S-P Interval (Distance to Epicenter)

Core idea

Overview

This equation calculates the distance between a seismic station and an earthquake's origin point by analyzing the time delay between the arrival of primary (P) and secondary (S) waves. It relies on the distinct velocity differences of these waves as they travel through the Earth's crust to estimate the proximity of the seismic event.

When to use: Apply this formula when you have recorded arrival times for both P-waves and S-waves from a single seismic station. It assumes the waves travel through a medium with relatively constant or well-averaged velocities, making it ideal for regional crustal earthquake analysis.

Why it matters: Determining the distance to an epicenter is the critical first step in locating an earthquake. By finding the distance from at least three different stations, seismologists can triangulate the exact location of the epicenter to issue tsunami warnings and direct emergency response efforts.

Symbols

Variables

d = Distance, $v_{p}$ = P-Wave Speed, $v_{s}$ = S-Wave Speed, $Δ$ t = Time Interval

d

Distance

km

v_{p}

P-Wave Speed

km/s

v_{s}

S-Wave Speed

km/s

Δ t

Time Interval

s

Walkthrough

Derivation

Formula: S-P Interval — Distance to Earthquake Epicentre

P-waves (primary) and S-waves (secondary) travel at different speeds. The time gap between their arrivals at a seismometer grows with distance, allowing the epicentre distance to be calculated.

Both wave speeds are approximately constant through the crust.
The S-P time interval Δt is measured from the seismogram.

1

Write travel-time equations:

P-waves arrive first because $v_{p}$ > $v_{s}$ . Each travel time is distance divided by wave speed.

t_{P} = \frac{d}{v _{p}}, t_{S} = \frac{d}{v _{s}}

2

Form the S-P time difference:

Subtract the P-wave arrival time from the S-wave arrival time.

Δ t = t_{S} - t_{P} = \frac{d}{v _{s}} - \frac{d}{v _{p}} = d (\frac{1}{v _{s}} - \frac{1}{v _{p}})

3

Solve for distance:

Rearrange to isolate d. The larger the S-P gap, the farther the epicentre.

d = \frac{Δ t}{\frac{1}{v _{s}} - \frac{1}{v _{p}}} = \frac{v _{p} v _{s}}{v _{p} - v _{s}} Δ t

Note: Three seismometer readings are needed to triangulate the exact epicentre location.

Result

d = \frac{Δ t}{\frac{1}{v _{s}} - \frac{1}{v _{p}}} = \frac{v _{p} v _{s}}{v _{p} - v _{s}} Δ t

Source: GCSE Geology — Seismology

Free formulas

Rearrangements

Solve for $Δ t$

Make Delta t the subject

Δ t = d \frac{v _{p} - v _{s}}{v _{p} v _{s}}

To make $Δ$ t the subject, multiply both sides by the reciprocal of the coefficient of $Δ$ t.

Difficulty: 2/5

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

Visual intuition

Graph

The graph is a straight line passing through the origin, representing a direct proportionality between distance and the time interval. For a geology student, this means a small time interval indicates the epicenter is nearby, while a large time interval signifies the earthquake occurred at a much greater distance. The most important feature is that the linear relationship means doubling the time interval always results in a doubling of the distance to the epicenter.

Graph type: linear

Why it behaves this way

Intuition

Imagine two runners, one fast (P-wave) and one slower (S-wave), starting simultaneously from the earthquake epicenter. The seismic station is the finish line.

d

Distance from the seismic station to the earthquake epicenter

This is the unknown distance we are trying to determine. It represents how far away the earthquake's origin point is from the recording station.

V_{p}

Average velocity of primary (P) waves through the Earth's crust

P-waves are the fastest seismic waves, traveling as compressions and dilations. They are the first to arrive at a seismic station, and their speed depends on the material's properties.

V_{s}

Average velocity of secondary (S) waves through the Earth's crust

S-waves are slower than P-waves, traveling as shear waves. They arrive after P-waves and cannot travel through liquids. Their speed also depends on the material's properties.

