Cross‐correlation of random fields: mathematical approach and applications

Abstract

International audienceRandom field cross-correlation is a new promising technique for seismic exploration, as it bypasses shortcomings of usual active methods. Seismic noise can be considered as a reproducible, stationary in time, natural source. In the present paper we show why and how cross-correlation of noise records can be used for geophysical imaging. We discuss the theoretical conditions required to observe the emergence of the Green's functions between two receivers from the cross-correlation of noise records. We present examples of seismic imaging using reconstructed surface waves from regional to local scales. We also show an application using body waves extracted from records of a small-scale network. We then introduce a new way to achieve surface wave seismic experiments using cross-correlation of unsynchronized sources. At a laboratory scale, we demonstrate that body wave extraction may also be used to image buried scatterers. These works show the feasibility of passive imaging from noise cross-correlation at different scales

Ocean tomography with acoustic daylight

“…The averaging time needed for a reliable retrieval of a deterministic timefront from the NCCF was found to range from several hours at horizontal separations of ∼0.5 km to ∼40 hours at the largest horizontal separation of 3.5 km. These values are larger than theoretical estimates [Gouédard et al, 2008;Zabotin and Godin, submitted manuscript, 2009] of ∼10 min and ∼8 hours, respectively, obtained assuming diffuse, isotropic noise. The difference is the result of a preponderance of localized noise sources at the experimental site.…”

Statistical properties of the seismic noise field: influence of soil heterogeneities

“…Even in case of partial equipartition it might be possible to extract exact Green's function information (Tsai 2010). In other words, inferring the Green's function is effective after the sufficiently selfaveraging process that is provided by the scattering and random spatial-temporal noise source distribution when sufficiently long time-series are considered (Campillo 2006;Gouédard et al 2008). This means that the required source density for reconstructing the full Green's function is determined by the position and size of the heterogeneous area and heterogeneities with respect to the sources (Derode et al 2003;Fan & Snieder 2009).…”

“…Using correlational forms of the reciprocity theorems, Wapenaar & Fokemma (2006) showed how inter-receiver Green's functions could be constructed by crosscorrelating recorded fields whereas other studies have also revealed the limitations of that method (Snieder 2004;Roux et al 2005;Gouédard et al 2008). For obtaining a diffuse field Halliday & Curtis (2008) analyse the requirements in terms of source distribution for the retrieval of scattered surface waves.…”

Long-range correlations of microseism-band pressure fluctuations in the ocean

“…Wavefields generated by spatially distributed random sources are known to remain partially coherent at points separated by distances that are large compared to the wavelength, with the twopoint cross-correlation function of the random wavefields approximating the Green's function, which describes deterministic wave propagation between the two observation points (Lobkis & Weaver 2001;Snieder 2004;Roux et al 2004;Wapenaar 2004;Godin 2006;Gouédard et al 2008). Cross-correlation functions of pressure fluctuations in the ocean have been investigated in the 0.5-30 mHz band, where the correlations characterize deep-water infragravity waves and their sources (Webb 1986;Godin et al 2014b), and at acoustic frequencies above 1 Hz, where geoacoustic parameters of the seafloor (Brown et al 2014), spatial (Godin et al 2010) and temporal (Woolfe et al 2015) variations of the sound speed in water and ocean current velocity (Godin et al 2014a) have been retrieved from noise cross-correlations.…”