What is the role of spectral whitening in the processing step done before cross-correlations in seismic interferometry?
I'm not an expert on this topic. But my understanding of seismic interferometry is that we try to obtain the 'Green's function' between various receivers. This Green's function is the equivalent of sending a spike pulse (i.e., delta function $\delta$) at one receiver, and recording the result at the other receiver. Once we have plenty of these Green's functions, ideally surrounding an area of interest, we can compute arbitrary Green's function recordings by evaluating a set of convolutions (rather than carrying out computations). It will be as if there is a recorder anywhere we want to record the signal.
In the time-domain, this $\delta$-spiked source is a source that's on for an infinitesimally small amount of time before switching off again. As you can maybe imagine, this source will have an infinitesimally small wavelength. We will thus be able to see every modulation of the signal as a single identifiable event in time: a scatter of amplitude $A_1$ happens after $t_1$ seconds, a reflection with amplitude $A_2$ happens after $t_2$ seconds, etc. Thus, we gain an incredibly high-resolution understanding of the medium in-between!
The frequency spectrum of such a $\delta$-spike has uniform amplitude. All frequencies are thus represented. It is called a 'broadband signal'.
In reality, we do not obtain $\delta$-spikes very often because the predominant frequencies in the data are provided by noise with a specific frequency, usually a very low one. This low-frequency, or long-wavelength, property makes the obtained Green's functions of very low frequency. Different events will smear and overlap in time. This makes it more difficult to interpret the medium from the obtained Green's functions.
Spectral whitening is a process that boosts frequencies with a low amplitude, to try and make any signal more $\delta$-pulse-like. See for example here https://twitter.com/DownUnderGeo/status/984324387720527872 for an example from active seismic exploration, how boosting the amplitude of higher frequencies (while retaining the low ones), doesn't just look nicer, but provides a considerable increase in identifiable events! The same principle applies to seismic interferometry. By filtering the data, we obtain a more $\delta$-like behavior, thus obtain a more broadband signal from a less-than-ideal signal.