Improvements in magnitude precision, using the statistics of relative amplitudes measured by cross correlation

Standard processing of seismic events for reporting in bulletins is usually done one-at-a time. State-of-the-art relative event methods, often involving cross correlation, are increasingly used and have improved estimates of event parameters for event detection, location and magnitude. This is because relative event techniques can simultaneously reduce measurement error and effects of model error. We show how cross correlation can be used to assign relative magnitudes for neighbouring seismic events distributed over a large region in east Asia and quantify to what extent the uncertainty in these values increases as waveform similarity breaks down. We find that cross correlation works well for magnitude comparison of two events when it is expected that they generate very similar signals even if these may be almost buried in large amounts of noise. This may be the case when investigating repeating earthquakes or nuclear explosions within a few kilometres of each other. Cross correlation is the optimal detector in these cases assuming noise is white and Gaussian, and also provides the least-squares solution for the relative amplitudes. However, when the waveform similarity of the underlying signals breaks down, due to interevent separation distance, source time function differences or focal mechanism differences, these assumptions are no longer valid and a bias is introduced into the relative magnitude measurement. This bias due to degradation of waveform similarity is modelled here with synthetics and an analytic expression for it is derived based on three terms—the cross-correlation coefficient (CC), and the signal-to-noise ratio (SNR) of the larger and smaller events. The analytic expression is a good match to the observed bias in the data. If the equation for relative magnitude is rewritten to correct for the bias due to the CC, a new equation results which is simply the log of the ratio of the L2 norms. The bias due to SNRs is still present because the observed waveforms inevitably contain both signal and noise. However, this bias is predicted to be minimal for typical detection thresholds. Making measurements of the ratio of the L2 norms is shown to remove the bias due to degradation of waveform similarity for real data. The scatter of these cross-correlation measurements of relative magnitude is much less than those obtained by differencing magnitudes in a traditional catalogue. Of 14 025 events in and near China, 34 per cent had over an order of magnitude reduction in the median standard deviation (0.0342 magnitude units) as compared to the estimated scatter in the catalogue (0.3454 magnitude units). And 78 per cent of the events show a factor 3 improvement or better in the precision of relative event size measured as the ratio of the L2 norms as compared to the precision of the catalogue for relative magnitudes. These results suggest that the ratio of the L2 norms is an appropriate measure of relative magnitudes for general seismicity of a monitoring region, when there is significant waveform dissimilarity for neighbouring events. This measure maintains a higher degree of measurement precision as compared to the catalogue.