Locate a compatible source from several stations

A distance does not supply a direction

Station A sits at (0, 0) km, B at (100, 0) km and C at (0, 100) km. The first coordinate is easting and the second is northing. A supplied S–P delay becomes a distance constraint under the selected travel-time rule, but a single distance leaves many possible directions.

For example, station A’s nominal distance is 72 km. On a full plane, every point on a circle of radius 72 km centered at A has that distance. Restricting the visible map to the positive-coordinate square clips the possibilities; it does not turn a single distance into a unique direction. The circle is a geometric constraint, not a wavefront observed in the browser.

Draw the constraints before searching

On paper, choose the same scale on both axes and mark the three stations. Convert their supplied delays using 12 km per second of lag. A gives 72 km, B gives 56.4 km and C gives 85.2 km. Draw or sketch circles at those nominal radii. Label the scale and station names so another learner can reconstruct your reasoning.

The printed delays are rounded, so the browser uses radial bands. Their ranges are 71.4–72.6 km for A, 55.8–57.0 km for B and 84.6–85.8 km for C. The radial half-width is 0.6 km in every case. At a small paper scale those bands may be thinner than your pencil line. Record the numerical intervals rather than treating the sketch’s thickness as a measurement.

Using distances from several stations is a classic way to constrain an earthquake’s location. USGS describes comparing observed and predicted arrival information across a network. The exercise here reduces that broader problem to a flat, zero-depth geometry with fixed speeds. USGS explanation of earthquake location.

Test a candidate against every station

Begin at (50, 50) km. This point is equally distant from all three stations: about 70.71 km. Its predicted S–P lag is therefore about 5.89 seconds at each station. The supplied lags differ, so the central position cannot fit all three merely because it looks balanced on the map.

Move the candidate toward B and consider the competing changes. B’s distance generally decreases along a direct move toward B, but the distances to A and C need not improve their matches at the same time. Capture a candidate that improves one residual while leaving another outside its band. Explain each residual from the relevant station distance.

A residual is predicted minus supplied lag. Positive means the candidate predicts too large a lag; negative means it predicts too small a lag. The sign is useful for reasoning, but a small residual at one station does not excuse a larger mismatch at another. All three constraints matter.

Use the overlap of the bands to guide your next candidate. The controls move in 5 km increments. Keep a short ledger of attempted coordinates and the station that rejects each attempt; a systematic search should leave evidence beyond the final marker position.

Interpret a compatible point cautiously

Inspect one compatible sampled candidate

At (60, 40) km the predicted lags are approximately 6.0093, 4.7140 and 7.0711 seconds for A, B and C. Each differs from the corresponding supplied delay by less than 0.05 seconds, so this sampled point satisfies all three conservative rounding bands. The check uses those constraints, not a hidden true-location answer.

A compatible point is an outcome of testing this model against these supplied observations. It does not prove an exact unique source in the real Earth. The source depth is fixed rather than inferred. For an actual earthquake, the hypocenter is the source location at depth and the epicenter is its surface projection; they coincide spatially only in this zero-depth simplification. USGS explanation of earthquake source terminology.

Do not generalize the exercise into “three stations always solve every three-dimensional earthquake.” Station geometry, arrival information, depth, velocity structure and uncertainty all affect the real inference.

Transfer: preserve two-station ambiguity

On an unrestricted plane, two stations lie at (0, 0) and (100, 0) km. A hypothetical source is 50√2 km from each. Test the two points (50, 50) and (50, −50). Can the two distance constraints distinguish them?

Check the earthquake-location transfer

No. Each point is 50 km horizontally and 50 km vertically from either station, giving distance √(50² + 50²) = 50√2 km. The two distances allow both positions. A suitably placed additional station can distinguish them; a plotting window that hides one point is not additional physical evidence.