Geology Reference
In-Depth Information
(a)
S
2
l
S
1
x
D
v
1
z
B
v
2
>
v
1
A
(b)
t
t
S
1
S
2
t
S
2
D
t
S
1
D
Fig. 5.12
The plus-minus method of refraction
interpretation (Hagedoorn 1959). (a) Refracted ray paths
from each end of a reversed seismic profile line to an
intermediate detector position. (b) Travel-time curves in the
forward and reverse directions.
S
1
x
c1
D
x
c2
S
2
For shallow dips,
x
ยข (unknown) is closely similar to the
offset distance
x
(known), in which case equation (5.18)
can be used in place of (5.19) and methods applicable to
a horizontal refractor employed. This approximation is
valid also in the case of an irregular refractor if the relief
on the refractor is small in amplitude compared to the
average refractor depth (Fig. 5.11(b)).
Delay times cannot be measured directly but occur in
pairs in the travel-time equation for a refracted ray from
a surface source to a surface detector. The
plus-minus
method
of Hagedoorn (1959) provides a means of solving
equation (5.18) to derive individual delay time values for
the calculation of local depths to an irregular refractor.
t
=
x v
+
d
+
d
SD
2
t
S
t
D
(5.21)
1
1
for the reverse ray, from shot point S
2
t
=-
(
l
x
)
v
++
d
d
(5.22)
SD
2
t
S
t
D
2
2
where
d
tD
is the delay time at the detector.
V
2
cannot be obtained directly from the irregular
travel-time curve of refracted arrivals, but it can be
estimated by means of Hagedoorn's
minus
term. This is
obtained by taking the difference of equations (5.21) and
(5.22)
t
2
-= -+-
=-
t
2
2
x v
l v
d
d
SD
S D
2
2
t
S
t
S
1
1
2
5.4.2 The plus-minus interpretation method
Figure 5.12(a) illustrates a two-layer ground model with
an irregular refracting interface. Selected ray paths are
shown associated with a reversed refraction profile line
of length
l
between end shot points S
1
and S
2
.The travel
time of a refracted ray travelling from one end of the line
to the other is given by
(
xl
)
v
+-
d
d
2
t
S
t
S
1
2
This subtraction eliminates the variable (geophone-
station dependent) delay time
d
t
D
from the above equa-
tion. Since the last two terms on the right-hand side of
the equation are constant for a particular profile line,
plotting the minus term (
t
S
1
D
-
t
S
2
D
) against the distance
(2
x
-
l
) yields a graph of slope 1/
v
2
from which
v
2
may
be derived. If the assumptions of the plus-minus method
are valid, then the minus-time plot will be a straight
line.Thus, this plot is a valuable quality control check for
the interpretation method. Often it can be difficult to
locate the crossover distances in real data, especially if
the refracted arrivals line is irregular due to refractor
topography. For minus-time points computed from ar-
rival times which are not from the same refractor, the
t
=++
l v
d
d
(5.20)
SS
t
S
t
S
12
1
2
where
d
t
S
1
and
d
t
S
2
are the delay times at the shot points.
Note that
t
S
1
S
2
is the reciprocal time for this reversed
profile (see Fig. 5.12(b)). For rays travelling to an inter-
mediate detector position D from each end of the line,
the travel times are, for the forward ray, from shot
point S
1
