Geology Reference
In-Depth Information
D
A
x
Direct ray
Fig. 5.1
Successive positions of the
expanding wavefronts for direct and
refracted waves through a two-layer
model. Only the wavefront of the first
arrival phase is shown. Individual ray paths
from source A to detector D are drawn as
solid lines.
z
θ
v
1
C
B
Refracted ray
v
2
>
v
1
geometries considered below are that the subsurface is
composed of a series of layers, separated by planar and
possibly dipping interfaces. Also, within each layer seis-
mic velocities are constant, and the velocities increase
with layer depth. Finally, the ray paths are restricted to a
vertical plane containing the profile line (i.e. there is no
component of cross-dip).
t
t
i
5.2.1 Two-layer case with horizontal interface
Figure 5.1 illustrates progressive positions of the wave-
front from a seismic source at A associated with energy
travelling directly through an upper layer and energy
critically refracted in a lower layer. Direct and refracted
ray paths to a detector at D, a distance
x
from the source,
are also shown.The layer velocities are
v
1
and
v
2
(>
v
1
) and
the refracting interface is at a depth
z
.
The direct ray travels horizontally through the top
of the upper layer from A to D at velocity
v
1
. The re-
fracted ray travels down to the interface and back up to
the surface at velocity
v
1
along slant paths AB and CD
that are inclined at the critical angle
q
, and travels along
the interface between B and C at the higher velocity
v
2
. The total travel time along the refracted ray path
ABCD is
x
crit
x
cros
x
Fig. 5.2
Travel-time curves for the direct wave and the head wave
from a single horizontal refractor.
Alternatively
12
)
(
2
2
x
v
2
zv
v
vv
-
t
=+
2
1
(5.2)
2
12
or
t
=++
t
t
t
AB
BC
CD
z
xz
v
-
2
tan
)
+
z
x
v
(
q
=
+
t
=+
2
t
(5.3)
i
v
cos
v
cos
q
q
1
2
1
Noting that sin
q
=
v
1
/
v
2
(Snell's Law) and cos
q
=
(1 -
v
2
/
v
2
)
1/2
, the travel-time equation may be ex-
pressed in a number of different forms, a useful general
form being
where, plotting
t
against
x
(Fig. 5.2),
t
i
is the intercept
on the time axis of a travel-time plot or
time-distance
plot
having a gradient of 1/
v
2
. The
intercept time t
i
,
is given by
x
v
2 cos
q
z
12
(
)
t
=+
2
2
2
(5.1)
2
zv
v
vv
-
2
1
v
t
i
=
(from (5.2))
1
12
