The optimal correlation detector?

Abstract

SUMMARY 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 C2 is typical. We perform a systematic comparison between the detection statistics C and C|C|, the latter having the same dynamic range as C2 but differentiating between correlation and anticorrelation. 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 reference signals. 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 C2, 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.