Separate rounding, grid resolution and model uncertainty

Printed precision creates an interval

Station A’s supplied lag is 6.0 seconds, rounded to the nearest 0.1 second. Treating it as exactly 6.000000 seconds would add precision that was never observed. The investigation uses the conservative inclusive interval 5.95–6.05 seconds, retaining both endpoints rather than depending on a particular tie-breaking convention.

Multiply both endpoints by the model’s 12 km per second of lag. The allowed radial interval is 71.4–72.6 km. Its half-width is 0.6 km and its full width is 1.2 km. These describe distance from station A; they are not automatically an east-coordinate uncertainty or a circular confidence region around the final candidate.

Repeat the calculation for B’s 4.7 seconds and C’s 7.1 seconds. The intervals become 55.8–57.0 km and 84.6–85.8 km. The common rounding precision produces the same radial half-width here because the same linear distance rule applies to all stations.

Read residuals before rounding them again

Inspect the candidate (60, 40) km. Its predicted delays are approximately 6.00925, 4.71405 and 7.07107 seconds. Subtract the supplied delays to obtain residuals approximately +0.00925, +0.01405 and −0.02893 seconds. All lie within the inclusive −0.05 to +0.05 second tolerance.

The signs retain information: A and B are slightly above their nominal targets, while C is slightly below. All three can still be compatible with rounding. Agreement does not require every residual to be exactly zero. Conversely, rounding a residual for display before checking it could hide a value just outside a threshold; interpret the check using the underlying value.

Capture this candidate and at least two neighboring 5 km-grid candidates. For each rejected neighbor, name the station whose band is violated and record the residual sign. Do not write only “wrong answer”: the useful evidence is which prediction conflicts with which supplied interval.

Keep grid resolution separate from data precision

The sliders offer 21 possible eastings and 21 possible northings, so there are 441 selectable pairs. This is a sampling grid chosen for interaction, not an instrument specification. The coordinate step is 5 km, whereas each radial rounding band has a 0.6 km half-width. Those numbers describe different things and should not be added together as though they were one error bar.

If one sampled point fits, that establishes at least one compatible sampled location. It does not prove that the continuous overlap contains exactly one mathematical point. Nearby positions between grid lines may also satisfy all three intervals. If no sampled point fit, that alone would not prove that no continuous solution exists; the grid could miss a narrow overlap.

Sketch a thin overlapping region between two grid lines to illustrate this distinction. Label the grid as a set of test locations and the region as possible locations under constraints. The sketch is a conceptual counterexample, not a computed uncertainty region for this particular dataset.

Distinguish model error from rounding

The displayed bands represent only rounding of the supplied S–P lags. They omit uncertain phase picks, phase-dependent timing errors, variable wave speeds, curved or refracted paths, station-position uncertainty and nonzero source depth. Real arrival-time location requires a suitable velocity model as well as observed arrival information. USGS account of the location procedure and crustal velocities.

A common offset added to both phase times at one station cancels in their difference. A mistaken S pick does not. Likewise, a perfectly measured lag can yield a biased distance when the assumed speeds are inappropriate. Narrow rounding bands therefore do not establish high real-world accuracy.

Make an evidence ledger with three headings: supplied data, modeling assumptions and untested uncertainty sources. Place the literal lag values in the first, straight rays and fixed speeds in the second, and uncertain depth in the third. State what additional observation or model would address one omitted source. Do not invent a numerical confidence level without an error model.

Location is not prediction

This investigation asks where an event could have occurred given its supplied arrivals. It has no mechanism for forecasting a future earthquake’s time, magnitude or rupture. Finding a compatible candidate does not assess hazard at that point, and a candidate mismatch does not show that a place cannot experience earthquakes.

Transfer: test a rounding boundary

For a supplied lag of 6.0 seconds and the stated inclusive convention, compare predicted lags of 6.049 and 6.051 seconds. Which is compatible? Does making the coordinate grid finer remove velocity-model uncertainty?

Check the location-uncertainty transfer

6.049 seconds has residual +0.049 and is compatible; 6.051 has residual +0.051 and is outside the rounding band. A finer grid tests more candidate positions but does not fix an incorrect velocity model. Search resolution and model accuracy are different limitations.