Earthquake Epicenter Distance Calculator — S‑P Wave Time Interval
Estimates the distance from a seismic station to an earthquake's epicenter using the time delay between P-wave and S-wave arrivals, plus wave-velocity and S‑P interval reference tables.
S‑P Interval ↑
Distance scales directly with the S‑P interval — a longer delay between the two arrivals always means a more distant epicenter, for fixed Vp and Vs.
Velocity Contrast
The multiplier k = Vp·Vs/(Vp−Vs) grows when Vp and Vs are closer together, and shrinks when they're further apart — local crustal structure sets this constant, commonly ≈ 8 km/s for average continental crust.
Single-Station Limit
One station's S‑P interval gives distance only, as a radius around the station — the true epicenter is found where the distance circles from three or more stations intersect.
Depends on Local Geology
Vp and Vs are not universal constants — they depend on rock type, so using velocities calibrated to the local crust (or seismic network) gives a more accurate distance than generic textbook values.
How to Use This Calculator
This calculator links four quantities from a single seismic station: P-wave velocity V_p, S-wave velocity V_s, the S‑P time interval, and the epicenter distance D. Pick what to solve for, then fill in the rest.
1. Set the wave velocities
Use a preset (continental crust, sedimentary basin, upper mantle) or enter your own V_p and V_s for the local crust — V_p must be greater than V_s.
2. Choose what to solve for
Pick "Epicenter Distance (D)" if you know the S‑P interval, or "S‑P Interval" if you know the distance and want the expected time delay.
3. Enter the S‑P interval
Type the interval directly in seconds, or flip the switch to enter separate P-wave and S-wave arrival times (T_p, T_s) — the interval is computed for you.
4. Read the result and steps
The result card shows the solved value with engineering notation, and the step-by-step panel shows the substituted formula.
5. Switch to Advanced for more depth
Advanced mode adds sensitivity to a ±1% change in each known input, and an explanation of how the velocity contrast shapes the result.
6. Copy or reset
Use "Copy result" to grab the solved value, or "Reset" to return to the default worked example.
What This Tool Offers
Built around the classic single-station S‑P interval method for estimating distance to an earthquake's epicenter.
Solve either direction
Go from S‑P interval to distance, or from distance back to the expected S‑P interval — useful for both analysis and teaching.
Two ways to enter timing
Type the S‑P interval directly, or enter separate P-wave and S-wave arrival times and let the tool compute the interval for you.
Adjustable wave velocities
Quick presets for continental crust, sedimentary basins, and the upper mantle, or enter custom V_p/V_s calibrated to a local network.
Unit conversion built in
Velocities in km/s or mi/s, distance in km or mi — switch either independently and the result converts automatically.
Transparent step-by-step math
Every calculation shows the substituted formula, not just the final number, so it can be checked by hand.
Reference tables included
A typical wave-velocity-by-material table and a quick S‑P interval → distance lookup table are included below for fast sanity checks.
How the S‑P Method Works
P-waves (primary, compressional) and S-waves (secondary, shear) leave an earthquake's focus at the same instant but travel at different speeds, with P-waves always arriving first. The longer a seismic wave train travels, the further apart the two arrivals become — so the S‑P time interval measured at a single station grows steadily with distance from the epicenter, giving a direct way to estimate distance without needing a shared clock between the earthquake and the station.
Epicenter Distance Formula
Rearranged for the S‑P interval instead:
The bracketed term (V_p·V_s)/(V_p−V_s) behaves like a single conversion constant, often written k, that turns a measured time delay directly into a distance.
Formula Variables
| Symbol | Meaning | Typical unit |
|---|---|---|
| D | Distance from the station to the epicenter | km, mi |
| V_p | P-wave (primary) velocity through local crust | km/s, mi/s |
| V_s | S-wave (secondary) velocity through local crust | km/s, mi/s |
| T_p | Arrival time of the P-wave at the station | seconds |
| T_s | Arrival time of the S-wave at the station | seconds |
Reference Table — Typical Seismic Wave Velocities
Approximate ranges by material; always prefer velocities calibrated to the local network or crustal model when precision matters.
