Stationary phase approximation in the ambient noise method revisited

The method of extracting Green's function between stations from cross correlation has proven to be effective theoretically and experimentally. It has been widely applied to surface wave tomography of the crust and upmost mantle. However, there are still controversies about why this method works. Snieder employed stationary phase approximation in evaluating contribution to cross correlation function from scatterers in the whole space, and concluded that it is the constructive interference of waves emitted by the scatterers near the receiver line that leads to the emergence of Green's function. His derivation demonstrates that cross correlation function is just the convolution of noise power spectrum and the Green's function. However, his derivation ignores influence from the two stationary points at infinities, therefore it may fail when attenuation is absent. In order to obtain accurate noise-correlation function due to scatters over the whole space, we compute the total contribution with numerical integration in polar coordinates. Our numerical computation of cross correlation function indicates that the incomplete stationary phase approximation introduces remarkable errors to the cross correlation function, in both amplitude and phase, when the frequency is low with reasonable quality factor <i<Q</i<. Our results argue that the distance between stations has to be beyond several wavelengths in order to reduce the influence of this inaccuracy on the applications of ambient noise method, and only the station pairs whose distances are above several (>5) wavelengths can be used.