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REVIEW 3 major objections 5 minor 33 references

Time window to constrain the corner value of the global seismic-moment distribution

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A few more decades of global earthquake records can tightly constrain the upper tail of the quake-size distribution.

desk verdict Useful statistical correction to Zöller's 200,000-year estimate, but the abstract's 'will shrink by 2097' only holds if the future maximum lands in a narrow window. read the letter →

arxiv 1908.02516 v1 pith:6XGX5NKS submitted 2019-08-07 physics.data-an physics.geo-ph

Alvaro Corral , Isabel Serra This is my paper · ORCID
classification physics.data-anphysics.geo-ph
keywords seismicmomentdistributioncornermagnitudeGutenberg-Richterlawextremevaluestatisticsmaximumorderstatistictruncatedpowertaperedearthquakecatalogs
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how many more years of global earthquake recording are needed before the tail of the earthquake-size distribution can be inferred. It argues that the answer is decades, not hundreds of thousands of years: the range of allowed values of the corner magnitude, the scale at which the Gutenberg-Richter power law bends downward, will narrow substantially by the end of this century once the next very large earthquake is observed. The argument uses the largest event in the global CMT catalog as the test statistic and compares it with the predicted distribution of the maximum of N earthquakes for three tail models: the truncated power law, the tapered Gutenberg-Richter law, and the truncated gamma. As of 2017 the compatible range of the corner magnitude is wide, but Table 1 projects, for example, that if the largest event by 2097 has magnitude 9.3, the tapered model's corner magnitude will be constrained to about 8.8-9.4. Because hazard assessments depend on how fast the tail decays, knowing when this parameter can be resolved affects the credibility of long-term seismic hazard estimates.

What carries the argument

The load-bearing object is the maximum-order-statistic identity Fmax(y)=Prob[Y <= y]=[F(y)]^N, the cumulative distribution of the largest of N independent seismic moments. For the truncated power-law model, the maximum Y is exactly the maximum-likelihood estimator of the truncation parameter, so inverting Fmax(yp)=p gives 95% probability intervals for the estimated corner. For the tapered and truncated-gamma models the same percentile inversion is performed through closed forms involving implicit functions (the Lambert W function for the tapered model and the inverse incomplete gamma for the truncated gamma), and the empirical maximum is compared with those intervals to decide which corner values are compatible with the data. The machinery also supplies the sample size N needed to shrink an interval to a chosen width by solving the width equation; this replaces the earlier 'well-sampledness' criterion, which the paper shows is not a valid statistical test because it treats the observed maximum as fixed while N grows.

What would settle it

Recompute the Table 1 ranges after removing aftershocks from the global CMT catalog; if declustering moves the 95% intervals by more than a few tenths of a magnitude or pushes the empirical maximum outside them, the independence assumption is the weak link. The long-run check is to record the actual maximum to 2097 and see which of the 9.1, 9.3, or 9.5 branches of Table 1 it selects.

Watch

Extended reading notes

Core claim

Contrary to an earlier analysis of the same catalog that concluded reliable estimation of the tapered corner magnitude would require about 200,000 years, the authors show that the correct use of the largest observed earthquake gives a much shorter horizon. The distribution of the maximum of N independent events is Fmax(y)=[F(y)]^N, so the observed largest magnitude can be tested against percentile intervals for each candidate model and each value of the corner magnitude. Inverting those percentiles yields, for each hypothetical true corner, the number N (or calendar year) at which the 95% interval narrows to a chosen width. Applying this to the 7,585 CMT events since 1977 leaves the corner magnitude unbounded above for all three models at the original catalog length, but already gives finite ranges by 2017, and under the paper's stated assumptions the projected 2097 ranges are much narrower: for the tapered model they are between about 0.5 and 1.0 magnitude wide, and for the truncated gamma between 0.5 and 1.4. The paper's Table 1 gives the predicted ranges for 2047 and 2097 conditional on the maximum magnitude observed in the interval (9.1, 9.3, or 9.5), showing that the outcome depends strongly on whether an event of magnitude 9.3 or larger occurs.

Load-bearing premise

The projected time windows assume the 7,585 catalogued earthquake magnitudes are independent, so the chance that the largest observed event stays below a value is the Nth power of the single-event chance; global catalogs contain aftershock clustering, and the paper states but does not test this dependence assumption.

