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Subsurface DC resistivity mapping: approximate 1-D interpretation
$1 LIEI Journal of Applied Geophysics34 (1996) 255-270
ELSEVIER
Subsurface DC resistivity mapping: approximate 1-D interpretation Youcef Meheni a, Roger Gu~rin b, Yves Benderitter
a, Alain Tabbagh
a,c,*
a Centre de Recherches Gdophysiques, CNRS, 58150 Garchy, France b Pewosystems, Compagnie G~n~rale de Gdophysique, 1 Rue Lg'on Migaux, 91341 Massy, France c D~.partement de G~ophysique Appliqu~e, Universil~ Paris 6, 75252 Paris, France
Received 30 March 1995; accepted 29 October 1995
Abstract Resistivity prospecting is the main tool used to investigate the shallow structure of the ground. A series of new techniques for determining the 2-D and 3-D geometry of the ground is now finding increasing use, but the light and simple Wenner prospecting technique remains a practical and efficient tool for rapidly mapping lateral variations in resistivity. When the resistivity changes are smooth, 1-D modelling can be used to interpret the data, and the criteria governing this approximation can be defined from synthetic data generated by a 3-D slab-model. For a Wenner array, two quadripole configurations can be used, Normal and Dipole-Dipole. For these two configurations the width of the transition zone, the apparent anisotropy effect and the precision of the resistivity values recovered from 1-D inversion differ. However the simultaneous inversion of both sets of data gives better results than for either configuration by itself. Two examples illustrate that in geological contexts where the thickness of the weathered zone causes the changes in the apparent resistivity value, this parameter can be recovered from 1-D inversion.
1. Introduction Resistivity prospecting is commonly used to describe the shallow structure of the ground. The method is well known. The different types of apparatus used are relatively cheap and the experience accumulated over the more than sixty years that the method has been ul;ed is very considerable. The electrical resistivity is very sensitive to granularity and porosity changes, knowledge of which is the major aim of the sut~,eys. Recently, new developments were proposed: AC
* Corresponding author.
resistivity meters that allow continuous measurements while moving (Hesse et al., 1986), the use of electrostatic poles rather than electrodes (Grard and Tabbagh, 1991, Tabbagh et al., 1993, Benderitter et al., 1994) to eliminate the ground-electrode contact problem, multi-electrode systems for profiling (Sorensen and Pedersen, 1992) and for rapid pseudo-sections with automatic switching (Griffiths and Turnbull, 1985, Li and Oldenburg, 1992) and direct inversion. All these improvements were supported by rapid advances in 2-D and 3-D modelling by purely numerical (Dey and Morrison, 1979, Sasaki, 1994) or surface and volume integral methods (Spahos, 1979, Poirmeur and Vasseur, 1988, Dabas et al., 1994). The strategy to be applied in the
0926-9851/96/$15.00 Copyright © 1996 Elsevier Science B. V. All rights reserved. SSDI 0926-9851(95)0(}024-0
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
256
field does, however, have to be considered with particular attention since a 3-D survey remains a major and quite expensive undertaking and the general 3-D interpretation a time consuming and complex process, that is not justified in many cases. Of the various solutions that have to be considered, the rapid extensive mapping of the apparent resistivity using a simple quadripole, supplemented with several soundings (to define the depth of the features that generate lateral variations) will continue to be an
important one. It can constitute either a first step prior to a 2-D or 3-D multi-electrode investigation at a particular location where detailed knowledge of the structure of the ground is needed, or a sufficient answer by itself in delineating significant features. In such a case, a chain of four electrodes can be dragged along the ground surface but, where this is not possible, the method using a simple Wenner quadripole with independent electrodes remains a cheap and light solution. As illustrated in Fig. 1 the
Meter
L---
P
P
Weaner N Weaner DD
N DD
I
I ~
I
I
I
P
i
JL
P
I ~
I
P
P
I
I
I
I "
I
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P
I
N
P
I
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bl
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i
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Fig. 1. Measurement method in both normal (N) and dipole-dipole (D-D) configuration using a switch and only one moving electrode .
Y. Meheni et al. / Journal of Applied Geophysics34 (1996) 255-270 spatial sampling step is equal to the inter-electrode separation, a, and two independant measurements can be made at e~ch quadripole position, in both normal (N) and dir.ole-dipole ( D - D ) Wenner configurations, by switching only one potential and one current electrode (]Jesse and Spahos, 1978). Two independant resistivity maps are then available. From these two maps it is possible to estimate qualitatively the variations of the electrical parameters of the ground taking into account the geological context. The purpose of this paper is to define under which conditions and what criteria a 1-D modelling inversion process can be applied to the apparent resistivity data obtained using simple Wenner quadripoles. The advantages of such an approximate inversion will then be illustrated by means of examples. For this purpose 1-D and 3-D modelling data for features of significant lateral e~:tent will first be compared. The 1-D modelling uses the calculation proposed by St6phanesco et al. (1930) and Flathe (1955) whereas the 3-D modelling employs the method of moments (Dabas et al., 1994). 2. Theoretical study of 1-D modelling limits in a simple 3-D context for a Wenner array A two-metre thick square slab embedded horizontally at a depth of 1.5 m is considered. The sides L of the slab can be of 6, 18 or 30 m (Fig. 2c). Its resistivity is 500 L)m when resistive and 20 g2m when conductive. The slab is located in a 100 gJm homogeneous ground. A Wenner array with a 5 m inter-electrode spacing for both N and D - D configurations is used. As the law of similarity states, multiplying each geometrical dimension by a constant factor would generate similar resttlts. The conclusions drawn from the present case can be extended to all similar models. The 1-D model corresponding to the slab has three layers: a first layer, Pl = 100 J2m, e 1 = 1.5 m; a second layer, P2 =: 500 J2m or 20 J2m, e 2 = 2 m; and a third layer, P3 = 100 O m . The apparent resistivity (half-)profiles obtained from "in-line" arrays are shown in Fig. 2a and b (the four electrodes "in-line" along the profile). The apparent resistivity maps are shown in Figs. 3 - 5 for the slabs with sides of 6, 18 and 30 m, respectively.
