THE SMOOTH SOUNDING GRAPH. A Manual for Field Work ... - BGR

11
to get as a contribution from electrode A to the voltage between M and N
U
( A)
MN
=
Iρ
⎛
⎜
2π
⎜
⎝
As the arrangement is symmetrical we get the same contribution from
electrode B.
This results into
U
MN
=
U
( A)
MN
+ U
( B)
MN
=
L
2
1
−
2 U
For the earth-resistivity ρ we then obtain
or
a
2
A
MN
−
L
2
=
1
+
a
2
⎞
⎟
⎠
Iρ
×
π (
L
2
)
2
a
− (
a
2
)
2
(7)
2 2
π ⎡⎛
L ⎞ ⎛ a ⎞ ⎤ U MN
ρ = ⎢⎜
⎟ − ⎜ ⎟ ⎥
(8)
a ⎢⎣
⎝ 2 ⎠ ⎝ 2 ⎠ ⎥⎦
I
U
ρ = K
,
I
K
=
2 2
π ⎡⎛
L ⎞ ⎛ a ⎞ ⎤
⎢⎜
⎟ − ⎜ ⎟ ⎥
a ⎢⎣
⎝ 2 ⎠ ⎝ 2 ⎠ ⎥⎦
i.e. the well-known formula used in calculating the resistivity ρ from the
current I between A and B and the voltage U between M and N.
K is called "factor of configuration" or "geometric factor". Its dimension is
[m].
We now have to study the function of this factor K. As a homogeneous
underground is concerned ρ be a constant. Working with a constant cur-
rent I - this is technically no problem because I depends after equation
(8) only on the quality of grounding the current electrodes A and B (con-
tact resistance) and not on their distance - the voltage U decreases by
enlarging L. This decrease is compensated by an increasing K. The regu-
lating function of K now shall be discussed regarding both electrode con-
figurations used nowadays in practical field work.