Hong Life CalculatorsUnderstanding the earthquake distance from S−P time — concept and principle
When an earthquake occurs, a seismograph first records the faster P wave and then the slower S wave. The gap between their arrivals, the S−P time, grows with distance, so it tells you how far away the earthquake was. This method is taught in every introductory geology course and is the basis for locating an epicentre by triangulation from three stations.
How it is calculated
d = Δt ÷ (1/vs − 1/vp)
- Δt is the S−P time in seconds, read from the seismogram.
- vp is the P-wave speed in km/s (about 6 km/s in the upper crust).
- vs is the S-wave speed in km/s (about 3.5 km/s).
- d is the distance from the station to the earthquake source in km.
The term 1/vs − 1/vp is the extra time the S wave needs per kilometre, so dividing the delay by it gives the distance.
The principle
Both waves leave the source at the same moment. Over a distance d, the P wave takes d/vp seconds and the S wave d/vs seconds. Their difference is Δt = d/vs − d/vp = d(1/vs − 1/vp), which rearranges to the formula above. With vp = 6 and vs = 3.5 km/s, 1/(1/3.5 − 1/6) = 8.4, so each second of S−P delay is about 8.4 km of distance. P waves are faster because they are compressional and depend on both the bulk and shear stiffness of rock, while S waves are shear waves and depend only on shear stiffness.
Worked example
- Inputs: Δt = 20 s, vp = 6.0 km/s, vs = 3.5 km/s.
- 1/vs = 0.285714 s/km and 1/vp = 0.166667 s/km.
- Difference: 0.285714 − 0.166667 = 0.119048 s/km.
- d = 20 ÷ 0.119048 = 168 km.
| S-wave speed (km/s) | Distance (km) |
|---|---|
| 0.875 | 20.48780488 |
| 1.75 | 49.41176471 |
| 2.62 | 93.33333333 |
| 3.5 | 168 |
| 4.38 | 323.0769231 |
| 5.25 | 840 |
| 7 | -840 |
How to read the result
The distance is the radius of a circle around the station on which the source lies. With three stations you draw three circles; where they intersect is the epicentre. A short S−P time of a few seconds means a nearby quake, often felt as a sharp jolt; a long one means a distant event. Note that the result is the straight-line distance to the hypocentre, which for a shallow quake is close to the epicentral distance.
Common mistakes
- Swapping vp and vs, which gives a negative or meaningless distance.
- Using crustal speeds for very distant earthquakes, where waves travel through the faster mantle; then travel-time tables are needed.
- Reading arrivals on a noisy record and picking the S wave too early or late; one second of error is about 8 km.
Frequently asked questions
Why does the S wave not travel through liquid?
Liquids have no shear stiffness, so shear waves cannot propagate. This is how the liquid outer core was discovered.
How do earthquake early warning systems use this?
They detect the P wave near the source and send a warning that can arrive before the stronger S waves reach more distant areas.
Is 8 km per second a good rule of thumb?
Many textbooks use about 8 km per second of S−P time for shallow local earthquakes; the exact factor depends on the speeds you assume.
Sources: S. Stein and M. Wysession, An Introduction to Seismology, Earthquakes, and Earth Structure, Blackwell, 2003 · P. M. Shearer, Introduction to Seismology, 2nd ed., Cambridge University Press, 2009