Editorial extensions

If this is right

  • By the end of 2017 the CMT data already give finite 95% ranges for the corner magnitude of the tapered (8.6-10.2), truncated power law (9.1-10.8), and truncated gamma (8.8-11.2) models, whereas the same data at mid-2012 allowed no upper bound.
  • If no earthquake larger than magnitude 9.1 occurs before 2047, the truncated power law narrows the corner to 9.1-9.5; if a 9.3 event occurs instead, the tapered model allows 8.8-9.95, so the observation of a very large event is the main information carrier.
  • The earlier conclusion that hundreds of thousands of years are needed for the tapered model is reversed; under the paper's assumptions, the 2097 ranges are narrow enough to distinguish among competing hypotheses about the tail.
  • Seismic hazard should be evaluated by mixing tail models over the range of compatible corner values (Eq. 12) rather than by relying on a point estimate or on asymptotic standard errors, which the paper argues misdescribe the uncertainty.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If aftershock clustering is strong enough to reduce the effective number of independent events below 7,585, the predicted narrowing dates would shift later than 2047 or 2097; this is testable because declustered catalogs would give wider percentile intervals.
  • The same maximum-based percentile logic transfers to any heavy-tailed record where the scale parameter is debated and only a handful of extremes exist, such as storm surges, floods, or geomagnetic disturbances, provided a candidate tail family is specified.
  • The calendar projections assume a constant global rate of 213.7 events per year and a fixed beta; a persistent rate change would move the dates but would not change the N-based widths, so the method is more robust than the specific years printed in Table 1.
  • The mixture over corner values in Eq. (12) is equivalent to a Bayesian posterior under a 1/M_c prior; making that prior explicit would let hazard calculators report full predictive distributions instead of range endpoints.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper revisits the question of how much additional earthquake recording is needed to constrain the corner value (Mc or mc) of the global seismic-moment distribution. Using the global CMT catalog (7,585 events with magnitude ≥5.75, 1977–2012.5, extended to 2017) and fixing β=0.67, the authors compute 95% probability intervals for the maximum of N independent observations under three models — truncated power law (TPL), tapered GR (Tap), and truncated gamma (TrG) — via Fmax(y)=[F(y)]^N. From this they derive currently compatible ranges of mc (e.g., 8.6–10.2 for Tap) and conditional future ranges for hypothetical maxima in 2047 and 2097, and they argue that these ranges will shrink substantially by 2097, contrary to earlier claims that hundreds of thousands of years would be needed. They also show that the 'well-sampledness' criterion used by Zöller (2013) is statistically flawed.

Significance. The paper makes a useful methodological correction: the earlier 'well-sampledness' criterion (Zöller 2013) is replaced by a proper extremal significance test based on the distribution of the maximum, and the forward calculation is not circular because mc is not fitted to the observed maximum. The central probability calculation under the stated independence assumption is standard and appears correct, and the resulting Table 1 gives concrete, falsifiable conditional predictions. However, the headline claim that the range 'will' shrink by 2097 is not established by the conditional analysis.

