1D complete calculation for electrostatic soundings interpretation

1D complete calculation for electrostatic soundingsinterpretation1

Alain Tabbagh2 and CeÂdric Panissod2

Abstract

Soundings achieved with the electrostatic method cannot be interpreted correctly with1D electrical programs if the extension of the array or the conductivity of the ground

are too large. A complete calculation taking into account the induction effect due to

the frequency must be performed. This paper presents the solutions for overcomingthe dif®culties encountered in this calculation: the iterative processes which make it

possible to determine the kernel functions of the Hankel transforms and the analytic

integration of the terms of the electric ®eld. Application to practical cases ®rst illus-trates the distortion of the electrostatic curves by reference to DC sounding curves.

The limitation in depth of investigation is then emphasized: in practice, the investigation

is limited by the skin depth corresponding to the frequency used. The examples of thesoundings obtained in the city of Alexandria (Egypt) demonstrate the importance of

using the complete calculation for very conductive grounds.

Introduction

The electrostatic method can be considered as a generalization to both dielectric andconductive media of the classical DC electrical method (Grard and Tabbagh 1991).

The use of electrostatic poles located in the air above ground level rather than electrodes

driven into the ground makes it possible to apply this method on insulating surfaces,such as tarmacs, hard rocks or very dry soils, where good electrical contacts are

dif®cult. This method is now commonly used in shallow depth surveying especially in

urban areas (Chevassu et al. 1992; Tabbagh, Hesse and Grard 1993; Mounir 1994;Panissod et al. 1998).

The application of this method for sounding, by regularly increasing the size of the

array, leads to the same limitations as those of the DC method: when the size of thearray is too large, a time-domain electromagnetic method might be preferable. However,

the electrostatic method constitutes a convenient solution to ®ll the gap between the

ground-penetrating radar method, which is commonly limited to a few metres, and theEM methods whose use is drastically limited in the presence of metallic disturbances.

q 2000 European Association of Geoscientists & Engineers 511

Geophysical Prospecting, 2000, 48, 511±520

1 Received November 1998, revision accepted September 1999.2 UniteÂ Mixte de Recherche 7619 ``Sisyphe'', UniversiteÂ de Paris 6 et Centre National de la Recherche

Scienti®que, Centre de Recherche GeÂophysique, 58150 Garchy, France.

Nevertheless, the frequency of the injected current is quite high and it would be

dif®cult to reduce the frequency to below 5 or 10 kHz, because this would lead to theuse of large and thus impractical poles. The magnitude of this parameter makes the

interpretation more dif®cult. Firstly, for high-resistivity soils, the dielectric permittivity

must be taken into account, and secondly, when the size of the array increases, theinduction number does not remain negligible and a correct interpretation necessitates

the use of a complete EM calculation (Tabbagh et al. 1993) rather than static

modelling. For example, a resistive ground of 5000 Qm having a relative permittivity of36 would correspond to an apparent resistivity of 2500 Qm if the frequency is equal to

100 kHz. For the same frequency, a 20-m-sided square quadripole over ground of

resistivity 100 Qm would correspond to a induction number of 3.16. Thus to interpretthe soundings correctly, a complete calculation was considered (Benderitter et al.1994). As ef®cient solutions to the different problems encountered in this calculation

have not yet been published in the open literature, we present the solutions here.

General formulation of the calculation

The existence of two injection poles, A and B, is equivalent to a line of dipoles joining A

to B, and the measurement between the two poles corresponds to an integration of the

electric ®eld from N to M. The calculation consists ®rst in determining the electric ®eldgenerated in the air above a layered ground by a horizontal electric dipole. Integration

must then be carried out over the line of dipoles and again over the measurement line.

We consider (Fig. 1) a Cartesian coordinate system centred at ground level with thez-axis positive downwards. The horizontal dipole is on the z-axis at an elevation dabove ground level; its direction is positive along the x-axis, its moment is Idx. The

ground comprises N layers with conductivity ji, relative permittivity ei and thickness ei

(i being the layer index). At each interface, the horizontal components of the electric

and magnetic ®elds are continuous. Following Sommerfeld (1926), all the components

of the ®elds can be calculated using a two-component potential verifying the Helmholtzequation. In the air, the expressions for this potential are thus,

Ax �eÿg0R

R�

�¥

0a0�l� eu0zJ0�lr�dl �

�¥

0F0�l; z� J0�lr�dl

and

Az �

�¥

0b0�l� eu0z ¶

¶xJ0�lr�dl �

�¥

0

¶¶x

G0�l; z� J0�lr�dl:

