Emergence rate of the time‐domain Green’s function from the ambient noise cross‐correlation function

Abstract

It has been demonstrated experimentally and theoretically that an estimate of the Green’s function between two receivers can be obtained from the time derivative of the long‐time average cross correlation of ambient noise between these two receivers. The emergence rate of these deterministic coherent arrival times of the cross‐correlation function (i.e., the Green’s function estimate) from the recordings of an isotropic distribution of random noise sources is derived by evaluating the amplitude of the variance of the cross‐correlation function. The knowledge of the emergence rate of NCF is essential for practical applications. To the first order, the variance is equal to the ratio of the product of the recorded energy by both receivers and the time–bandwidth product of the recordings. The variance of the time derivative of the correlation function is shown to have a similar dependency. These simple analytic formulas show a good agreement with the variance determined experimentally for the correlation of ocean ambient noise recorded in shallow water at a depth of 21 m, in the frequency band [300–530 Hz] for receiver separation up to 28 m and averaging time from 1 to 33 min.