Geology Reference
In-Depth Information
Fig. 9.7
Wavefront
propagation (
dashed lines
)
and ray refraction through
a seismic discontinuity
separating two
homogeneous media.
Higher velocity in the
lower layer determines an
increase of spacing —
Fig. 9.8
Formation of a
head wave at the turning
point interface. This is
formed by the envelope of
secondary spherical waves
generated along the
discontinuity (
red lines
)
of both the spacing between wavefronts and the
incidence angle, so that —
2
>—
1
and ™
2
>™
1
.
Conversely, for
s
2
>
s
1
we would have downward
bending of the seismic rays. This phenomenon is
termed
seismic refraction
and is analogous to the
refraction in optics. From (
9.40
)wehave:
wave propagates horizontally (in the ideal limit
of plane waves) through the lower layer with
velocity 1/
s
2
asshowninFig.
9.8
.
During its propagation, the transmitted wave
excites the interface surface between the upper
and lower layer, determining by Huygens's prin-
ciple the formation of a
head wave
(sometimes
called a
bow wave
) that travels upwards in the
direction of the Earth's surface. The seismic rays
associated with this secondary source have the
same incidence angle, ™
c
, as the downgoing rays
(Fig.
9.8
). A complete description of the head
waves is not possible in the context of ray theory
and requires an analysis in terms of waves rather
than seismic rays. It is important to note that
the velocity of propagation of the head wave in
Fig.
9.8
is 1/
s
1
, while the boundary perturbation
always moves at higher velocity 1/
s
2
along the in-
terface. Therefore, any spherical wavefront gen-
erated along the discontinuity surface and propa-
gating through the upper layer will be overtaken
by its source, so that each new wavefront will
start beyond the last one. The resulting envelope
wave is
V
-shaped and tangent to all the emitted
spherical wave fronts, as shown in Fig.
9.8
.Let
us consider now a layered material, such that the
p
D
s
1
sin ™
1
D
s
2
sin ™
2
(9.45)
Therefore, the incidence angle of the transmit-
ted wave will be given by:
™
2
D
arcsin
s
1
s
2
sin ™
1
(9.46)
A
critical incidence angle
, ™
c
,isdefinedasan
incidence angle such that the transmitted ray has
™
2
D
90
ı
. In these case, we say that the ray is at
its
turning point
.From(
9.46
) we easily obtain:
™
c
D
arcsin
s
2
s
1
(9.47)
At the turning point, we have that the slowness
coincides with the ray parameter and the ray
direction becomes horizontal. The corresponding
