For a crust of thickness over a faster mantle half-space ( , , , )

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

The Solid Earth, Ch Surface waves, free oscillations and seismic tomography

Problem

For a crust of thickness HH over a faster mantle half-space (vS1=3.5 km s−1v_{S1}=3.5\,\mathrm{km\,s^{-1}}, vS2=4.5 km s−1v_{S2}=4.5\,\mathrm{km\,s^{-1}}, ρ1=2800 kg m−3\rho_1=2800\,\mathrm{kg\,m^{-3}}, ρ2=3300 kg m−3\rho_2=3300\,\mathrm{kg\,m^{-3}}), the fundamental Love-mode group velocity at T=40 sT=40\,\mathrm{s} period has been computed for several thicknesses. Data: for H=25,30,35,40,45 kmH=25, 30, 35, 40, 45\,\mathrm{km} the group velocity is U=3.99,3.83,3.70,3.59,3.51 km s−1U=3.99, 3.83, 3.70, 3.59, 3.51\,\mathrm{km\,s^{-1}} respectively. At H=35 kmH=35\,\mathrm{km} the same mode has phase velocity c=4.17 km s−1c=4.17\,\mathrm{km\,s^{-1}} at T=40 sT=40\,\mathrm{s}. A seismologist analyses a 40 s40\,\mathrm{s} Love wave recorded at epicentral distance Δ=2100 km\Delta = 2100\,\mathrm{km} and argues: "Assuming H=35 kmH=35\,\mathrm{km}, the wave train should arrive at t=Δ/c=2100 km/4.17 km s−1=504 st = \Delta/c = 2100\,\mathrm{km}/4.17\,\mathrm{km\,s^{-1}} = 504\,\mathrm{s} after the origin time. But the wave train is observed to arrive at t=568 st=568\,\mathrm{s}. Since Δ/t=2100/568=3.70 km s−1\Delta/t = 2100/568 = 3.70\,\mathrm{km\,s^{-1}} is slower than assumed, the crust must be thicker than 35 km35\,\mathrm{km}, and the estimate should be revised upward." ((a)) Find the error in this reasoning. ((b)) Give the correct estimate of HH from the observation, using the table.

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