1. 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 (§2.6)
© A.R. Lowry 2017

2. 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

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

4. … 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’.

5. Source Seismology
2016 New Zealand
earthquake, Mw 7.9

6. 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

7. 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.

8. 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

9. 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.