Geology Reference
In-Depth Information
Solving for refractor depth
x
A
F
θ
13
tvv
vv
i12
z
=
12
(
)
2
2
-
2
z
1
v
1
2
1
A useful way to consider the equations (5.1) to (5.3) is
to note that the total travel time is the time that would
have been taken to travel the total range
x
at the refractor
velocity
v
2
(that is
x
/
v
2
), plus an additional time to allow
for the time it takes the wave to travel down to the refrac-
tor from the source, and back up to the receiver. The
concept of regarding the observed time as a refractor
travel-time plus
delay times
at the source and receiver is
explored later.
Values of the best-fitting plane layered model para-
meters,
v
1
,
v
2
and
z
, can be determined by analysis of
the travel-time curves of direct and refracted arrivals:
•
v
1
and
v
2
can be derived from the reciprocal of the
gradient of the relevant travel-time segment, see Fig. 5.2
• the refractor depth,
z
, can be determined from the
intercept time
t
i
.
At the crossover distance
x
cros
the travel times of direct
and refracted rays are equal
B
E
θ
23
z
2
v
2
> v
1
C
D
v
3
> v
2
Fig. 5.3
Ray path for a wave refracted through the bottom layer
of a three-layer model.
an offset distance
x
, involving critical refraction at the
second interface, can be written in the form
x
v
2
z
cos
2
z
cos
q
q
1
13
2
23
t
=+
+
(5.5)
v
v
3
1
2
12
)
(
2
2
x
v
x
v
2
zv
v
vv
-
cros
cros
2
1
=
+
where
1
2
12
-
1
-
1
=
sin
vv
)
;
=
sin
v v
)
q
(
q
(
Thus, solving for
x
cros
13
13
3
23
and the notation subscripts for the angles relate directly
to the velocities of the layers through which the ray
travels at that angle (
q
13
is the angle of the ray in layer 1
which is critically refracted in layer 3).
Equation (5.5) can also be written
12
vv
vv
+
-
È
Í
˘
˙
2
1
x
cros
=
2
z
(5.4)
2
1
From this equation it may be seen that the crossover
distance is always greater than twice the depth to the
refractor. Also the crossover distance equation (5.4)
provides an alternative method of calculating
z
.
x
v
t
=++
3
t
t
(5.6)
12
5.2.2 Three-layer case with
horizontal interface
The geometry of the ray path in the case of critical refrac-
tion at the second interface is shown in Fig. 5.3.The seis-
mic velocities of the three layers are
v
1
,
v
2
(>
v
1
) and
v
3
(>
v
2
).The angle of incidence of the ray on the upper inter-
face is
q
13
and on the lower interface is
q
23
(critical angle).
The thicknesses of layers 1 and 2 are
z
1
and
z
2
respectively.
By analogy with equation (5.1) for the two-layer case,
the travel time along the refracted ray path ABCDEF to
where
t
1
and
t
2
are the times taken by the ray to travel
through layers 1 and 2 respectively (see Fig. 5.4).
The interpretation of travel-time curves for a three-
layer case starts with the initial interpretation of the top
two layers. Having used the travel-time curve for rays
critically refracted at the upper interface to derive
z
1
and
v
2
, the travel-time curve for rays critically refracted at
the second interface can be used to derive
z
2
and
v
3
using
equations (5.5) and (5.6) or equations derived from
them.