In the ith layer, the expressions for the potential are

Ax �

�¥

0�ai�l� eÿuiz

� di�l� euiz� J0�lr�dl �

�¥

0Fi�l; z� J0�lr�dl

and

Az �

�¥

0�bi�l� eÿuiz

� z i�l� euiz�

¶¶x

J0�lr�dl �

�¥

0

¶¶x

Gi�l; z� J0�lr�dl:

512 A. Tabbagh and C. Panissod

q 2000 European Association of Geoscientists & Engineers, Geophysical Prospecting, 48, 511±520

In these expressions the constant m0Idx/4p is omitted for simpli®cation, J0 is thezeroth-order Bessel function, r2

� x2� y2, R2

� r2� z2, g0

2�ÿe0m0q

2, u02�g0

2� l2,

gi2� ijim0qÿ e0eim0q

2 and ui2� gi

2� l2, q being the angular frequency. The con-

tinuity at the different horizontal interfaces of the horizontal components of the ®eldsallows us to determine the functions ai, bi, di and zi, and thus to determine the potential

and the ®elds everywhere. F and G are the kernel functions which are dependent on the

electrical properties of the model.In the air, the ®nal expressions of the horizontal components of the electric ®eld are

Ex �1

iqe0m0

¶¶x

¶Ax

¶x�

¶Az

¶z

� �ÿ g2

0Ax

� �and

Ey �1

iqe0m0

¶¶y

¶Ax

¶x�

¶Az

¶z

� �:

Ex is divided into two terms. The ®rst one, Ex1, is a derivative of the divergence of

the potential, and the other one, the `induced term', is Ex2�ÿiqAx. The problemsencountered in integrating this second term are different from those encountered in

integrating the ®rst one. The solution of both problems will be presented below.

Iterative calculation of F0 and G0

Different methods can be used to determine the expressions of the components of thepotential in the air. The most elegant one, commonly applied in EM for layered ground

solutions and in electrical sounding (Koefoed 1968), is an iterative process. It will be

used here, but two different functions have to be determined.

Electrostatic soundings interpretation 513


Figure 1. Reference model for the calculation of the ®elds generated by an electrical dipoleabove a layered ground.

The continuity of the horizontal ®elds implies that Ax, ¶Ax /¶z, Az and (1/(ji�

iqe0ei))(¶Ax /¶x� ¶Az /¶z) are continuous. The continuity of Ax and of ¶Ax /¶z are ®rstused to calculate F0. We consider the ratio, Ri� (¶Fi/¶z)/Fi, at each interface. At the

lower one, RN�ÿuN. An iterative expression can be established between the ratio at

the top of the ith layer and the ratio at its bottom: Ri� ui(Ri�1ÿ ui tanh(uiei))/(uiÿRi�1 tanh(uiei)). From this relationship, R1 at ground level can be determined

and we obtain a0� (l/u0)(u0�R1)/(u0ÿR1) eÿu0d.

In order to calculate G0(l, z) we consider the functions Qi(l, z)�Gi(l, z)� 1/l2

(¶Fi(l, z)/¶z). The functions are continuous at each interface and (1/(ji� iqe0ei))

(¶Qi /¶z) are also continuous. At the lower interface, the ratio of this last expression

to QN is PN�ÿZN, where ZN� uN /(jN� iqeNe0). The iterative relationship is Pi�

Zi(Pi�1ÿZi tanh(uiei))/(ZiÿPi�1 tanh(uiei)), from which we can deduce P1. From

the relationship P1� (¶G0/¶z)� (u20/l2)F0/(G0� (1/l2)(¶F0/¶z)), the expression for

(a0� u0b0) eu0z can be deduced which allows us to determine Ex1 and Ey.

Integration of the divergence term

The expressions for Ex1 and Ey can be transformed using (¶Ax/¶x� ¶Az/¶z)�( ¶/¶x)

(Ax�D), where D�R¥

0 u0F0(l, z)J0(lr) dl. For a series of dipoles extending from A to

B, Ey (or Ex1) can be expressed by Ey �R

xBxA

¶2/¶x¶y (Ax � D) dx, where dx refers to thesource point, while ¶/¶x refers to the point where the ®eld is calculated. We thus have

Ey� [¶/¶y(Ax�D)]xAÿ [¶/¶y(Ax�D)]xB

.

The calculation of the voltage between the two measuring poles corresponds to theformula,

VM ÿ VN �

�MN

E ? dl �

�MN

Ex1 dx � Ey dy;

thus

VM ÿ VN � ��Ax � D�xAÿ �Ax � D�xB

�xM ; yMÿ ��Ax � D�xA

ÿ �Ax � D�xB�xN ; yN

:

Integration of the induced term, Ex2

Numerical integration methods can be used without any problems if the measurementpoints are not on the segment AB. For example, there is no dif®culty in calculating this

term for a square or a dipole±dipole array (we used the Gaussian quadrature with 24

points). For a Schlumberger or a normal Wenner array, we have to consider the electric®eld at a place where a dipole source also exists. Numerical calculation is not possible

here and an analytical integration of the induced term is the only solution.