257
2.1. Magnitude and lateral extensions of anomalies and of parameters recovered by 1-D modelling The apparent resistivity profiles are presented in Fig. 2a and b, the thickness of the bodies recovered by 1-D inversion in Fig. 2c and the lateral variations of the resistivity recovered by 1-D inversion in Fig. 2d. All these figures and Table 1 show that while the agreement is not good for the 6 m slab data, the 1-D modelling works well for the data in the central parts of the 18 m and 30 m slabs. This suggests a general explanation: the 1-D approximation is good if the sides of the slab are longer than the array size. But the width of the transition zone depends mainly on the array and requires more comment. It is of comparable extent for both a resistive and a conductive slab in each type of configuration. The D - D configuration however gives shorter transitions, indicating the advantage of the D - D configuration in locating lateral resistivity changes. The slopes of the curves are roughly similar when L = 18 m and L = 30 m. It can therefore be concluded that when L is greater than the array, the size and configuration of the latter are the only parameters determining the width of the transition zone. This width can be defined as the distance separating the point where the apparent resistivity value equals that of the homogeneous layer plus ten percent (respectively - 1 0 % for a conductive slab) and the point where it equals that of the three layers minus ten percent. This width equals 3a for the N configuration and a for the D - D one in the case of in-line profiles (Fig. 2a-c). Consequently the central quasi-flat zone has a width of around L-a in the D - D and around L - 3 a in the N Wenner configuration. The following criterion can be adopted: if the lateral extent of a horizontal layer equals or exceeds the total size of the array (3a), the application of a 1-D inversion is justified. 2.2. Apparent anisotropy As can be observed (Figs. 4 and 5) the transition zone is often different depending on whether the side of the slab is parallel or perpendicular to the array. In this case the central flat area is not square. This is called an "apparent anisotropy effect" and it consti-
Y. Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
258
200.
apparent
resistivity(G.m)
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Y. Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
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Fig. 3. Apparent resistivity maps for a 6 X 6 m slab and an a = 5 m Wenner array: (a) conductive slab and N configuration, (b) conductive slab and D - D configuration, (c) resistive slab and N configuration, (d) resistive slab and D - D configuration.
Fig. 2. (a) Apparent resistivity profiles for three different slabs using an in-line N Wenner quadripole, (b) Apparent resistivity profiles for three different slabs using; an in-line D - D Wenner quadripole, (c) Recovered thickness of conductive slabs (18 m and 30 m square) after 1-D inversion (bold line for N configuration, thin line for D - D configuration), (d) Recovered resistivity of conductive slabs after 1-D inversion.
260
Y. Meheni et al. // Journal of Applied Geophysics 34 (1996) 255-270
tutes a severe difficulty in electrical or EM resistivity mapping. There are various methods to correct for this effect. The effect does not exist for a 1-D tabular
field and can be a criterion for inversion. The effect is very marked when the sides of the slab are smaller than the array size: in Fig. 3 only the response of a
(a)
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(c)
96. 10
(d)
121. 143'. 170'. 20m
214'.
n.m
96. 10
121. 143'. 170'. 20m
Fig. 4. Apparent resistivity maps for an 18 x 18 m slab.
214'. ~.m
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
resistive slab to a Wenner D - D configuration suggests that the width and length of the buried body are of equivalent size. Contrary to the case of Sligram EM prospecting
(Gudrin et al., 1995), applying this criterion is not easy and in practice will probably not be used because rotating the array by 90 ° in the field is time consuming.
(a)
37. 10
:::::::::::::::::::::::: ::: :::::::::::.:::::::: ::::::::::::::::::::::::: :::::::::::::::::::::::::
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(b)
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(e) 96.
261
(d)
120. 143. 169'. 20m
212'.
fl.m
96. i0
20m
Fig. 5. Apparent resistivity maps for a 30 X 30 m slab.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
262
Table 1 Ratio of the apparent resistivity calculated directly above the slab centre to the apparent resistivity for the case of a 3-layer I - D model
6× 6 m 18 × 18 m 30 × 3 0 m
N Wenner
D-D Wenner
Conductive
1.58
2.01
Resistive
0.60
0.77
Conductive
1.13
0.90
Resistive
0.91
1.05
Conductive
1.01
0.97
Resistive
1.01
1.00
• a conductive layer of thickness e and resistivity p embedded between two resistant ones can be replaced by one or several conductive layer(s) such that: E e i / P i = e / p = S ("conductance") i
The fact that the apparent anisotropy effect is less pronounced for the D - D configuration than for the N one must be emphasised.
where e i and pi are the thickness and resistivity respectively of the remaining layer(s) a resistive layer embedded between two conductive ones can be replaced by one or several resistive layer(s) such that:
~e~.p~=e.p=T i
( "transverse resistance").
2.3. 1-D inversion using 3-D theoretical data The 1-D inversion is subject to the rules of equivalence (Maillet, 1947) which state that:
It can be concluded from these rules that the relative sensitivity of the inversion process to resistivity or thickness variations will be equivalent for
Table 2 Resistivity P2-, thickness e 2 , c o n d u c t a n c e a n d transverse resistance of the second layer obtained by 1-D i n v e r s i o n c o m p a r e d with the resistivity P2 a n d the thickness e 2 of the slab 3 0 × 30 m slab
18 × 18 m slab
N Wenner (a = 5 m)
Conductive
Resistive
Conductive
Resistive
e~ ( m )
1.51
1.46
1.95
2.09 515
p~ ( O m )
24
395
20
S'(s)
0.08
-
0.1
-
AS'/S
- 20%
-
0%
-
T ' ( O m 2)
-
790
-
1030
AT'/T
-
- 21%
-
+ 3%
Ae~/e 2
- 24%
-- 2 7 %
- 2.5%
+ 4.5%
Ap'2/P2
+ 20%
- 21%
0%
+ 3%
D-D Wenner
e~ ( m )
2.53
2.61
2.13
1.95
(a = 5 m)
p~ ( g 2 m )
17
580
19
495
S'(s) /tS'/S
0.118
-
0.105
-
+ 18%
-
+ 5%
-
T ' ( g 2 m 2)
-
1160
--
990
AT'/T
-
+ 16%
-
- 1%
Ae~/e 2
+ 26%
+ 30%
+ 6.5%
- 2.5%
Ap'z/p2
- 15%
+ 16%
-5%
- 1%
Simultaneous inversion
%e~ (m)
1.88
1.84
2.03
2.04
N and D-D
p~ ( O m )
21
477
20
505
S'(s) AS'/S
0.095 - 5%
-
0.1 0%
-
T ' ( O m 2)
-
954
-
1010
AT'/T
--
--4.5%
--
+ 1%
Ae~ 1 / e 2
-- 6 %
- 8%
+ 1.5%
+ 2%
Ap'2/P2
Jr- 5 %
-- 4 . 5 %
0%
+ 1%
Wenner
E Meheni et al./Journal of Applied Geophysics 34 (1996) 255-270
both a resistive and a conductive slab, contrary to the case for the EM situation. Table 2 shows the results obtained at the center of the slab, first for conductance and transverse resistance and then for the resistivity P2 and thickness e 2 of the slab, when the other parameter is fixed at the correct value. The results of the inversion when using both configuration data simultaneously are shown at the bottom of the table. As can be expected from the previous figures, the results are good for the 30 × 30 m slab and only barely correct for the 18 × 18 m one. In all cases the precision for P2 is better than for e 2. For the 18 X 18 m slab, the N configuration reduces the conductance and the transmittance while the D - D configuration increases these parameters. The very good results achieved when inverting simultaneously the data of both configurations should be emphazised. For the 18 × 18 m slab the relative error is reduced from around 20% to around 5%. This conclusion is confirmed by the mapping of P2 after inversion (Figs. 6 and 7): the anisotropy effect is reduced (compared to Figs. 4 and 5) and the shape of the anomaly is more representative of the heterogeneity. When the simultaneous inversion is applied to the 6 × 6 m slab, the e~ or P2 values remain inexact but the shape of the anc,maly is correct (Fig. 8) and the anisotropy effect considerably reduced. This type of inversion could be an interesting general technique to correct for appare, nt anisotropy when both N and D - D Wenner data are available.
