Geology Reference
In-Depth Information
C 2
Â
n
I
r
p
1
k
r
Ê
Á
ˆ
˜
1
V
P =
+
2
(8.13)
2
r
0
n
=
1
n
r 2
where
C 1
4 z
r 1
2 z
2
2
r
=
r
(
2
nz
)
n
0
C 0
r 0
P
The first term in the brackets of equation (8.13) refers
to the normal potential pertaining if the subsurface were
homogeneous, and the second term to the disturbing
potential caused by the interface. The series is conver-
gent as the dimming factor, or reflection coefficient,
k is less than unity (
ρ 1
z
2 z
r 1
ρ 2
C 1
4 z
r 2
k =-
)
+
)
, cf. Section
(
rrrr
2
(
1
2
1
3.6.1).
Knowledge of the potential resulting at a single point
from a single current electrode allows the computation
of the potential difference D V between two electrodes,
resulting from two current electrodes, by the addition
and subtraction of their contribution to the potential at
these points. For the Wenner system with spacing a
C 2
Fig. 8.10 Parameters used in the calculation of the potential due
to a single surface electrode above a single horizontal interface
using the method of images.
Table 8.1 Distribution and intensity of electrical sources due to a
single horizontal interface.
I
r
p
1
Source
Intensity
Depth/height
Distance
D V
=
14
+
F
(8.14)
(
)
2
a
C 0
I
0
r 0
C 1
k I
2 z
r 1
where
C ¢ 1
k I
2 z
r 1
C 2
k 2 I
4 z
r 2
Â
C ¢ 2
k 2 I
4 z
r 2
1
14
1
44
Ê
Ë
ˆ
¯
n
F
=
k
-
(8.15)
etc
22 2
22 2
+
nz a
+
nz a
n
=
1
Relating this to the apparent resistivity r a measured by
the Wenner system (equation (8.10))
C 2 is the image of C 1 in the medium 1/air interface at
height 2 z ,C 2 is the image of C 1 in the medium 1/2
interface at depth 4 z , etc. Each image in the medium 1/2
interface is reduced in intensity by a factor k , the reflec-
tion coefficient of the interface. (There is no reduction
in intensity of images in the medium 1/air interface, as its
reflection coefficient is unity.) A consequence of the pro-
gressive reduction in intensity is that only a few images
have to be considered in arriving at a reasonable estimate
of the potential at point P. Table 8.1 summarizes this
argument.
The potential V P at point P is the sum of the contribu-
tions of all sources. Employing equation (8.6)
rr
a
=+
1 14 F
(
)
(8.16)
Consequently the apparent resistivity can be computed
for a range of electrode spacings.
Similar computations can be performed for multi-
layer structures, although the calculations are more easily
executed using recurrence formulae and filtering tech-
niques designed for this purpose (see later). Field data can
then be compared with graphs (master curves) represent-
ing the calculated effects of layered models derived by
such methods, a once-important but now little-used
technique known as curve matching . Figure 8.11 shows an
interpretation using a set of master curves for vertical
electrical sounding with a Wenner spread over two
horizontal layers. The master curves are prepared in
I
2
2
kI
2
2
kI
r
2
2
2
k
i
r
p
r
r
r
p
1
0
1
1
1
1
V
=
+
+
+
...
+
+
...
P
2
r
r
r
p
p
2
i
Thus
 
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