We must calculate the integral,

S �

�x0M

x0N

�xB

xA

�¥

0F0�l; z� J0�lr�dl dx dx0;

where F0(l, z)�a0(l) eu0z� (l/u0) eÿu0|z� d|.



It is possible to exchange the order of integration between x and l and to perform the

analytic integration by noting that for these pole arrays (when the four poles are on thesame line), we have r� |x|. We can thus calculate

R¥0 (1/l)F0(l, z)

RxBxA

J0(lx) d(lx) dl,

using the relationship,R

x0 Jv(u) du� 2S¥

k� 0 J2k� v� 1(x). The same process is repeated

for the second integration, and we ®nally obtain

S �

�¥

0

F0�l; z�

l2

4X¥

n� 1

n�

J2n�l�xM ÿ xA�� ÿ J2n�l�xM ÿ xB��

ÿ J2n�l�xN ÿ xA�� J2n�l�xN ÿ xB��

�!dl:

This expression is calculated by numerical integration and the induced term is added tothe preceding voltage expression.

Application

Type of distortion of the sounding curve

As observed in the ®rst experiments (Benderitter et al. 1994), the sounding curves for

electrostatic measurements at 22 kHz tend to differ from those of DC electrical

measurements when the size of the array increases. This can be interpreted as the effectof the induction term when the induction number increases. However, these experiments

were only achieved with both square and normal Wenner (a-con®guration) arrays for

which this effect corresponds to an increase in the apparent resistivity. For other arrays,the distortion of the end of the curve can be different. For example, for the dipole±

dipole Wenner array (b-con®guration), where the con®guration of the poles is A, B, N,

M, the apparent resistivity decreases. This is illustrated in Fig. 2 for the four-layerground model observed in the ®rst experiments. The existence of such a difference

between different arrays can constitute a test: (i) for evidence of the effect of the

induction term (in a Wenner array it is easy to exchange the B and M poles), (ii) tode®ne an upper limit to the array extent, and (iii) for the application of DC sounding

interpretation programs.

Reduction of the depth of investigation

The bending observed on the end of the curves also corresponds to a loss of information.

This part of the curve is practically the same whatever the resistivity of the deepest

layers. In the four-layer example already considered here, the resistivity of the fourthlayer as determined by DC soundings was 360 Qm. If this layer is replaced by a 50 Qm

layer, the change in the curve is very small (Fig. 3) and would be impossible to observe

from experimental results. In fact, the limit in depth of the third 85 Qm layer is totallyunknown from electrostatic data at a frequency of 22 kHz.

The existence of the skin depth explains this phenomenon: for a 100 Qm homo-

geneous ground which corresponds to the order of magnitude of the resistivity of the





Figure 2. (a) Electrostatic (bold line) and electrical (thin line) sounding curves for a normalWenner array over a four-layer ground; r1� 38 Qm, e1� 0.4 m, r2�113 Qm, e2� 4.2 m, r3�

85 Qm, e3� 28 m and r4� 360 Qm at 22 kHz. (b) Electrostatic (bold line) and electrical (thin line)sounding curves for a dipole±dipole Wenner array over the same layered ground as in (a).

®rst three layers, the skin depth for a frequency of 22 kHz is 34 m. As a rule of thumb,the skin depth corresponds roughly to the limit in depth beyond which it is impossible

to obtain any information.

Results in a very conductive environment: the example of Alexandria (Egypt)

The electrostatic method is mainly dedicated to arid area and urban zone exploration.The most unfavourable conditions for its use correspond to very conductive grounds

where the skin depth is small. This situation was found in Alexandria (Egypt) when

searching for the location of the archaeological remains of the Heptastadium in theisthmus joining the Pharos island to the mainland. There, electrostatic (using a 23 kHz

injection frequency) and electromagnetic (EM31 Geonics) pro®les were performed

together with a series of electrostatic soundings and other geophysical tests with seismicrefraction and ground-penetrating radar (Hesse et al. 1998).

