For a magnitude event

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

The Solid Earth, Ch The earthquake source: moment tensor, radiation and spectra

Problem

For a magnitude Mw=5.0M_w = 5.0 event, M0=4.0×1016 N mM_0 = 4.0\times10^{16}\,\mathrm{N\,m} and the corner frequency is fc=0.72 Hzf_c = 0.72\,\mathrm{Hz}; at constant stress drop fc∝M0−1/3f_c \propto M_0^{-1/3}. The corner crosses the 20 s20\,\mathrm{s} measurement period (0.05 Hz0.05\,\mathrm{Hz}) at M0=1.19×1020 N mM_0 = 1.19\times10^{20}\,\mathrm{N\,m} (Mw=7.3M_w = 7.3); well above this moment the 20 s20\,\mathrm{s} spectral amplitude grows only about as M01/3M_0^{1/3} instead of as M0M_0, and near it the full ω−2\omega^{-2} spectrum M0/[1+(f/fc)2]M_0/[1+(f/f_c)^2] is needed. A second earthquake has M0=2.0×1021 N mM_0 = 2.0\times10^{21}\,\mathrm{N\,m}. ((a)) Find its corner frequency, and confirm it lies below 0.05 Hz0.05\,\mathrm{Hz}. ((b)) Find its moment magnitude MwM_w, using Mw=23(log⁡10M0−9.1)M_w = \tfrac23(\log_{10}M_0 - 9.1) with M0M_0 in N m\mathrm{N\,m}. ((c)) Using the ω−2\omega^{-2} spectrum, find the ratio of its 20 s20\,\mathrm{s} amplitude to that of the Mw=7.3M_w = 7.3 crossing event. Compare it with the ratio of their moments and with the M01/3M_0^{1/3} estimate, and say why the estimate falls short.

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