263
(a)
19. 0
10
31.
44.
63'.
i01'. Q.m
20m
2.4. Theoretical conclusions
The 1-D inversion can be validly applied when the size of the 3-D feature is larger than the total array length. The transition zone is wider in the N Wenner configuration than in the D - D one. When seeking one parameter, the simultaneous inversion of both data gives better results than two separate inversions.
(b) 99.
3. Experimental cases
The application of the approximate 1-D inversion to current data implies two types of assumptions.
0
10
158. 225'. 319'.
510. fl.m
20m
Fig. 6. Maps of the resistivity of the second layer after a 1-D simultaneous inversion for a 30 × 30 m slab: (a) conductive slab, (b) resistive slab.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
264
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45'.
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63'.
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Fig. 8. Maps of the resistivity of the second layer after a I-D simultaneous inversion for a 6 X 6 m slab.
Y. Meheni et a l . / Journal of Applied Geophysics 34 (1996) 255-270
The first assumption,as discussed in the previous section, is that the lateral resistivity changes slowly. The second assumption is more complex: with one (or two) series of apparent resitivity data available, only one (or two) parameter(s) can be calculated.
265
Therefore, the use of the proposed inversion is limited to geological situations where lateral variations are likely to be explained by variations in one (or two) parameter(s) only. This case is common in near-surface investigations and corresponds to situa-
(a)
.
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K Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
266
I
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Fig. 10. Thickness of the conductive layer at "Le Gu4 Goujard" after simultaneous reversion using both N and D - D Wenner data.
tions where the sound bedrock is covered by a weathered layer itself covered by a topsoil. The second layer often has a rather stable and low resis-
1.5 0
25
50m
.7
tivity. Its thickness may vary with the degree of weathering, which then directly effects the observed apparent resistivity variations. Such situations appear
'.4
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/
Fig. 1 h Thickness of the conductive layer at "Le Gu6 Goujard" after approximate 1-D inversion using N Wenner data, D - D Wenner data and EM 31 resistivities.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
267
in carbonate areas of the Paris Basin. The groundwater circulation in the middle Jurassic limestone (Roy, 1983) is governed by fractures in a sound rock of relatively high resistivity (250-1000 g2m). These fractures are often too narrow and too deep to be directly detected. However, the alteration is more pronounced and the weathered layer thicker above these fractures. This layer contains an abundance of fine particles, generating a good conductivity. Under these conditions resistivity mapping is a useful technique for locating the fracture zones which play an important part in ground-water ciculation (Benderitter and Robin, 1987, Meheni et al., 1996). The two following examples illustrate the application of the approximate 1-D inversion in such a geological carbonate environment. (a) N
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3.1. Le Gu£ Goujard (parcel 8), St Martin sur Nohain, Ni~vre, France
Bois
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array, a = 5 m.
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This parcel is located near a river and presents a gently sloping surface from NE to SW. The survey was carried out using a 5 m square grid and a 5 m inter-electrode distance. The conductive layer above the unweathered limestone becomes thicker in three linear zones (Fig. 9a and b). Two of these zones are parallel to the slope, the third lies in the centre of the map and is orientated SW-NE. The resistivity exceeds 300 g2m in the S and NW of the map which suggests that the limestone is directly below the topsoil. The information from both maps is very coherent, the variations being more pronouced in the D - D configuration. Two electrical soundings led to the following results: p~ = 50 J2m and e I = 0.3 m for the topsoil layer, P2 = 25 g2m for the conductive layer and P3 = 250 /2 m for the limestone. The map of the thickness of the conductive layer e 2 (Fig. 10) is the result of the simultaneous 1-D inversion using both N and D - D Wenner data. The thickness is less than 0.1 m in the resistive zones and reaches 1.7 m in the third conductive one. The variations are smoother on the e 2 map than on the apparent resistivity map. The values seem very realistic ones and in agreement with the results obtained by inverting the EM 31 resistivity map (Gu4rin et al., 1995). The simultaneous inversion of the three ap-
Y. Meheni et a l . / Journal of Applied Geophysics 34 (1996) 255-270
268
parent resistivity data sets (Fig. 11) gives similar results.
3.2. Le Bois Rond, Garchy, Nibvre, France The surveyed area is located on a flat plateau. Both resistivity maps (Fig. 12a and b), obtained with a 5 m square grid and a 5 m inter-electrode distance, show very strong variations from 40 S2m to 400 g2m. The high resistivity values very clearly delineate an area of limestone outcrop. Five soundings performed on different zones, two in the NE of area, one in the more conductive zone and two on the resistive one, indicate a three layer system with a conductive weathered layer of varyiable thickness. The map of this thickness (Fig. 13), which varies between 0 and approximately 5 m, was produced using the following values: Pl = 54 J2m, e~ = 0.4 m; P2 = 30 g2m and P3 = 450 O m . Once again, the simultaneous inversion of both N and D - D configuration data reduces the high frequency variations and
[
::::::::::::::::::::::
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the results are coherent with those for the EM 31 apparent resistivity (Fig. 14).
4. Conclusion
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In near-surface surveys the interpretation of apparent resistivity data can be improved in cases where a complete 3-D inversion process is too involved for practical use or the density of data insufficient to allow it. The proposed approximate 1-D inversion is a very quick and simple method that allows a first interpretation in cases where only one parameter varies significantly. The results are significant when the geological context justifies this major assumption. This is the case in a carbonate environment as demonstrated in the examples. It would therefore be interesting to test its application in crystalline bedrock areas where the main parameter is also the thickness of the weathered overburden layer.
