Omori Utsu Law

Omori–Utsu Law — modified Omori aftershock decay

The dominant short-horizon signal. After a large earthquake, the rate of aftershocks decays as a power law in time — fast at first, then ever more slowly, often detectable for months or years. The modified Omori (Omori–Utsu) law is the quantitative form of that decay, and it is the baseline for "what happens tomorrow" after a felt earthquake. It is the temporal kernel embedded in both the Reasenberg–Jones operational model and the self-exciting ETAS-Model.

Honest framing. The Omori–Utsu law describes the average decay of a population of aftershocks; it is not a prediction of any individual event. It produces a conditional rate, which — combined with the Gutenberg–Richter magnitude term — yields a bounded, calibrated probability, never an alarm. A larger aftershock (even one exceeding the mainshock) is always possible during a sequence; the law quantifies how its probability decays, it does not forbid it.

Conventions. $t$ is time elapsed since the mainshock; magnitudes are homogenized to $M_w$; $M_c$ is the magnitude of completeness; $b$ is the Gutenberg–Richter slope.

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Table of contents

1. Intuition and history
2. Governing equation
3. The parameters $K$, $c$, $p$
4. Cumulative form (derivation)
5. Parameter estimation by maximum likelihood
6. Short-term incompleteness — the c-value trap
7. From decay rate to a forecast probability
8. Assumptions and failure modes
9. Role in operational earthquake forecasting
10. Worked illustration
11. Decay and incompleteness (diagram)
12. References

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1. Intuition and history

In 1894 Fusakichi Omori, studying aftershocks of the 1891 Nōbi earthquake in Japan, found that the frequency of aftershocks per day fell off roughly as the reciprocal of the time since the mainshock: twice as long after the mainshock, half as many aftershocks per day. His original law was

$$n(t) = \frac{K}{t + c} \qquad (\text{Omori, 1894}),$$

with $c$ a small constant that keeps the rate finite at $t = 0$.

Six decades later Tokuji Utsu (1957, 1961) showed that the decay is generally steeper or shallower than the exact $1/t$ of Omori, and generalized the exponent to an adjustable $p$:

$$n(t) = \frac{K}{(t + c)^{p}} \qquad (\text{modified Omori / Omori–Utsu}).$$

The intuition is one of relaxation: the mainshock loads its neighborhood with stress; that stress is shed by a cascade of smaller ruptures, and the rate of shedding decays as the most critically loaded patches fail first and the population of near-failure patches is depleted. Crucially, the same power-law decay falls out from first principles of rate-and-state friction (Dieterich 1994), which is the mechanistic bridge between Coulomb stress transfer and the empirical Omori law (see Models-Classical §9). The Omori–Utsu law is thus both an empirical regularity and a prediction of fault mechanics — one of the reasons it is so reliable a short-horizon signal.

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2. Governing equation

The modified Omori (Omori–Utsu) law gives the rate of aftershocks at time $t$ after a mainshock:

$$\boxed{,n(t) = \frac{K}{(t + c)^{,p}},}$$

• $n(t)$ — the rate of aftershocks (events per unit time, above a completeness $M_c$) at elapsed time $t$.
• $K$ — the productivity constant; it scales the total number of aftershocks.
• $c$ — a small time offset (hours to a day) that keeps the rate finite at $t \to 0$.
• $p$ — the decay exponent, typically $p \in [0.9, 1.5]$, varying sequence to sequence.

Omori's 1894 law is the special case $p = 1$, $n(t) = K/(t+c)$; Utsu's generalization frees $p$.

Reference. Utsu, T., Ogata, Y. & Matsu'ura, R. S. (1995), The centenary of the Omori formula for a decay law of aftershock activity, J. Phys. Earth 43(1), 1–33, doi:10.4294/jpe1952.43.1.

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3. The parameters $K$, $c$, $p$

$K$ — productivity. $K$ sets the amplitude of the sequence: how many aftershocks there are overall. It scales with the mainshock magnitude (larger mainshocks produce far more aftershocks), which in the ETAS-Model is made explicit through the Utsu productivity law $k(m) = K,e^{\alpha(m - M_0)}$.

