How can three stations help locate an earthquake?

A fault slips; the disturbance travels

Rocks can store elastic strain energy as they deform. Slip releases some of that stored energy. The two-block illustration separates permanent offset from the small local oscillation in a passing wave.

Start underground

The hypocenter is where rupture starts. The epicenter is its projection onto the surface. A buried front needs time to reach the surface; its map ring does not begin spreading from the epicenter immediately at rupture.

Follow a material marker

An ideal P wave moves material parallel to its local travel direction. S motion is perpendicular. The gold marker stays near its own equilibrium position while the disturbance moves through the medium.

Read the arrival, not just the peak

A station record tells a timing story. In our authored display pulses, the first motion begins at a calculated onset. A later, larger peak is not that onset. Adjust the picks and keep their precision explicit.

Distance is a clue, not a direction

Under the chosen constant speeds, a bigger S−P gap means a larger source-to-station distance. One station leaves many directions possible. At a known depth, its distance sphere becomes a circle of possible horizontal positions.

Add independent clues

Two stations can leave two locations. A third non-collinear station can distinguish them in this exact model. If the data or model do not agree, keep that disagreement visible rather than moving circles until they meet.

Test the assumptions

A late S pick changes a gap. A station clock error can shift both phases while preserving its gap. A different assumed medium changes the interpretation. Several different depths and origins can fit three P arrivals exactly.

Look at real records

Real Earth structure, propagation and instruments are richer than this local model. The historic PMM recording and Loma Prieta comparison retain their original annotations. They are evidence with their own scales and limitations.

A homogeneous direct-arrival model

This lesson uses a flat zero-elevation surface and one isotropic solid with vP=6.2 km/s and vS=3.57 km/s. Those chosen speeds are traceable to the Canadian EPB location model described in USGS OFR82-777. Our fictional event is not a reconstruction of its New Brunswick aftershocks. The original study included elevation corrections; this model does not.

Coordinates and propagation

Coordinates are local kilometres: x east, y north and h positive downward. R²=(x−xᵢ)²+(y−yᵢ)²+h². A source at origin time t₀ gives P=t₀+R/vP and S=t₀+R/vS. The 3D map uses [x,−h,−y] with one common scale.

The gap removes common origin time

S−P=R(1/vS−1/vP). With the stated speeds, the coefficient is approximately0.118821722 s/km and R≈8.415969582 times the gap in seconds. A5.00 s gap gives42.07984791 km. It supplies a source distance, not a direction.

Source distance is not the map radius

At known depth h, the map radius is ρ=√(R²−h²). The default source is(30,40,12) km, origin20 s; stationA is at(0,0,0). Its source distance is51.419841 km but its horizontal radius is50 km. If R<h, no exact circle exists at that depth.

The surface front has an emergence time

After elapsed time τ, a direct spherical front has radius vτ. Its surface intersection exists only for τ≥h/v and has radius√((vτ)²−h²). At12 km, P emerges at1.935484 s and S at3.361345 s. The initial apparent ring speed is a geometric intersection effect, not a faster material wave.

Three complete pairs versus three P arrivals

With three non-collinear surface stations and exact P/S gaps, squared-range differences solve x and y; R²−ρ² gives h². Selecting h≥0 chooses the below-plane branch. Three P times alone leave four unknowns(x,y,h,t₀), and the reference examples give distinct depth/origin combinations with the same P arrivals.

Fit a common origin, then inspect residuals

For a candidate source, t₀*=Σwⱼ(tⱼ−Tⱼ)/Σwⱼ. RMS=√[Σwⱼ(tⱼ−t₀*−Tⱼ)²/Σwⱼ]. The examples use equal positive weights on six arrivals. RMS is in seconds; it depends on the model and picks. Small residuals do not prove spatial accuracy or predict another event.

Reading bounds make bands

Authored P and S reading bounds of±0.05 and±0.10 s give a worst-case gap bound±0.15 s and source-range bound±1.262395437 km. At a fixed depth these become annuli. The continuous overlap is not a rectangle or a statistical confidence interval. Separate coordinate extrema are computed from boundary intersections and circle extrema.

Sampling and rounding are different

The map’s filled cells sample the band overlap at1 km cell centers; absence of a sampled cell is not proof that a smaller overlap is absent. Rounded0.01 s paper times instead have±0.005 s time-rounding bounds and±0.01 s worst-case gap rounding. That is different from the larger reading-bound exercise.

A fixed observation set for each assumption test

Generation controls recompute synthetic records after moving the fictional source or station geometry. The reference audit freezes its original observations before changing a pick, clock or assumed speed. The proposed fourth station compares two fixed P-only source candidates; it does not create a new earthquake or forecast.

