EEPAS Model

EEPAS Model — Every Earthquake a Precursor According to Scale

EEPAS turns a counter-intuitive but well-documented observation into a forecast: every earthquake is, to some degree, a long-term precursor of a larger one to follow — and the bigger the precursor, the bigger, later, and farther the event it foreshadows. Where Reasenberg–Jones and STEP forecast aftershocks on a horizon of hours to days, EEPAS forecasts future mainshocks on a horizon of months to years, from the accumulating population of smaller events. This page is a deep, self-contained treatment of that one model: the precursory-scale-increase phenomenon it is built on, the scaling relations and rate-density equations, parameter estimation, assumptions, strengths and limitations, its place in operational forecasting, a structural diagram, and references.

Honest framing. EEPAS produces a medium-term rate density — a smooth, bounded forecast of where and at what rate larger events are elevated, never a date, a place, or a magnitude certainty. It is a forecast, with probabilities in $(0,1)$, not a prediction (Jordan et al., 2011). It sits outside this product's 1-week primary window and is used as a feature / context source, not a short-term core model. A single outcome neither confirms nor refutes it. See Honest-Limits.

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Table of contents

1. Intuition and history
2. The precursory scale increase phenomenon
3. The predictive scaling relations
4. The forecast rate density
5. Parameter estimation
6. Assumptions and failure modes
7. Strengths and limitations
8. Role in operational earthquake forecasting
9. Worked illustration
10. How EEPAS is used in this product
11. References

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1. Intuition and history

The aftershock models on this wiki run the clock forward from a big event: a mainshock occurs, and its aftershocks decay. EEPAS, developed by David Rhoades and Frank Evison in New Zealand, runs the logic the other way. They observed that minor-to-moderate earthquakes tend to cluster in space and time before a larger event — a long-term, low-level swarm-like increase in activity that precedes the mainshock rather than following it. They called this the precursory scale increase, denoted $\Psi$ (Psi).

The radical move in EEPAS is to refuse to identify which small events are precursors. Instead, every earthquake is treated as a precursor "according to its scale" — each event contributes a smooth bump of elevated probability for a larger future event, with the bump's magnitude, timing, and footprint all scaled to the precursor's own size. Summing these contributions over the whole recent catalog produces a forecast of where larger events are now more likely over the coming months to years.

Rhoades & Evison (2004) formalized this as a space–time–magnitude point process and entered it into the international CSEP forecasting experiments, where it has demonstrated genuine, statistically significant medium-term forecasting skill in New Zealand, Japan, California, and Italy.

Why "according to scale" is the load-bearing phrase. EEPAS does not need to know which earthquakes are "real" precursors — an unanswerable question in real time. By scaling every event's contribution and summing, it converts an unidentifiable signal into a calibratable rate density. That is the conceptual trick that makes it testable.

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2. The precursory scale increase phenomenon

The empirical foundation is the $\Psi$ phenomenon: ahead of a mainshock of magnitude $M_m$, there is a period of increased rate of moderate earthquakes over an area, and the magnitude, duration, and area of that precursory increase all scale predictably with the size of the eventual mainshock. Larger mainshocks are preceded by precursory increases that involve larger precursor events, last longer, and cover a wider area.

EEPAS inverts this: observing a precursor event of magnitude $M_p$ now, it predicts the distribution of the mainshock magnitude $M_m$, the elapsed time $T_P$ until it, and the area $A$ over which it may occur — all as functions of $M_p$.

flowchart TD
    EQ["Each earthquake i<br/>(magnitude M_i)"] --> M["Predicts mainshock<br/>magnitude: normal,<br/>mean a_M + b_M·M_i"]
    EQ --> T["Predicts elapsed time:<br/>lognormal, median<br/>10^(a_T + b_T·M_i)"]
    EQ --> A["Predicts area:<br/>bivariate-normal,<br/>scale 10^(a_A + b_A·M_i)"]
    M --> CONTR["Contribution of event i<br/>= f(m)·g(t)·h(x,y), weighted w_i"]
    T --> CONTR
    A --> CONTR
    CONTR --> SUM["Sum over all events<br/>+ baseline μ·λ₀"]
    SUM --> RD["Medium-term<br/>rate density λ(t,m,x,y)"]

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3. The predictive scaling relations

For a precursor of magnitude $M_p$, the expected mainshock magnitude $M_m$, the precursor time $T_P$, and the precursory area $A$ scale linearly (in the appropriate variable) with $M_p$:

$$M_m = a_M + b_M, M_p \qquad (b_M \text{ fixed to } 1),$$ $$\log_{10} T_P = a_T + b_T, M_p,$$ $$\log_{10} A = a_A + b_A, M_p,$$

each with an associated spread $\sigma_M, \sigma_T, \sigma_A$ that becomes the width of the corresponding density below.

