How much averaging is necessary to cancel out cross-terms in noise correlation studies?

We present an analytical approach to jointly estimate the correlation window length and number of correlograms to stack in ambient noise correlation studies to statistically ensure that noise cross-terms cancel out to within a chosen threshold. These estimates provide the minimum amount of data necessary to extract coherent signals in ambient noise studies using noise sequences filtered in a given frequency bandwidth. The inputs for the estimation process are (1) the variance of the cross-correlation energy density calculated over an elementary time length equal to the largest period present in the filtered data and (2) the threshold below which the noise cross-terms will be in the final stacked correlograms. The presented theory explains how to adjust the required correlation window length and number of stacks when changing from one frequency bandwidth to another. In addition, this theory provides a simple way to monitor stationarity in the noise. The validity of the deduced expressions have been confirmed with numerical cross-correlation tests using both synthetic and field data.