Y. Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
As noticed in combining both theoretical and experimental data, this type of processing opens new approaches to correcting for the apparent anisotropy effect and obtaining unbiased maps of lateral variations, even if the ~zalues of the restored parameter can not be considered to be exact.
References Benderitter, Y., Jolivet, A., Mounir, A. and Tabbagh, A., 1994. Application of the electrostatic quadripole to sounding in the hectometric depth range. J. Appl. Geophys., 31: 1-6. Benderitter, Y. and Robin, G., 1987. Approche singulibre des discontinuitts de perratabilit6 en milieu carbonatt. Hydrogtologie, 3: 161-168. Dabas, M., Tabbagh, A. and Tabbagh, J., 1994. 3-D inversion in subsurface electrical surveying - I. Theory. Geophys. J. Int., 119: 975-990. Dey, A. and Morrison, H.F., 1979. Resistivity modelling for arbitrarily shaped two-dimensional structures. Geophys. Prospect., 27: 106-136. Flathe, H., 1955. A praclical method of calculating model graphs for horizontally stratified media. Geophys. Prospect., 3: 268294. Griffiths, D.H. and Turnbull, J., 1985. A multi-electrode array for resistivity surveying. First Break, 3: 16-20. Grard, R. and Tabbagh, A., 1991. A mobile four electrode array and its application to ~.he electrical survey of planetary grounds at shallow depth. J. Geophys. Res., 96 - B3: 4117-4123. Gutrin, R., Meheni, Y., Rakotondrasoa, G. and Tabbagh, A., 1995. Interpretation of Slingram conductivity mapping in near surface geophysics: u~sing a simple parameter fitting with I-D model. Geophys. Prospect., 44: 233-249.
269
Hesse, A. and Spahos, Y., 1978. Evaluation of Wenner and dipole-dipole resistivity measurements and the use of a new switch for archaeological field work. Archaeophysika, 10: 647-655. Hesse, A., Jolivet, A. and Tabbagh, A., 1986. New prospects in shallow depth electrical surveying for archaeological and pedological applications. Geophysics, 51: 585-594. Li, Y. and Oldenburg, D.W., 1992. Approximate inverse mapping in DC resistivity problems. Geophys. J. Int., 109: 343-362. Maillet, R., 1947. The fundamental equations of electrical prospecting. Geophysics, 12: 529-556. Meheni, Y., Benderitter, Y., Gutrin, R. and Tabbagh, A., 1996. Comparaison de difftrentes m&hodes pour la cartographie d&aillte de la rtsistivit6 ~lectrique du sous-sol dans la gamme de profondeur 2-20 m. Gtologues, 109: 25-36. Poirmeur, C. and Vasseur, G., 1988. Three-dimensional modelling of a hole-to-hole electrical method: application to the interpretation of a field survey. Geophysics, 53: 402-414. Roy, B., 1983. Gtologie, gtophysique, hydrogtologie des formations jurassique moyen dans le Nord-Ouest de la Nibvre. Th~se de 3brae cycle, Universit6 de Dijon, 174 pp. Sasaki, Y., 1994. 3-D resistivity inversion using the finite-element method. Geophysics, 59: 1839-1848. Sorensen, K. and Pedersen, F.F., 1992. Slaebe-geoelektrik, Miljoministeriet. Skov-og. Naturstyrelsen, 2, 44 pp. Spahos, Y., 1979. Calculs sur module et rtle du quadriptle en prospection ~lectrique de subsurface. Application ~t la dttection archtologique, Th~se de 3~me cycle, Universit6 P and M Curie, Pads. Sttphanesco, S., Schlumberger, C. and Schlumberger, M., 1930. Sur la distribution 61ectrique potentielle autour d'une prise de terre ponctuelle dans un terrain ~ couches horizontales, homogbnes et isotropes. J. Phys. Radium, 7: 132-140. Tabbagh, A., Hesse, A. and Grard, R., 1993. Determination of electrical properties of the ground at shallow depth with an electrostatic quadripole: field trials on archaeological sites. Geophys. Prospect., 41: 579-597.
ELSEVIER
Subsurface DC resistivity mapping: approximate 1-D interpretation Youcef Meheni a, Roger Gu~rin b, Yves Benderitter
a, Alain Tabbagh
a,c,*
a Centre de Recherches Gdophysiques, CNRS, 58150 Garchy, France b Pewosystems, Compagnie G~n~rale de Gdophysique, 1 Rue Lg'on Migaux, 91341 Massy, France c D~.partement de G~ophysique Appliqu~e, Universil~ Paris 6, 75252 Paris, France
Received 30 March 1995; accepted 29 October 1995
Abstract Resistivity prospecting is the main tool used to investigate the shallow structure of the ground. A series of new techniques for determining the 2-D and 3-D geometry of the ground is now finding increasing use, but the light and simple Wenner prospecting technique remains a practical and efficient tool for rapidly mapping lateral variations in resistivity. When the resistivity changes are smooth, 1-D modelling can be used to interpret the data, and the criteria governing this approximation can be defined from synthetic data generated by a 3-D slab-model. For a Wenner array, two quadripole configurations can be used, Normal and Dipole-Dipole. For these two configurations the width of the transition zone, the apparent anisotropy effect and the precision of the resistivity values recovered from 1-D inversion differ. However the simultaneous inversion of both sets of data gives better results than for either configuration by itself. Two examples illustrate that in geological contexts where the thickness of the weathered zone causes the changes in the apparent resistivity value, this parameter can be recovered from 1-D inversion.
1. Introduction Resistivity prospecting is commonly used to describe the shallow structure of the ground. The method is well known. The different types of apparatus used are relatively cheap and the experience accumulated over the more than sixty years that the method has been ul;ed is very considerable. The electrical resistivity is very sensitive to granularity and porosity changes, knowledge of which is the major aim of the sut~,eys. Recently, new developments were proposed: AC
* Corresponding author.