• Magnitude ($M_m = a_M + b_M M_p$, with $b_M = 1$). Fixing $b_M = 1$ encodes that a precursor foreshadows a mainshock a roughly constant number of magnitude units larger than itself — the scale-invariance at the heart of the model.
• Time ($\log_{10} T_P = a_T + b_T M_p$). The precursor time grows with precursor magnitude: bigger precursors anticipate events further in the future — the basis of EEPAS's months-to-years horizon.
• Area ($\log_{10} A = a_A + b_A M_p$). The precursory footprint widens with precursor magnitude.

The six constants $a_M, b_M, a_T, b_T, a_A, b_A$ (plus the three spreads) are the model's free parameters, fit to a region's catalog.

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4. The forecast rate density

EEPAS assembles a space–time–magnitude rate density as a baseline term plus a weighted sum over every prior earthquake, with each event contributing a product of three densities — normal in magnitude, lognormal in elapsed time, bivariate-normal in location:

$$\lambda(t, m, x, y) = \mu, \lambda_0(m, x, y)

• \sum_{i:,t_i < t} w_i; f(m \mid M_i); g(t - t_i \mid M_i); h(x, y \mid x_i, y_i, M_i),$$

where each factor follows from the corresponding scaling relation of §3:

• $f(m \mid M_i)$ — magnitude: a normal density with mean $a_M + b_M M_i$ and standard deviation $\sigma_M$. The precursor of magnitude $M_i$ predicts a mainshock about $a_M$ units larger, smeared by $\sigma_M$.
• $g(t - t_i \mid M_i)$ — time: a lognormal density in elapsed time with median $10^{,a_T + b_T M_i}$ and log-standard-deviation $\sigma_T$. Lognormal (not exponential) because the precursor time is positive and multiplicatively scaled — the contribution rises, peaks at the predicted lead time, and tapers, rather than decaying monotonically like an aftershock.
• $h(x, y \mid x_i, y_i, M_i)$ — location: a bivariate-normal density centred on the precursor, with variance scaling as the predicted area $A \propto 10^{,a_A + b_A M_i}$ — bigger precursors spread their probability over a wider footprint.
• $w_i$ — weight: down-weights events that are themselves likely to be aftershocks (so the model forecasts future mainshocks, not the decay of past ones), and normalizes the contributions.
• $\mu, \lambda_0(m, x, y)$ — baseline: a mixing term that blends EEPAS with a time-invariant baseline (a Proximity-to-Past-Earthquakes / smoothed-seismicity model). $\mu$ is the mixing weight between the precursory signal and the baseline.

Contrast the time kernel with the aftershock models: Reasenberg–Jones and ETAS use an Omori power-law that decays from $t = 0$, forecasting events that follow a trigger; EEPAS uses a lognormal that rises to a peak at the predicted precursor lead time, forecasting events that the precursor anticipates. That single difference is what makes EEPAS a medium-term anticipatory model rather than a short-term reactive one.

Honest caveat — pin the constants, trust the structure. The published EEPAS density constants contain known typos across papers, acknowledged by the authors, and reference implementations differ slightly in detail. If EEPAS is used in practice, pin the exact constants and normalizations to a reference implementation (pyCSEP / floatCSEP) rather than transcribing them from any single paper. The structure above — baseline plus a weighted sum of normal × lognormal × bivariate-normal contributions — is the load-bearing, well-established part.

