measuring the electrical resistivity of the earth using a fabricated resistivity meter

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Measuring the electrical resistivity of the Earth using a fabricated resistivity meter

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2005 Eur. J. Phys. 26 501

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INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 26 (2005) 501–515 doi:10.1088/0143-0807/26/3/015

Measuring the electrical resistivity ofthe Earth using a fabricated resistivitymeter

J A Olowofela1, V O Jolaosho1 and B S Badmus2

1 Department of Physics, University of Ibadan, Ibadan, Nigeria2 Department of Physics, University of Agriculture, Abeokuta, Nigeria

Received 6 October 2004, in final form 7 February 2005Published 5 April 2005Online at stacks.iop.org/EJP/26/501

AbstractA four-electrode equipment which can be made available in the physicslaboratory was designed and fabricated for measuring current and potentialvalues so as to obtain the resistivity values. Readings taken from our chosenlocation using both Wenner and Schlumberger arrays were analysed. In theresistivity method, the Wenner configuration discriminates between resistivitiesof different geoelectric lateral layers while the Schlumberger configuration isused for the ‘depth sounding’. The theory behind the resistivity method wasintroduced with some degree of sophistication. Experimentally determinedvalues of depth to surface were correlated with the values determined fromthe excavated site and this gave a good correlation. It was observed thatthe resistivity value in the chosen location increases with the depth and theSchlumberger method was seen to have a greater penetration than the Wenner.

1. Introduction

What is the electrical conductivity of the Earth? This question, though of much interest togeophysicists, physicists and geologists, is misphrased. This is because zones of homogeneityexist within the Earth and it would be wrong to try and measure the conductivity ofseveral layers of the Earth. The Earth could be said to contain several unconsolidated andconsolidated layers being the weathered layers. Below this layer are several consolidatedunweathered layers. An accurate answer to the above question will assist in unravelling thetypes, quantity, quality and positions of minerals below the Earth’s surface.

The point of contact between physics and geology has shown that the Earth’s conductivityis a key to its interaction with both terrestrial and extraterrestrial electric and magnetic fields.Because its primary constituents are insulating silicon oxide (SiO2), one might think that thecrust is a poor conductor of electricity. However, large quantities of surface and undergroundwater put the Earth’s outer electrical conductivity within a few orders of magnitude.

0143-0807/05/030501+15$30.00 c© 2005 IOP Publishing Ltd Printed in the UK 501

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502 J A Olowofela et al

(a)

(c)

(b)

Figure 1. (a) Electrode with the sink assumed to be at infinity. (b) A point source of the current atthe surface of a uniform ground. (c) Current systems of two electrodes.

1.1. Current flow in the earth

Let us suppose that an electrode whose effective dimension is zero supplies current at the rateof 1 A to a uniform ground at a point P on the surface. To avoid violation of the conservationprinciple, we must also assume that current is withdrawn from the ground at the same rateelsewhere, but for the present we may assume that the sink is located far enough away thatit does not disturb the field near the source (figure 1(a)) everywhere. Furthermore, sinceall conditions are symmetrical with respect to the point P, we may surmise that φ (i.e. thepotential) will be a function of R = √

r2 + z2, the distance from P. Since ∇2(

1R

) = 0. Letus assume a solution of the form φ(r, z) = C/R. If with this simple function we are able tosatisfy the boundary conditions, the uniqueness theorem guarantees that this is the correct andonly solution to the problems.

