Sensitivity function for various geoelectric arrays
F. Chitea1, Georgescu P 1
1
University of Bucharest, Faculty of Geology and Geophysics
Resistivity measurements are commonly used for investigating various types of
environmental problems. As a general principle, the acquisition techniques consist in measuring,
between two electrodes (M, N), the difference of potential generated by the electrical current
which is injected into the ground via two other electrodes (the transmitting line AB). Various
type of electrode combinations (A,B,M,N) can be used when investigating a particular area
(Schlumberger, Wenner –α, Wenner –β, Wenner - γ, pole-pole, pole-dipole, dipole-dipole,
gradient), but the selection of the proper geoelectric array to be used for a specific investigation
has to be performed by considering the estimated depth of the target and its characteristics, the
signal strength, the presence or absence of noise sources, the main expected variation in the
electrical resistivity of the geological medium (i.e. along the vertical, along the horizontal, or in
both directions), and the sensitivity of the array.
As expected, the sensitivity function of geoelectric arrays depends on the relative
positions of the electrodes. In terms of resolution and of depth of investigation, each of the above
mentioned arrays has its own specific benefits and limitations. Moreover, for certain arrays, the
field set-up is another element to be carefully considered, in order to avoid violating the
resistivity method basic principles.
Sensitivity can be described by the Frechet derivative. Calculating the sensitivity consists
in determining the alteration that a resistivity variation, generated within a small volume of soil,
induces in the potential drop measured between the MN electrodes. The higher the value of the
sensitivity function, the larger is the recorded influence due to the resistivity change applied to
the small volume of soil (Loke, 2004).
The Frechet derivative, obtained by considering the injection of the electrical current by
means of just one non-linear injection electrode, and by also taking into account only one
receiving electrode, is given by equation (1)(Loke, 2004):
F 3D (x, y, z) =
1
⋅
x( x − a ) + y 2 + z 2
(1)
1.5
1.5
+ y 2 + z 2 ( x − a) 2 + y 2 + z 2
a- electrodes separation
(x,y,z ) indicates the location of the small volume being subject to the resistivity change
(as shown in Fig. 1)
4π 2
[x
2
] [
]
Fig. 1. 2D view of the soil layered model, indicating the configuration of the electrodes
and the volume for which the sensitivity function is calculated.
For quadripolar arrays, a summing accounting for the contribution of the other intervening
electrodes has to be performed, in order to obtain the overall sensitivity.
By studying the 1D distribution of the sensitivity function, Loke has shown that the
Wenner array has a better vertical resolution than the pole-pole array.
In the following, 2D variations in the sensitivity function shall be discussed, specifically
addressing, for different arrays, their lateral and vertical resolutions. Generally speaking, the
contour pattern of the represented sensitivity function is different for the various arrays.
However, a common feature of all the investigated quadripolar and tripolar array types is the
systematic occurrence of the highest sensitivity values in the immediate vicinity of the electrodes.
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Fig.2. Sensitivity distributions for Wenner and Dipole-Dipole acquisition systems
Positive values indicate the regions where the presence of a body which is more resistive
than its background will result in an enhancement of the measured resistivity, while in the areas
with negative sensitivities, the same body will induce a reduction of the measured resistivity.
By analyzing the sensitivity function for various geoelectric arrays (Wenner,
Schlumberger, Dipole-Dipole), noticeable advantages and limitation have been identified. In the
following, we will underline only recommendations to be considered in choosing an array bestsuited to site-specific characteristics:
Wenner array has the strongest signal strength, and this feature recommends the
use of this particular electrode configuration when the survey is carried out in
the presence of geoelectrical noise.
Dipole-dipole array is sensitive when a lateral resistivity variation is
encountered, and it is therefore the best choice when mapping vertical structures.
Pole-pole has the deepest depth of investigation: it is therefore very practical
when the target is located at large depth and displays a resistivity which is highly
contrasting with that of the background.
Not only the benefits, but also shortcomings related to the utilization a specific array can
be deduced by analyzing the sensitivity function. As an example, the pole-pole array has the
widest horizontal coverage and the deepest depth of investigation, but it can pick up a large
amount of telluric noise and has the poorest resolution (Looke, 2004).
When multi-electrode measuring systems are available, one can use hybrid electrode
configurations, in order to minimize the specific weakness of each individual array. Widely used
hybrid configurations are Wenner-Schlumberger and Gradient-Dipole, highly recommended
when complex resistivity distributions are expected (Chitea et al., 2009).