$p$ — decay exponent. $p$ sets the shape of the decay. $p = 1$ is the classic Omori $1/t$; $p > 1$ decays faster (the sequence quiets sooner); $p < 1$ decays more slowly (a long-lived sequence). $p$ varies with the tectonic setting and, physically, with crustal temperature and heterogeneity. It is not a universal constant and must be fit per sequence.

$c$ — time offset. $c$ is the most subtle and most contested parameter. Mathematically it regularizes the rate at $t = 0$. Physically, whether $c$ is a real, finite time-scale of the nucleation process or merely an artifact of short-term catalog incompleteness has been debated for decades — because immediately after the mainshock the smallest aftershocks are unrecorded, flattening the apparent early rate and inflating the fitted $c$ (see §6). A robust pipeline treats a large fitted $c$ with suspicion and checks it against the completeness time.

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4. Cumulative form (derivation)

For a forecast over a finite window we need the expected number of aftershocks between times $t_1$ and $t_2$ — the integral of the rate:

$$N(t_1, t_2) = \int_{t_1}^{t_2} \frac{K}{(t + c)^{,p}}, \mathrm{d}t .$$

For $p \neq 1$, integrate the power directly:

$$\int (t+c)^{-p},\mathrm{d}t = \frac{(t+c)^{1-p}}{1-p},$$

so

$$\boxed{,N(t_1, t_2) = \frac{K}{1 - p}\Big[(t_2 + c)^{1-p} - (t_1 + c)^{1-p}\Big] \quad (p \neq 1),}$$

For the special case $p = 1$ the integral of $1/(t+c)$ is a logarithm:

$$N(t_1, t_2) = K\big[\ln(t_2 + c) - \ln(t_1 + c)\big] \quad (p = 1).$$

A key qualitative consequence: for $p \le 1$ the cumulative count diverges as $t_2 \to \infty$ (the sequence never formally ends), whereas for $p > 1$ it converges to a finite total. This is the temporal analogue of the branching-ratio convergence condition in the ETAS-Model.

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5. Parameter estimation by maximum likelihood

The aftershock times form a non-homogeneous Poisson point process with intensity $n(t)$. The parameters $(K, c, p)$ are estimated by maximizing the point-process log-likelihood (Ogata 1983). For events at times $t_1 < t_2 < \dots < t_N$ observed over $[S, T]$,

$$\ln L(K, c, p) = \sum_{i=1}^{N} \ln n(t_i) - \int_{S}^{T} n(t),\mathrm{d}t ,$$

where the first term rewards placing high rate where events actually occurred and the second ("compensator") term penalizes total predicted rate — it is what makes the fit calibrated rather than a curve-fit. Substituting $n(t) = K/(t+c)^p$ and the cumulative form from §4 gives a three-parameter optimization solved numerically (Newton/quasi-Newton). Standard tooling: the R ETAS family, bayesianETAS, and pyetas.

Why MLE, not log–log least squares. Binning the rate and fitting a line in log–log space is biased: the bins are correlated, empty late-time bins are mishandled, and the choice of bin width changes the slope. The point-process MLE uses the event times directly and is the standard.

Reference. Ogata, Y. (1983), Estimation of the parameters in the modified Omori formula for aftershock frequency by the maximum likelihood procedure, J. Phys. Earth 31(2), 115–124, doi:10.4294/jpe1952.31.115.

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6. Short-term incompleteness — the c-value trap

This is the most important operational caveat, and it bites exactly when the forecast matters most.

The problem. In the minutes-to-days right after a large mainshock, the seismic network is saturated: many small aftershocks are buried in the coda of the mainshock and of each other, so the catalog's completeness magnitude $M_c$ spikes for hours to days. The observed early rate is therefore artificially flat — not because aftershocks are scarce, but because the small ones are not recorded.

The consequences.

• Fitting a flat $M_c$ to this incomplete early window inflates $c$ (the apparent rate is flat at the start) and depresses $p$ (the apparent decay looks shallower), biasing the whole fit.
• A naive fit then underestimates productivity $K$ — i.e. it under-forecasts aftershocks at the highest-stakes, highest-traffic moment, when the public most needs the number.