The electrical structure of the ground corresponds to a two-layer model with a ®rst

unsaturated layer of varying thickness and of resistivity ranging from 20 to 70 Qm, anda second seawater-saturated layer of resistivity ranging from 1 to 10 Qm, depending on

its porosity. In such conditions, the skin depth for a 23 kHz frequency is 3 m in a 1 Qm

homogeneous ground, 11 m in a 10 Qm ground and 20 m in a 30 Qm ground. Thus, theexpected maximum depth of investigation is about 10 m. On the apparent-resistivity



Figure 3. Electrostatic sounding curves when r4� 360 Qm (upper curve) and r4� 50 Qm (lowercurve), the other parameters being identical to those in Fig. 2.

curves, the bending-up may appear for pole separations as low as 4 m in the normal

Wenner con®guration. If we consider, for example, the sounding acquired in Abou elMellah Street (Fig. 4), the correct interpretation (with ES formulation) corresponds to

a 26.3 Qm and 1 m thick layer above a 5 Qm layer, whereas the DC interpretation of the

electrostatic measurements would correspond to a three-layer structure limiting the5 Qm layer to a 5 m thickness and introducing a 500 Qm layer underneath. Even when

the bending-up is not clearly observable in the raw data and when a sounding would

also be interpreted by a two-layer model in DC interpretation, the resistivity of thesecond conductive layer is incorrect with this last interpretation. In Khaled KoraõÈm

Street (Fig. 5), the resistivity of the second layer would be interpreted as 13.3 Qm

whereas its correct value is only half as much, i.e. 6.7 Qm.

Conclusions

Even in the case of a seawater-saturated layer, electrostatic sounding and pro®ling

constitute a relevant tool to explore the terrain. In the case of Alexandria, the depth of

investigation was in fact limited to 10 m or less for porous substrata. Such values arequite unusual but, in environmental research, conductive soils may correspond to



Figure 4. Electrostatic sounding curves obtained in Abou el Mellah Street in Alexandria (Egypt).The complete calculation corresponds to a two-layer model where r1� 26.3 Qm, e1� 1 m andr2� 5 Qm; the DC electrical interpretation would have given a three-layer model where e2� 5 mand r3� 500 Qm. The stars denote the ES measurement points, the thin line corresponds to theinterpretation using the DC formulation, the thick line corresponds to the interpretation usingthe ES formulation.

highly polluted areas where the electrostatic method can also be the most relevant one.

For the more usual resistivity values of above 10 Qm, the depth of investigation issuf®cient for most environmental and engineering applications in places where other

methods, such as electrical sounding (due to the nature of the ground surface),seismics (due to urban environment) or radar (due to a lack of penetration) are

impractical.

References

Benderitter Y., Jolivet A., Mounir A. and Tabbagh A. 1994. Application of the electrostatic

quadripole to sounding in the hectometric depth range. Journal of Applied Geophysics 31, 1±6.

Chevassu G., Lagabrielle R., Mounir A. and Tabbagh A. 1992. Apparent resistivity pro®ling on a

road with an electrostatic quadrupole. In: No Trenches in Town (eds J.P. Henry and M. Mermet),

pp. 203±205. Balkema, Rotterdam.

Grard R. and Tabbagh A. 1991. A mobile four electrode array and its application to the electrical

survey of planetary grounds at shallow depths. Journal of Geophysical Research 96, 4117±4123.

Hesse A., Andrieux P., Atya M., Benech C., Camerlynck C., Dabas M. et al. 1998. Geophysical

investigations for the location of the Heptastadium in Alexandria (Egypt). Proceedings of the 4th

Meeting of the Environmental and Engineering Geophysical Society, Barcelona, Spain, pp. 715±718.



Figure 5. Electrostatic sounding curves obtained in Khaled KoraõÈm Street in Alexandria (Egypt).The complete calculation corresponds to a two-layer model where r1� 33.3 Qm, e1� 2.5 m andr2� 6.6 Qm; the DC electrical interpretation would also have given a two-layer model, but withe1� 2.14 m and r2� 13.3 Qm. The stars denote the ES measurement points, the thin linecorresponds to the interpretation using the DC formulation, the thick line corresponds to theinterpretation using the ES formulation.

Koefoed O. 1968. The application of kernel function in interpreting geoelectrical resistivity

measurements. Geoexploration Monographs, Series 1, No. 2. GebruÈder Borntraeger.

Mounir A. 1994. Application du quadripole eÂlectrostatique aÁ la prospection sur la gamme de

profondeur 0±100 m. TheÁse, UniversiteÂ de Paris 6.

Panissod C., Dabas M., Hesse A., Jolivet A., Tabbagh J. and Tabbagh A. 1998. Recent

developments in shallow depth electrical and electrostatic prospecting using mobile arrays.

Geophysics 63, 1542±1550.

Sommerfeld A. 1926. UÈ ber die Ausbreitung der Wellen in der drahtlosen Telegraphie. Annalen

der Physik 4, 1135±1153.

Tabbagh A., Hesse A. and Grard R. 1993. Determination of electrical properties of the ground at

shallow depth with an electrostatic quadrupole: ®eld trials on archaeological sites. Geophysical

Prospecting 41, 579±597.



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1D complete calculation for electrostatic soundings interpretation

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