How many correlograms and how much stacking are necessary to ensure that noise crossterms cancel out in seismic interferometry?

Abstract:

In seismic interferometric (SI) studies, the empirical Green's functions are retrieved by stacking a number of correlograms of a prescribed length. When processing SI data, the choice of the correlation window length and the number of correlograms are based on some empirical or pre-stablished values based on experience. Here, 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 estimation relies on two quantities extracted from the data itself: (a) variance of the cross-correlation energy density calculated over an elementary time length equal to the largest period present in the filtered data and (b) the threshold below which the noise cross-terms will be in the final stacked correlograms. Our approach is also able to explain how to tune the required correlation window length and number of stacks when changing from one frequency bandwidth to another. A bonus from our approach is that we are also able to monitor noise stationarity. The validity of the expressions from our theory have been demonstrated with numerical cross-correlation tests using synthetic and field data.