The fix. Either start the fit after the time at which completeness recovers, or — the method the product uses — adopt an incompleteness-aware likelihood with a time-dependent completeness $M_c(t)$ that rises right after the mainshock and relaxes back. This corrects the productivity and the early decay rather than discarding the early data (see Gutenberg-Richter-Law §$M_c$ and Models-Employed). This is a decided method, not an open question: under-forecasting at this moment is both a credibility failure and an indirect safety-communication failure.

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7. From decay rate to a forecast probability

The Omori–Utsu law gives a rate of events $\ge M_c$. To publish a probability of an event above a target magnitude $M^\ast$ over a horizon $[t_1, t_2]$, combine it with the Gutenberg–Richter exceedance factor and the Poisson assumption.

Expected number of events $\ge M^\ast$ in the window:

$$N_{\ge M^\ast}(t_1, t_2) = \underbrace{\frac{K}{1-p}\Big[(t_2+c)^{1-p} - (t_1+c)^{1-p}\Big]}_{\text{Omori–Utsu count} \ge M_c} \cdot \underbrace{10^{-b(M^\ast - M_c)}}_{\text{GR exceedance } \Phi(M^\ast)} .$$

Under a non-homogeneous Poisson process the probability of one or more such events is

$$\boxed{,P(\ge 1\ \text{event} \ge M^\ast) = 1 - e^{-N_{\ge M^\ast}(t_1, t_2)},}$$

This is precisely the Reasenberg–Jones construction (Gutenberg–Richter magnitude term × modified-Omori time decay) and the public-facing formula used throughout the product (see Models-Employed). Only the quality of the rate changes as the model improves; the $1 - e^{-N}$ wrapper is fixed.

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8. Assumptions and failure modes

• Single mainshock. The plain Omori–Utsu law describes decay relative to one triggering event. Real sequences have secondary aftershocks (aftershocks of aftershocks), bursts, and multiple mainshocks, which a single power law cannot capture — this is exactly the gap the self-exciting ETAS-Model fills by summing an Omori–Utsu kernel over every past event.
• Stationary completeness. A flat $M_c$ is violated by short-term incompleteness (§6); use $M_c(t)$.
• $c$ is fragile. Treat a large fitted $c$ as a likely incompleteness artifact until checked.
• No spatial term. The law gives "how many over time," not "where." Spatial structure requires the ETAS spatial kernel or smoothed-seismicity background.
• Power-law tail. For $p \le 1$ the sequence formally never ends (count diverges); long-horizon extrapolation must be reported honestly with this in mind.

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9. Role in operational earthquake forecasting

The Omori–Utsu law is the single most reliable short-horizon (hours-to-weeks) signal in statistical seismology, and it appears in the product in three nested ways:

1. Directly, as the temporal decay in the transparent Reasenberg–Jones aftershock model — the operational ancestor of USGS aftershock forecasting and a transparent sanity-check baseline.
2. As a kernel inside the ETAS-Model, where the temporal trigger of every past event is an Omori–Utsu term $g(t - t_i) = (p-1)/c,\big(1 + (t-t_i)/c\big)^{-p}$, summed over the whole catalog — turning the single-mainshock law into a full self-exciting process.
3. As a mechanistic prediction of rate-and-state friction (Dieterich 1994): a Coulomb stress step produces an Omori-like $1/t$ decay of triggered seismicity from first principles, linking the empirical law to fault physics (see Models-Classical §9).

Because the short horizons (1–7 days) the product forecasts are dominated by aftershock decay, getting Omori–Utsu right — especially its post-mainshock incompleteness handling — is where most of the realizable short-term forecasting skill lives.

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10. Worked illustration

A felt $M_w,6.4$ mainshock occurs. A regional/generic fit gives productivity $K = 30$ events/day (above $M_c = 2.5$, in the normalization $n(t) = K/(t+c)^p$ with $t$ in days), offset $c = 0.05$ d (≈ 1.2 h), and decay $p = 1.1$. We want the probability of at least one $M \ge 5.0$ aftershock during the first 7 days, with $b = 0.95$.

