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.

freq
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.
Figure 1 |

5/3/2005