resistivity meters that allow continuous measurements while moving (Hesse et al., 1986), the use of electrostatic poles rather than electrodes (Grard and Tabbagh, 1991, Tabbagh et al., 1993, Benderitter et al., 1994) to eliminate the ground-electrode contact problem, multi-electrode systems for profiling (Sorensen and Pedersen, 1992) and for rapid pseudo-sections with automatic switching (Griffiths and Turnbull, 1985, Li and Oldenburg, 1992) and direct inversion. All these improvements were supported by rapid advances in 2-D and 3-D modelling by purely numerical (Dey and Morrison, 1979, Sasaki, 1994) or surface and volume integral methods (Spahos, 1979, Poirmeur and Vasseur, 1988, Dabas et al., 1994). The strategy to be applied in the
0926-9851/96/$15.00 Copyright © 1996 Elsevier Science B. V. All rights reserved. SSDI 0926-9851(95)0(}024-0
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
256
field does, however, have to be considered with particular attention since a 3-D survey remains a major and quite expensive undertaking and the general 3-D interpretation a time consuming and complex process, that is not justified in many cases. Of the various solutions that have to be considered, the rapid extensive mapping of the apparent resistivity using a simple quadripole, supplemented with several soundings (to define the depth of the features that generate lateral variations) will continue to be an
important one. It can constitute either a first step prior to a 2-D or 3-D multi-electrode investigation at a particular location where detailed knowledge of the structure of the ground is needed, or a sufficient answer by itself in delineating significant features. In such a case, a chain of four electrodes can be dragged along the ground surface but, where this is not possible, the method using a simple Wenner quadripole with independent electrodes remains a cheap and light solution. As illustrated in Fig. 1 the
Meter
L---
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P
Weaner N Weaner DD
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I
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i
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Y. Meheni et al. / Journal of Applied Geophysics34 (1996) 255-270 spatial sampling step is equal to the inter-electrode separation, a, and two independant measurements can be made at e~ch quadripole position, in both normal (N) and dir.ole-dipole ( D - D ) Wenner configurations, by switching only one potential and one current electrode (]Jesse and Spahos, 1978). Two independant resistivity maps are then available. From these two maps it is possible to estimate qualitatively the variations of the electrical parameters of the ground taking into account the geological context. The purpose of this paper is to define under which conditions and what criteria a 1-D modelling inversion process can be applied to the apparent resistivity data obtained using simple Wenner quadripoles. The advantages of such an approximate inversion will then be illustrated by means of examples. For this purpose 1-D and 3-D modelling data for features of significant lateral e~:tent will first be compared. The 1-D modelling uses the calculation proposed by St6phanesco et al. (1930) and Flathe (1955) whereas the 3-D modelling employs the method of moments (Dabas et al., 1994). 2. Theoretical study of 1-D modelling limits in a simple 3-D context for a Wenner array A two-metre thick square slab embedded horizontally at a depth of 1.5 m is considered. The sides L of the slab can be of 6, 18 or 30 m (Fig. 2c). Its resistivity is 500 L)m when resistive and 20 g2m when conductive. The slab is located in a 100 gJm homogeneous ground. A Wenner array with a 5 m inter-electrode spacing for both N and D - D configurations is used. As the law of similarity states, multiplying each geometrical dimension by a constant factor would generate similar resttlts. The conclusions drawn from the present case can be extended to all similar models. The 1-D model corresponding to the slab has three layers: a first layer, Pl = 100 J2m, e 1 = 1.5 m; a second layer, P2 =: 500 J2m or 20 J2m, e 2 = 2 m; and a third layer, P3 = 100 O m . The apparent resistivity (half-)profiles obtained from "in-line" arrays are shown in Fig. 2a and b (the four electrodes "in-line" along the profile). The apparent resistivity maps are shown in Figs. 3 - 5 for the slabs with sides of 6, 18 and 30 m, respectively.
257
2.1. Magnitude and lateral extensions of anomalies and of parameters recovered by 1-D modelling The apparent resistivity profiles are presented in Fig. 2a and b, the thickness of the bodies recovered by 1-D inversion in Fig. 2c and the lateral variations of the resistivity recovered by 1-D inversion in Fig. 2d. All these figures and Table 1 show that while the agreement is not good for the 6 m slab data, the 1-D modelling works well for the data in the central parts of the 18 m and 30 m slabs. This suggests a general explanation: the 1-D approximation is good if the sides of the slab are longer than the array size. But the width of the transition zone depends mainly on the array and requires more comment. It is of comparable extent for both a resistive and a conductive slab in each type of configuration. The D - D configuration however gives shorter transitions, indicating the advantage of the D - D configuration in locating lateral resistivity changes. The slopes of the curves are roughly similar when L = 18 m and L = 30 m. It can therefore be concluded that when L is greater than the array, the size and configuration of the latter are the only parameters determining the width of the transition zone. This width can be defined as the distance separating the point where the apparent resistivity value equals that of the homogeneous layer plus ten percent (respectively - 1 0 % for a conductive slab) and the point where it equals that of the three layers minus ten percent. This width equals 3a for the N configuration and a for the D - D one in the case of in-line profiles (Fig. 2a-c). Consequently the central quasi-flat zone has a width of around L-a in the D - D and around L - 3 a in the N Wenner configuration. The following criterion can be adopted: if the lateral extent of a horizontal layer equals or exceeds the total size of the array (3a), the application of a 1-D inversion is justified. 2.2. Apparent anisotropy As can be observed (Figs. 4 and 5) the transition zone is often different depending on whether the side of the slab is parallel or perpendicular to the array. In this case the central flat area is not square. This is called an "apparent anisotropy effect" and it consti-
Y. Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
258
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:::::~:::::::::::~:~:::::::::::::::::::::::::::::::::::::::::::::::::::::~:::::::::::::::~::::::::::::::::::: ::::::::::::::::::::::::::::::::::::~:~:::::::::::::::~::::::::::::::::::::~::::~::::::::::~::;:::::::~::::~::
(c) I
0
i:~i~m;m:#~ m~- ' 97. 111. 122. 135'. I0
20m
(d) I
154'. fl,m
!ri!i~!:~!i!~!i!i!:!!!~!~--~~,
97. 0
I0
iii. 122. 135'.
154'. fl.m
20m
Fig. 3. Apparent resistivity maps for a 6 X 6 m slab and an a = 5 m Wenner array: (a) conductive slab and N configuration, (b) conductive slab and D - D configuration, (c) resistive slab and N configuration, (d) resistive slab and D - D configuration.
Fig. 2. (a) Apparent resistivity profiles for three different slabs using an in-line N Wenner quadripole, (b) Apparent resistivity profiles for three different slabs using; an in-line D - D Wenner quadripole, (c) Recovered thickness of conductive slabs (18 m and 30 m square) after 1-D inversion (bold line for N configuration, thin line for D - D configuration), (d) Recovered resistivity of conductive slabs after 1-D inversion.
260
Y. Meheni et al. // Journal of Applied Geophysics 34 (1996) 255-270
tutes a severe difficulty in electrical or EM resistivity mapping. There are various methods to correct for this effect. The effect does not exist for a 1-D tabular
field and can be a criterion for inversion. The effect is very marked when the sides of the slab are smaller than the array size: in Fig. 3 only the response of a
(a)
(b)
I 36. I0
50.
63.
81'.
