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On the three laws of earthquake physics

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that Omori's law ends in a source bifurcation, that the Gutenberg-Richter $b$ value varies sharply by triad type, and that Bath's magnitude gap depends on mainshock magnitude.

desk verdict A synoptic review with one genuinely new empirical candidate (the mirror Bath law for foreshocks) and a useful six-type triad taxonomy, but the load-bearing catalog fits lack uncertainties, completeness control, and declustering. read the letter →

arxiv 2506.06504 v1 pith:SXBONRJL submitted 2025-06-06 physics.geo-ph

A.V. Guglielmi , A.D. Zavyalov , O.D. Zotov , B.I. Klain This is my paper
classification physics.geo-ph
keywords tectonicearthquaketriadOmorilawGutenberg-RichterBath'ssourcedeactivationepochforeshocksaftershocks
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

The paper argues that the three classical statistical laws of earthquake physics—Omori's aftershock decay, Gutenberg-Richter magnitude-frequency, and Bath's magnitude gap—are each real but carry hidden qualifications that standard catalog practice overlooks. Using a classification of main shocks into six types of foreshock-mainshock-aftershock triads, it claims that Omori's law holds only during a bounded 'Omori epoch' of roughly 10 to 100 days and then ends in a source bifurcation, that the Gutenberg-Richter $b$ value differs strongly by triad type with $b=1$ atypical for main shocks, and that Bath's gap depends on the mainshock magnitude. It also reports a mirror analogue of Bath's law that bounds the largest foreshock. A sympathetic reader would care because the claims, if confirmed, would redefine when and how the laws can be used for aftershock and foreshock diagnostics.

What carries the argument

The load-bearing object is the tectonic earthquake triad and its six-type classification, based on whether foreshock count $N_-$ and aftershock count $N_+$ in equal windows satisfy $N_-<N_+$, $N_-=N_+$, or $N_->N_+$, and whether either count is zero. The load-bearing identity is the definition of the source deactivation coefficient, $\sigma=-(1/n^2)(dn/dt)$, which the paper postulates and then inverts on smoothed aftershock data; constancy of $\sigma$ is shown equivalent to Omori's hyperbolic decay. The 'proper time' of the source generalizes that construction to non-constant $\sigma$, and the Omori epoch is the time interval, found empirically to last 10-100 days, during which $\sigma$ stays constant before the source bifurcates.

What would settle it

Recompute the six triad types, the $b$ values in Table 2, and the Bath and mirror-Bath slopes using a different global catalog or systematically varied window lengths and epicentral-radius rules; if the reported differences in $b$ and in the slopes disappear or change sign, the central claims fail.

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Extended reading notes

Core claim

The paper's central discovery claim is that the earthquake source relaxes in two stages: during an Omori epoch the deactivation coefficient $\sigma$ is constant, so the classical Omori law holds and aftershock activity is predictable, and then a bifurcation-like transition moves the source into a non-stationary stage where $\sigma$ changes erratically and neither the Omori nor the Hirano-Utsu decay fits. On the statistical side, the paper claims that main shocks sort into six triad types—classical, symmetrical, and mirror, each complete or shortened—and that the Gutenberg-Richter $b$ value varies by type: lowest in complete classical triads, highest in incomplete symmetrical triads, with $b=1$ typical only for the rare mirror incomplete type. It further claims that the Bath gap $\Delta M$ grows with mainshock magnitude for classical triads and full mirror triads, and that foreshocks obey a mirror Bath law (called Zotov's law) with slopes that differ between triad classes.

Load-bearing premise

The whole classification and all fitted parameters rest on the assumption that the USGS/NEIC catalog, equal-duration foreshock and aftershock windows, and the 24-hour epicentral-radius rule cleanly separate genuine foreshocks and aftershocks from background and unrelated earthquakes for $M \geq 5.8$ events at 0-250 km depth.

Editorial extensions

If this is right

  • Aftershock forecasts based on Omori decay are valid only inside the Omori epoch; after roughly 10-100 days the source enters a regime where the decay law no longer applies, so extrapolating Omori curves beyond that window is unreliable.
  • The Gutenberg-Richter $b$ value cannot be treated as a single regional constant: the paper's Table 2 gives $b$ values that differ by more than a factor of two between triad types for foreshocks, main shocks, and aftershocks.
  • Bath's law as classically stated is incomplete: for classical triads the paper finds $\Delta M = 0.48 M_0 - 1.85$, so the largest aftershock magnitude depends on the mainshock magnitude.
  • A mirror Bath law for foreshocks, if verified, would give a statistical upper bound on foreshock magnitude in complete classical triads via $\Delta M = 0.67 M_0 - 2.94$.
  • The six-type classification means that nearly half of large main shocks belong to shortened triads with no foreshocks or no aftershocks, so any model of main-shock generation should reproduce this ratio.