What ideal P and S motion leave out

The local marker illustration uses a chosen sinusoid and envelope. Its direction respects the inclined ray, but its frequency, amplitude, repeated timing and display length are authored. Real paths include refraction, reflection, conversions, surface waves, attenuation, radiation patterns and finite rupture. Propagating shear S waves do not traverse a liquid; no interface solver is implemented here.

Magnitude and local intensity

Magnitude estimates event size using a specified method. Shaking intensity varies with location. Propagation, local geology, radiation and instruments affect recordings. A gain control on an arbitrary synthetic pulse cannot establish a real magnitude or damage level.

A network turns timing into location

Seismic stations contribute observations to a common inference. Their geometry and clocks matter alongside the travel-time model. An added station is useful when its evidence distinguishes plausible alternatives.

Read model fit critically

The fixed observation audit shows why matching some data can leave an assumption unresolved. A P-only fit can hide depth/origin tradeoffs; a common clock offset can preserve a same-station gap.

Separate event size from local experience

Historic recordings connect one event to different local responses. This helps explain why a location or magnitude alone does not describe everyone’s shaking.

What distinction does the fault illustration teach?

• Rocks travel from the source to every station
• Slip releases stored strain energy; waves cause local motion
• The wavefront is a moving crack in every direction

An ideal P wave travels to the right in a solid. Which local motion matches it?

• Left and right
• Only up and down
• No material motion

A5.00 s S−P gap gives42.08 km under these speeds. What does that tell you?

• The direction is north
• Source distance, without a unique direction
• The earthquake magnitude

R=51.419841 km and known depth h=12 km. What is the horizontal map-circle radius?

• 51.419841 km
• 50 km
• 12 km

A and B allow both(30,40) and(30,−40). May we choose the northern point just because it looks plausible?

• Yes, maps prefer north
• No, both satisfy those data
• Yes, two stations always locate one point

Do three P arrival times automatically determine east, north, depth and origin time?

• Yes, three is always enough
• No, different depths and origins can share those P arrivals
• Yes, if the display gain is high

If central circles miss a common point, what should we do?

• Move the circles until they meet
• Keep the data, show reading bounds and inspect assumptions
• Call the residual the next earthquake probability

Different stations record different amplitudes from one earthquake. What follows?

• The event must have several physical sizes
• Local conditions and recording systems can affect the signals
• The tallest arbitrary pulse measures the magnitude

Earthquake mechanism, hypocenter, epicenter and local P/S motion

Lisa Wald/USGS, mechanism, recording and location sections. The lesson adds explicit buried-source and depth-plane geometry.

Elastic rebound and permanent displacement

USGS historical account of Reid’s elastic rebound interpretation after the1906 earthquake. The original two-pose illustration is not a quantitative rupture model.

Traceable chosen P/S speeds

Cranswick et al., USGS Open-File Report82-777, Results: EPB used a homogeneous half-space with6.2 and3.57 km/s plus elevation corrections. This lesson supplies a separate flat-surface fictional geometry.

Location uses travel-time models and measured arrivals

USGS description includes comparison of measured/predicted times and least-squares location. Depth and origin are inferred quantities.

Depth, origin and timing-fit definitions

ComCat documentation for depth, depthError, rms and time. The lesson’s synthetic equal-weight RMS is not an official catalog uncertainty.

P/S propagation in Earth’s different materials

Lisa Wald USGS diagram explanation: P can propagate through solids/liquids; a propagating shear S wave does not cross a liquid. Linked context; image not bundled.

Separate spherical-model prediction example

TauP3.2.0 documentation: PREM, depth200 km,57.4°, P566.77 s and S1028.61 s. The461.84 s difference is arithmetic from rounded documented predictions; no TauP run or measurement is claimed.

Historic observed PMM seismogram identity

USGS Teleseisms example: April23,2000 Argentina M6.9 Mw event, approximately600 km depth, origin09:27:23.1 UTC. Annotated PMM P/S times09:38:55 and09:48:25 yield570 s. Original wrapped display retained.

Reading wrapped seismogram displays

USGS explanation of display conventions. The local GIF is not raw calibrated waveform data.

Magnitude versus local shaking

USGS distinguishes magnitude, energy release and location-dependent shaking intensity.

Actual Loma Prieta site-response composite and rights

USGS Public Domain media record:1989 M6.9 event, approximately equal station distances and bedrock/mud/sand-gravel displays. Original800×306 composite unchanged; no calibrated amplitude axis provided here.

Reuse basis for USGS-produced recording display

USGS-produced information generally public domain unless third-party rights are indicated. PMM source/image has no third-party credit shown; the Loma Prieta page has its own explicit Public Domain field.

Location science is not future prediction

USGS distinguishes earthquake prediction from existing scientific capabilities. No future-event claim is part of this lesson.