Empirical aftershock-decay relation
Omori-Utsu Law
Aftershock occurrence rates commonly decline approximately as an inverse power of elapsed time after a mainshock.
n(t) = K / (t + c)^p
n(t) is the expected event rate at elapsed time t; K scales productivity, c regularizes the earliest interval, and p controls decay. Parameters depend on the sequence, catalog, magnitude threshold, region, and fitting method.
The decay laboratory uses K = 1,000, c = 0.5 day, and adjustable p. It describes an expected rate over a population of sequences, not the time or magnitude of the next earthquake.
(events/day)
Dots are one simulated catalog around the expected rate, not scheduled events. The log-log inset tests whether the tail follows an approximately straight power-law relation.
- CHANGE
- Days since the mainshock
- WATCH
- aftershock rate
- MEANING
- The decay laboratory uses K = 1,000, c = 0.5 day, and adjustable p. It describes an expected rate over a population of sequences, not the time or magnitude of the next earthquake.
A dense beginning relaxes into a long tail.
Events arrive rapidly just after the mainshock, then become progressively more separated. Log axes reveal why the sequence is a power-law decay rather than an exponential clock.
What it actually says
The modified Omori law models the average temporal rate of aftershocks. It does not say aftershocks occur at regular intervals: individual events remain stochastic around a rate that generally falls with elapsed time.
The exponent p is often near one, but it is not universal. K, c, and p interact with magnitude completeness, spatial windows, secondary triggering, background seismicity, and the chosen start time. Operational forecasts therefore combine Omori-Utsu decay with magnitude and spatial models.
"A useful law compresses a pattern. It does not erase the conditions that make the pattern true."
How the idea developed
The modern form emerged through observation, argument, and later refinement. The timeline separates the first insight from the version now used in textbooks and practice.[1]
Fusakichi Omori describes an inverse-time decline in aftershock frequency.
Tokuji Utsu introduces the modified form with a general decay exponent p.
The epidemic-type aftershock sequence model combines Omori-Utsu triggering across generations.
Operational earthquake forecasting fits sequence-specific rate, magnitude, and spatial components with explicit uncertainty.
How the pattern works
The relation becomes useful only when its mechanism, measurement process, and operating range are visible.
A mainshock changes stress and failure conditions on nearby faults.
Aftershocks can trigger their own descendants, producing overlapping generations.
The population of highly susceptible fault patches diminishes and stress-driven rates relax.
Immediately after large events, overlapping waveforms hide small shocks and bias c and early rates.
n(t) is the expected event rate at elapsed time t; K scales productivity, c regularizes the earliest interval, and p controls decay. Parameters depend on the sequence, catalog, magnitude threshold, region, and fitting method.
Where it earns its keep
Applications are strongest when the law changes a decision, measurement, model, or experiment rather than merely providing an analogy.
Update short-term earthquake forecasts
ApplicationSequence-specific decay helps estimate how activity changes over hours, days, and months.
Communicate probability ranges and magnitude thresholds, not deterministic countdowns.
Plan instrument and inspection deployment
ApplicationExpected rate decline informs when dense observations and safety precautions are most valuable.
Large aftershocks remain possible even as the average rate falls.
Compare triggering hypotheses
ApplicationFitted p, c, productivity, and residual structure constrain statistical and physical models.
Catalog completeness and spatial selection must be tested before interpretation.
Where it stops working
A single Omori-Utsu curve may be inadequate when secondary mainshocks, swarms, changing completeness, spatial migration, or time-varying background rates are present.
The law governs occurrence rate, not magnitude. Magnitude-frequency behavior is modeled separately, commonly with Gutenberg-Richter, and forecast uncertainty remains substantial.
"The rate reaches zero after a fixed date"
Better: Power-law tails decline gradually and have no universal cutoff."p is always exactly one"
Better: It is fitted and varies across catalogs, windows, and sequences."A lower rate means no large aftershock"
Better: Rate and magnitude distribution are distinct components."c is purely a fault property"
Better: Early catalog incompleteness can strongly influence its estimate.Sources and further reading
Original publications and serious secondary scholarship are prioritized over summaries.
- USGS - Onset of aftershocks: Constraints on the Rate-and-State modelModern USGS study stating and testing the Omori-Utsu rate form.https://www.usgs.gov/publications/onset-aftershocks-constraints-rate-and-state-model
- Utsu, Ogata, and Matsuura - The Centenary of the Omori FormulaHistorical and technical review of the formula and its extensions.https://doi.org/10.4294/jpe1952.43.1
- Ogata - Statistical Models for Earthquake OccurrencesFoundational treatment of epidemic-type aftershock sequence modeling.https://doi.org/10.1029/GM037p0013
- USGS - Aftershocks of the 2018 M6.9 Hawaii earthquakeOperational explanation of aftershock decay and long sequence duration.https://www.usgs.gov/observatories/hvo/news/volcano-watch-aftershocks-2018-magnitude-69-earthquake-expected-continue