Infer distance from seismic arrival times

Two arrivals provide a useful difference

A timing card reports a P arrival at 10 seconds and an S arrival at 16 seconds on the same station clock. Mark both times on a paper number line. The separation is 16 − 10 = 6 seconds. That difference is the S–P lag; it is not the time since the earthquake began unless the origin time is independently known.

P waves involve compression and expansion with particle motion along the propagation direction. S waves involve shear with particle motion perpendicular to propagation. In a solid, P waves travel faster than S waves. S waves do not propagate through fluids as shear body waves. Wave propagation carries a disturbance; it does not mean a parcel of ground travels from source to station. British Geological Survey explanation of seismic waves.

The browser supplies fictional lags rather than recording ground motion. It does not teach or assess picking arrivals from noisy waveforms. Your task is to reason from the provided timings while keeping that distinction explicit.

Derive the distance rule

The selected teaching model has straight paths through a homogeneous material: the properties are the same everywhere. P speed is 6 km/s and S speed is 4 km/s. If the path distance is d kilometers, P travel time is d/6 seconds and S travel time is d/4 seconds. Travel time is distance divided by speed.

Subtract the travel times: S–P = d/4 − d/6 = d(1/4 − 1/6) = d/12. The fractions 1/4 and 1/6 have units seconds per kilometer. Their difference is 1/12 second per kilometer. Therefore d = 12 × lag, with the multiplier expressed in kilometers per second of lag.

For the timing card, 6 seconds gives a nominal distance of 72 km. The corresponding travel times are 12 seconds for P and 18 seconds for S. Those travel times need not equal the clock readings 10 and 16 seconds; an unknown origin time shifts both arrival readings together. The equal six-second separation is the information used here.

Do not multiply the lag by 6 − 4. The difference of speeds is not the reciprocal-speed difference in the travel-time equation. Check the arithmetic with a nearby practice distance of 48 km: P takes 8 seconds, S takes 12 seconds, and the four-second lag returns 48 km.

Identify what subtraction cancels

Add 20 seconds to both times on the opening card. The new readings are 30 and 36 seconds; their difference remains six seconds. A common clock offset at that station cancels, as does the common earthquake origin time. An error affecting only one phase does not cancel: if S is picked one second late, the estimated lag increases by one second.

Make two columns on paper: common shifts and phase-dependent changes. Record one example in each and calculate the resulting lag. Explain why subtraction solves only the first kind of problem. Correct arithmetic cannot repair a misidentified arrival or an unsuitable wave-speed model.

Inspect the candidate’s predicted times

At the default candidate, inspect each station’s predicted P time, S time and lag. These are times since the hypothetical origin, calculated from the candidate’s distance. Compare the predicted lag with the supplied value: A is 6.0 seconds, B is 4.7 seconds and C is 7.1 seconds. Move one coordinate and predict which station becomes closer before checking its time changes.

The controls describe a local flat map in kilometers. Source depth is fixed at zero, so the modeled distance lies in that plane. This is a deliberate simplification, not a claim that natural earthquakes generally start at the surface. Real location methods use arrival observations with travel-time models that account for additional source and Earth properties. USGS overview of earthquake location.

The nominal distance is not exact: the supplied lags were rounded to the nearest tenth of a second. The map consequently shows narrow distance bands rather than treating every printed delay as infinitely precise.

Transfer to another timing card

A different station reports P at 41.2 seconds and S at 45.7 seconds. Using only this lesson’s speeds, calculate the lag and nominal distance. Then add seven seconds to both readings and repeat.

Check the seismic-arrival transfer

The lag is 4.5 seconds and the nominal distance is 54 km. Adding a shared seven-second offset changes the readings to 48.2 and 52.7 seconds, but the lag and inferred distance stay unchanged. This cancellation does not address phase-picking errors or a wrong velocity model.