Correcting for bias due to noise in coda wave interferometry

Coda wave interferometry (CWI) utilizes multiply scattered waves to diagnose small changes in a medium by using the scattering medium as an interferometer. Since the medium is usually stationary over the duration of a seismic experiment, different (non-overlapping) time windows in the coda allow for independent estimates of the medium perturbation. If the seismograms are contaminated with noise, only those time windows can be used for which the amplitude of the coda is significantly above the ambient noise level. This limits the usable number of independent time windows. Here, we show how bias due to noise in CWI can be accounted for, by deriving a correction factor for the cross-correlation coefficient, under the assumptions that the stochastic processes underlying the noise realizations in both signals are mutually uncorrelated and stationary with zero mean. This correction factor allows more time windows further into the decaying coda to be used, and hence allows for a reduction of the error bars on the medium perturbation estimates. We demonstrate the validity of this correction factor by using data from a numerical experiment and field measurements. These experiments involve the displacement of point scatterers and a change in the source location, respectively. The application of our correction factor is not limited to CWI, but can be used to correct for bias induced by noise in any application that uses cross-correlation between different signals that are contaminated with noise.