Omori's Law Equation, Derivation & Rearrangements

Core idea

Overview

Omori's Law is an empirical formula that describes the temporal decay of aftershock frequency following a major earthquake. It establishes that the rate of aftershocks decreases roughly with the reciprocal of time elapsed since the mainshock.

When to use: Apply this equation when modeling the expected frequency of aftershocks in a seismic sequence over time. It is most effective in the days and weeks following a mainshock, assuming the geological setting remains relatively consistent without new major ruptures.

Why it matters: Predicting aftershock decay is vital for public safety, as it allows engineers and emergency responders to estimate the window of high risk for structural collapse. It also provides a baseline for seismologists to detect anomalies, such as a potential second large earthquake disguised as an aftershock.

Symbols

Variables

n(t) = Aftershock frequency, K = Productivity constant, c = Time offset constant, t = Time since mainshock

n(t)

Aftershock frequency

events/day

K

Productivity constant

Variable

c

Time offset constant

days

t

Time since mainshock

days

Walkthrough

Derivation

Understanding Omori's Law

Describes the hyperbolic decay of aftershock frequency with time after a mainshock.

The aftershock sequence follows a simple power-law decay.
The mainshock time is known precisely.

1

State the modified Omori law:

The rate of aftershocks n at time t after the mainshock decays hyperbolically. K is a productivity constant, c a small time offset, and p ≈ 1.

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

2

Simplified form (p = 1):

With p = 1 (the original Omori law), aftershock rate is inversely proportional to time.

n (t) = \frac{K}{c + t}

Note: This is one of the oldest empirical laws in seismology (1894). It is used in earthquake forecasting to estimate how long aftershock hazard persists.

Result

n (t) = \frac{K}{c + t}

Source: University Seismology — Aftershock Statistics

Free formulas

Rearrangements

Solve for n(t)

Make n the subject

n (t) = \frac{K}{c + t}

Exact symbolic rearrangement generated deterministically for n.

Difficulty: 3/5

Solve for $K$

Make K the subject

K = n (t) (c + t)

Exact symbolic rearrangement generated deterministically for K.

Difficulty: 2/5

Solve for $c$

Make c the subject

c = - t + \frac{K}{n ( t )}

Exact symbolic rearrangement generated deterministically for c.

Difficulty: 3/5

Solve for $t$

Make t the subject

t = - c + \frac{K}{n ( t )}

Exact symbolic rearrangement generated deterministically for t.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph displays a hyperbolic curve where aftershock frequency (n) is plotted against time (t). As time increases, the frequency decreases rapidly before leveling off toward a horizontal asymptote at zero, reflecting the inverse relationship between the two variables.

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine a rapidly decaying curve, starting high and quickly dropping, representing the diminishing rate of seismic tremors as the Earth's crust gradually settles after a major rupture.

n(t)

The instantaneous frequency (rate) of aftershocks per unit time.

This value tells us how many aftershocks we expect to observe in a short period at a specific time 't' after the mainshock.

t

Time elapsed since the mainshock.

As this value increases, the aftershock activity generally decreases because the crust is slowly returning to a more stable state.

K

A constant reflecting the overall productivity or intensity of the aftershock sequence.

A larger 'K' means the mainshock generated a more vigorous aftershock sequence, leading to a higher frequency of aftershocks at any given time.

c

A constant often called the 'offset time' or 'delay constant.'

This constant ensures the aftershock frequency remains finite and realistic immediately after the mainshock, preventing the formula from predicting an infinite rate at t=0.

Signs and relationships

1/(c+t): The inverse relationship with (c+t) means that as time 't' increases, the denominator grows, causing the overall aftershock frequency n(t) to decrease.

Free study cues

Insight

Canonical usage

Units for time (t and c) must be consistent, and n(t) will be in units of count per that time unit, with K in units of count.

Common confusion

The most common mistake is using inconsistent time units for 't' and 'c'. For example, if 't' is in days, 'c' must also be in days.

Unit systems

n(t)count/time · Represents the number of aftershocks occurring per unit of time (e.g., aftershocks/day).

$t$ time · Time elapsed since the mainshock (e.g., seconds, hours, days).

$K$ count · An empirical constant representing the total productivity of the aftershock sequence. Its unit is typically 'aftershocks' or 'events'.

$c$ time · An empirical constant, often interpreted as an initial offset time or a measure of the duration of the initial high aftershock rate. Its unit must match that of 't'.

One free problem

Practice Problem

After a magnitude 7.2 earthquake, a seismologist determines the productivity constant K is 150 and the time offset c is 0.5 days. Calculate the expected frequency of aftershocks exactly 2.5 days after the mainshock.

Productivity constant150

Time offset constant0.5 days

Time since mainshock2.5 days

Solve for: $n$

Hint: Add the time offset to the elapsed time before dividing the productivity constant by the result.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Following a magnitude 7.0 earthquake, a seismologist uses Omori's Law to estimate how many detectable aftershocks will occur on the third day compared to the first day.

Study smarter

Tips

The constant c is a small value that accounts for the delay in detecting shocks immediately after the main event.
The value of K represents the overall productivity or amplitude of the aftershock sequence.
Always ensure that the units for time (t) and frequency (n) are consistent, such as days and shocks per day.

Avoid these traps

Common Mistakes

Confusing the rate of aftershocks (n) with the magnitude of the aftershocks.
Ignoring the 'c' constant when calculating values close to t = 0.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Describes the hyperbolic decay of aftershock frequency with time after a mainshock.

Apply this equation when modeling the expected frequency of aftershocks in a seismic sequence over time. It is most effective in the days and weeks following a mainshock, assuming the geological setting remains relatively consistent without new major ruptures.

Predicting aftershock decay is vital for public safety, as it allows engineers and emergency responders to estimate the window of high risk for structural collapse. It also provides a baseline for seismologists to detect anomalies, such as a potential second large earthquake disguised as an aftershock.

Confusing the rate of aftershocks (n) with the magnitude of the aftershocks. Ignoring the 'c' constant when calculating values close to t = 0.

Following a magnitude 7.0 earthquake, a seismologist uses Omori's Law to estimate how many detectable aftershocks will occur on the third day compared to the first day.

The constant c is a small value that accounts for the delay in detecting shocks immediately after the main event. The value of K represents the overall productivity or amplitude of the aftershock sequence. Always ensure that the units for time (t) and frequency (n) are consistent, such as days and shocks per day.

References

Sources

Wikipedia: Omori's Law
Britannica: Omori's Law
Omori, F. (1894). On the after-shocks of earthquakes. Journal of the College of Science, Imperial University of Tokyo, 7, 111-200.
An Introduction to Seismology, Earthquakes, and Earth Structure (Stein & Wysession)
Stein, S., & Wysession, M. (2003). An Introduction to Seismology, Earthquakes, and Earth Structure (2nd ed.). Blackwell Publishing.
University Seismology — Aftershock Statistics

Omori's Law

Overview

Variables

Derivation

State the modified Omori law:

Simplified form (p = 1):

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Gutenberg-Richter Law

Frequently Asked Questions

Sources