Δ t

The time difference between the arrival of the P-wave and the S-wave at a seismic station

This is the directly measured observable. A longer time interval between the P-wave and S-wave arrival indicates that the waves have traveled a greater distance, allowing the faster P-wave to pull further ahead of the S-wave.

Signs and relationships

V_p - V_s: This term represents the difference in velocities between P-waves and S-waves. It appears in the denominator because a larger difference means the P-wave gains a lead over the S-wave more quickly.

Free study cues

Insight

Canonical usage

This equation requires consistent units for velocity (e.g., km/s or m/s) and time (e.g., s) to yield a distance in the corresponding unit (e.g., km or m).

Common confusion

A common mistake is mixing length units (e.g., using velocities in m/s but expecting distance in km, or vice versa) or time units (e.g., using minutes for time difference when velocities are in seconds).

Unit systems

$d$ km or m - The resulting distance unit (e.g., km or m) will match the length unit used in the velocity terms.

$V_{p}$ km/s or m/s - P-wave velocity. Must be in consistent units with V_s and \Delta t.

$V_{s}$ km/s or m/s - S-wave velocity. Must be in consistent units with V_p and \Delta t.

$Δ t$ s - The time difference between S-wave and P-wave arrival. Must be in units consistent with the time component of the velocity units (e.g., seconds when velocities are in km/s or m/s).

One free problem

Practice Problem

A seismic station records a P-wave arrival at 08:00:10 and an S-wave arrival at 08:00:34. If the P-wave velocity is 8.0 km/s and the S-wave velocity is 4.5 km/s, calculate the distance from the station to the earthquake epicenter.

P-Wave Speed8 km/s

S-Wave Speed4.5 km/s

Time Interval24 s

Solve for: $d$

Hint: Subtract the P-wave arrival time from the S-wave arrival time to find the interval, then use the product-over-difference velocity formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In P-wave arrives at 12:00:00, S-wave at 12:00:10, S-P Interval (Distance to Epicenter) is used to calculate Distance from P-Wave Speed, S-Wave Speed, and Time Interval. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Ensure Vp is always larger than Vs, as P-waves are faster and arrive first.
Time units for deltaT must match the time denominator in velocity units (usually seconds).
The term (Vp ×Vs) / (Vp - Vs) is often treated as a single coefficient known as the Omori constant.

Avoid these traps

Common Mistakes

Using the wrong wave speeds for the local rock type.
Convert units and scales before substituting, especially when the inputs mix km, km/s, s.
Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

P-waves (primary) and S-waves (secondary) travel at different speeds. The time gap between their arrivals at a seismometer grows with distance, allowing the epicentre distance to be calculated.

Apply this formula when you have recorded arrival times for both P-waves and S-waves from a single seismic station. It assumes the waves travel through a medium with relatively constant or well-averaged velocities, making it ideal for regional crustal earthquake analysis.

Determining the distance to an epicenter is the critical first step in locating an earthquake. By finding the distance from at least three different stations, seismologists can triangulate the exact location of the epicenter to issue tsunami warnings and direct emergency response efforts.

Using the wrong wave speeds for the local rock type. Convert units and scales before substituting, especially when the inputs mix km, km/s, s. Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

In P-wave arrives at 12:00:00, S-wave at 12:00:10, S-P Interval (Distance to Epicenter) is used to calculate Distance from P-Wave Speed, S-Wave Speed, and Time Interval. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Ensure Vp is always larger than Vs, as P-waves are faster and arrive first. Time units for deltaT must match the time denominator in velocity units (usually seconds). The term (Vp ×Vs) / (Vp - Vs) is often treated as a single coefficient known as the Omori constant.

References

Sources

Wikipedia: P-wave
Wikipedia: S-wave
Wikipedia: Epicenter
Britannica: Seismology
USGS Earthquake Glossary
British Geological Survey (BGS) educational resources
An Introduction to Seismology, Earthquakes, and Earth Structure by Stein and Wysession, 3rd Edition
Fundamentals of Geophysics by Lowrie, 3rd Edition

Overview

Variables

Derivation

Write travel-time equations:

Form the S-P time difference:

Solve for distance:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Richter Magnitude

Frequently Asked Questions

Sources