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5. Parameter estimation

• Scaling constants and spreads $(a_M, a_T, b_T, a_A, b_A, \sigma_M, \sigma_T, \sigma_A)$: estimated by maximizing the space–time–magnitude point-process log-likelihood

  $$\ln L = \sum_i \ln \lambda(t_i, m_i, x_i, y_i) ;-; \int \lambda(t, m, x, y), dt, dm, dx, dy,$$

  over a region's catalog above a completeness magnitude $M_c$ — the same likelihood form used for every point-process model in this stack (Methodology-History). The compensator (integral) term is what calibrates the density.

• Mixing weight $\mu$: fit jointly, controlling how much weight the precursory signal carries versus the smoothed-seismicity baseline.

• Weights $w_i$: assigned to down-weight likely aftershocks (so EEPAS forecasts independent future mainshocks, not aftershock decay).

• Completeness $M_c$ and $b$-value: as for every model here, derive $M_c$ and the Aki–Utsu $b$-value on a rolling basis; never hard-code them (Models-Classical §1).

• Tooling: EEPAS is implemented in the CSEP testing ecosystem (pyCSEP / floatCSEP); use those reference implementations to avoid the documented constant-transcription pitfalls.

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6. Assumptions and failure modes

Assumption	What breaks it
The $\Psi$ precursory scale increase is a real, region-stable phenomenon.	Regions or periods where the precursory signal is weak or absent; the model reduces toward its baseline.
Scaling relations are log-linear with stable constants.	Constants drift between regions/catalogs; cross-region transfer is invalid without refitting.
Magnitude / time / area densities are separable products.	Real precursory geometry is not perfectly separable; the product form is a tractable approximation.
The catalog is complete above $M_c$.	Incompleteness distorts the population of precursors and biases the fit.
Aftershocks can be down-weighted cleanly via $w_i$.	Imperfect declustering leaks aftershock signal into the "precursor" sum.

The honest limitation is horizon: EEPAS forecasts months to years ahead, so it cannot answer the short-term "tomorrow" question at all — and its medium-term skill, while real and CSEP-verified, is modest and best realized in a multi-year evaluation, not a single forecast.

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7. Strengths and limitations

Strengths

• Anticipatory, not reactive — the only model in this family that forecasts future mainshocks from precursors, complementing the aftershock-focused R–J/STEP/ETAS.
• No precursor identification required — treating every event as a scaled precursor sidesteps the unanswerable "is this a foreshock?" question and yields a calibratable density.
• CSEP-verified skill — demonstrated, statistically significant medium-term forecasting skill in New Zealand, Japan, California, and Italy.
• Smooth and bounded — a rate-density surface, naturally combinable with other models in an ensemble or as a feature.

Limitations

• Medium-term only — months-to-years horizon; outside this product's 1-week core window.
• Modest skill — real but not large; meaningful only over long evaluation periods.
• Constant fragility — published constants have documented typos; must be pinned to a reference implementation.
• Region-specific — scaling constants do not transfer between tectonic settings without refitting.

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8. Role in operational earthquake forecasting

EEPAS occupies a distinct niche in the operational landscape:

• It is a medium-term complement to the short-term aftershock systems. Operational forecasting spans horizons — hours/days (aftershocks: R–J, STEP, ETAS) up to months/years (background and precursory: EEPAS, smoothed seismicity, renewal). EEPAS supplies the medium-term layer.
• It has been a standing participant in CSEP forecasting experiments across multiple regions, one of the few medium-term models with prospectively-tested skill — which is what distinguishes it from the discredited precursor schemes (AMR, LURR, RTL) discussed in Models-Classical §10. The difference is that EEPAS makes a smooth, fully-specified, testable rate density rather than an alarm, and it has been scored honestly and passed.
• In an ensemble, EEPAS's medium-term density can be blended with short-term clustering models and a smoothed-seismicity background, each model carrying the horizon it forecasts best.

EEPAS is the standing example, in this product's reference set, that a precursory idea can be made honest and operational — by refusing to identify precursors, building a smooth testable density, and submitting it to prospective CSEP scoring rather than declaring alarms.

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9. Worked illustration

Illustrative only — schematic numbers, not a forecast.