Step 1 — expected count $\ge M_c$ over $[0, 7]$ days (using $p \neq 1$ cumulative form):

$$N(0,7) = \frac{K}{1-p}\big[(7+c)^{1-p} - (0+c)^{1-p}\big] = \frac{30}{-0.1}\big[(7.05)^{-0.1} - (0.05)^{-0.1}\big] .$$

Numerically $(7.05)^{-0.1} = e^{-0.1\ln 7.05} \approx e^{-0.1955} \approx 0.8224$ and $(0.05)^{-0.1} = e^{-0.1\ln 0.05} \approx e^{+0.2996} \approx 1.3493$, so

$$N(0,7) = -300 ,(0.8224 - 1.3493) = -300,(-0.5269) \approx 158\ \text{events} \ge M_c .$$

Step 2 — apply the GR exceedance factor to magnitude $5.0$:

$$\Phi(5.0) = 10^{-b(M^\ast - M_c)} = 10^{-0.95(5.0 - 2.5)} = 10^{-2.375} \approx 4.22\times10^{-3}.$$

So the expected number of $M \ge 5$ aftershocks in the first week is $N_{\ge 5} = 158 \times 4.22\times10^{-3} \approx 0.667$.

Step 3 — probability of at least one:

$$P(\ge 1,\ M\ge 5,\ \text{week 1}) = 1 - e^{-0.667} \approx 49% .$$

So after a substantial mainshock there is roughly a coin-flip chance of an $M\ge 5$ aftershock in the first week — large, and correctly so. Repeating the calculation for day 1 alone ($N(0,1)$) gives a much higher daily rate at the start of the sequence that then decays; this state-dependence (large during an active sequence, near-baseline when quiet) is exactly what the forecast must convey. Note the strong sensitivity to incompleteness: if the early catalog were under-counted and $K$ under-fit by, say, 40 %, this 49 % would collapse toward ~35 % — under-warning at the worst moment, the reason §6 matters.

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11. Decay and incompleteness (diagram)

flowchart TD
    M[Mainshock at t = 0] --> R[Aftershock rate<br/>n of t = K / t + c ^ p]
    R --> EARLY[Early window<br/>hours to days]
    R --> LATE[Late window<br/>days to months]
    EARLY --> INC{Network saturated?<br/>small events buried}
    INC -- yes --> BIAS[Mc spikes -> flat apparent rate<br/>inflates c, depresses p, under-fits K]
    BIAS --> FIX[Incompleteness-aware likelihood<br/>time-dependent Mc of t]
    INC -- no --> FIT[Stable Omori–Utsu fit]
    FIX --> FIT
    LATE --> FIT
    FIT --> N[Expected count over horizon<br/>N = K/1-p of t2+c^1-p - t1+c^1-p]
    N --> GR[x GR exceedance<br/>Phi of M* = 10^- b M* - Mc]
    GR --> P[Probability<br/>P = 1 - exp - N x Phi]

Loading

The decay rate $n(t)$, integrated over the horizon and multiplied by the Gutenberg–Richter exceedance factor, becomes the published probability; the same kernel, summed over every event, is the temporal heart of the ETAS-Model.

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References

1. Omori, F. (1894), On the aftershocks of earthquakes, J. Coll. Sci. Imp. Univ. Tokyo 7, 111–200.
2. Utsu, T. (1961), A statistical study on the occurrence of aftershocks, Geophys. Mag. 30, 521–605.
3. Utsu, T., Ogata, Y. & Matsu'ura, R. S. (1995), The centenary of the Omori formula for a decay law of aftershock activity, J. Phys. Earth 43(1), 1–33, doi:10.4294/jpe1952.43.1.
4. Ogata, Y. (1983), Estimation of the parameters in the modified Omori formula for aftershock frequency by the maximum likelihood procedure, J. Phys. Earth 31(2), 115–124, doi:10.4294/jpe1952.31.115.
5. Reasenberg, P. A. & Jones, L. M. (1989), Earthquake hazard after a mainshock in California, Science 243(4895), 1173–1176, doi:10.1126/science.243.4895.1173.
6. Dieterich, J. (1994), A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res. 99(B2), 2601–2618, doi:10.1029/93JB02581.
7. Jordan, T. H. et al. (2011), Operational earthquake forecasting: state of knowledge and guidelines for utilization (ICEF Report), Annals of Geophysics 54(4), 315–391, doi:10.4401/ag-5350.

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See also: Gutenberg-Richter-Law · ETAS-Model · Models-Classical · Models-Employed · Evaluation-and-Tests · Glossary