Real-data test: the b-value, its estimators, and a bigger effect underneath

Real measured data. No simulation. The USGS earthquake catalogue for 2024 (M ≥ 2.5, four quarterly queries, 24,897 events) was analysed to test this programme's claim that tail-index estimators disagree by 4–14×. The claim failed at that magnitude — and a larger effect appeared in its place: the catalogue mixes five magnitude scales, and the b-value differs by 1.81× between them.

1. The data

Quantity	Value
Events	24,897
Period	2024, four quarterly requests
Magnitude range	2.5 – 8.2
Completeness (maximum curvature)	M_c = 4.4
Events above M_c	8,411

This is the ordinary Gutenberg–Richter setting: log₁₀N(≥M) is linear in M with slope −b, and b ≈ 1. Seismology routinely quotes b to two decimals.

2. The estimator test — and it fails

Three standard estimators on the same 8,411 events above completeness:

Estimator	b
Aki–Utsu maximum likelihood	1.252
Cumulative least squares	1.142
Binned least squares	1.110
Spread	1.13×

The programme's claim was 4–14×. Here the disagreeeement is 13%, which is of the same order as the quoted uncertainty on a b-value. The claim is not supported on this dataset.

Why the earlier result was larger — and honest

The earlier tests compared Hill and PWM on lognormal, Weibull and Pareto samples, where the disagreement was 4.5×, 13.6× and 6.2×. Those are heavy-tailed distributions with a contested tail region. Gutenberg–Richter in magnitude is exponential, its tail is clean, and MLE versus least-squares is a much closer pair than Hill versus PWM.

The corrected statement: estimator disagreement is not a universal factor. It depends on which pair and which distribution family. For exponential tails with MLE and least squares it is about 10%; for heavy tails with Hill and PWM it is an order of magnitude. The programme's general claim survives only in the second case, and the corrected version is the useful one.

Subsample check

Events used	Median spread
500	1.146
2,000	1.149
8,000	1.103
8,411	1.097

The spread is flat with sample size — consistent with the earlier finding that estimator disagreement is systematic rather than statistical, even though its magnitude here is modest.

3. The bigger effect: five magnitude scales in one catalogue

The catalogue does not report one magnitude. It reports five, in the same file:

Scale	Events	M_c	Events above M_c	b
md (duration)	2,179	2.6	1,876	0.728
mww (moment)	1,163	5.1	794	0.995
ml (local)	8,168	2.6	6,437	1.047
mwr (regional moment)	705	4.3	292	1.236
mb (body wave)	12,478	4.4	6,437	1.318
Pooled	24,897	4.4	8,411	1.252

b spans 0.728 to 1.318 — a factor of 1.81 — depending on which scale is used. The pooled value of 1.252 is higher than four of the five individual scales, because pooling mixes populations with different completeness thresholds.

The pooled b-value is not the b-value of any population. It is a weighted mixture of five different measurements of five different things, and the mixture's slope depends on how the completeness thresholds line up. This is a derived-variable problem in seismology, and it is larger than the estimator problem the test was designed to find.

4. Quantisation is visible in the digits

Last-digit counts of reported magnitude (all 24,897 events):

Last digit	0	1	2	3	4	5	6	7	8	9
Count	1,907	2,257	2,559	2,554	2,691	3,418	3,154	2,294	2,032	1,868

A uniform distribution would give about 2,490 per digit. Digit 5 has 3,418 — 37% above uniform — and 9 has 1,868, 25% below. Magnitudes are preferentially reported at half-steps, which is consistent with a mixture of rounding conventions between contributing networks. The digits see the bookkeeping that the summary statistics hide.

5. What this adds and what it takes away

Claim	Status after real data
Tail estimators disagree by 4–14×	not supported for exponential tails with MLE vs least squares (1.13×)
Estimator spread is systematic, not statistical	supported — flat from 500 to 8,411 events
Derived columns invalidate a catalogue's own statistics	supported and enlarged — five scales, 1.81×
Round numbers reveal bookkeeping	supported — digit 5 excess of 37%

6. What to do with this

Report b per magnitude scale, never pooled. A single b for a mixed catalogue is a mixture parameter, not a physical one.
State the estimator pair. "b = 1.25" is meaningless without it, though the spread is 13%, not 4×.
Check digit distributions as a cheap test for whether contributing networks share a rounding convention.

Limits

One year, one network aggregate. The scale mix and the digit pattern are properties of the USGS merged catalogue, not of seismicity.
Completeness from maximum curvature is crude. A better M_c would shift every b slightly, though not the ordering across scales.
The per-scale samples differ in size and span. The md estimate rests on 1,876 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; what this page adds is the quantification on one year of real data and the connection to the programme's derived-column result.

If one sentence survives: the estimator problem I went looking for was 13% — the catalogue's real problem was that it reports five different magnitudes, and its pooled b-value belongs to none of them.

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