Fig. 3 – Resistivity pseudosections obtained on the synthetic model using:
a) Wenner-Schlumberger array b) Dipole-Gradient array
Results obtained on the synthetic model by using hybrid acquisition techniques (WennerSchlumberger and Gradient-Dipole) are presented as resistivity pseudosections in Fig. 3. By
comparing the resistivity pseudosections displayed in figure 3, which were obtained by using
different acquisition arrays, there can be noticed that the Dipole-Gradient array secures a better
coverage of the in-depth investigation. In areas where both types of geological structures (vertical
and horizontal) are expected to occur, choosing between Wenner and the dipole-dipole array will
severely decrease the quality of the results. Under such circumstances, better results might be
obtained by using the Wenner-Schlumberger array. The signal strength of this hybrid array is
weaker than the one provided by the Wenner array, but the depth of investigation is increased and
lateral coverage is improved when using Wenner-Schlumberger, again by comparison with
Wenner. As compared to the dipole-dipole array, there can be noticed that the signal strength of
the Wenner-Schlumberger is higher, but the laterally covered area is smaller. By comparing the
sensitivity function for various arrays, specific disadvantages can be outlined, and there can be
also issued recommendations for using one set-up or another, such an analysis being
recommended before the beginning of the acquisition campaign.
Electrodes separation is another important issue which has to be considered when geoelectric
measurements are designed. When the depth of the target is large, the length of the transmission
line (AB) is extended, in order to reach the desired investigation depth. Recent studies have
shown great interest for shallower targets as well. The sensitivity function shows that for smaller
electrode separations, the resolution in the vicinity of the surface is increased, but some other
important aspects must be alternatively considered, when smaller electrodes separations are
required.
Electrical prospecting method uses metallic electrodes which are inserted into the ground up to a
depth which depends on the field conditions and on the type of array. Consequently, current
electrodes, as well as potential electrodes, can only approximately be described as pointelectrodes. That circumstance may induce certain difficulties in the interpretation of the resulted
geological data, for which the usual approach relies on theoretical models devised under the
assumption of point sources/sinks of current (Geprgescu P., 1982). In order to illustrate the
difference between values computed under the assumption of a point-electrode, and values
obtained under the more realistic assumption that a real electrode, of finite length, was utilized,
there has been computed the potential generated in a certain point M , by a current source of
length a , that injected a current of intensity I into a homogeneous and isotropic half-space of
resistivity ρ . Assuming that the current distribution was uniform along the electrode length a ,
the corresponding expression is:
V =
I .ρ
r2 + a2 + a
ln
4π a
r2 + a2 − a
(2)
When using a Schlumberger array, the apparent resistivity corresponding to a point-electrode
would be equal to the resistivity of the considered half-space, while the apparent resistivity
corresponding to the line-electrode is:
ρa =
r.ρ
(3)
r2 + a2
The ratio between the apparent resistivity corresponding to the line-electrode and the apparent
resistivity corresponding to a point-electrode is:
F
ξ
ξ2 +1
(4)
F
F=
where ξ =
r
a
1
0.9
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0.3
0.2
0.1
0
F
0
1
2
3
4
5
6
7
8
9
10
x
Fig. 4. Variation of function F
The diagram illustrating the F function variation (fig.4) indicates that the apparent resistivity
computed in the case of a line-electrode is equal to the apparent resistivity corresponding to a
point-electrode only if the distance between the current source and the point M (where the
electrical potential is measured) is much larger that the electrode burying-depth a . The distance
r between the current source and the point M has to be at least four times larger than the
electrode burying-depth, in order to secure differences smaller than 5% between the actually
computed resistivity values and values existing in the ideal case. In addition, by taking into
account the circumstance that both the current electrodes, and the potential electrodes, are
actually linear, the electrodes separation must be at least 5-7 times larger than their buryingdepth. The latter observation is of utmost importance in carrying out resistivity tomographies.
One has to select an optimum value for the ratio of electrodes-separation versus burying-depth, in
order to secure appropriate results, especially when taking into account the circumstance that
within the various electrodes combinations utilized by the multi-electrode measuring systems,
electrodes separations are variable.
References:
Chitea F., Ioane D., Kodom K, (2009) Geoelectrical evaluation of soil properties. Geophysical
Research Abstracts, Vol. 11, EGU2009-11624-4.
Georgescu Paul, (1982), Prospectiuni electrice, Universitatea Bucuresti
Loke, M.H. (2004), Tutorial: 2D and 3D electrical imagining survey.