major comments (3)
  1. [Abstract/Conclusions; Table 1 in 'Proper constraining of the corner seismic-moment: Tap and TrG cases'] The abstract and Conclusions assert that 'under reasonable assumptions, the range will be substantially reduced by the end of this century,' but Table 1 only lists compatible mc ranges conditional on hypothetical future maxima of 9.1, 9.3, and 9.5; no probabilities are assigned to these maxima. This is load-bearing, because models still compatible with current data can produce a future maximum that breaks the claimed reduction. For instance, the Tap model with mc=10.2 (the upper end of the 2017 compatible range) has S(10)≈3.2×10^-5, giving an expected 0.83 events of magnitude ≥10 in 120 years and a probability >50% of at least one; under a 10.0 maximum, the Tap compatible range shifts to roughly mc≈9.7–∞, so the range is not substantially reduced. The unconditional claim therefore needs either to be weakened to a conditional one or derived by integrating over the Poisson-distributed future maximum using a prior over mc.
  2. [Eq. (5); 'Proper constraining of the corner seismic-moment: TPL case'] Eq. (5) and all subsequent percentile calculations assume the 7,585 events are independent. The paper acknowledges this once ('assuming ... there is no dependence between the magnitudes') but does not quantify the effect of aftershock clustering on the 2047/2097 ranges. Since N enters Eqs. (10)–(11) and Table 1 through p^{1/N}, a modest reduction in the effective number of independent events can materially widen the reported intervals; a sensitivity analysis (e.g., using a declustered catalog or an effective N) is needed before the projections can be considered robust.
  3. [Throughout; Figs. 2–3 and Table 1] All entries use β=0.67, stated to be 'very close to the maximum-likelihood solution,' but no uncertainty in β is propagated. The compatible ranges of mc are monotone in β through the percentile formulas (10)–(11), so an error of, say, 0.05 in β could shift the bounds by several tenths of a magnitude. The paper needs at least a sensitivity analysis for β or a justification for treating it as known.
minor comments (5)
  1. [Discussion, Eq. (12)] Eq. (12) writes Stpl(x|Mc) in the integrand while the surrounding text is about the tapered model; this should be Stap (or a generic S_model).
  2. [Discussion, first sentence] The word 'assessment' is misspelled as 'assesment'.
  3. [Figure 3 caption] The caption uses the notation 'mpm' and '(mp, mp+0.95)' without defining which interval is symmetric and which is of minimum width; a brief explanation would improve readability.
  4. [SI captions, Figs. S4–S6] The captions misspell 'corresponding' as 'corresponing'.
  5. [Section 'Proper constraining of the corner seismic-moment: TPL case', last paragraph] The sentence 'we need to wait about 30 years to chose between these three answers' contains a typo: 'to chose' should be 'to choose'.

Circularity Check

0 steps flagged · score 0.0 of 10

Forward calculation, not circular: future corner-value ranges are conditional prediction-interval inversions, and the abstract's overstatement is a robustness issue, not a derivation that reduces to its inputs.

full rationale

The paper's core derivation is a forward calculation: from Eq. (5), Fmax(y)=[F(y)]^N, and the three model CDFs (Eqs. 2-4), it constructs 95% intervals for the maximum of N independent draws, then inverts the condition that a specified observed or hypothetical maximum lies inside that interval to obtain the compatible mc range shown in Table 1. This is a standard prediction-interval inversion, not a fit of mc to the data renamed as a prediction; the future maxima 9.1, 9.3, and 9.5 are explicitly hypothetical scenarios, as the paper says: 'This table also explores the values of these ranges in the future, depending on the hypothetical value of the maximum magnitude observed.' The abstract's unqualified wording that the range 'will be substantially reduced' is stronger than the conditional table, and the independence assumption behind Eq. (5) is acknowledged only in passing ('assuming that the TPL were the right model, that there is no dependence between the magnitudes, and that the long-term global earthquake rate and β were constant'). Those are correctness or robustness concerns, not circularity: the derivation does not assume what it sets out to show. The only numerical input plausibly taken from earlier same-author work is β=0.67, which is presented as a fixed value ('with β fixed to 0.67') and whose exact choice affects the numbers but not the logic; the self-citation to [20] is used to contrast previously computed standard errors, not to justify the present conclusion. No step in the derivation is equivalent by construction to its inputs, so no circular step can be exhibited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis is a forward statistical calculation given the model family and the fixed parameters. The main outside inputs are the shape parameter β, the rate Ra, the threshold a, and the independence assumption. None of these are derived in this paper; β comes from the authors' earlier fit (Ref. 20), Ra from the catalog itself, and the independence assumption is asserted with a maximum-entropy citation. The scenario maxima (9.1, 9.3, 9.5) are not fitted values but hypothetical inputs.

