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Probability-based magnitude of completeness

PMC is how this server turns “the station was recording and did not pick” into a number. Two tools use it: predict_detectability returns a per-station detection probability, and explain_silence asks whether the silences are what that probability expected.

The name comes from a published method. The payload is a parametric form of that idea, not a completeness map.

The published method

Schorlemmer and Woessner (2008) estimate detection from past earthquakes instead of from a Gutenberg–Richter sample.

  1. Triplets. For each station, while it was recording: magnitude M, hypocentral distance L, and whether any phase was determined. A station that was off is not a trial.
  2. Per-station detection probability PD(M, L), non-parametric in the magnitude–distance plane, non-decreasing in magnitude and non-increasing in distance.
  3. Network detection probability at a point: one minus the product, over stations that were on, of “this station missed”.
  4. Completeness MP: the smallest magnitude at which that network probability reaches a chosen level (often 0.999). No frequency–magnitude relation is required.

That fourth step is a map. No tool here returns MP at a point.

What the tools evaluate

The server stores a fitted export and evaluates one origin against it. Detection is still per station, and a plus is still “any phase”, not a named P or S. The probability is not the non-parametric PD(M, L) surface. Magnitude and distance are folded into an expected signal-to-noise ratio, and the station’s detection probability is a logistic in that SNR:

log10(A) = a + b·M + c·log10(R_hyp) + s_station
expected_snr_db = 20·log10(A) − noise_p50_db
p = 1 / (1 + exp(−(α_station + β_station · expected_snr_db)))

A is an equivalent amplitude in the same frame as the weekly noise baseline. It is not a Wood–Anderson nanometre and it is not the amplitude inside the local-magnitude formula. M is the origin’s local magnitude. Rhyp is hypocentral distance. noise_p50_db is the station-week noise percentile, not the noise in the ten seconds around the pick.

Coefficients are data (model_version on the payload). They are not hard-coded in the tool.

Silence becomes a likelihood the same way a detection does. Over delivering stations inside the distance cutoff:

log P(data | origin) = Σ_detected log p + Σ_silent log(1 − p)

explain_silence returns that sum as log_likelihood (natural log) and the number of terms as n_trials. Stations that were absent or silent in the inventory, and stations beyond the cutoff, are not terms.

How to read a result

detection_probability is a fitted probability. Treat it as odds, not as a fact that the station would or would not have seen the event.

explain_silence classifies each station. below_noise means p < 0.5: the model puts the odds against a detection. It does not mean the trace in this window sits under the noise floor. should_have_detected means the station was delivering, in range, had no pick, and p ≥ 0.5.

One such silence can be a noise burst or a gap inside a day the availability table still calls delivering. Several, at stations closer than those that did detect, is the strongest evidence these tools can give that the solution is wrong. The event may still be real and mislocated: relocate, then score again.

log_likelihood is comparable across origins only when n_trials is comparable. When n_trials is 0 the sum is 0.0 — there were no trials, not a perfect score.

What this section does not give you

  • A completeness map, a Gutenberg–Richter Mc, or a statement that the catalogue is complete above some magnitude.
  • A per-phase probability. P and S are not separate trials.
  • Depth as its own feature. Depth enters only through Rhyp.
  • The noise at the predicted arrival. The baseline is a weekly percentile.
  • A verdict. The tools return p, a class, and a likelihood. They do not decide what the event is.

References

  • Schorlemmer, D. and Woessner, J. (2008). Probability of detecting an earthquake. Bulletin of the Seismological Society of America. doi:10.1785/0120070105
  • Mignan, A. and Woessner, J. (2012). Estimating the magnitude of completeness for earthquake catalogs. Community Online Resource for Statistical Seismicity Analysis. doi:10.5078/corssa-00180805
  • Arora, N. S., Russell, S. and Sudderth, E. (2013). NET-VISA: Network processing vertically integrated seismic analysis. Bulletin of the Seismological Society of America. doi:10.1785/0120120107