A Misfit Circle in a Four-Station Epicenter Solution
A regional seismic network records a magnitude 4.6 earthquake. Four stations report S-minus-P arrival-time intervals: RKV 5.0 s, BLM 7.5 s, PTR 10.0 s and OAK 6.2 s. The network uses a local rule of thumb of 8 km of epicentral distance per second of S-P interval, giving radii of 40, 60, 80 and 50 km. A candidate draws circles of those radii around the four stations. The RKV, BLM and PTR circles intersect in a small region about 5 km across; the OAK circle misses that region by roughly 25 km. OAK is installed on 300 m of unconsolidated basin sediment; the other three stations sit on bedrock. The draft report states that the epicenter is at the intersection of the three consistent circles and that the earthquake was about 34 km deep, 'because the circles did not all meet at a single point.' The network manager asks for a review before the report is released.
1. Which station's data should be re-examined rather than simply plotted, and what physical and instrumental explanations should be checked?
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OAK is the outlier, and it is the station whose data should be scrutinized first, but not discarded outright. Three bedrock stations agree within about 5 km, which is well inside the uncertainty of the 8 km/s rule; a single station on 300 m of basin sediment disagrees by 25 km.
Thick, low-velocity sediment amplifies and delays seismic waves, and the effect is larger for S waves than for P waves. A delayed S arrival inflates the S-P interval and therefore the apparent distance, pushing the circle outward exactly as observed. That is a station site-response effect, not an epicenter.
Before applying a correction, the analyst should also rule out instrument problems: a drifted or unsynchronized clock, an incorrect instrument response, a mis-picked phase on a noisy record, or a wrong station coordinate. The practical steps are to inspect OAK's seismogram and phase picks, verify timing against a known event or GPS-disciplined clock, and, if the site effect is real, derive an empirical station correction from well-located nearby events instead of trusting the raw S-P interval.
2. Evaluate the report's depth conclusion of about 34 km. What actually controls the size of the region where the circles fail to meet?
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The depth claim does not follow from the data. On a flat map, each S-P interval gives an epicentral distance only. The circles are drawn from those distances, so their scatter measures horizontal location uncertainty - picking error, the crude 8 km/s conversion, lateral velocity variation and station corrections. A 5 km across mismatch of three circles says the horizontal solution is good to a few km; it says nothing quantitative about focal depth.
Depth is constrained by how travel times change with distance and by the arrival of depth phases such as pP and sS, which require modeling waveforms, not just reading S-P intervals. With a handful of stations at 40 to 80 km and no near-field recordings, depth is weakly resolved at best.
A further problem is that the 8 km/s rule is an approximation for a particular local structure. If the true velocity structure differs, distances are systematically wrong and the residuals carry a distance-dependent pattern. The correct route is to invert the arrival times with a layered velocity model, report a formal depth uncertainty, and state honestly that depth is poorly constrained if the inversion is unstable.
3. List the concrete actions the network should take before publishing the location and depth.
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First, fix the input data: verify clock accuracy at all four stations, re-pick P and S arrivals on the raw records, confirm station coordinates, and check that each station's response is current.
Second, quantify the site effect. Compare OAK's residuals for several well-located previous events to derive a station correction; if the correction removes the misfit, OAK can be retained with a stated uncertainty.
Third, replace the rule of thumb with a proper inversion: fit hypocentral coordinates and origin time using a layered velocity model appropriate to the region, weighting stations by their residuals and reporting confidence ellipses rather than a single point.
Fourth, use additional data to pin the depth - more stations at a wider range of distances, waveform modeling of depth phases, and first-motion polarities to help constrain the focal mechanism and depth. Publish the depth with an explicit uncertainty and note the sensitivity to the velocity model.
Finally, state clearly in the report that the original 34 km figure was an artifact and should be withdrawn.