Magnitude-scale mixing and the apparent stability of the Gutenberg–Richter b-value

1. Introduction

The Gutenberg–Richter relation describes the size distribution of earthquakes as
log₁₀ N(≥M) = a − bM, with b typically near 1 (Gutenberg & Richter, 1944).
It is one of the most widely quoted numbers in seismology and is used to forecast event rates,
to characterise source regions and to compare tectonic environments (Schorlemmer et al., 2005).

A merged catalogue does not report a single magnitude. Different networks, era and magnitude
range contribute local magnitude M_L (Richter, 1935), body-wave magnitude
m_b, surface-wave magnitude M_s, duration magnitude M_d
and moment magnitude M_w (Kanamori, 1977). Each maps the physical size of an event
to a number through a different empirical rule. The effect of mixing such scales on a
frequency–magnitude distribution has long been recognised qualitatively, but a routine
per-scale comparison across a large modern catalogue is not something we found reported in the
form we present here.

This paper makes three narrow, checkable points: (i) the pooled b is stable across years
while the per-scale values are not; (ii) the per-scale spread is about twice the spread between
standard estimators; (iii) a boundary comparison in the completeness selection can move b
by more than either effect.

2. Data

Events were retrieved from the USGS ComCat FDSN event service for calendar years 2022, 2023 and
2024, with a minimum magnitude of 2.5 and the default network aggregation. The catalogue returns
the preferred magnitude and its type per event. Totals are given in Table 1. No events were added,
removed or re-associated by us.

Table 1. Catalogue sizes and pooled estimates.

3. Methods

3.1 Completeness

Completeness magnitude M_c was estimated by maximum curvature: the modal bin of
the 0.1-unit magnitude histogram, with a 0.05 correction added (Woessner & Wiemer, 2005;
Wiemer & Wyss, 2000). This is a crude estimator and is known to underestimate
M_c in some settings; it is used because it is transparent and identical for every
subsample, so comparisons between scales are internally consistent.

3.2 Estimators

Three estimators were applied to the same data above M_c: the Aki–Utsu maximum
likelihood estimator (Aki, 1965),

b = log₁₀(e) / ( M̄ − (M_c − ΔM/2) ),

a least-squares fit to cumulative log N(≥M), and a least-squares fit to the binned density.
The comparison follows the concern that tail or size-distribution indices are estimator-dependent
(Clauset et al., 2009).

3.3 Per-scale analysis

Events were partitioned by their reported magnitude type, and the same completeness and
estimation procedure applied within each partition. Partitions with fewer than 500 events were
excluded, leaving four or five scales per year.

4. Results

4.1 The pooled value is stable; the per-scale values are not

Table 2. b-value by magnitude scale. The pooled
value is outside the range of four of the five scales in every year.

The ordering is consistent across all three years, which argues that the differences are
properties of how the scales are constructed rather than of any year’s seismicity. The pooled
value of about 1.22 is not the b-value of any population in the file; it is the slope of a
mixture whose weights depend on how the five completeness thresholds happen to align.

4.2 The estimator effect is smaller

On the 2024 data above completeness, the three estimators returned 1.252, 1.142 and 1.110 — a
spread of 1.13×. Subsamples of 500, 2,000, 8,000 and 8,411 events gave median spreads of 1.146,
1.149, 1.103 and 1.097, i.e. flat in sample size. The estimator disagreement is therefore
systematic rather than statistical, but it is roughly half the size of the scale-mixing effect.

4.3 A boundary comparison larger than either

Completeness was computed as Mc = bin_edge + 0.05. In binary, 4.35 + 0.05 evaluates
to 4.400000000000001, so the selection m >= Mc excluded every event at exactly
magnitude 4.4 — the modal bin. For 2024 this removed 2,060 events and changed b from 1.252
to 0.980, a 28% difference. Neither the data nor the estimator changed. The defect was found only
because two independently written scripts disagreed on the same quantity.

4.4 Reported magnitudes are quantised

Last-digit counts over 24,897 magnitudes show a 37% excess at digit 5 (3,418 observed against
about 2,490 expected under uniformity) and a 25% deficit at 9 (1,868). This is consistent with
contributing networks using different rounding conventions, and it is a caveat on any analysis
that treats the reported magnitude as a continuous variable.

5. Discussion

The stability of the pooled b across three years is real but uninformative: a mixture
parameter can be stable while its components move, and here the components differ by more than a
factor of two. We therefore suggest that stability of a pooled estimate is not, on its own,
evidence that the catalogue is well behaved.

The practical recommendation is small: report b per magnitude scale alongside the pooled
value, and state the estimator. The reporting cost is one extra table row; the benefit is that a
reader can see whether a reported change is physical or compositional.

The boundary error in §4.3 is worth reporting separately because it is not a seismological
problem. Any selection at a threshold computed by floating-point addition will exclude the
threshold value itself. Because the modal bin of a magnitude distribution sits at the completeness
threshold, the excluded bin is the one that matters most, and the resulting error is maximal.

6. Limitations

• Three years and one network aggregate. The scale mix is a property of the ComCat merged
  catalogue, not of seismicity in general.
• Maximum curvature is a crude completeness estimator; a better M_c would shift
  every b slightly, though the ordering across scales is unlikely to change.
• The per-scale samples differ greatly in size and magnitude range. M_d rests on
  about 2,500 events over a narrow range, which is why its value is furthest from the others.
• We did not attempt to correct for the different physical meaning of each scale, nor to compare
  against a relocated or moment-only catalogue. Both would strengthen or overturn the interpretation.
• A systematic literature search has not yet been completed. Mixed-magnitude effects on
  frequency–magnitude distributions are certainly discussed in the seismological literature, and a
  reviewer may well identify earlier treatments of the same point.

7. Data and code

Events are public and were retrieved from the USGS ComCat FDSN event service; the query
parameters are given in §2. Analysis scripts and the machine-readable results accompany this
manuscript as plain Python and JSON, and depend only on NumPy. All random seeds are fixed and
stated in the scripts.

References

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