Windowing in the time domain is equivalent to subsampling in the dual domain. The first trace in Figure reffig:freq is a time signal that might correspond to several identical subsurface sources below a single layer in the earth. It is many thousands of samples long. The second trace is its autocorrelation calculated using all data in the frequency domain multiplication. We recognize the auto-correlation to be symmetric, and could have truncated the result after half of the samples. Also, because each event correlates with all the others, many correlation peaks appear at late time. These late events contain information about similarity of the earth's impulse response recovered by each independent source function. Mathematically, this is a chain of many convolutions of complicated functions and very difficult to understand or utilize. They do not fit into the framework of passive seismic imaging by representing the kinematics of a reflection gather. So I discard them. In this case, the ``deepest subsurface reflector of interest'' is the second peak of the output correlation. The correlations after this could be what Schuster et al. (2004) call other terms.
Instead of truncating the auto-correlation in the time-domain, the last trace was calculated by decimating the Fourier transform of the signal before multiplication. The trace was padded with zeros to facilitate plotting with the previous traces. The number of frequencies used to produce the second autocorrelation was 8 times fewer than the Fourier representation of the input. As long as the level of decimation maintains support for the time window desired for the final result, we can subsample the input after transforming to the frequency domain and before subsequent processing.
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freq
Figure 1 Idealized signal of three identical subsurface sources, its autocorrelation, and autocorrelation after subsampling the frequency axis by a factor of 4. Right panel is zoomed in view. |  |