Reading between the lines

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

  • One testable extension is to recompute the six triad types and all fitted parameters under different catalog completeness cutoffs; if the $b$-value ordering across triad types survives, the claim that it reflects source properties is strengthened, and if it shifts, the result is a catalog artifact.
  • The same inverse-problem machinery that extracts $\sigma$ from an aftershock sequence could be applied to other clustered point processes, such as induced seismicity or volcanic tremor swarms, where a constant-decay epoch followed by bifurcation would be observable.
  • If the Omori epoch duration grows with mainshock magnitude, the end of the epoch could serve as an objective marker of source healing and might sharpen operational aftershock-hazard statements in the first weeks after a large event.
  • The mirror Bath law, though the paper flags it as unverified, suggests a natural check: measure the maximum foreshock magnitude across many complete classical triads and test whether the reported slope $a=0.67$ and intercept $-2.94$ reproduce in independent catalogs.
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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

4 major / 6 minor

Summary. The paper is a synoptic review of the authors' long-term work on earthquake phenomenology. It introduces a classification of tectonic earthquake triads (foreshock-main shock-aftershock) into three classes (classical, symmetrical, mirror) and two types per class (complete, incomplete), reporting on the basis of USGS/NEIC catalog data that these six types differ in Gutenberg-Richter b-values, in the fitted parameters of a generalized Bath's law, and in a proposed mirror analogue of Bath's law for foreshocks. The paper also formulates an 'elementary theory' of aftershock evolution built on a deactivation coefficient sigma, claims that Omori's law holds only during a finite 'Omori epoch' that ends in a bifurcation, and introduces the notion of proper time of an earthquake source. The central claims are that the three classical laws, while reliable, carry hidden qualifications: Omori's law is temporally limited, Gutenberg-Richter b depends on triad type, and a foreshock analogue of Bath's law exists.

Significance. If the empirical claims were established with proper statistical support, this paper would be significant: it would qualify three of the most widely used empirical laws in seismology and could motivate new catalog-based tests. The conceptual apparatus of earthquake triads and deactivation coefficient is potentially useful for organizing aftershock and foreshock studies. The authors are honest in Section 5 in labeling the new propositions as hypotheses requiring independent verification, and the axiomatic part of the paper is explicit and self-consistent. However, the load-bearing empirical content is currently under-determined: no uncertainties, completeness analysis, background control, declustering, or sensitivity tests are provided for any of the catalog-based fits, and the 'Omori epoch' is illustrated with a single event. The paper's value is therefore more as a research summary than as a demonstration of the claimed new qualifications of the three laws.