111'. ~ .m
i~ii:~iiii!~;~i!~;! i ~ 36.
20m
i0
::::::::::::::::~::::::::::::::::::~::::.:::::::::::::::::~:m::::::::::::::::~:~::::::::::::::::::::~:È:::
50.
~ 63.
81'.
111'. ~ .m
20m
:::::~:::::::::::::::::::::::::::::::~::::::~:~:::~::::::::::::.:.~.::::~::::::.:::::::::::::::::~::::::.::::
::::::.::.:::::::::.::::::::::::..:::::::::::.::::.:::::.:::::m.:::::::::.:.:.::::::::::::::::x:::::::::::: ::::::::::::::::::::::::::::.:::::::::::::.:::::::::x::::::::.::::::::::.::::::::.::::::::::::::..::::::::.::
~ :::::.:::~::::::::'::':::::::~:::::::::::::::::::~:
........
::::x.::::::::::::::.::::-::.:::::::::::::::::.:.:
:::::::::::::::::::::::::::::::u:_:::::::::::::::::::::.,:.: x:::::::::::::::::::::::::: , :~:.::::::::::::: :~.:::::::~4:::::::::::::::::::.~::::~:::.:~::~:.::::::::::::::~.::::.::::::::~.::::::::::::::.:::.::: -
(c)
96. 10
(d)
121. 143'. 170'. 20m
214'.
n.m
96. 10
121. 143'. 170'. 20m
Fig. 4. Apparent resistivity maps for an 18 x 18 m slab.
214'. ~.m
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
resistive slab to a Wenner D - D configuration suggests that the width and length of the buried body are of equivalent size. Contrary to the case of Sligram EM prospecting
(Gudrin et al., 1995), applying this criterion is not easy and in practice will probably not be used because rotating the array by 90 ° in the field is time consuming.
(a)
37. 10
:::::::::::::::::::::::: ::: :::::::::::.:::::::: ::::::::::::::::::::::::: :::::::::::::::::::::::::
50.
63.
(b)
79'.
106.
~.m
20m
37. 0
I0
50.
79.
106.
~.m
10
63.
120. 143. 169.
212.
f~.m
20m
::::::.:::::::::::. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::: ===================================================================================
(e) 96.
261
(d)
120. 143. 169'. 20m
212'.
fl.m
96. i0
20m
Fig. 5. Apparent resistivity maps for a 30 X 30 m slab.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
262
Table 1 Ratio of the apparent resistivity calculated directly above the slab centre to the apparent resistivity for the case of a 3-layer I - D model
6× 6 m 18 × 18 m 30 × 3 0 m
N Wenner
D-D Wenner
Conductive
1.58
2.01
Resistive
0.60
0.77
Conductive
1.13
0.90
Resistive
0.91
1.05
Conductive
1.01
0.97
Resistive
1.01
1.00
• a conductive layer of thickness e and resistivity p embedded between two resistant ones can be replaced by one or several conductive layer(s) such that: E e i / P i = e / p = S ("conductance") i
The fact that the apparent anisotropy effect is less pronounced for the D - D configuration than for the N one must be emphasised.
where e i and pi are the thickness and resistivity respectively of the remaining layer(s) a resistive layer embedded between two conductive ones can be replaced by one or several resistive layer(s) such that:
~e~.p~=e.p=T i
( "transverse resistance").
2.3. 1-D inversion using 3-D theoretical data The 1-D inversion is subject to the rules of equivalence (Maillet, 1947) which state that:
It can be concluded from these rules that the relative sensitivity of the inversion process to resistivity or thickness variations will be equivalent for
Table 2 Resistivity P2-, thickness e 2 , c o n d u c t a n c e a n d transverse resistance of the second layer obtained by 1-D i n v e r s i o n c o m p a r e d with the resistivity P2 a n d the thickness e 2 of the slab 3 0 × 30 m slab
18 × 18 m slab
N Wenner (a = 5 m)
Conductive
Resistive
Conductive
Resistive
e~ ( m )
1.51
1.46
1.95
2.09 515
p~ ( O m )
24
395
20
S'(s)
0.08
-
0.1
-
AS'/S
- 20%
-
0%
-
T ' ( O m 2)
-
790
-
1030
AT'/T
-
- 21%
-
+ 3%
Ae~/e 2
- 24%
-- 2 7 %
- 2.5%
+ 4.5%
Ap'2/P2
+ 20%
- 21%
0%
+ 3%
D-D Wenner
e~ ( m )
2.53
2.61
2.13
1.95
(a = 5 m)
p~ ( g 2 m )
17
580
19
495
S'(s) /tS'/S
0.118
-
0.105
-
+ 18%
-
+ 5%
-
T ' ( g 2 m 2)
-
1160
--
990
AT'/T
-
+ 16%
-
- 1%
Ae~/e 2
+ 26%
+ 30%
+ 6.5%
- 2.5%
Ap'z/p2
- 15%
+ 16%
-5%
- 1%
Simultaneous inversion
%e~ (m)
1.88
1.84
2.03
2.04
N and D-D
p~ ( O m )
21
477
20
505
S'(s) AS'/S
0.095 - 5%
-
0.1 0%
-
T ' ( O m 2)
-
954
-
1010
AT'/T
--
--4.5%
--
+ 1%
Ae~ 1 / e 2
-- 6 %
- 8%
+ 1.5%
+ 2%
Ap'2/P2
Jr- 5 %
-- 4 . 5 %
0%
+ 1%
Wenner
E Meheni et al./Journal of Applied Geophysics 34 (1996) 255-270
both a resistive and a conductive slab, contrary to the case for the EM situation. Table 2 shows the results obtained at the center of the slab, first for conductance and transverse resistance and then for the resistivity P2 and thickness e 2 of the slab, when the other parameter is fixed at the correct value. The results of the inversion when using both configuration data simultaneously are shown at the bottom of the table. As can be expected from the previous figures, the results are good for the 30 × 30 m slab and only barely correct for the 18 × 18 m one. In all cases the precision for P2 is better than for e 2. For the 18 X 18 m slab, the N configuration reduces the conductance and the transmittance while the D - D configuration increases these parameters. The very good results achieved when inverting simultaneously the data of both configurations should be emphazised. For the 18 × 18 m slab the relative error is reduced from around 20% to around 5%. This conclusion is confirmed by the mapping of P2 after inversion (Figs. 6 and 7): the anisotropy effect is reduced (compared to Figs. 4 and 5) and the shape of the anomaly is more representative of the heterogeneity. When the simultaneous inversion is applied to the 6 × 6 m slab, the e~ or P2 values remain inexact but the shape of the anc,maly is correct (Fig. 8) and the anisotropy effect considerably reduced. This type of inversion could be an interesting general technique to correct for appare, nt anisotropy when both N and D - D Wenner data are available.
