Geology Reference
In-Depth Information
a quantity known as the
apparent velocity
which is higher
than the refractor velocity when recording along a pro-
file line in the updip direction from the shot point and
lower when recording downdip.
The conventional method of dealing with the possible
presence of refractor dip is to
reverse
the refraction ex-
periment by firing at each end of the profile line and
recording seismic arrivals along the line from both shots.
In the presence of a component of refractor dip along
the profile direction, the
forward
and
reverse
travel time
plots for refracted rays will differ in their gradients and
intercept times, as shown in Fig. 5.5(b).
The general form of the equation for the travel-time
t
n
of a ray critically refracted in the
n
th dipping refractor
(Fig. 5.6; Johnson 1976) is given by
t
x
n
-
Â
1
x
sin
h
cos
+
cos
b
(
a
b
)
Fig. 5.4
Travel-time curves for the direct wave and the head
waves from two horizontal refractors.
1
i
i
i
t
=
+
(5.8)
n
v
v
1
i
=
1
i
where
h
i
is the vertical thickness of the
i
th layer beneath
the shot,
v
i
is the velocity of the ray in the
i
th layer,
a
i
is
the angle with respect to the vertical made by the down-
going ray in the
i
th layer,
b
i
is the angle with respect to the
vertical made by the upgoing ray in the
i
th layer, and
x
is
the offset distance between source and detector.
Equation (5.8) is comparable with equation (5.7), the
only differences being the replacement of
q
by angles
a
and
b
that include a dip term. In the case of shooting
downdip, for example (see Fig. 5.6),
a
i
=
q
in
-
g
i
and
b
i
=
q
in
+
g
i
, where
g
i
is the dip of the
i
th layer and
q
in
=
sin
-1
(
v
1
/
v
n
) as before. Note that
h
is the vertical thick-
ness rather than the perpendicular or true thickness of a
layer (
z
).
As an example of the use of equation (5.8) in inter-
preting travel-time curves, consider the two-layer case
illustrated in Fig. 5.5.
Shooting downdip, along the forward profile
5.2.3 Multilayer case with
horizontal interfaces
In general the travel time
t
n
of a ray critically refracted
along the top surface of the
n
th layer is given by
n
=+
=
-
Â
2
1
1
x
v
z
cos
q
i
in
t
(5.7)
n
v
n
i
i
where
-
1
=
sin
vv
)
q
in
(
i
n
Equation (5.7) can be used progressively to compute
layer thicknesses in a sequence of horizontal strata repre-
sented by travel-time curves of refracted arrivals. In prac-
tice as the number of layers increases it becomes more
difficult to identify each of the individual straight-line
segments of the travel-time plot. Additionally, with in-
creasing numbers of layers, there is less likelihood that
each layer will be bounded by strictly planar horizontal
interfaces, and a more complex model may be necessary.
It would be unusual to make an interpretation using this
method for more than four layers.
x
sin
h
cos
+
cos
b
(
a
b
)
t
=
1
+
1
2
v
v
1
1
x
sin
(
+
)
+
h
cos
(
-
)
qg
qg
12
1
1
12
1
=
v
v
1
1
h
cos
+
(
qg
)
1
12
1
+
5.2.4 Dipping-layer case with
planar interfaces
v
1
x
sin
+
)
+
2
h
cos
cos
(
qg
q g
12
1
1
12
1
=
In the case of a dipping refractor (Fig. 5.5(a)) the value of
dip enters the travel-time equations as an additional un-
known.The reciprocal of the gradient of the travel-time
curve no longer represents the refractor velocity but
v
v
1
1
x
sin
+
2
z
cos
(
qg
)
+
q
12
1
12
=
(5.9)
v
v
1
1
