Geology Reference
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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.
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