The Optimal Correlation Detector?

Correlation detectors are now used routinely in seismology to detect occurrences of signals bearing close resemblance to a reference waveform. They facilitate the detection of low-amplitude signals in significant background noise that may elude detection using energy detectors, and they associate a detected signal with a source location. Many seismologists use the fully normalized correlation coefficient $C$ between the template and incoming data to determine a detection. This is in contrast to other fields with a longer tradition for matched filter detection where the theoretically optimal statistic $C^2$ is typical. We perform a systematic comparison between the detection statistics $C$ and $C|C|$, the latter having the same dynamic range as $C^2$ but differentiating between correlation and anti-correlation. Using a database of short waveform segments, each containing the signal on a 3-component seismometer from one of 51 closely spaced explosions, we attempt to detect P- and S- phase arrivals for all events using short waveform templates from each explosion as a reference event. We present empirical statistics of both $C$ and $C|C|$ traces and demonstrate that $C|C|$ detects confidently a higher proportion of the signals than $C$ without evidently increasing the likelihood of triggering erroneously. We recall from elementary statistics that $C^2$, also called the coefficient of determination, represents the fraction of the variance of one variable which can be explained by another variable. This means that the fraction of a segment of our incoming data that could be explained by our signal template decreases almost linearly with $C|C|$ but diminishes more rapidly as $C$ decreases. In most situations, replacing $C$ with $C|C|$ in operational correlation detectors may improve the detection sensitivity without hurting the performance-gain obtained through network stacking. It may also allow a better comparison between single-template correlation detectors and higher order multiple-template subspace detectors which, by definition, already apply an optimal detection statistic.