Suppose a single precursor event of magnitude $M_p = 4.5$ is observed, and (for illustration) the fitted scaling gives $a_M = 1.0$ (so $M_m = 1.0 + M_p$), $a_T = 0.5,\ b_T = 0.3$, and a region-scaled area constant. EEPAS then contributes:

1. Magnitude: a normal density centred at $M_m = 1.0 + 4.5 = 5.5$, smeared by $\sigma_M$ — i.e. this precursor most strongly elevates the probability of an $\approx M,5.5$ future event.
2. Time: a lognormal in elapsed time with median $10^{,0.5 + 0.3 \times 4.5} = 10^{1.85} \approx 71$ days — the contribution rises toward a peak around two-to-three months out, then tapers. (A larger precursor would push the peak years out.)
3. Location: a bivariate-normal centred on the precursor, with a footprint set by the area-scaling constant — wider for larger precursors.
4. Sum and baseline: this one bump is added to the contributions of every other recent event, plus the smoothed-seismicity baseline $\mu,\lambda_0$, producing the full medium-term rate density over the region.

No single bump is a forecast; the sum over the whole catalog is. The map that results is a smooth field of medium-term elevated probability, brightest where recent moderate events cluster.

Read it honestly. A region showing elevated medium-term rate is more likely, not due — the density is scored over many regions and years (see Evaluation-and-Tests), never read as a countdown for one place.

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10. How EEPAS is used in this product

In CAOS_SEISMIC, EEPAS is a medium-term feature / context source, explicitly not a short-term core forecaster (see Models-Employed and Methodology-History):

• The product's primary horizon is ~1 day to ~1 week, where ETAS, Reasenberg–Jones clustering, and the STEP-style gridded output dominate. EEPAS sits above that window.
• EEPAS's precursory-scale rate density (and the smoothed-seismicity baseline it builds on) can be supplied as a medium-term context feature that informs, but does not by itself set, the short-term number.
• If EEPAS is computed, its constants are pinned to a reference implementation (pyCSEP / floatCSEP) rather than transcribed from papers, per the documented-typo caveat, and refit to the operating region — never reused across tectonic settings.
• Any EEPAS contribution is CSEP-scored (Evaluation-and-Tests) before it can influence a public number, and the output is always a bounded, calibrated rate density — never an alarm. See Honest-Limits.

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References

1. Rhoades, D. A. & Evison, F. F. (2004). Long-range earthquake forecasting with every earthquake a precursor according to scale. Pure and Applied Geophysics 161(1), 47–72. doi:10.1007/s00024-003-2434-9
2. Rhoades, D. A. (2007). Application of the EEPAS model to forecasting earthquakes of moderate magnitude in Southern California. Seismological Research Letters 78(1), 110–115. doi:10.1785/gssrl.78.1.110
3. Rhoades, D. A. & Evison, F. F. (2006). The EEPAS forecasting model and the probability of moderate-to-large earthquakes in central Japan. Tectonophysics 417(1–2), 119–140. doi:10.1016/j.tecto.2005.05.051
4. Evison, F. F. & Rhoades, D. A. (2004). Demarcation and scaling of long-term seismogenesis. Pure and Applied Geophysics 161(1), 21–45. doi:10.1007/s00024-003-2433-x
5. Schorlemmer, D., Gerstenberger, M. C., Wiemer, S., Jackson, D. D. & Rhoades, D. A. (2007). Earthquake likelihood model testing. Seismological Research Letters 78(1), 17–29. doi:10.1785/gssrl.78.1.17
6. Rhoades, D. A., Schorlemmer, D., Gerstenberger, M. C., Christophersen, A., Zechar, J. D. & Imoto, M. (2011). Efficient testing of earthquake forecasting models. Acta Geophysica 59(4), 728–747. doi:10.2478/s11600-011-0013-5
7. Jordan, T. H., Chen, Y.-T., Gasparini, P., Madariaga, R., Main, I., Marzocchi, W., Papadopoulos, G., Sobolev, G., Yamaoka, K. & Zschau, J. (2011). Operational earthquake forecasting: state of knowledge and guidelines for utilization. Annals of Geophysics 54(4), 315–391. doi:10.4401/ag-5350

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Related pages: Models-Classical · Reasenberg-Jones-Model · STEP-Model · Models-Employed · Methodology-History · Evaluation-and-Tests · Honest-Limits · Glossary.