GG304 Lecture 19
1
LECTURE 19: SEISMIC WAVE PROPAGATION
Huygen’s principle describes the
advance of a wavefront as a set of
individual spherical waves advancing
from point sources along a previous
location of the wavefront – the tangent
to these secondary wavelets defines
the new wavefront.
When a wavefront encounters a media
with different elastic properties (wave
speeds α1 vs. α 2 ) some of the energy from the incident wave will be reflected
by the boundary and some will be refracted across the boundary.
Huygen’s principal applied to the
reflected waves shows that triangles
ABC and ABD are congruent, which
defines the law of reflection:
The angle of incidence (i) is equal to the angle of reflection (i')
Huygen’s principal applied to the
refracted waves shows that the wave
travels the segment AE=ABsin(r) in
medium 2 during the time that that it
traverses CB=ABsin(i) in medium 1.
This defines the law of refraction:
Clint Conrad
19-1
University of Hawaii
GG304 Lecture 19
sini α1
=
sinr α 2
2
which is known as Snell’s law.
Huygen’s principle allows seismic waves to
bend around the edges of an object. This is
known as diffraction .
Fermat’s principle states that seismic rays
follow the path that gives the shortest travel
h
i i'
times between points. For the intersection of a
ray path with an interface at position x, these
r
travel times can be written as:
"
2%
Reflected waves: t = $ h 2 + x 2 + h 2 + (d − x ) ' α1
#
&
2
Refracted waves: t = h 2 + x 2 α1 + h 2 + (d − x ) α 2
x
h
d-x
Minimizing these travel times as a function of x yields the laws of reflection and
refractions, as above.
An incident P-wave deflects an interface in
directions both parallel to and
perpendicular to the propagation direction
in the second medium, generating refracted
P- and SV-waves, respectively, as well as
reflected counterparts. Fermat’s principle
allows us to calculate their incident angles:
sini p sini s sinrp sinrs
=
=
=
α1
β1
α2
β2
Similarly, an incident SV-wave will induce reflected and refracted SV- and Pwaves. However, a SH-wave does not deform the interface in the vertical
direction, and thus produces only reflected and refracted SH-waves.
Clint Conrad
19-2
University of Hawaii
GG304 Lecture 19
3
If α 2 > α1 , then the refracted angle r
will be greater than the incident
angle i. If the incident angle is equal
to a critical angle ic, then r=90°,
and the refracted ray travels along
the interface. The critical angle can
α
be determined as:
sini c = 1
α2
For incident angles > ic, energy is not lost to refraction, so reflections with large
amplitudes are recorded for distances d > 2h0 sini c (the critical distance).
A travel time diagram shows the first arrivals of seismic energy as a function of
distance away from a
seismic source.
For a slow layer over a
faster halfspace:
Close to the source, the
first arrival will be a direct
P-wave traveling through
the top layer.
Farther from the source, Pwaves refracted through
the faster lower layer will arrive first. From this diagram, we can determine:
V0 and V1 are determined from the inverse of slopes of the travel time curves.
The critical angle is then: i c = sin−1 (V0 V1)
The intercept T01 represents the time for the P-wave to travel through the
upper layer: T 01 = 2h0 cosi c V0
Clint Conrad
which allows us to estimate the thickness h0.
19-3
University of Hawaii