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(a)
(b)
S
x
R
t
Reflected wave
a
1
z
1
Head wave
slope 1/
a
2
Direct wave
slope 1/
a
1
A
C
B
a
x
x
x
2
c
cross
Figure 4.34.
(a) Ray paths for seismic energy travelling from source S to receiver R
in a simple two-layer model. The P-wave velocity is
α
1
for the upper layer and
α
2
for
the lower layer, where
α
2
>α
1
. The direct wave takes ray path SR in the upper layer.
The reflected wave takes ray path SCR in the upper layer. The head wave takes ray
path SABR.
(b) Travel-time-distance plots for the model in (a).
aray that travelled down to the interface at the
critical angle i
c
,
8
then along the
interface with the velocity of the lower layer and then back up to the seismometer,
again at the critical angle. (Head waves are second-order waves not predicted by
geometrical ray theory.)
Direct wave
The time taken for energy to reach the receiver directly through the top layer is
simply
x
α
1
t
=
(4.28)
This is the equation of a straight line when time is plotted against distance.
Reflected wave
For the reflected-ray path, the travel time is
SC
α
1
+
CR
α
1
t
=
(4.29)
where SC and CR are the lengths of the ray paths shown in Fig. 4.34. Since
the top layer is uniform, the reflection point C is midway between source S and
seismometer R. Using Pythagoras' theorem, we can write
z
1
+
x
2
4
SC
=
CR
=
(4.30)
where
z
1
is the thickness of the top layer.
8
Snell's law for the interface between two media where
i
is the angle of incidence and
r
the angle of
refraction is
sin
i
sin
r
=
α
1
α
2
When
r
is 90
◦
, the angle of incidence
i
sin
−
1
(
2
)iscalled the
critical angle
.For angles
of incidence greater than the critical angle, no energy is refracted into the second layer.
=
i
c
=
1
/
