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

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.