Geoscience Reference
In-Depth Information
Figure 4.35.
(a) A
three-layer model in
which
α
3
>α
2
>α
1
. (b)
The travel-time-distance
plot for the model in (a).
Only the direct wave
(slope 1/
α
1
) and the two
head waves (slopes 1/
(a)
(c)
HIDDEN LAYER
LOW-VELOCITY LAYER
S x R
S x R
a
1
a
1
z
1
z
1
α
2
a
2
z
2
and 1/
α
3
) are shown. For
distances less than the
critical distance, the
head-wave arrivals are
extended back to the time
axis by dashed lines. For
this particular structure,
the head wave from the
second layer is always a
second arrival; so this
layer cannot be detected
using first-arrival times
alone. It is a hidden layer.
(c) A three-layer model in
which
α
3
>α
1
>α
2
. (d)
The travel-time-distance
plot for the model in (c).
Notice that the reflection
from the interface at the
base of the low-velocity
layer is delayed with
respect to the direct wave.
When there is no
low-velocity zone, this
reflection and the direct
wave are asymptotic
(compare this with Fig.
4.32(b)).
a
2
z
2
a
3
a
3
(b)
(d)
t
t
slope 1/
slope 1/
a
1
slope 1/
a
1
a
2
a
3
slope 1/
slope 1/
a
3
x
x
Note that, although all of the equations have used P-wave velocities
α
1
and
α
2
, they all hold for shear waves as well (velocities would then be
β
2
).
Shear waves are not usually studied in detail in crustal seismology, partly because
of the extra expense of recording horizontal components as well as the vertical
component but mainly because they are second arrivals; it is often very difficult
to pick travel times accurately enough to make a good shear-wave velocity-depth
model.
β
1
and
4.3.3 A multilayered model
The travel times for a model consisting of
n
uniform horizontal layers of thick-
ness
z
j
and P-wave velocity
α
j
are determined in exactly the same way as for
the two-layer model. The only extra matter to be remembered is that the rays
bend according to Snell's law as they cross each interface (sin
i
/
is a constant
along each ray). The travel time for a wave refracted along the top of the
m
th
layer is
α
+
m
−
1
α
2
j
2
z
j
α
j
x
α
m
t
=
1
−
(4.41)
α
m
j
=
1
α
j
<α
m
.
provided that
