Reconstruct a source from explicitly modeled S–P ranges

1. Define the evidence and convention

P and S waves travel at different speeds, so their arrival-time difference can constrain distance under an assumed propagation model. In a homogeneous medium with speeds Vp and Vs, S−P=r(1/Vs−1/Vp). The source origin time cancels, but velocity structure and correct phase identification do not.

2. Connect mechanism and model

The original noiseless packet fixes Vp=6 and Vs=3.5 km/s, depth 10 km and stations at (0,0),(60,0),(0,60) km. Source east/north coordinates determine hypocentral ranges. A flat map circle must use horizontal range √(r²−depth²), not the full slanted hypocentral distance.

Subtracting squared station-range equations removes the common depth and quadratic source terms. Thus x=(r1²−r2²+3600)/120 and y=(r1²−r3²+3600)/120. This derivation explains how geometry constrains location rather than merely drawing three decorative circles.

The model knows the source because it generated the times; that knowledge is not an independent observation. Actual locations require uncertain picks, realistic travel-time models, station geometry and often joint depth/origin estimation. A poor fit should not be repaired by silently changing the recorded arrival times.

3. Reconstruct the worked packet

For source (20,20) km at 10 km depth, ranges are 30,√2100,√2100 km. S−P times are about 3.57143,5.45545,5.45545 s. The squared-range differences recover x=y=20 km; the nearest horizontal radius is √800≈28.2843 km, not 30.

4. Predict and investigate

Predict first: Why is an epicentral map circle smaller than a hypocentral range for a source below the surface?

Reconstruct ranges from the generated time differences and recover x/y from squared differences. Compare projected and slanted distances. State how an unknown depth or layered velocities would alter the inference.

5. Test a changed case

Apply the evidence to a changed case

Place the source at (0,0) with the same 10 km depth. What are the first station’s hypocentral range and horizontal projected radius?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

The station directly above the source has range 10 km and projected radius zero. Its nonzero S−P interval is 25/21≈1.19048 s; an epicenter at the station is not a zero-length interior path.

6. Retain a defensible evidence notebook

Preserve first predictions, map orientation, intermediate calculations, units, assumptions and alternatives. These maps, numerical packets and diagrams are original invented instructional cases, not surveyed sites or current hazard forecasts. Independent practical review and genuinely unfamiliar evidence remain separate learner gates.

Mechanism reference, checked 30 October 2026. Original prose, diagrams and numerical exercises. No physical loading, pressurization or field hazard exposure is requested.