Introduction to geophysics
Topics
1. Seismology
2. Gravity
3. Geomagnetism
(Introduce the web-page, grading policies, etc.)
Seismic Rays and Snell's Law:
Reminder of waves:
Wavelength, 
Frequency, f
Period, T=1/f
Velocity, V=f
Wavefront
When a sudden stress is applied (or released), the corresponding
strain propagates out from the source in the form of waves.
2D waveforms are lines (surfaces in 3D) that connect points of
equal travel times.
Schematically illustrated wavefronts:
Numerically simulated wavefronts:
Huygen’s principle
This is a geometrical construction that tells us how wavefronts will
move (but not why). It states: “Every point on a wavefront acts as
a source of spherical secondary wavelets such that after some
time dT the primary wavefront lies on the envelope defined by all
the secondary wavelets. The radius of the secondary wavelets will
be V dT, where V is the wave speed.
(Note: the principle is deficient in that it fails to account for the directionality of the wave
propagation in time, i.e., it doesn't explain why the wavefront at time T + dT in the above
figure is the upper rather than the lower envelope of the secondary wavelets. Why does
an expanding spherical wave continue to expand outward from its source, rather than reconverging inward back toward the source?)
There are different types of wavefronts
Body Waves: P-wave
- P stands for ''Primary'' or ''Pressure'’.
- Particles undergo coaxial volume change.
- Similar to sound waves traveling through air.
Animation courtesy of Dr. Dan Russell, Kettering University
Body Waves: S-wave
- P stands for '’Secondary'' or '’Shear'’.
- Particles undergo non-coaxial with no volume change.
Animation courtesy of Dr. Dan Russell, Kettering University
Typical propagation velocities of various materials:
P and S - velocities are a function of the rock density, , and it's
elastic modulus, C:
V
(why elasticity?)
C

Hook’s law (highly simplified):
stress  elastic modulus strain
Shear modulus (or rigidity):


shear stress
shear strain
Bulk modulus:

K

pressure
volume change
P-velocity:
S-Velocity:

VP 
K
4

3


VS 

• VP>VS (since both bulk and shear modulus are positive).

• For liquids and
gases =0, therefore VS=0, and VP is reduced.
• P and S velocities are reduced if the rock is fractured of porous.
Given that:
elastic constant
V
density
how does body waves vary with depth?
Preliminary Reference Earth Model (PREM):
Seismic velocities of denser
rocks are higher, since the
elastic moduli K and  are
also dependent on density.
Surface waves:
Reyleigh waves:
Love:
Animation courtesy of Dr. Dan Russell, Kettering University
•Surface wave velocities are slower than body wave velocities.
•Surface waves are more destructive.
A few seismograms:
Local:
Note that the amplitude of the surface waves is greater than that
of the body waves.
Teleseismic
Note that the P-wave is more visible in the vertical component.
On the other hand, the S-wave is clearly more visible in the
horizontal components.
Sesimic waves and seismic rays
A wave front is a surface connecting all points of equal travel time
from the source.
Rays are normal to the wavefront, and they point in the direction
of wave propagation.
While the mathematical description of the wavefronts is rather
complex, that of the rays is simple. For many applications is it
convenient to consider rays rather than wavefronts.
Fermat’s principle and Snell’s law:
According to Fermat’s principle, a wave propagating from point P
to Q follows a path of minimum time.
P
1
2
Q
What path will take a ray from P to Q?
The travel time from P to Q is:
TPQ
d e
a2  x 2
b 2  (c  x) 2
 


V1 V2
V1
V2
Since the wave speed in each layer is constant, the ray path is
completely defined by the position of P, Q and the point where the

ray crosses
the interface. Since this is a path for which the traveltime is minimum:
dTPQ
x
cx
0

dx
V1 a2  x 2 V2 b2  (c  x) 2
Note that:


x
a x
2
2
 sin( 1) and
cx
b  (c  x)
2
2
 sin(  2 )
Thus we get Snell’s law:
Sin(1) Sin(2 )

P
C1
C2

P is called the ray parameter. Even in a multilayered or a highly
heterogeneous medias, the ray parameter remains constant along
the ray path
Snell’s law
Just as in optics
The angle of reflection equals the angle of incidence, and the
angle of refraction is related through the velocity ratio:
incoming
reflected
sin(  air
) sin(  air
) sin( glass


Vair
Vair
Vglass
refracted
)
P
Seismic rays too obey Snell's law. But conversions from P to S
and vice versa can also occur.
Extended form of Snell’s law
The incidence angle of the reflected and transmitted waves are
controlled by an extended form of the Snell's law:
sin i
1

sin 
1
sin i sin 


P
2
2
Sesimic waves and seismic rays
It is important to understand that the two approaches are not
exactly equivalent.
Consider a planar wavefront passing through a slow anomaly.
Can this anomaly be detected by a seismic network located on the
opposite side?
With increasing distance from the anomaly, the wavefronts
undergo healing. This effect is often referred to as the Wavefront
Healing.
QuickTime™ and a
Compact Video decompressor
are needed to see this picture.
On the other hand, according to the ray theory the travel time from
point A to B is given by:
T 
B
A
B

A
1
dS
C(S)
where dS is the distance measured along the ray, and C is the

seismic velocity.
Thus, a ray traveling through a slow anomaly will arrive after a ray
traveling through the rest of the medium.