Geology Reference
In-Depth Information
Fig. 9.9
Seismic ray
geometry for a layered
spherical Earth model,
such that the seismic wave
velocity
c
increases with
the layer depth. The
existence of a turning point
does not always depend
from the achievement of a
critical incidence angle
(Eq.
9.47
). In fact, in this
example ™
4
< 90
ı
Fig. 9.10
Seismic rays from a source point
S
for a spher-
ical Earth model such that the seismic wave velocity
c
increases linearly with depth. Both the
range
(horizontal
angular distance travelled by the ray) and the turning point
depth increase as the take-off angle decreases
seismic velocity is constant within each layer
L
k
(
k
D
1,2, :::,
n
) but increases progressively travel-
ling into deeper layers. In this instance, the seis-
mic rays will be broken lines that bend
upwards
until the critical angle is reached. According to
Snell's law (
9.40
)wehavethat:
In a spherical Earth, the upward propagation
of seismic waves and the existence of turning
points is a consequence of the spherical Snell's
law (
9.42
). In this instance, for a velocity
c
that
continuously raises with depth, the downgoing
segment of a seismic ray will bend progressively
through increasing incidence angles, as far as
a turning point depth
z
max
is reached. This is
illustrated in Fig.
9.9
and does not require the for-
mation of head waves. In general, the maximum
depth
z
max
of a seismic ray and the maximum
angular distance, , from the source depend from
the take-off angle ™
0
(Fig.
9.10
).
We shall face now the problem of determining
the range and the arrival time of a seismic ray. Let
us consider first the flat Earth approximation and
assume to have placed a number of receivers (i.e.,
seismometers) at distances
X
1
,
X
2
, :::,
X
n
from a
seismic source
S
. If the velocity
c
D
c
(
z
)is
a monotonically increasing function of depth,
then a measurement of the corresponding arrival
times
T
1
,
T
2
, :::,
T
n
allows to fit these data by a
p
D
s
1
sin ™
1
D
s
2
sin ™
2
D
:::
D
s
n
sin ™
c
D
s
nC1
(9.48)
where ™
1
<™
2
< ::: <™
c
. Therefore, if
s
n
is
the slowness within the layer where the incidence
angle assumes the critical value ™
c
, then the
slowness in the underlying layer will coincide
with the ray parameter
p
. In the limit case that the
velocity increases continuously with depth, we
still have a turning point at some depth
z
D
z
max
and it results:
s
(
z
max
)
D
p
. Head waves form an
important class of seismic waves in exploration
geophysics, where the depths reached by artificial
waves do not exceed a few tens km and the flat
Earth approximation is effective.