major comments (4)
  1. [2.1, 2.2, Fig. 3, Table 2] The six-type classification and the reported b-value differences rest on raw synchronous summation of USGS/NEIC events with no statement of the equal-duration window length, no completeness magnitude, no background subtraction, no declustering, and no uncertainty estimates. The text gives no basis for judging whether extreme values such as b=1.7 for TM2 foreshocks or b=0.4 for TC1 main shocks reflect source physics or are artifacts of small samples and catalog incompleteness. To support the central claim that the Gutenberg-Richter b value differs by triad type and that b=1 is not typical for main shocks, the authors must provide confidence intervals, a completeness analysis, and a robustness check against alternative window choices and against an M>=5.8 event lying inside another sequence's aftershock zone.
  2. [2.3, 4.2, Figs. 4 and 9] The fits of Mmax as a linear function of mainshock magnitude for both Bath's law (a=0.48, b=-1.85 for classical triads; a=0.44, b=-1.61 for mirror triads) and the proposed Zotov law for foreshocks (a=0.67, b=-2.94) depend on the 24-hour epicentral-radius selection rule of Zavyalov and Zotov (2021), but no standard errors, goodness-of-fit measures, or sensitivity tests are reported. The 24-hour window and fixed-radius rule can bias the maximum aftershock magnitude downward for larger mainshocks if early aftershocks are incompletely recorded, which would directly produce the positive slope in Eq. (6). The claims that Bath's law magnitude gap depends on M0 and that a mirror analogue exists therefore require a demonstration that the fitted slopes are robust to the selection procedure.
  3. [3.1, Eqs. (7)-(10)] The 'theorem' that Omori's law holds if and only if the deactivation coefficient is constant is a definitional equivalence: sigma in Eq. (7) is defined exactly so that sigma=const is the differential form of the Omori law, and Eq. (10) is merely the quadrature solution of dn/dt = -sigma(t) n^2. The subsequent statement that replacing world time with proper time in Eq. (10) gives a natural generalization is also definitionally empty unless independent physical or observational content is supplied for sigma(t). This section is internally consistent, but it should be presented as a definitional framework for diagnosing aftershock decay, not as a derivation of a physical law.
  4. [3.2, Figs. 5-7] The two-stage relaxation mode and the Omori epoch are illustrated with a single event (M=6, Northern California, 1984), with an unspecified smoothing procedure in Eq. (11), no description of how the spline approximation was made, no uncertainty interval on the resulting sigma(t), and an operational criterion for the '1% accuracy' yellow bar that is not defined. The general claim that the Omori epoch lasts from 10 to 100 days and tends to increase with mainshock magnitude is asserted without a multi-event analysis. This is a load-bearing point for the paper's revision of Omori's law; it needs either a reproducible statistical analysis over many sequences or a clear downgrade of the claim to a preliminary observation.
minor comments (6)
  1. [2.1] There is a typo in the sentence 'the main beats in the shortened triads are several times more than in the full triads'; 'beats' should be 'shocks'.
  2. [5] The text contains an obvious typo: 'this pfper' should read 'this paper'.
  3. [References] The reference list has an inconsistent rendering of the journal name in Bath (1965) and misspells 'Abercrombie' as 'Abercombie' in Mori and Abercrombie.
  4. [Figures 3 and 4, Table 2] All b-values and fitted Bath/Zotov parameters are presented without error bars, confidence intervals, or sample sizes; these should at least be supplied in the captions or in the table.
  5. [3.2] The notation for proper time is confusing: the text says 'the world time t is replaced by the so-called proper time of the earthquake source t', but both times use the same symbol t; a distinct symbol would improve clarity.
  6. [2.1] The magnitude range for the main shocks is given as '0 58M = -', which is ambiguous; the paper should state explicitly which magnitudes are included in each analysis.

Circularity Check

2 steps flagged · score 6.0 of 10

The aftershock 'theory' is a definitional tautology: sigma is defined so that sigma=const is equivalent to Omori's law, and the proper-time reformulation reparametrizes the general solution; the six-type classification and Bath/Zotov fits are independent empirical analyses.

  1. self definitional [Section 3.1, Eq. (7) and the theorem following it]
    "The deactivation coefficient is calculated using the formula σ = -(1/n^2) dn/dt (7). From (7) follows a statement (theorem), the content of which makes the choice of postulate obvious. Namely, the classical Omori law (1) is satisfied if and only if the deactivation coefficient is independent of time. Indeed, if σ = const, then (1) follows from (7)."

    Equation (7) defines σ as the logarithmic derivative of 1/n. The statement 'σ = const' is exactly the differential equation dn/dt = -σ n^2, whose solution is n = k/(c+t), i.e. Omori's law (1). Thus the 'theorem' restates the definition of σ rather than deriving Omori's law from an independent physical premise. Any claim that the axiomatic theory 'explains' or 'predicts' Omori's law from the deactivation postulate is a tautology: the postulate was chosen precisely so that constancy of σ is equivalent to the target law. The later 'experimental discovery' of an Omori epoch then reduces to locating time intervals where the data happen to follow a hyperbola, not to a prediction from independent theory.

  2. renaming known result [Section 3.1, Eq. (10); Section 3.2, proper-time discussion]
    "Formula (10) flexibly models the evolution of aftershocks and is a natural generalization of Omori's law [Guglielmi, 2016]. ... The only difference between the law of aftershock evolution (10) and the Omori law in its classical formulation (1) is that the world time t is replaced by the so-called proper time of the earthquake source τ."

