The geometry of signal space: a case study of direct mapping between seismic signals and event distribution

Under favourable circumstances, seismic waveforms corresponding to an ensemble of events related by a common, spatially distributed process collectively exhibit a regular, signal-space geometry. When events in the ensemble have a common, or nearly common, source mechanism, this geometry is a distorted image of the distribution of events in the source region. The signal-space image can be visualized using a relatively simple waveform alignment and projection operation. Ensemble waveform correlation measurements can be inverted to estimate the distribution of the events in the source region, up to an arbitrary rotation, reflection and scaling, with residual distortion. We demonstrate these concepts with synthetic waveforms and with observations of long-wall mining induced seismicity for which substantial ground truth information is available. Our experience with these data has implications for location, correlation detection and machine learning and possible application to studies of repeating events in induced, volcanic and glacial seismicity. Our results place limits on the widely held assumption that waveform correlation is a useful measure of event separation. We suggest that the constraints on event separation need to be evaluated in the context of a population of related events, whose waveforms sample the signal space image of the source region. A better indicator of event separation is the length of the shortest path in signal space along the image.