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arxiv: 2512.06572 · v3 · submitted 2025-12-06 · ⚛️ physics.geo-ph

Calendar Time Local Earthquake Forecasts from Earthquake Nowcasts: A Do-It-Yourself (DIY) Ensemble Method

John B Rundle , Ian Baughmann , Andrea Donnellan , Lisa Grant Ludwig , Geoffrey C Fox , Kazuyoshi Nanjo This is my paper

Pith reviewed 2026-05-17 01:02 UTC · model grok-4.3

classification ⚛️ physics.geo-ph
keywords earthquake forecastingnowcastingGutenberg-Richter relationROC analysisensemble methodcalendar time forecastspositive predictive valueseismic hazard
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The pith

A count of small earthquakes since the last major one, combined with Gutenberg-Richter statistics from surrounding regions, directly yields skillful calendar-time forecasts for large local events.

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

The paper develops a method that builds an ensemble of earthquakes drawn from larger surrounding regions to match the observed count of small events in a target local area. The forecast probability for a large target-magnitude earthquake is then read off directly as the positive predictive value from the associated receiver operating characteristic curve, without assuming any parametric probability distribution. Skill is reported to increase as time since the previous major earthquake grows, and the approach is checked against existing UCERF3 forecasts for the Los Angeles and San Francisco boxes before being applied to a 125 km radius around Los Angeles after the 1994 Northridge event.

Core claim

By assuming only that the Gutenberg-Richter magnitude-frequency statistics of a local area match those of larger surrounding regions, an ensemble of earthquakes can be assembled from the surrounding data; conditioning on the number of small earthquakes n(t) observed since the last large event then supplies a probability for the next large event that equals the positive predictive value of the corresponding ROC curve, and this probability exhibits significant skill that improves with elapsed time.

What carries the argument

The ensemble of earthquakes constructed from larger surrounding regions using the Gutenberg-Richter relation to match the local small-earthquake count n(t), from which the forecast probability is read directly as the positive predictive value of the ROC curve.

If this is right

  • One-year and five-year forecasts become available for any chosen local circle once the count of small events since the last large event is known.
  • The same procedure can be repeated for any other region by defining a local area and drawing the ensemble from its surroundings.
  • Forecast skill is expected to rise steadily as the time interval since the preceding major earthquake lengthens.
  • Validation against UCERF3 for the Los Angeles and San Francisco boxes supplies a concrete benchmark for the method's performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the assumption holds across many regions, the approach could be implemented with publicly available catalogs and minimal additional modeling.
  • The same counting-plus-ensemble logic might be tested on other phenomena that obey power-law size distributions, such as landslides or forest fires.
  • A natural extension would be to examine whether the skill improvement with time persists when the method is applied to sequences outside California.

Load-bearing premise

The Gutenberg-Richter magnitude-frequency statistics in the local forecast area are the same as those measured in the larger surrounding regions used to build the ensemble.

What would settle it

An independent data set in which the positive predictive values computed from the ROC curves for successive values of n(t) fail to match the observed fraction of target-magnitude earthquakes that actually occur.

Figures

Figures reproduced from arXiv: 2512.06572 by Andrea Donnellan, Geoffrey C Fox, Ian Baughmann, John B Rundle, Kazuyoshi Nanjo, Lisa Grant Ludwig.

Figure 2
Figure 2. Figure 2: Conditional Receiver Operating Characteristic (ROC) diagrams for an ensemble of 30 regions from 3.6o to 6.5o surrounding Los Angeles, CA, at 0.1o interval half-widths, with a forecast time of TF = 5 years. a) ROC diagram computed after no small earthquakes have occurred. b) ROC diagram computed after 150 small earthquakes have occurred. c) ROC diagram computed after 300 small earthquakes have occurred. d) … view at source ↗
read the original abstract