    Equation (10) is the general solution of the evolution equation (9), dn/dt = -σ(t) n^2. For any monotone decreasing n(t) one can choose a rescaled time τ = k/n(t) - c so that n = k/(c+τ), which is Omori's law in the new coordinate. Presenting the general solution as a 'generalization of Omori's law' and attributing the change to 'proper time' is therefore a coordinate reparametrization of the general solution, not a new physical law. The empirical content is not increased; the claim that Omori's law holds in proper time is true by construction for any decaying sequence.

full rationale

The genuine circularity is confined to the axiomatic aftershock theory in Sections 3.1 and 3.2. Equation (7) defines σ so that σ=const is equivalent to Omori's law, making the theorem immediately following it a restatement of the definition. Similarly, equation (10) is the general solution of the assumed evolution equation, and the 'proper time' reformulation is a reparametrization that can make any monotone decay look like Omori's law. These steps make the theoretical derivation partially circular by construction. The paper's more prominent empirical claims — the six-type triad classification, the b-value differences in Figure 3 and Table 2, and the Bath/Zotov and mirror-Bath linear fits in Figures 4 and 9 — are not circular: they are estimates from USGS/NEIC data using stated counting rules, and their outputs are not identical by construction to the input counts. They may be methodologically fragile (window lengths, completeness, background contamination, missing uncertainties), but fragility is a correctness risk, not a circularity. The self-citations to the group's own prior work (Zavyalov & Zotov 2021 for epicentral radii; Guglielmi & Zotov 2021 for the σ(M) prediction) are load-bearing as authority, but I do not count them as definitional circularity because the empirical values remain externally falsifiable against the catalog. Score 6 reflects the definitional circularity of the theoretical aftershock derivation while recognizing the independent empirical content of the classification and fits.

Assumptions & free parameters 13 free parameters · 7 assumptions · 3 invented entities

The central empirical results depend on several fitted parameters: Bath and Zotov slope and intercept pairs for four data subsets, b values per triad type, the Omori epoch criterion, and data-selection choices. The theoretical section depends on axioms that are partly definitional (sigma as a one-dimensional state variable, Omori law as constancy of sigma) and partly unverified data assumptions (catalog completeness, clean foreshock and aftershock separation). The invented concepts (sigma, proper time) are operationally defined but provide no independent falsifiable handle beyond the smoothed aftershock counts they were constructed from.

free parameters (13)
  • Bath law slope a, classical triads = 0.48
    Linear fit to Delta M versus M0 for classical triads (Section 2.3, Fig. 4).
  • Bath law intercept b, classical triads = -1.85
    Same fit; used to claim Delta M depends on M0.
  • Bath law slope a, full mirror triads = 0.44
    Fit for full mirror triads (Section 2.3).
  • Bath law intercept b, full mirror triads = -1.61
    Same fit for full mirror triads.
  • Zotov law slope a, complete classical foreshocks = 0.67
    Mirror Bath law fit for foreshocks (Section 4.2, Fig. 9).
  • Zotov law intercept b, complete classical foreshocks = -2.94
    Same fit for complete classical foreshocks.
  • Zotov law slope a, complete symmetric foreshocks = 0.47
    Second mirror Bath law fit (Section 4.2).
  • Zotov law intercept b, complete symmetric foreshocks = -2.0
    Same fit for complete symmetric foreshocks.
  • Gutenberg-Richter b per triad type = TC1: 0.4; TC2: 0.5; TS1: 0.8; TS2: 1.3; TM1: 0.7; TM2: 1.0 (mainshocks; Table 2)
    b values fitted from USGS catalog; differences drive the claim that b=1 is not typical.
  • Omori epoch duration threshold = about 20 days in Fig. 7; range 10-100 days
    Chosen by visual criterion sigma constant within 1%; the epoch boundary is not determined by a statistical test.
  • Aftershock selection parameters = 24 h window; epicentral radius from Zavyalov and Zotov (2021)
    Data-processing choice that defines which events count as aftershocks and affects all subsequent fits.
  • Smoothing procedure for inverse problem = unspecified
    Formula (11) uses angle-bracket smoothing; the kernel or regularization is not described, so extracted sigma(t) is not uniquely defined.
  • Mainshock selection thresholds = M >= 5.8, depth 0-250 km
    Data selection criterion for all catalog analyses (Section 2.1).
assumptions (7)
  • domain assumption The aftershock frequency n(t) is a smooth, substantially positive function of time.
    Section 3.1 takes smoothness and positivity as basic; real catalogs are discrete counts with gaps, so smoothing is needed.
  • domain assumption The earthquake source state is fully described by a single scalar deactivation coefficient sigma(t).
    Section 3.1 introduces sigma as the only state variable; no mechanism or independent estimate is given.
  • ad hoc to paper Omori's law is equivalent to constancy of sigma, by definition (7).
    This is not an empirical discovery but a definitional equivalence; the 'theorem' immediately after (7) restates the definition.
  • ad hoc to paper The proper time tau is defined so that equation (10) takes Omori's form.
    Section 3.2 and Conclusion: replacing world time by proper time makes any monotone n(t) look like Omori decay; no independent clock or falsifiable prediction.
  • domain assumption USGS/NEIC catalog is complete for M >= 5.8 and depths 0-250 km over 1973-2019.
    Used to construct Figures 2-4; completeness is not checked, which could bias foreshock and aftershock counts and b values.
  • domain assumption Foreshocks and aftershocks are correctly identified by equal-duration windows before and after the mainshock and by a 24-hour epicentral-radius rule.
    Sections 2.1 and 2.3; no separation from background or triggered events is demonstrated.
  • standard math Standard ODE solution methods for dn/dt = -sigma n^2.
    Formula (10) is a standard solution; not at issue.
invented entities (3)
  • Source deactivation coefficient sigma(t) independent evidence
    purpose: Phenomenological state variable that controls aftershock decay; constant sigma gives Omori's law.
    Operationally defined by inversion of aftershock counts (7)-(11) and claimed to decrease with mainshock magnitude; but the inversion depends on arbitrary smoothing so the handle is partly internal to the authors' method.
  • Proper time of the earthquake source (tau)
    purpose: A rescaled time variable chosen so that aftershock evolution looks like Omori's law in the new time.
    Section 3.2 and Conclusion: it is a reparametrization of world time, not a new observable; any monotonic decay can be mapped to Omori form by choosing tau appropriately.
  • Omori epoch independent evidence
    purpose: Name for the time interval after a main shock during which the deactivation coefficient is nearly constant and Omori's law applies.
    It is a data-derived phase (Fig. 7), but its identification uses a 1% constancy criterion chosen by the authors and needs independent replication.

