The Optimal Correlation Detector?

Late in 2005 I wrote a paper together with Frode Ringdal called "The detection of low magnitude seismic events using array-based waveform correlation". It was published on April 1, 2006, in Geophysical Journal International (ominous date!) https://doi.org/10.1111/j.1365-246X.2006.02865.x and, to this day, it is by far the most highly cited paper I have been involved in. The idea is very simple: if you record an earthquake or explosion from a given location on a given seismic station, the signal it generates can be used as a template to detect subsequent events at or very close to the location of the template event - at far lower signal amplitudes than you would need to detect them using more traditional detection methods. The method is even more powerful if you have a seismic array or network as you can do this on each channel and stack the detection statistic traces with no loss (!), increasing the signal-to-noise ratio of the peak in the detection statistic greatly. You can also automatically screen out false alarms by checking the alignment of the peaks on the different detection statistic traces.

The paper received relatively little interest in the beginning - we were monitoring for underground nuclear explosions using CTBT/IMS-type seismic arrays and there simple weren't that many people doing that kind of work. In later years, citations increased enormously (the opposite of what usually happens with a paper: many citations to start with before its relevance fades ... ).

I think there were two main reasons for this. Firstly there was a huge expansion in dense seismic networks for monitoring of induced seismicity. Secondly there was a wave of machine learning approaches for detecting and classifying seismic signals and matched filter type methods were an obvious baseline comparison.

Shortly after the paper, an applied mathematician asked why on earth was I using a fully normalized correlation coefficient to detect a signal, instead of C^2 like the theory tells you. I replied that it was because of the nice properties of the "detection statistic signals" - they showed both correlation and anti-correlation and you could stack them much more nicely - the positives and negatives would cancel out like they do in classical beamforming; you can't see that with an "only positive" detection trace.

A couple of years later and we were looking for explosions in North Korea and replaced the C trace with C * abs( C ) - which is actually cheaper to calculate as you do not need to take square roots! (See "Seismic Monitoring of the North Korea Nuclear Test Site Using a Multichannel Correlation Detector" https://doi.org/10.1109/TGRS.2011.2170429 ) - although we never really looked at the difference in performance.

In later years, I realized that people were still citing the 2006 paper and using the C detection statistic. I got an uneasy feeling that the seismological community had inherited bad practice in matched filter detections - and that I was probably somewhat to blame for this. Working at a new place, I coded up a simple signal detector - to compare the two statistics using a set of freely available signals from repeating Ground Truth explosions (see "A benchmark case study for seismic event relative location" https://doi.org/10.1093/gji/ggaa362 )

The message was clear, a correlation detector using the C * abs( C ) detection statistic is vastly more effective than using C. It follows from basic statistics - C^2 is the coefficient of determination and tells you what fraction of the detection statistic is due to your signal. If you detect using the square root of this value then the fraction is much smaller and your detections will be needlessly closer to the noise level. The details are given in "The optimal correlation detector?" (GJI, 2022, https://doi.org/10.1093/gji/ggab344).

In April this year, I was at the EGU General Assembly in Vienna for the first time in 5 years (thanks COVID!) and I saw some superb posters on matched filter detectors for seismic signals - some really great work(!) - but again, I saw references to the 2006 paper and the C statistic in use! I talked to some of the authors and nobody had given it much thought. Correlation detectors - subspace detectors, cone detectors, matched field detectors - have come a long way since 2006 but bad habits die hard - and I feel partly responsible for this!

So please give the 2022 paper a little read - and change your codes to calculate C|C| instead of C. It really is worth it - nothing to lose - you will detect more relevant signals without necessarily increasing your false alarm rate (and the false alarm scanning procedures work equally well with C|C|!).

Happy signal detection!

(PS. All the articles mentioned in this piece are #OpenAccess ... just click on the links for the pdf files!)