This paper presents a new technical method for computing calendar time forecasts in a local area for large earthquakes of a target magnitude MT using a count small earthquakes MS < MT in the area, together with the Gutenberg-Richter (GR) magnitude-frequency relation. The GR relation states that for every large target earthquake of magnitude greater than MT , there are on average NGR small earthquakes of magnitude MT > M >= MS. The only assumption is that the GR statistics of the local area are the same as in the larger surrounding regions. This assumption is used to construct an ensemble of earthquakes in larger surrounding regions to be used in computing the forecast. The method has significant skill, as defined by the Receiver Operating Characteristic (ROC) test, which improves as time since the last major earthquake increases. The probability is conditioned on the number of small earthquakes n(t) that have occurred since the last large earthquake. There is no need to assume a probability model, as the probability is instead computed directly as the Positive Predictive Value (PPV) associated with the ROC curve. The method is validated by comparison to the UCERF3 forecasts for the UCERF3-defined geographic boxes centered on Los Angeles and San Francisco. The method is then applied to a 125-KM radius circular area around Los Angeles, California, following the January 17, 1994 magnitude M6.7 Northridge earthquake, and short term forecasts (1 year and 5 year ) are computed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper presents a DIY ensemble method for calendar-time local forecasts of large earthquakes (target magnitude MT) based on the count n(t) of smaller events (MS < MT) since the last large event. Using the Gutenberg-Richter relation, it scales n(t) by an average NGR to construct an ensemble from larger surrounding regions under the assumption that local GR statistics match those of the surrounding areas. Probabilities are computed directly as the positive predictive value (PPV) from ROC curves rather than from an assumed probability model. The method claims significant ROC skill that improves with time since the last major event, is validated by comparison to UCERF3 forecasts in geographic boxes around Los Angeles and San Francisco, and is applied to generate 1-year and 5-year forecasts in a 125 km radius around Los Angeles following the 1994 Northridge M6.7 earthquake.

Significance. If the GR spatial-invariance assumption holds and the reported ROC skill proves robust, the approach offers a simple, low-parameter method for generating empirical forecasts that avoids explicit probability models and derives probabilities directly from observed counts via PPV. The direct validation against UCERF3 and the concrete application to the post-Northridge period add practical value. The absence of fitted parameters and the use of ensemble construction from surrounding data are methodological strengths that could make the technique accessible for operational or DIY use, provided the key assumption is shown to be tenable at the relevant spatial scales.

major comments (3)
  1. [Abstract] Abstract: The claim that GR statistics equivalence between the local area and larger surrounding regions is 'the only assumption' is load-bearing for the ensemble construction used to compute PPV and ROC skill. No quantitative check (e.g., separate b-value or productivity fits for the 125 km LA circle versus the surrounding domains) or sensitivity runs are described, which leaves open the possibility that spatial variation in GR parameters biases the effective threshold for large events and the reported skill improvement with time since last event.
  2. [Validation section] Validation against UCERF3: The manuscript states that ROC skill improves with time since the last M≥MT event and is validated for UCERF3-defined boxes around Los Angeles and San Francisco, yet provides no quantitative error bars on the ROC curves, no description of ensemble sampling procedure, and no discussion of data exclusion rules or sensitivity to the choice of surrounding region. These omissions make it difficult to evaluate whether the claimed skill improvement is statistically significant or robust.
  3. [Application to Northridge] Northridge application: The 1-year and 5-year forecasts for the 125 km radius area around Los Angeles after the 1994 M6.7 event rest on the same untested GR equivalence assumption used to scale n(t). Without sensitivity tests on this assumption or on the choice of surrounding regions, the reliability of the specific numerical forecasts cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract and method description could more explicitly state the precise procedure for selecting and weighting the surrounding regions used to build the ensemble, including any distance or magnitude cutoffs.
  2. [Figures] Figure captions and axis labels for the ROC curves should include the exact definition of the positive class and the number of events or trials underlying each curve to improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We have carefully considered each point and provide point-by-point responses below. Where appropriate, we will revise the manuscript to address the concerns raised.

read point-by-point responses

  1. Referee: [Abstract] The claim that GR statistics equivalence between the local area and larger surrounding regions is 'the only assumption' is load-bearing for the ensemble construction used to compute PPV and ROC skill. No quantitative check (e.g., separate b-value or productivity fits for the 125 km LA circle versus the surrounding domains) or sensitivity runs are described, which leaves open the possibility that spatial variation in GR parameters biases the effective threshold for large events and the reported skill improvement with time since last event.