263
(a)
19. 0
10
31.
44.
63'.
i01'. Q.m
20m
2.4. Theoretical conclusions
The 1-D inversion can be validly applied when the size of the 3-D feature is larger than the total array length. The transition zone is wider in the N Wenner configuration than in the D - D one. When seeking one parameter, the simultaneous inversion of both data gives better results than two separate inversions.
(b) 99.
3. Experimental cases
The application of the approximate 1-D inversion to current data implies two types of assumptions.
0
10
158. 225'. 319'.
510. fl.m
20m
Fig. 6. Maps of the resistivity of the second layer after a 1-D simultaneous inversion for a 30 × 30 m slab: (a) conductive slab, (b) resistive slab.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
264
(a)
20. 10
32.
45'.
(a)
63'.
I
I00'.
!~!~i~!~i~i~i~i~i!i!i!i!!~(~ ~,
~ .m
56.
20m
66.
i0
::..
75.
85.
100.
~.m
20m
:::::.
:x:.:
.......
.....
-..:::
::::.
:
:
.
:
:
.
v
:
:
.
: : i i i i i [ : iiiiiii!::::iii!!i!:::iiiii!i!!:ii:iiiiiii:
:.: .....
:......
: ......... v...:::
:.::
...... . : : : .
:.
: : x : .....
....... . : : : .
:::...
:....
:..:v • ...:::
.::v..
iili :!!:iiiiii ::iiiii![:: :::i[[[!!: :[[[[i!
7115:
i i?iiiiiii!iiii
::i: :
[[[!:~=.ii [i[:::i i
[ [ i
:::::
::::::
i[:::hl
: : : i l h :i:
::::.:iii
[[[!
i i i i i ill?
:iii]i :
:
! i i i il i ili2iill i)i i i ii i il !iiii ??i!:!i:i:?i:?:~:i:!:!'?!':ilil i iii?))ii?i?i~illl)ili~ii?i?)iii?!i
i i?iii!)iill
ih: ......
:hh
:i :::
: ]h::
::':]ii.
"':
.ilh :
::iiiii ....... :
:
iii~
!i i ill
::::i: .....
x:
liiiiiiiiiiiiliiiiililiiiiiiiiiiiiiiiiii! :::::
:::
"iiiii
i i!
::
:i
:::::
i
:::::::
hKi:
: : : :
:
i/"
:::
.....
:
. ::::
::ii:
. :IK
=::::
iiiii
:
i
(b)
(b) l 99. 0
10
155.
218.
305.
99.
478'. ~.m
20m
Fig. 7. Maps of the resistivty of the second layer after a 1-D simultaneous inversion for an 18 × 18 m slab.
0
i0
121.
141.
164'.
200'.
n.m
20m
Fig. 8. Maps of the resistivity of the second layer after a I-D simultaneous inversion for a 6 X 6 m slab.
Y. Meheni et a l . / Journal of Applied Geophysics 34 (1996) 255-270
The first assumption,as discussed in the previous section, is that the lateral resistivity changes slowly. The second assumption is more complex: with one (or two) series of apparent resitivity data available, only one (or two) parameter(s) can be calculated.
265
Therefore, the use of the proposed inversion is limited to geological situations where lateral variations are likely to be explained by variations in one (or two) parameter(s) only. This case is common in near-surface investigations and corresponds to situa-
(a)
.
70.
116.
170.
.
N
250.
415.
Q.m
/
292'.
556.
~ .m
/
I - -
0
25
50m
(b)
~m-0
58. 25
iii.
180.
50m
Fig. 9. A p p a r e n t resistivity m a p s at " L e Gud G o u j a r d " : (a) N W e n n e r array, a = 5 m; (b) D - D W e n n e r array, a = 5 m.
K Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
266
I
i ~ ~ ~~ ~i~i~iii~i~ili~i~iiiii!i!i~ili~ ¸
1.5 0
25
.7
.4
.2
.lm
50m
/
N
Fig. 10. Thickness of the conductive layer at "Le Gu4 Goujard" after simultaneous reversion using both N and D - D Wenner data.
tions where the sound bedrock is covered by a weathered layer itself covered by a topsoil. The second layer often has a rather stable and low resis-
1.5 0
25
50m
.7
tivity. Its thickness may vary with the degree of weathering, which then directly effects the observed apparent resistivity variations. Such situations appear
'.4
'.2
'.1 m
/
Fig. 1 h Thickness of the conductive layer at "Le Gu6 Goujard" after approximate 1-D inversion using N Wenner data, D - D Wenner data and EM 31 resistivities.
Y. Meheni et al./ Journal of Applied Geophysics 34 (1996) 255-270
267
in carbonate areas of the Paris Basin. The groundwater circulation in the middle Jurassic limestone (Roy, 1983) is governed by fractures in a sound rock of relatively high resistivity (250-1000 g2m). These fractures are often too narrow and too deep to be directly detected. However, the alteration is more pronounced and the weathered layer thicker above these fractures. This layer contains an abundance of fine particles, generating a good conductivity. Under these conditions resistivity mapping is a useful technique for locating the fracture zones which play an important part in ground-water ciculation (Benderitter and Robin, 1987, Meheni et al., 1996). The two following examples illustrate the application of the approximate 1-D inversion in such a geological carbonate environment. (a) N
37.
74.
126. 212".
426'. fl.m
(b) I
~~ ii~ ziizi~iiiili:-~i~:~r~---.
32. 0 Fig.
25 12. A p p a r e n t
Wenner
69. 123. 220.
475.
¢~.m
50m resistivity
map
a r r a y , a = 5 m ; (19) D - D
at " L e
Wenner
3.1. Le Gu£ Goujard (parcel 8), St Martin sur Nohain, Ni~vre, France
Bois
Rond":
array, a = 5 m.
(a) N
This parcel is located near a river and presents a gently sloping surface from NE to SW. The survey was carried out using a 5 m square grid and a 5 m inter-electrode distance. The conductive layer above the unweathered limestone becomes thicker in three linear zones (Fig. 9a and b). Two of these zones are parallel to the slope, the third lies in the centre of the map and is orientated SW-NE. The resistivity exceeds 300 g2m in the S and NW of the map which suggests that the limestone is directly below the topsoil. The information from both maps is very coherent, the variations being more pronouced in the D - D configuration. Two electrical soundings led to the following results: p~ = 50 J2m and e I = 0.3 m for the topsoil layer, P2 = 25 g2m for the conductive layer and P3 = 250 /2 m for the limestone. The map of the thickness of the conductive layer e 2 (Fig. 10) is the result of the simultaneous 1-D inversion using both N and D - D Wenner data. The thickness is less than 0.1 m in the resistive zones and reaches 1.7 m in the third conductive one. The variations are smoother on the e 2 map than on the apparent resistivity map. The values seem very realistic ones and in agreement with the results obtained by inverting the EM 31 resistivity map (Gu4rin et al., 1995). The simultaneous inversion of the three ap-
Y. Meheni et a l . / Journal of Applied Geophysics 34 (1996) 255-270
268
parent resistivity data sets (Fig. 11) gives similar results.
