Geology Reference
In-Depth Information
plot will curve away from a central straight section. Also,
any lateral change of refractor velocity
v
2
along the pro-
file line will show up as a change of gradient in the minus
term plot.
For the valid range of detectors determined from the
minus-time plot, the delay times can now be calculated.
Adding equations (5.21) and (5.22)
Δ
x
S
1
D
2
D
1
S
2
v
1
v
2
>
v
1
t
2
+=+++
t
l v
2
d
d
d
Fig. 5.13
The generalized reciprocal method of refraction
interpretation (Palmer 1980).
SD
S D
2
t
S
t
S
t
D
1
1
2
Substituting equation (5.20) in the above equation yields
t
+=
t
t
2
d
SD
S D
SS
t
D
1
2
1
2
When computing the plus term for each detector, the
refractor is assumed to be planar between the points of
emergence from the refractor of the forward and reverse
rays, for example between A and B in Fig. 5.12(a) for rays
arriving at detector D.
Hence
1
2
(
)
=
t
+
t
-
t
(5.23)
d
t
D
SDSD
S
1
2
1
2
This delay time is the
plus
term of the plus-minus
method and may be used to compute the perpendicular
depth
z
to the underlying refractor at D using equation
(5.17).
v
2
is found from the minus-time plot and
v
1
is
computed from the slope of the direct ray travel-time
plot (see Fig. 5.12(b)). Note that the value of all delay
times depends on the
reciprocal time
. Errors in this time,
which is recorded at maximum range along the profile,
and often with the lowest signal-to-noise ratio, intro-
duce a constant error into all delay times. Great care must
be taken to check the errors in this value.
A plus term and, hence, a local refractor depth can be
computed at all detector positions at which head wave
arrivals are recognized from both ends of the profile line.
In practice, this normally means the portion of the pro-
file line between the crossover distances; that is, between
x
c1
and
x
c2
in Fig. 5.12(b).
Where a refractor is overlain by more than one layer,
equation (5.17) cannot be used directly to derive a re-
fractor depth from a delay time (or plus term). In such a
case, either the thickness of each overlying layer is com-
puted separately using refracted arrivals from the shal-
lower interfaces, or an average overburden velocity is
used in place of
v
1
in equation (5.17) to achieve a depth
conversion.
The plus-minus method is only applicable in the case
of shallow refractor dips, generally being considered
valid for dips of less than 10°. With steeper dips,
x
¢ be-
comes significantly different from the offset distance
x
.
Further, there is an inherent smoothing of the inter-
preted refractor geometry in the plus-minus method.
5.4.3 The generalized reciprocal method
This problem of smoothing is solved in the
generalized
reciprocal method (GRM)
of refraction interpretation
(Palmer 1980) by combining the forward and reverse
rays which leave the refractor at approximately the same
point and arrive at different detector positions separated
by a distance
D
x
(see Fig. 5.13). The method uses a
velocity analysis function
t
v
given by
(
)
2
t
=
t
+
t
-
t
(5.24)
v
S DS DS S
11
2 2
12
the values being referred to the mid-point between
each pair of detector positions D
1
and D
2
. For the
case where D
1
= D
2
= D (i.e.
D
x
= 0), equation (5.24)
reduces to a form similar to Hagedoorn's minus term
(see above). The optimal value of
D
x
for a particular
survey is that which produces the closest approach to
a linear plot when the velocity analysis function
t
v
is
plotted against distance along the profile line, and is
derived by plotting curves for a range of possible
D
x
values. The overall interpretation method is more
complex than the plus-minus method, but can deliver
better velocity discrimination, greater lateral resolution
and better depth estimates to boundaries. The method
also demands denser data coverage than the plus-minus
method. The principles of the method, its imple-
mentation and example datasets are clearly laid out in
Palmer's topic (Palmer 1980), but beyond the scope of
this one.