    Authors: We agree that explicitly demonstrating the robustness of the GR equivalence assumption would strengthen the paper. While the successful validation against UCERF3 forecasts provides some indirect support for the assumption at the scales considered, we will add a new section or appendix with quantitative comparisons of b-values and productivity rates between the local 125 km radius and the surrounding regions used in the ensemble. We will also include sensitivity tests by varying the surrounding region definitions and showing the impact on the ROC curves and forecasts. revision: yes

  2. Referee: [Validation section] The manuscript states that ROC skill improves with time since the last M≥MT event and is validated for UCERF3-defined boxes around Los Angeles and San Francisco, yet provides no quantitative error bars on the ROC curves, no description of ensemble sampling procedure, and no discussion of data exclusion rules or sensitivity to the choice of surrounding region. These omissions make it difficult to evaluate whether the claimed skill improvement is statistically significant or robust.

    Authors: We will revise the validation section to include bootstrap-derived error bars or confidence intervals on the ROC curves to assess statistical significance. We will also provide a detailed description of the ensemble sampling procedure, including how surrounding regions are selected and the number of realizations used. Additionally, we will discuss data exclusion rules (e.g., aftershocks handling) and present sensitivity analyses to different choices of surrounding regions to demonstrate robustness of the skill improvement. revision: yes

  3. Referee: [Application to Northridge] The 1-year and 5-year forecasts for the 125 km radius area around Los Angeles after the 1994 M6.7 event rest on the same untested GR equivalence assumption used to scale n(t). Without sensitivity tests on this assumption or on the choice of surrounding regions, the reliability of the specific numerical forecasts cannot be assessed.

    Authors: We acknowledge this limitation in the current presentation. In the revised manuscript, we will incorporate sensitivity tests for the Northridge application by varying the surrounding regions and GR parameters within reasonable bounds, showing how the 1-year and 5-year probability forecasts change. This will allow readers to better assess the reliability of the numerical values provided. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical PPV from ROC on conditioned n(t) counts is independent of GR ensemble construction

full rationale

The paper constructs forecasts by counting observed small earthquakes n(t) since the last large event, then computes probabilities directly as the Positive Predictive Value (PPV) associated with the ROC curve on those counts. The GR relation is invoked solely to build an external ensemble from larger surrounding regions under the explicit assumption of spatial invariance in b-value and productivity; this assumption is presented as the sole modeling choice rather than a fitted parameter or self-referential definition. No equations in the derivation reduce the reported ROC skill or PPV probabilities to the input counts or GR parameters by algebraic construction. No self-citations are used as load-bearing uniqueness theorems, and the method is validated against the independent UCERF3 forecasts, keeping the central claim self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on one explicit domain assumption about Gutenberg-Richter statistics being transferable from surrounding regions to the local area; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Gutenberg-Richter magnitude-frequency relation statistics of the local area are the same as in the larger surrounding regions
    Used to construct the ensemble of earthquakes from surrounding regions for the forecast.

pith-pipeline@v0.9.0 · 5589 in / 1383 out tokens · 69506 ms · 2026-05-17T01:02:03.627803+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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    At right is the plot of 447 small earthquakes that have occurred since the 1/17/1994 Northridge earthquake. Acknowledgements. Research by JBR was supported in part by a grant from the Southern California Earthquake Center grant #SCON-00007927 to UC Davis, and by the John LaBrecque fund, a generous gift from John LaBrecque to the University of California, ...

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    The Cumulative Distribution Function (CDF) is derived from the histogram. The magenta curves close to the red curve represent the 1s deviation from the histogram, again using a bootstrap method. The dashed blue line is the accumulation function, computed from equation (2). b) The red "thermometer" has the same value as the CDF (for easy reference), repres...

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