1. Geology 5660/6660 Applied Geophysics 20 Jan 2016
Last time: Brief Intro to Seismology &
derivation of the Seismic Wave Equation:
Four types of seismic waves:
 P (“Primary” = sound; a body wave)
 S (“Secondary” = shear; also a body wave)
 Surface waves (Love & Rayleigh: at free surface only)
 Normal Modes (“Resonant tones” = standing waves)
An abbreviated Derivation of the elastic wave equation:
 Stress, strain and displacement waves propagate
in a medium
 Rheology is linear elastic:  = c (Hooke’s Law)
 Defined several elastic constants (e.g., E, , K)
Read for Fri 17 Jan: Burger (Ch 2.2.2–2.6)
© A.R. Lowry 2016

2. Geology 5660/6660 Applied Geophysics 20 Jan 2016
Last time cont’d: The Wave Equation
• The elastic wave equation:
 Assumes an isotropic solid
 Stress/strain relations assume infinitesimal strain
so that strain is the derivative of displacement:  = ∂u/∂x
 Results in the wave equation:
• Velocities are more sensitive to  &  than to ;
are sensitive to porosity, rock composition, cementation,
pressure, temperature, fluid saturation
Read for Fri 22 Jan: Burger (Ch 2.2.2–2.6)
© A.R. Lowry 2016

3. Instrumentation spring frame mass dashpot
Seismic ground motions are recorded by a seismometer or
geophone.
Basically these consist of:
• A frame, hopefully well-coupled to the Earth,
• Connected by a spring or lever arm to an
• Inertial mass.
• Motion of the mass is damped, e.g., by a dashpot.
• Electronics convert mass movement to a recorded signal
(e.g., voltage if mass is a magnet moving through a
wire coil or vice-versa).

4. Commonly-used by industry, less often for academic, seismic
isometric view
Geophone:
Commonly-used by industry, less
often for academic, seismic
reflection studies
Often vertical component only
Often low dynamic range
10 Hz “natural frequency”
Undamped response
of mechanical system
Response after
electronic damping 10 20
100
200
500
A Seismometer differs mostly in cost/
componentry… 3-c, > dynamic range
cross-sectional view

5. Understanding the Frequency Domain:
Recall that an idealized mass on a spring is a
harmonic oscillator: Position x of the
mass follows the form x = A cos (t + )
where A is amplitude, t is time,  is the natural
frequency of the spring, and  is a phase
constant (tells us where the mass was at
reference time t = 0). x
T = 2/ A
In the frequency domain this
is a delta-function: 

6.  Signal recorded by a seismometer is a convolution of the
wave source, the Earth response, and the seismometer
response.
Wave
Source
Earth
Response
Seismometer
Response  
where  denotes convolution:
Example:  =
So, want seismometer response to
look as much as possible like a
single delta-function in time: 
t = 0 

7. Seismometer response is given by:
where i is current, 0 is “natural frequency” of the
spring-mass system oscillation, K is electromagnetic
resistance to movement of the coil, R is electrical
resistance to current flow in the coil, & x is movement
of the coil relative to the mass.
The damping factor h is given by:
where  is the mechanical
damping factor.
Hence we choose K and R to give a time response that
looks as much as possible like a delta-function (= a flat
frequency response):

8. Seismometers typically are designed to be slightly
This corresponds to 0/h = 1: Critical damping
Underdamped
Critically Damped
Overdamped
Seismometers typically are designed to be slightly
overdamped (0/h = 0.7).

9. An explosion at a depth of 1 km & t = 0
Seismometer Damping
Source Function
An explosion at a depth of 1 km & t = 0
is recorded by a seismometer at the
surface with the damping response
shown. What will the seismogram look
like?
V = 5 km/s

10. (Note: for really big signals, can get more robust operation
and lower frequencies from other types of instruments…
E.g. GPS!)
4 April 2010 M7.2 Baja California earthquake

11. (Note: for really big signals, can get lower frequencies from
other types of instruments… E.g. GPS!)
GPS Displacement
Seismometer Displacement
10 August 2009 M7.6 earthquake north of Andaman

12. Huygen’s Principle:
Every point on a wavefront can be treated as a point source
for the next generation of wavelets. The wavefront at a time
t later is a
surface tangent
to the furthest
point on each
of these
wavelets.
We’ve seen
this before…
This is useful
because the
extremal points
have the greatest constructive interference

13. Fermat’s Principle (or the principle of least time):
The propagation path (or raypath) between any two points
is that for which the travel-time is the least of all possible
paths.
Recall that a ray is normal to a wavefront at a given time:
A key principle because most of our applications will involve
a localized source and observation at a point (seismometer).