arxiv: 2604.03383 · v2 · submitted 2026-04-03 · ⚛️ physics.geo-ph · physics.comp-ph· physics.soc-ph

Recognition: no theorem link

Exceedance Probabilities for Large Earthquakes From DIY Local Earthquake Ensemble Nowcasting and Forecasting

Andrea Donnellan, Geoffrey Fox, Ian Baughman, John B Rundle, Kazuyoshi Nanjo, Lisa Grant Ludwig

Pith reviewed 2026-05-13 17:59 UTC · model grok-4.3

classification ⚛️ physics.geo-ph physics.comp-phphysics.soc-ph

keywords earthquake forecastingnowcastingGutenberg-Richter relationROC analysisexceedance probabilityseismic ensemblelocal earthquake probability

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The pith

Earthquake probabilities can be forecast from counts of small quakes since the last major event using the Gutenberg-Richter relation.

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

The paper introduces a method to compute local probabilities for large earthquakes by counting smaller ones since the previous major quake. It relies on the Gutenberg-Richter relation to create an ensemble from surrounding areas and uses data mining to derive forecasts. These forecasts exhibit skill according to ROC tests, which strengthens as time since the last large earthquake grows. The approach is validated against UCERF3 models for Los Angeles and San Francisco areas and then applied to generate short-term forecasts around Los Angeles after the 1994 Northridge quake. A sympathetic reader would care because it suggests a simple, accessible way to anticipate seismic risks in specific locations without complex simulations.

Core claim

The central discovery is that exceedance probabilities for target magnitude earthquakes can be calculated directly as the positive predictive value from an ROC curve based on the number of small earthquakes n(t) occurring since the last large event, with the ensemble defined using the Gutenberg-Richter relation in larger surrounding regions, and that this method shows significant skill that improves with increasing time since the last major earthquake.

What carries the argument

The Receiver Operating Characteristic (ROC) curve applied to the count of small earthquakes since the last major event, from which the Positive Predictive Value (PPV) gives the exceedance probability.

If this is right

The method produces 1-year and 5-year forecasts for circular areas around points of interest.
Forecast skill, as measured by ROC, increases with time since the last major earthquake.
The probabilities can be directly compared to those from UCERF3 for the same geographic boxes.
Application to the area around Los Angeles after the Northridge earthquake yields specific short-term probability estimates.

Where Pith is reading between the lines

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

If the Gutenberg-Richter relation holds, this could enable rapid updates to local forecasts whenever new small earthquakes are recorded.
The approach might be extended to other regions with similar seismic catalogs to test its generality beyond California.
By focusing on time since last major event, it highlights the importance of clustering or triggering effects in earthquake sequences.

Load-bearing premise

The Gutenberg-Richter magnitude-frequency relation holds in the local and surrounding regions.

What would settle it

A verification showing that the ROC-derived positive predictive values do not increase with time since the last major earthquake, or fail to match observed large earthquake rates in the Los Angeles area after the Northridge event, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.03383 by Andrea Donnellan, Geoffrey Fox, Ian Baughman, John B Rundle, Kazuyoshi Nanjo, Lisa Grant Ludwig.

**Figure 1.** Figure 1: Left: Regional seismicity (small dots) used in this paper. Large earthquakes are shown as red circles. A region of radius 125 KM around the city of Los Angeles is shown as a blue circle. Right: Green bars show the histogram of the number of small earthquakes between large earthquakes. Solid red stair-step line is the Cumulative Distribution Function (CDF) corresponding to the histogram. Magenta dashed line… view at source ↗

read the original abstract

This paper focuses on the problem of anticipating the local occurrence of future large earthquakes. "Local" is defined as the probability of a large earthquake occurring with a defined circle of arbitrary radius surrounding a point of interest. The main (and for that matter, the only) assumption for all these works is that the Gutenberg-Richter (GR) magnitude-frequency relation holds. Here we describe a 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. Using the idea that the GR relation is valid throughout the surrounding region, we define an ensemble of earthquakes in larger surrounding regions to be used in computing the forecast. What follows is simple data mining. 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. The probability is 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.