free parameters (3)
  • β (power-law exponent minus 1) = 0.67
    Shape parameter of the seismic-moment distribution, fixed to 0.67 (so 1+β=1.67) and stated to be close to the maximum-likelihood solution. The time windows in Table 1 and the N calculations in Fig. 2 use this value; no uncertainty is propagated.
  • Ra (annual rate above threshold) = 213.7 events per year
    Used to convert the required number of events N into calendar years for 2047 and 2097 projections. Estimated from the CMT catalog and assumed constant in time.
  • a (lower cut-off seismic moment) = 5.31 x 10^17 N·m (moment magnitude 5.75)
    Catalog completeness threshold; fixed, but all model calculations depend on it.
assumptions (4)
  • domain assumption The true global seismic-moment distribution belongs to one of the three families TPL, Tap, or TrG, with a single corner parameter Mc.
    The inference only checks compatibility of Mc within these models; a different tail shape would change the percentiles and the time windows. Entered in Section 'Probabilistic models'.
  • domain assumption The N observations are independent, so the maximum has CDF Fmax(y)=[F(y)]^N.
    Eq. (5). Independence is asserted via a maximum-entropy argument (Ref. 23), but earthquake catalogs contain aftershock sequences; violations affect the maximum distribution and the projected constraints.
  • domain assumption The annual rate of events above the threshold is constant at 213.7 events per year.
    Used in Table 1 to extrapolate from 2017 to 2047/2097. Completeness and rate changes over the 80-year window would shift the number of events.
  • domain assumption The lower cut-off a and the shape parameter β are known without error.
    β is taken from a previous fit by the same authors (Ref. 20); the paper does not propagate its uncertainty into the time-window estimates. Stated in Section 'Proper constraining TPL'.

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Cite this review

Pith. "Pith review of Time window to constrain the corner value of the global seismic-moment distribution." pith.science (2026). https://pith.science/paper/6XGX5NKS

@misc{pith2026190802516,
  author       = {Pith},
  title        = {Pith review of: Time window to constrain the corner value of the global seismic-moment distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XGX5NKS}},
  note         = {Machine review of arXiv:1908.02516}
}
read the original abstract

It is well accepted that, at the global scale, the Gutenberg-Richter (GR) law describing the distribution of earthquake magnitude or seismic moment has to be modified at the tail to properly account for the most extreme events. It is debated, though, how much additional time of earthquake recording will be necessary to properly constrain this tail. Using the global CMT catalog, we study how three modifications of the GR law that incorporate a corner-value parameter are compatible with the size of the largest observed earthquake in a given time window. Current data lead to a rather large range of parameter values (e.g., corner magnitude from 8.6 to 10.2 for the so-called tapered GR distribution). Updating this estimation in the future will strongly depend on the maximum magnitude observed, but, under reasonable assumptions, the range will be substantially reduced by the end of this century, contrary to claims in previous literature.

Figures

Figures reproduced from arXiv: 1908.02516 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Number of years necessary to obtain a reliable estimation of the truncation parameter Mc for the TPL model with β = 0.67 as a function of the hypothetical true value of Mc (represented by mc), according to Ref. [16] (decreasing curve) and according to our results [inverting Eq. (11), increasing curves], assuming an average rate of 213.7 events per year. In the latter case we impose that 95%-probability intervals hav… view at source ↗
Figure 3
Figure 3. 95%-probability intervals, represented by the starting and ending points (mp, mp+0.95) for the truncation parameter Mc of a TPL distribution with N = 7, 585 earthquakes (in terms of the corresponding truncation magnitude mc), as a function of the hypothetical true values of mc. The value of the exponent is 1 + β = 1.67. Two kinds of intervals are shown: symmetric (r = 1/2 in Eq. (9)) and of minimum width (the r that… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: (Fig. S1) ccdf S(x) and pdf f(x) of TPL distribution with β = 0.67, a corresponding to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. August 8, 2019 19/24 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: (Fig. S2) ccdf S(x) and pdf f(x) of Tap distribution with β = 0.67, a corresponding to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. August 8, 2019 20/24 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: (Fig. S3) ccdf S(x) and pdf f(x) of TrG distribution with β = 0.67, a corresponding to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. August 8, 2019 21/24 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: (Fig. S4) ccdf Smax(y) and pdf fmax(y) of the maximum of 7,585 TPL observations with β = 0.67, a corresponing to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. Critical values at the 95% confidence level are shown as horizontal lin…
Figure 8
Figure 8. Figure 8: (Fig. S5) ccdf Smax(y) and pdf fmax(y) of the maximum of 7,585 Tap observations with β = 0.67, a corresponing to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. Critical values at the 95% confidence level are shown as horizontal lin…
Figure 9
Figure 9. Figure 9: (Fig. S6) ccdf Smax(y) and pdf fmax(y) of the maximum of 7,585 TrG observations with β = 0.67, a corresponing to moment magnitude 5.75, and Mc corresponding to the values of mc shown in the legend. Critical values at the 95% confidence level are shown as horizontal lin…

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Reviewed August 14, 2026 · model on record in the stance chip above.