On estimating attenuation from the amplitude of the spectrally whitened ambient seismic field

Measuring attenuation on the basis of interferometric, receiver–receiver surface waves is a non-trivial task: the amplitude, more than the phase, of ensemble-averaged cross-correlations is strongly affected by non-uniformities in the ambient wavefield. In addition, ambient noise data are typically pre-processed in ways that affect the amplitude itself. Some authors have recently attempted to measure attenuation in receiver–receiver cross-correlations obtained after the usual pre-processing of seismic ambient-noise records, including, most notably, spectral whitening. Spectral whitening replaces the cross-spectrum with a unit amplitude spectrum. It is generally assumed that cross-terms have cancelled each other prior to spectral whitening. Cross-terms are peaks in the cross-correlation due to simultaneously acting noise sources, that is, spurious traveltime delays due to constructive interference of signal coming from different sources. Cancellation of these cross-terms is a requirement for the successful retrieval of interferometric receiver–receiver signal and results from ensemble averaging. In practice, ensemble averaging is replaced by integrating over sufficiently long time or averaging over several cross-correlation windows. Contrary to the general assumption, we show in this study that cross-terms are not required to cancel each other prior to spectral whitening, but may also cancel each other after the whitening procedure. Specifically, we derive an analytic approximation for the amplitude difference associated with the reversed order of cancellation and normalization. Our approximation shows that an amplitude decrease results from the reversed order. This decrease is predominantly non-linear at small receiver–receiver distances: at distances smaller than approximately two wavelengths, whitening prior to ensemble averaging causes a significantly stronger decay of the cross-spectrum.