Desk Editor's Note private letter to a colleague

The paper gives a replicable local nowcasting method that turns small-quake counts into ROC-derived probabilities, with skill improving as time since the last large event grows, but the validation stays within the same catalog and compares only to UCERF3.

read the letter

The main takeaway is a practical procedure for local exceedance probabilities around any chosen point. You pick a circle radius, count small events since the last large one, and convert that count into a probability using the positive predictive value from an ROC curve built on an ensemble drawn from surrounding areas under the Gutenberg-Richter assumption. They show the ROC skill rising with elapsed time and apply the method to a 125 km circle around Los Angeles after the 1994 Northridge quake for 1- and 5-year forecasts, plus a side-by-side check against UCERF3 boxes for Los Angeles and San Francisco. That comparison and the explicit DIY setup for arbitrary circles are the concrete new pieces. The approach is simple enough that someone with catalog access could code it up quickly, and the direct use of PPV avoids extra parameters. The soft spot is the validation. The ROC curves and skill numbers are derived from the same data used to define the ensembles, so aftershock clustering or rate changes in the catalog could inflate the apparent performance. They test against another model rather than against actual future events held out in time, and no error propagation or sensitivity runs on the circle size or magnitude thresholds appear in the description. The GR relation is used throughout but not checked here. This is for seismologists or risk analysts who want a lightweight local tool they can run on public data for specific sites. A reader already deep in operational forecasting might try the method on their own catalogs to see how it behaves. It has enough working code-like steps and an external model comparison to justify sending it to a serious referee, though reviewers will likely press on the in-sample nature of the ROC results and ask for prospective checks. I would recommend peer review with that focus.

Referee Report

3 major / 2 minor

Summary. The paper proposes a DIY local ensemble nowcasting method for exceedance probabilities of large earthquakes (M ≥ MT) within a user-defined radius around a point of interest. It assumes the Gutenberg-Richter magnitude-frequency relation holds throughout the region, constructs an ensemble from small-event counts n(t) since the last large event in surrounding areas, and computes the conditional probability directly as the positive predictive value (PPV) associated with an ROC curve derived from the empirical data. The method is claimed to exhibit significant ROC skill that increases with time since the last major event; it is validated via comparison to UCERF3 forecasts in fixed geographic boxes centered on Los Angeles and San Francisco, then applied to produce 1-year and 5-year forecasts for a 125 km radius around Los Angeles following the 1994 Northridge earthquake.

Significance. If the reported ROC skill generalizes prospectively, the approach supplies a transparent, parameter-light (beyond the GR assumption) ensemble method for local nowcasting that can be implemented with public catalogs and directly compared to operational models such as UCERF3. The explicit use of PPV from ROC curves and the reported improvement with elapsed time since the last large event constitute falsifiable, data-driven predictions that could be tested in real time.

major comments (3)

[Abstract] Abstract and validation section: the ROC skill and PPV values are derived from the same catalog used to define the local ensemble and thresholds, without a held-out prospective test period; this leaves the reported AUC improvement with time-since-last-event vulnerable to in-sample bias from aftershock clustering and non-stationary rates, as the skeptic note correctly flags.
[Validation] Validation paragraph: skill is assessed solely by comparison to UCERF3 forecasts for fixed boxes rather than by direct hit-rate evaluation against subsequently observed M ≥ MT events in an independent time window; this weakens the claim that the n(t)-conditioned PPV constitutes an independent forecasting advance.
[Method] Method description: the GR assumption is stated as the sole modeling premise, yet no sensitivity analysis or robustness checks against deviations from GR (e.g., b-value variation or catalog incompleteness) are presented; because the ensemble construction and PPV mapping rest entirely on this assumption, its untested status is load-bearing for the central probability claim.

minor comments (2)

[Method] Clarify how the ensemble radius and magnitude thresholds MS, MT are chosen in practice and whether they are fixed a priori or data-mined; the current description leaves the degree of user discretion ambiguous.
[Results] Add explicit error bars or bootstrap estimates on the reported PPV values and AUC scores to quantify sampling uncertainty in the empirical counts n(t).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive review of arXiv:2604.03383. We address each major comment point by point below, offering clarifications on the retrospective nature of the analysis while indicating revisions where they strengthen the presentation without altering the core method.

read point-by-point responses

Referee: [Abstract] Abstract and validation section: the ROC skill and PPV values are derived from the same catalog used to define the local ensemble and thresholds, without a held-out prospective test period; this leaves the reported AUC improvement with time-since-last-event vulnerable to in-sample bias from aftershock clustering and non-stationary rates, as the skeptic note correctly flags.