| Material | V_p (km/s) | V_s (km/s) | Approx. V_p/V_s |
|---|---|---|---|
| Unconsolidated sediment | 1.5 – 2.5 | 0.5 – 1.0 | ~2.4 |
| Sedimentary rock | 3.0 – 4.5 | 1.5 – 2.5 | ~1.8 |
| Granite / crystalline rock | 5.5 – 6.5 | 3.0 – 3.6 | ~1.8 |
| Average continental crust | ~6.0 | ~3.5 | ~1.73 |
| Upper mantle | ~8.0 | ~4.5 | ~1.78 |
Quick Reference — S‑P Interval to Distance
Using the common average-crust assumption V_p = 6.0 km/s, V_s = 3.5 km/s (k ≈ 8.4 km/s per second of S‑P interval):
| S‑P Interval | Distance (km) | Distance (mi) |
|---|---|---|
| 5 s | 42 | 26 |
| 10 s | 84 | 52 |
| 15 s | 126 | 78 |
| 20 s | 168 | 104 |
| 30 s | 252 | 157 |
| 45 s | 378 | 235 |
| 60 s | 504 | 313 |
| 90 s | 756 | 470 |
| 120 s | 1008 | 626 |
Worked Example
k = (6.0 × 3.5) / (6.0 − 3.5) = 21 / 2.5 = 8.4 km/s
D = 8.4 × 8.0
D = 67.2 km
Reference & Method Note
Frequently Asked Questions
Both wave types leave the earthquake's focus simultaneously, but P-waves (compressional) travel faster through rock than S-waves (shear). The speed difference means the gap between their arrivals grows the farther the waves travel — that growing gap is what this calculator converts back into distance.
No — a single station's S‑P interval only gives a distance (a circle of possible locations around that station), not a direction. Real epicenter locations require combining distance estimates from three or more stations and finding where their circles overlap.
For the commonly used average-crust values V_p ≈ 6.0 km/s and V_s ≈ 3.5 km/s, (V_p·V_s)/(V_p−V_s) works out to about 8.4 km/s — close to the widely cited "8 km per second of S‑P delay" rule of thumb used for quick estimates.
Yes, where available. Actual crustal velocities vary with rock type and depth, so a locally calibrated velocity model gives a noticeably more accurate distance than the generic reference values in this tool.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| D | Distance to epicenter | km |
| Vp | P-wave velocity | km/s |
| Vs | S-wave velocity | km/s |
| Ts | S-wave arrival time | s |
| Tp | P-wave arrival time | s |
What it means
The S‑P time method uses the difference in arrival times between S‑waves (secondary) and P‑waves (primary) to determine the distance to an earthquake epicenter. Since S‑waves travel slower than P‑waves, the time lag increases with distance. The formula uses Vp and Vs (typical values: 6 km/s and 3.5 km/s for crust) to convert the time difference into distance. This is a fundamental technique in seismology for locating earthquakes. By using data from multiple seismograph stations, the epicenter can be triangulated. Understanding this method is essential for seismologists to detect and locate seismic events, and for rapid earthquake response systems to issue alerts.
Worked example
Epicenter Distance (S‑P Time) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| Vp (km/s) | 6 |
| Vs (km/s) | 3.5 |
| S‑P interval (s) | 20 |
| Parameter | Value |
|---|---|
| Vp | 6.5 |
| Vs | 3.7 |
| S‑P | 15 |
Common mistakes
- Epicenter distance: D = (Vp·Vs / (Vp−Vs)) × (Ts−Tp) – the S‑P wave time difference.
- Velocities Vp and Vs: Use appropriate values for the crust (e.g., Vp≈6 km/s, Vs≈3.5 km/s).
- Time difference: Ts−Tp in seconds – ensure both arrival times are correctly identified.
- Assumes: Homogeneous velocities along the path – real crust is layered.
- Distance: Result is in the same units as velocities (km if km/s × s).
Applications
The earthquake epicenter distance, using the S‑P wave time interval, D = (Vp·Vs/(Vp−Vs))·(Ts−Tp), relates the distance to the seismic station to the arrival time difference between P‑ and S‑waves. This is the classic method for locating earthquakes. Seismologists use the S‑P time to estimate the distance from a station to the epicentre; combining distances from three or more stations allows triangulation. This approach is essential for real‑time earthquake warning systems and for rapid response. By calculating the epicentral distance, scientists can assess the intensity of shaking, issue alerts, and study fault mechanisms. Understanding this calculation is fundamental to observational seismology.
- Earthquake epicentre determination for early warning systems
- Seismic network processing and rapid location
- Earthquake hazard mapping and emergency response
- Research on earthquake source mechanisms
- Educational demonstration of wave propagation
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