob E, Snieder R. Unified green’s function retrieval by cross correlation[J]. Physical Review Letters, 2006, 97: 234301. doi: 10.1103/PhysRevLett.97.234301 [102] Wapenaar K, Slob E, Snieder R, et al. Tutorial on seismic interferometry: Part 2- Underlying theory and new advances[J]. Geophysics, 2010, 75(5): 75A211–75A227. doi: 10.1190/1.3463440 [103] Watson A. 1966. Treatise on the Theory of Bessel Functions, 2nd[M]. Cambridge, UK: Cambridge University Press. [104] Weaver R L, Lobkis O I. Ultrasonics without a source: Thermal fluctuation correlations at mHz frequencies[J]. Physical Review Letters, 2001, 87:134301. doi: 10.1103/PhysRevLett.87.134301 [105] Weaver R L, Froment B, Campillo M. On the correlation of non-isotropically distributed ballistic scalar diffuse waves[J]. Journal of the Acoustical Society of America, 2009,126(4):1817-1825. doi: 10.1121/1.3203359 [106] Weaver R L. On the amplitudes of correlations and the inference of attenuations, specific intensities and site factors from ambient noise[J]. Comptes Rendus Geoscience, 2011,343(8): 615-622. [107] Webb S C. The Earth’s ‘hum’ is driven by ocean waves over the continental shelves[J]. Nature, 2007, 445(7129): 754– 756. doi: 10.1038/nature05536 [108] Weemstra C, Boschi L, Goertz A, et al. Seismic attenuation from recordings of ambient noise[J]. Geophysics, 2013,78(1):Q1-Q14. doi: 10.1190/geo2012-0132.1 [109] Weemstra C, Westra W, Snieder R, et al. On estimating attenuation from the amplitude of the spectrally whitened ambient seismic field[J]. Geophysical Journal International, 2014,197(3):1770-1788. doi: 10.1093/gji/ggu088 [110] 吴华礼, 陈晓非, 潘磊.基于频率-贝塞尔变换法的关东盆地S波速度成像[J]. 地球物理学报, 2019, 62(9): 3400-3407.doi: 10.6038/cjg2019N0205Wu H L, Chen X F, Pan L. S-wave velocity imaging of the Kanto basin in Japan using the frequency-Bessel transformation method[J]. Chinese Journal of Geophysics, 2019, 62(9): 3400-3407(in Chinese). doi: 10.6038/cjg2019N0205. [111] Yang Y, Ritzwoller M H. Characteristics of ambient seismic noise as a source for surface wave tomography[J]. Geochemistry, Geophysics, Geosystems, 2008, 9: Q02008. [112] 杨振涛, 陈晓非, 潘磊等.基于矢量波数变换法(VWTM)的多道Rayleigh波分析方法[J]. 地球物理学报, 2019, 62(1): 298-305. doi: 10.6038/cjg2019M0641.Yang Z T, Chen X F, Pan L, et al. Multi-channel analysis of Rayleigh waves based on the Vector Wavenumber Tansformation Method (VWTM)[J]. Chinese Journal of Geophysics, 2019, 62(1): 298-305(in Chinese). doi: 10.6038/cjg2019M0641. [113] Yao H, Van Der Hilst R D, De Hoop M V. Surface-wave array tomography in SE Tibet from ambient seismic noise and twostation analysis, I: phase velocity maps[J]. Geophysical Journal International, 2006, 166: 732–744. doi: 10.1111/j.1365-246X.2006.03028.x [114] Yao H, Van Der Hilst R D. Analysis of ambient noise energy distribution and phase velocity bias in ambient noise tomography, with application to SE Tibet[J]. Geophysical Journal International, 2009, 179: 1113–1132. doi: 10.1111/j.1365-246X.2009.04329.x [115] Yao H, Gouedard P, Collins J A, et al. Structure of young East Pacific Rise lithosphere from ambient noiss correlation analysis of fundamental- and higher-mode Scholte-Rayleigh waves[J]. Comptes Rendus Geoscience, 2011,343:571–583. doi: 10.1016/j.crte.2011.04.004 [116] 姚振兴. 层状介质、非轴对称震源情况下的反射法[J]. 地球物理学报, 1979, 22(2):181-194. doi: 10.3321/j.issn:0001-5733.1979.02.006Yao Z. Generalized Reflection coefficients for a layered medium and asymmetrical source[J]. Chinese Journal of Geophysics, 1979, 22(2):181-194(in Chinese). doi: 10.3321/j.issn:0001-5733.1979.02.006 [117] Yokoi T, Margaryan S. Consistency of the spatial autocorrelation method with seismic interferometry and its consequence[J]. Geophysical Prospecting, 2008, 56: 435–451. doi: 10.1111/j.1365-2478.2008.00709.x -