• ISSN 2097-1893
  • CN 10-1855/P

基于平面波模型重访地震背景噪声互相关及空间自相关(SPAC)

Revisiting the cross-correlation and SPatial AutoCorrelation (SPAC) of the seismic ambient noise based on the plane wave model

  • 摘要: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: Since Aki (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.

     

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