Models Classical

Models — Classical Theories (index)

The analytical, physics-informed and statistical models that define the field-standard baseline for short-term probabilistic seismic forecasting. This page is an index: each model now lives on its own deep sub-page with its governing equation(s), parameter estimation, assumptions and failure modes, role in operational earthquake forecasting, a worked illustration, a diagram, and a References section with DOIs. A well-tuned ETAS remains the de-facto operational baseline that any candidate model — including every neural model — must beat in prospective Evaluation-and-Tests before it can claim forecasting skill.

Honest framing. Every model below produces, at best, a conditional rate or probability, never a deterministic prediction. A prediction is a deterministic statement that an event will or will not occur; a forecast gives a probability strictly in $(0,1)$ (Jordan et al., 2011).

Conventions used throughout. Magnitudes are homogenized to moment magnitude $M_w$ where possible; $M_c$ is the magnitude of completeness; $b$ is the Gutenberg–Richter slope; $\beta = b\ln 10$. The history available at time $t$ is $\mathcal{H}_t = {(t_i, x_i, y_i, m_i): t_i < t}$.

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How these models fit together

The product is a conditional estimator: given recent observations $\mathcal{H}_t$, it outputs a bounded probability of one-or-more target events in a region over horizons of ~1 day, ~2 days, ~1 week. The classical models play three distinct roles — the forecasting baselines an enhanced model must beat, the building blocks of the conditional intensity, and the calibration scaffold that keeps the output honest.

Layer	Model(s)	Forecast role
Magnitude–frequency	Gutenberg-Richter-Law	Converts a rate of events $\ge M_{\text{ref}}$ into a rate $\ge$ any $M$; the magnitude term of every forecast.
Aftershock decay	Omori-Utsu-Law, Reasenberg-Jones-Model	The dominant short-horizon (1–7 day) signal; the baseline for "tomorrow's earthquakes."
Self-exciting clustering	ETAS-Model	State-of-the-art physics-free short-term baseline; the model to beat.
Long-term smoothed rate	Smoothed-Seismicity	The time-independent spatial background $\mu(x,y)$ ETAS needs; also the mandatory null.
Medium-term precursory	EEPAS-Model	Months-to-years scale; outside the 1-week window, useful as a feature/context source.
Operational hybrid	STEP-Model	Production reference for the "next-day" probabilistic map output shape.
Renewal / recurrence	Brownian-Passage-Time	Long-term, fault-specific; conditions the background over years.
Physics-based stress	Rate-and-State-and-Coulomb	Mechanistic priors on where triggering is enhanced or suppressed.

flowchart LR
    GR[Gutenberg-Richter<br/>magnitude term] --> P[Published probability<br/>P = 1 - exp - N Phi]
    OM[Omori-Utsu<br/>aftershock decay] --> ET[ETAS<br/>self-exciting clustering]
    SM[Smoothed seismicity<br/>background mu] --> ET
    ET --> P
    RJ[Reasenberg-Jones] --> ST[STEP<br/>operational map]
    OM --> RJ
    RS[Rate-and-state + Coulomb<br/>stress priors] -. informs .-> ET
    BPT[Brownian Passage Time<br/>renewal] -. conditions .-> SM
    EEPAS[EEPAS<br/>medium-term] -. feature/context .-> ET

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The classical models — one line each

• Gutenberg-Richter-Law — the frequency–magnitude relation $\log_{10} N(\ge M) = a - b,M$; supplies the exponential magnitude density that distributes any forecast rate over magnitude, and sets the large-event tail via $b$ and $M_{\max}$.
• Omori-Utsu-Law — the modified Omori law $n(t) = K/(t+c)^p$ for aftershock-rate decay; the dominant short-horizon signal, with productivity, $c$, $p$ and post-mainshock incompleteness.
• ETAS-Model — the Epidemic-Type Aftershock Sequence: a self-exciting (Hawkes) point process where every event triggers its own Omori aftershocks; the de-facto operational baseline to beat.
• Reasenberg-Jones-Model — the original operational aftershock model (Omori × Gutenberg–Richter) that produces calibrated next-day/next-week aftershock probabilities; basis of USGS OAF.
• STEP-Model — Short-Term Earthquake Probability: a daily hybrid that adds the time-varying clustering rate to a smoothed background, producing the canonical next-day probability map shape.
• EEPAS-Model — Every Earthquake a Precursor According to Scale: a medium-term model in which each event raises the rate of larger future events via predictive scaling relations.
• Smoothed-Seismicity — the time-independent spatial background $\mu(x,y)$ built by kernel smoothing of past epicentres; ETAS's background term and the project's mandatory Poisson null.
• Brownian-Passage-Time — the BPT (inverse-Gaussian) renewal model for quasi-periodic recurrence of characteristic earthquakes on a single fault; long-term, time-dependent hazard.
• Rate-and-State-and-Coulomb — Dieterich rate-and-state friction and Coulomb stress change $\Delta\mathrm{CFS}$; the mechanistic layer giving physical priors on where triggering is favored.

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From a conditional intensity to the published probability

Every model above feeds one object — a conditional intensity (rate) $\lambda$. The product converts the expected count $N$ over a horizon and the magnitude exceedance factor $\Phi(M^\ast) = 10^{-b(M^\ast - M_c)}$ into the single public exceedance probability

$$P(\ge 1\ \text{event} \ge M^\ast) = 1 - e^{-N,\Phi(M^\ast)} .$$

This formula never changes; only the quality of $\lambda$ improves. See Models-Employed for the exact models CAOS_SEISMIC runs, and Evaluation-and-Tests for how each is scored prospectively.

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See also: Models-ML · Models-Employed · Temporal-Point-Processes · Methodology-History · Evaluation-and-Tests · References · Glossary