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Pith. "Pith review of On the three laws of earthquake physics." pith.science (2026). https://pith.science/paper/SXBONRJL

@misc{pith2026250606504,
  author       = {Pith},
  title        = {Pith review of: On the three laws of earthquake physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXBONRJL}},
  note         = {Machine review of arXiv:2506.06504}
}
read the original abstract

The paper provides a synoptic overview of a series of works carried out by a group of researchers at the Institute of Physics of the Earth RAS with the aim of finding new approaches to the problems of earthquake physics. The fundamental laws of Omori, Gutenberg-Richter and Bath have served as a constant support and reference point in the course of many years of research. The concept of the tectonic earthquake triad as a natural trinity of foreshocks, main shocks and aftershocks is used in the article to organise the thematic material. A classification of main shocks into six types of triads found in experience is given. The parameters appearing in the three laws for different types of triads are given. The axiomatic theory of the evolution of aftershocks is outlined. The concepts of source deactivation, Omori epoch and source bifurcation are introduced, and the notion of the proper time of unsteady lithospheric processes is introduced. Convergence of foreshocks and divergence of aftershocks are mentioned. The general conclusion is that the Omori, Gutenberg-Richter and Bath laws are reliable tools in the experimental and theoretical study of earthquakes. The laws have a depth of content that has been demonstrated by the ability to enrich the original formulations of the discoverers with interesting and important additional statements.

Figures

Figures reproduced from arXiv: 2506.06504 by the authors.

Figure 3
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Relativity of Time in Earthquake Physics

    physics.geo-ph 2025-09 conditional novelty 4.0 of 10

    Ordering the 2011 Tohoku foreshocks by event number reveals two phases: a linear (constant-rate) phase followed by an exponential phase, with a deactivation coefficient jumping from 0 to 0.065.

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Works this paper leans on

3 extracted references · 1 canonical work pages · cited by 1 Pith paper

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    arXiv doi:10.48550/arxiv.2108.02438 2017

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    Tectonophysics and Current Issues in Earth Sciences

    V. 413, Iss. 1–2. P. 13-23. Omori F. On the aftershocks of earthquake // J. Coll. Sci. Imp. Univ. Tokyo. 1894. P. 111–200. Rodrigo M.R. A spatio -temporal analogue of the Omori -Utsu law of aftershock sequences // Cornell University Library: arXiv:2111.02955v1 [physics.geo- ph], s ubmitted on 21 Oct 2021, pp. 1-12. https://doi.org/10.48550/arXiv.2111.0295...

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