Why Can Noise Cross-correlations Replace Seismograms in Extracting Dispersion Relations and Travel Times?

Since introduced to seismic tomography about a decade ago, the technique of retrieving dispersion relations and travel times in noise cross-correlations has been one of the most rapidly developing method in seismology. Many seismologists have used this technique, and among them some have made significant discoveries. This method was originally introduced from acoustics into seismology. It was verified for acoustic waves that the noise cross-correlations resemble the Green's functions. However, since seismic waves are vector waves, and the noise sources are concentrated on the free surface, the conclusion in the acoustic case cannot be taken for granted in the seismological case. Thus, whether noise cross-correlations can replace seismograms in extracting the dispersion relations and the travel times should be studied thoroughly. In our study, we attempt to investigate the relation between the dispersion relations and travel times extracted in the noise cross-correlations and those extracted in the seismograms. We start with the general equation of noise cross-correlation, and write it in the form of normal-mode summation after assuming that the earth structure is spherically symmetric and the distribution of noise sources is laterally uniform. Then we use asymptotic analysis (traveling wave decomposition and mode-sum to ray-sum transformation) to write the surface-wave and body-wave representation of the noise cross-correlation, analyzing which we theoretically prove that as long as the earth model is spherically symmetric and the distribution of the noise sources is laterally uniform, the dispersion relations and the travel times extracted in noise cross-correlations are identical to those extracted in seismograms. In addition, numerical results are provided to verify our proof. This proposition establishes the feasibility theoretically, although only under very strict assumptions. Furthermore, we investigate the case in which the structure of the earth and the distribution of noise sources can varies laterally. In this more general case, we combine theoretical analysis and numerical approaches together, and try to explain why we can obtain good results when we use noise cross-correlations to retrieve dispersion relations and travel times.