Infer path speed from summed slowness contributions

1. Define the evidence and frame

A travel time measures a sum of path length divided by speed along the ray. Even in two simple vertical layers, the effective speed is not generally the arithmetic mean of layer speeds. Each segment contributes time, and the total path divided by total time gives the corresponding effective speed.

2. Connect mechanism and model

The original packet has upper thickness a at speed b and lower thickness 30 km at 8 km/s. The one-way time is a/b+30/8 seconds. This is a vertical ray with known geometry; the refraction lesson’s oblique path length must not be silently substituted. The same units must be used in every segment.

Inversion works backward from observations to model parameters. One travel time can be matched by multiple thickness–speed combinations. Holding thickness known helps constrain speed, but an unknown origin time would add another ambiguity. More rays and independent geological constraints can reduce, but do not automatically eliminate, nonuniqueness.

A seismic velocity boundary need not be a simple change of chemical composition: pressure, temperature, phase and porosity also affect elastic behavior. Crust–mantle composition and lithosphere–asthenosphere mechanical behavior describe different classifications of the interior. A useful interpretation keeps these distinctions rather than labeling every velocity contrast as a new chemical layer.

3. Reconstruct the worked packet

For upper 20 km at 4 km/s, times are 5 and 3.75 s, totaling 8.75 s. Total path 50 km gives 5.71429 km/s. The unweighted mean of 4 and 8 is 6 km/s and does not reproduce the time.

4. Predict and investigate

Predict first: Which layer contributes more time per kilometer, and why?

Hold total geometry fixed while changing upper speed. Reconstruct both segment times before computing effective speed. Then identify two hypothetical changes that could offset each other in a single observed travel time.

5. Test a changed case

Apply the evidence to a changed case

At upper thickness 30 km and speed 6 km/s, find total travel time and effective speed.

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

Compare a worked response

Times are 5 and 3.75 s; 60 km over 8.75 s gives 6.85714 km/s. Equal path lengths produce a harmonic, not arithmetic, speed average.

6. Retain a defensible evidence notebook

Preserve first predictions, full units, geometry, source provenance, assumptions and rejected alternatives. Synthetic models are not field measurements. The explicitly labeled USGS packet is actual catalog evidence, but working through a public example is not independent practical competence or an operational hazard assessment.

Mechanism and data reference, checked 30 October 2026. Original teaching prose, diagrams and exercises; actual USGS rows retain their agency and provenance.