Authors: The ROC curves and PPV values are intentionally constructed from the full historical catalog because the nowcasting method is data-driven and relies on observed small-event counts n(t) to map directly to conditional probabilities. The improvement in skill with elapsed time since the last large event is an empirical pattern visible in the data and constitutes a falsifiable claim for prospective testing. We will revise the abstract and validation section to explicitly state the retrospective character of the ROC analysis and discuss possible influences from aftershock clustering. revision: partial
Referee: [Validation] Validation paragraph: skill is assessed solely by comparison to UCERF3 forecasts for fixed boxes rather than by direct hit-rate evaluation against subsequently observed M ≥ MT events in an independent time window; this weakens the claim that the n(t)-conditioned PPV constitutes an independent forecasting advance.

Authors: Direct hit-rate evaluation against future M ≥ MT events is the ultimate test but is impractical on short timescales given the rarity of target events. UCERF3 is an independent, comprehensive reference model for the same California regions; agreement between our simple n(t)-based PPV and UCERF3 forecasts therefore provides a meaningful benchmark. We maintain that this comparison supports the method as a transparent, low-parameter ensemble nowcast and do not plan to alter the validation approach. revision: no
Referee: [Method] Method description: the GR assumption is stated as the sole modeling premise, yet no sensitivity analysis or robustness checks against deviations from GR (e.g., b-value variation or catalog incompleteness) are presented; because the ensemble construction and PPV mapping rest entirely on this assumption, its untested status is load-bearing for the central probability claim.

Authors: The Gutenberg-Richter relation is the sole modeling premise and is adopted because it is a well-established empirical regularity across seismic catalogs. To strengthen the manuscript we will add a short robustness subsection that examines the effect of modest b-value variations and catalog completeness thresholds on the resulting PPV values, confirming that the reported skill remains stable within plausible ranges. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical PPV from observed n(t) counts and ROC on real data, with external UCERF3 validation

full rationale

The paper's central derivation computes exceedance probabilities directly as the positive predictive value (PPV) from ROC curves conditioned on observed small-earthquake counts n(t) since the last large event. This is presented as simple data mining on the catalog, not as an algebraic reduction or fitted parameter renamed as a prediction. The GR magnitude-frequency relation is invoked only as the sole external assumption to define the local ensemble region, not as a self-referential input. Validation proceeds by direct comparison to the independent UCERF3 model forecasts for fixed geographic boxes, providing an external benchmark rather than an in-sample identity. No step equates the reported skill or probability to its own construction via self-citation chain, uniqueness theorem, or ansatz smuggling; the result remains falsifiable against subsequently observed events.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests entirely on the validity of the Gutenberg-Richter relation as the sole stated assumption, with the forecast emerging from empirical counts and ROC analysis rather than new theoretical derivation.

axioms (1)

domain assumption Gutenberg-Richter magnitude-frequency relation holds throughout the surrounding region
Explicitly identified as the main and only assumption for the entire method.

pith-pipeline@v0.9.0 · 5599 in / 1282 out tokens · 95867 ms · 2026-05-13T17:59:43.717826+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

and Fox, G.C., 2026

Rundle, J.B., Baughman, I., Donnellan, A., Grant Ludwig, L. and Fox, G.C., 2026. From local earthquake nowcasting to natural time forecasting: A simple do-it-yourself (DIY) method. Earth and Space Science, 13(1), p.e2025EA004820. 17. Fox, G.C., Rundle, J.B., Donnellan, A. and Feng, B., 2022. Earthquake nowcasting with deep learning. Geohazards, 3(2), pp.1...

work page 2026
[2]

and Mehta, A., 2018

Bhatia, A., Pasari, S. and Mehta, A., 2018. Earthquake forecasting using artificial neural networks. The international archives of the photogrammetry, remote sensing and spatial information sciences, 42, pp.823-827. 35. Pasari, S., 2018, November. Stochastic modeling of earthquake interevent counts (Natural Times) in Northwest Himalaya and adjoining regio...

work page arXiv 2018
[3]

Nowcast Function

Field, E.H., Arrowsmith, R.J., Biasi, G.P., Bird, P., Dawson, T.E., Felzer, K.R., Jackson, D.D., Johnson, K.M., Jordan, T.H., Madden, C. and Michael, A.J., 2014. Uniform California earthquake rupture forecast, version 3 (UCERF3)—The time-independent model. Bulletin of the Seismological Society of America, 104(3), pp.1122-1180. 52. Field, E.H., Dawson, T.E...

work page 2014