3.2. Le Bois Rond, Garchy, Nibvre, France The surveyed area is located on a flat plateau. Both resistivity maps (Fig. 12a and b), obtained with a 5 m square grid and a 5 m inter-electrode distance, show very strong variations from 40 S2m to 400 g2m. The high resistivity values very clearly delineate an area of limestone outcrop. Five soundings performed on different zones, two in the NE of area, one in the more conductive zone and two on the resistive one, indicate a three layer system with a conductive weathered layer of varyiable thickness. The map of this thickness (Fig. 13), which varies between 0 and approximately 5 m, was produced using the following values: Pl = 54 J2m, e~ = 0.4 m; P2 = 30 g2m and P3 = 450 O m . Once again, the simultaneous inversion of both N and D - D configuration data reduces the high frequency variations and
[
::::::::::::::::::::::
4.8 0
25
. .~
1.6
.7
.3
.I m
50m
Fig. 14. Thickness of the conductive layer at "Le Bois Rond" after I-D simultaneous inversion using N Wenner data, D - D Wenner data and EM 31 resistivities.
the results are coherent with those for the EM 31 apparent resistivity (Fig. 14).
4. Conclusion
N
I
?iii!i!!~i~i~i~:.~("ili. . . . . 5.6
1.8
.7
. .3
.I m
1
Fig. 13. Thickness of the conductive layer at "Le Bois Rond" after simultaneous inversion using N and D - D resitivity data.
In near-surface surveys the interpretation of apparent resistivity data can be improved in cases where a complete 3-D inversion process is too involved for practical use or the density of data insufficient to allow it. The proposed approximate 1-D inversion is a very quick and simple method that allows a first interpretation in cases where only one parameter varies significantly. The results are significant when the geological context justifies this major assumption. This is the case in a carbonate environment as demonstrated in the examples. It would therefore be interesting to test its application in crystalline bedrock areas where the main parameter is also the thickness of the weathered overburden layer.
Y. Meheni et al. / Journal of Applied Geophysics 34 (1996) 255-270
As noticed in combining both theoretical and experimental data, this type of processing opens new approaches to correcting for the apparent anisotropy effect and obtaining unbiased maps of lateral variations, even if the ~zalues of the restored parameter can not be considered to be exact.
References Benderitter, Y., Jolivet, A., Mounir, A. and Tabbagh, A., 1994. Application of the electrostatic quadripole to sounding in the hectometric depth range. J. Appl. Geophys., 31: 1-6. Benderitter, Y. and Robin, G., 1987. Approche singulibre des discontinuitts de perratabilit6 en milieu carbonatt. Hydrogtologie, 3: 161-168. Dabas, M., Tabbagh, A. and Tabbagh, J., 1994. 3-D inversion in subsurface electrical surveying - I. Theory. Geophys. J. Int., 119: 975-990. Dey, A. and Morrison, H.F., 1979. Resistivity modelling for arbitrarily shaped two-dimensional structures. Geophys. Prospect., 27: 106-136. Flathe, H., 1955. A praclical method of calculating model graphs for horizontally stratified media. Geophys. Prospect., 3: 268294. Griffiths, D.H. and Turnbull, J., 1985. A multi-electrode array for resistivity surveying. First Break, 3: 16-20. Grard, R. and Tabbagh, A., 1991. A mobile four electrode array and its application to ~.he electrical survey of planetary grounds at shallow depth. J. Geophys. Res., 96 - B3: 4117-4123. Gutrin, R., Meheni, Y., Rakotondrasoa, G. and Tabbagh, A., 1995. Interpretation of Slingram conductivity mapping in near surface geophysics: u~sing a simple parameter fitting with I-D model. Geophys. Prospect., 44: 233-249.
269
Hesse, A. and Spahos, Y., 1978. Evaluation of Wenner and dipole-dipole resistivity measurements and the use of a new switch for archaeological field work. Archaeophysika, 10: 647-655. Hesse, A., Jolivet, A. and Tabbagh, A., 1986. New prospects in shallow depth electrical surveying for archaeological and pedological applications. Geophysics, 51: 585-594. Li, Y. and Oldenburg, D.W., 1992. Approximate inverse mapping in DC resistivity problems. Geophys. J. Int., 109: 343-362. Maillet, R., 1947. The fundamental equations of electrical prospecting. Geophysics, 12: 529-556. Meheni, Y., Benderitter, Y., Gutrin, R. and Tabbagh, A., 1996. Comparaison de difftrentes m&hodes pour la cartographie d&aillte de la rtsistivit6 ~lectrique du sous-sol dans la gamme de profondeur 2-20 m. Gtologues, 109: 25-36. Poirmeur, C. and Vasseur, G., 1988. Three-dimensional modelling of a hole-to-hole electrical method: application to the interpretation of a field survey. Geophysics, 53: 402-414. Roy, B., 1983. Gtologie, gtophysique, hydrogtologie des formations jurassique moyen dans le Nord-Ouest de la Nibvre. Th~se de 3brae cycle, Universit6 de Dijon, 174 pp. Sasaki, Y., 1994. 3-D resistivity inversion using the finite-element method. Geophysics, 59: 1839-1848. Sorensen, K. and Pedersen, F.F., 1992. Slaebe-geoelektrik, Miljoministeriet. Skov-og. Naturstyrelsen, 2, 44 pp. Spahos, Y., 1979. Calculs sur module et rtle du quadriptle en prospection ~lectrique de subsurface. Application ~t la dttection archtologique, Th~se de 3~me cycle, Universit6 P and M Curie, Pads. Sttphanesco, S., Schlumberger, C. and Schlumberger, M., 1930. Sur la distribution 61ectrique potentielle autour d'une prise de terre ponctuelle dans un terrain ~ couches horizontales, homogbnes et isotropes. J. Phys. Radium, 7: 132-140. Tabbagh, A., Hesse, A. and Grard, R., 1993. Determination of electrical properties of the ground at shallow depth with an electrostatic quadripole: field trials on archaeological sites. Geophys. Prospect., 41: 579-597.