By cross-correlating noise traces recorded at the surface we can construct the wavefield that would be recorded at one of the locations if there was an impulsive source at the other.
Cole 1995 tested this conjecture on synthetic data generated by a phase shift method, modeling the background noise as many random plane waves coming from all directions in the subsurface. However, his results were not conclusive, and his tests on field data were unsuccessful.
Rickett and Claerbout 1996 used phase shift modeling of random incident plane waves to show that cross-correlating noise traces gives the correct kinematics for horizontally layered and point diffractor models. The length of the time series before cross-correlation was shown to be the critical parameter in determining the signal to noise ratio; whereas the number and spatial distribution of plane waves determined the coherency of events. They also suggested that a half-differential filter, , be applied to the output (in the 2-D case) to boost higher frequencies that may have been suppressed by the stacking of many plane waves.