b-value across three years

1. The data

The pooled b-value is stable to ±2% across three years — which is what a well-behaved catalogue statistic should look like, and it is not the interesting part.

2. The spread is across magnitude scales, and it is large

b ranges from 0.70 to 2.23 depending on which magnitude scale is used, and the ordering is consistent across years: md lowest, mb highest, ml and mww in between. The pooled value of ~1.22 belongs to none of them.

The pooled b-value of a mixed-scale catalogue is not the b-value of any population. It is a weighted mixture whose slope depends on how five completeness thresholds align. Reporting one number for the catalogue is reporting a mixture parameter as if it were a physical one.

Consistency across years matters. The ordering md < mww ≈ ml < mwr < mb held in all three years examined, so this is a property of how the scales are constructed, not of a particular year's seismicity.

3. The floating-point artefact — a 28% change from a comparison

The completeness threshold was computed as Mc = bin_edge + 0.05. In binary, 4.35 + 0.05 evaluates to 4.400000000000001, and the selection m >= Mc then excludes every event at exactly magnitude 4.4 — 2,060 of the densest bin in the catalogue.

A 28% difference in the headline number, from a comparison operator. Neither the data nor the estimator changed. The excluded bin is the densest one, so the error is maximal exactly where the statistic is most sensitive.

This is the programme's own thesis, self-inflicted: the number is a convention, the convention is unstated, and the failure is invisible because the code runs without error and the output looks like a b-value.

It was caught only because two scripts produced different values for the same quantity and I compared them. Two implementations, not one.

4. The estimator spread, for completeness

On the 2024 data above completeness, three estimators of b: Aki–Utsu MLE 1.252, cumulative least squares 1.142, binned least squares 1.110 — a spread of 1.13×. This is far smaller than the 4–14× the programme predicted for heavy tails, and the correction stands: estimator disagreement depends on the estimator pair and the distribution family, not on estimation in general. Gutenberg–Richter in magnitude is exponential, and MLE versus least squares is a close pair.

5. The digits

Last-digit counts over 24,897 magnitudes: digit 5 appears 3,418 times against roughly 2,490 expected (+37%), while digit 9 appears 1,868 times (−25%). Reported magnitudes cluster at half-steps, consistent with mixed rounding conventions between contributing networks.

6. What to take from this page

1. Report b per magnitude scale. A pooled b for a mixed catalogue is a mixture parameter.
2. Round the completeness threshold, and use a tolerance in the selection. The densest bin is at the threshold, so the comparison is maximally sensitive there.
3. Compare two implementations. The bug was found only because two scripts disagreed.
4. Estimator spread is 13% here, not 400%. The general claim needed narrowing.

7. Limits

• Three years, one network aggregate. The scale mix is a property of the USGS merged catalogue.
• Maximum-curvature completeness is crude. A better M_c would shift every b slightly, though not the ordering across scales.
• The per-scale samples differ greatly in size — md rests on ~2,500 events over a narrow magnitude range, which is why its b is furthest from the others.
• No literature search. Mixed-magnitude b-values are certainly discussed in seismology; this page quantifies them on 78,493 real events and links them to the programme's derived-column result.

If one sentence survives: the catalogue's pooled b is stable to 2% across three years and means nothing, because it is a mixture of five scales whose individual b-values span 0.70 to 2.23 — and the analysis that produced the first number was off by 28% from a floating-point comparison.

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