A refraction line is shot in both directions across a crust whose base

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Rays and travel times: from arrivals to a layered planet

Problem

A refraction line is shot in both directions across a crust whose base, the mantle interface, is not flat but dips gently. The direct wave in both shots gives the same crustal speed, v1=6.00 km s−1v_1 = 6.00\,\mathrm{km\,s^{-1}}. Shooting down-dip (into thickening crust) gives an apparent mantle speed v2d=7.20 km s−1v_{2d} = 7.20\,\mathrm{km\,s^{-1}}; shooting up-dip (into thinning crust) along the same line gives v2u=9.30 km s−1v_{2u} = 9.30\,\mathrm{km\,s^{-1}}. These are related to the true critical angle ici_c and the dip δ\delta by ic+δ=sin⁡−1(v1/v2d)i_c+\delta=\sin^{-1}(v_1/v_{2d}) and ic−δ=sin⁡−1(v1/v2u)i_c-\delta=\sin^{-1}(v_1/v_{2u}). ((a)) Find the true critical angle ici_c, the dip δ\delta, and the true mantle speed v2v_2. ((b)) A geologist, working only from the down-dip shot, takes v2dv_{2d} itself as the mantle speed. Is this an overestimate or an underestimate of the true v2v_2, and by roughly what fraction here? ((c)) Explain, in a sentence or two, why a single one-way profile can never separate the true mantle speed from the interface's dip, however precisely its intercept and crossover are measured.

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