Introduction to Seismology Geology 5640/6640 Introduction to Seismology 27 Feb 2017 Last time: Seismic Source Modeling • The solution to the spherical wave equation: is undefined as r  0… So, we define • Dirac delta function: Heaviside step function: For an earthquake, the f(t) in the source term of the wave equation: is a (tensor) moment rate of energy release: (& moment is M = sA!) Read for Wed 1 Mar: S&W 75-86 (§2.6) © A.R. Lowry 2017

Source Seismology 2011 Christchurch earthquake, M6.3, after a larger M7.0 eq further west in 2010… 2010 M7.0 2011 M6.3 2 injured NZ $4B Difference is proximity… 185 dead NZ $15B

Can use this to get other interesting pieces of information about the earthquake rupture process…

… Including our growing recognition that many large earthquakes involve complex rupture simultaneously on several faults that may have completely different dip and orientation.* 2010 M7.0 Haiti 2002 M7.9 Denali Crone et al., BSSA, 2004 Hayes et al., Nat. Geosci., 2010 *Lesson for Utah! Ya think future Wasatch rupture won’t cross segment boundaries? Ya got another think comin’.

Source Seismology 2016 New Zealand earthquake, Mw 7.9

Seismic Wave Energy Partitioning With Snell’s Law in our tool-belt, we’re ready to consider what happens to seismic amplitudes when an incoming wave arrives at a change in properties (and hence, conversions occur). One obvious thing that has to happen is conservation of energy, i.e., reflected energy + transmitted energy = energy of the incoming wave As you might expect, energy is related to amplitude of the wave. incoming P A q

Impedance Contrast: Thus far we’ve focused much of the discussion on concepts related to velocity & travel-time, but seismic waves also have amplitude, A, of the particle displacements: Amplitudes of reflections & refractions are determined by energy partitioning at the boundary. A normally-incident (= 0) P-wave with amplitude Ai produces a reflected P with amplitude: incoming P A q (reflection coefficient) (transmission coefficient) and a refracted P: where Zi = iVi is the impedance in layer i.

The energy E in a wave is directly proportional to the amplitude A, and for this example, sign (i.e. propagation direction) matters. We’ll use the sign of the z-component (positive-down) of propagation. Then we have: Displacement must be continuous at the boundary so: Ai + Arfl = Arfr And: 1 + R = T Note however for the P-wave depicted here, this applies only to the case where i = 0°… (Why?) incoming P A q

An SH-wave, it turns out, produces no P or SV conversions either at a horizontal boundary. So we can consider an example in which S- velocities are 1, 2; densities are 1, 2; and angles of incidence are j1, j2: In this instance, the only non-zero displacement is uy. The wave equation must satisfy: We’ve derived solutions to similar equations, and S&W write a solution for this specific case (p66) as where r = kz/kx is just a ratio of wavenumbers.