If the air above the ground has no conductivity (and this is a good approximation) thenthe condition to be satisfied at the surface is ∂φ

∂z= 0 when z = 0. This is obviously true for

φ = C/R. We consider a small hemisphere with the centre at P and bounded by the surfaceR = a and the plane Z = 0 (figure 1(b)). If the resistivity of the ground is ρ in ohm-metres,then the total current that flows outward across a unit area of this hemispherical surface isgiven by the current density function J = I

ρ

(∂φ

∂R

)R=0 = (

Cρa2

). The total current that flows

from P, therefore is 2πa2J , and since this must be equal to 1, we find that C = Iρ/2π . Hence

φ(r, z) = Iρ

2πR. (1)

To obtain the expression for the potential due to a pair of current electrodes (figure 1(c)), weneed only superimpose two of these solutions. Because Laplace’s equation is linear, the sum

Measuring the electrical resistivity of the Earth using a fabricated resistivity meter 503

of two solutions must also be a solution. Thus the potential at a point P, distance R1 from thesource and R2 from the sink, will be (figure 1(c))

φ(P ) = Iρ

2π

(1

R1− 1

R2

). (2)

This leads immediately to the formula for the ground resistivity approximate to any electrodeconfiguration.

If two electrodes are inserted in the ground, and if an external voltage is applied acrossthem, there will be a flow of current through the Earth from one electrode to the other. Thelines of flow are always perpendicular to lines along which the potential is constant, the latterbeing referred to as equipotential lines. This relationship is illustrated by the cross sectionshown in figure 2(a). The potential difference (or the voltage) applied across electrodes Aand B is distributed along the space between them as indicated by the dotted lines. In ahomogeneous conductor, the potential with respect to A along a vertical plane cutting thesurface at C, which is midway between A and B, will be half as large as its value at B. If onecould measure the underground potential, one would observe that the potential is the same asat any surface point such as D whenever the ratio of the distance from the point to A and thatto B respectively is the same as the ratio at the surface point. In the case of D, this ratio isone-third. The full line extending downward from D and bending back under A is the trace ofthis equipotential surface on the vertical plane containing A and B. Figure 2(b) shows where afamily of such surfaces intersects the horizontal plane containing A and B. The equipotentiallines must always be perpendicular to the lines of current flow, since no component of thecurrent at any point can flow along the equipotential line at that point.

The configuration of equipotential lines in the Earth, although not observable directly, canbe deduced by measurements with potential electrodes at the surface. Lateral inhomogeneitiesin the conductivity of subsurface materials cause distortions in the current flow which giverise to corresponding irregularities in the potential lines. Figure 2 illustrates how bodies ofanomalously high or low resistivity deflect the flow of current in their vicinity. Changes inthe potential associated with these deflections can be observed by potential electrodes at thesurface. This is the basis for the equipotential line method, a qualitative technique which issometimes employed to locate ore bodies having anomalous resistivity. Variations of resistivityin the vertical directions attest to the pattern of equipotential lines, causing distortions whichare likely to show up as anomalies. It is not feasible, however, to map such variations simply bydetermining the equipotential line configuration at the surface. A more quantitative approachmust be used, one that is based on the ratio between the current introduced into the Earth andthe potential difference it causes between selected points along the surface. The resistivitymethod, which involves this type of measurement, is designed to allow three-dimensionalmapping of the resistivity within the Earth. If the variation of resistivity is entirely vertical(as when the pattern consists of a number of discrete layers, each of different resistivity,separated by horizontal interfaces), the technique is relatively simple even though there willbe limitations in the precision and reliability with which the actual resistivity pattern can bedetermined. If discrete laterally bounded bodies of anomalous resistivity are to be detectedand their depths and shapes determined, the method becomes difficult to apply.

2. Purpose and scope of the work

Electrical prospecting is far more diversified than other geophysical methods. Electricalmethods are much more frequently used in searching for metals and minerals than they


inputcurrent=1

surface

inputelectrode

area= 2 π r2 hemispherical

equipotential surface

source sink

equipotentials

surface surface

U1

U2

U1

U2

U1>U2 U1<U2

(a)

(b) (c)

Figure 2. Electric field lines and equipotential surfaces around a single electrode at the surfaceof a uniform half-space: (a) hemispherical equipotential surfaces, (b) radially outward field linesaround a source and (c) radially inward field lines around a sink.

are in exploring for petroleum [4]. In geophysics, some electrical techniques, for instance,self-potential, telluric current and magneto-telluric, depend on measuring the influence ofnatural fields.

Certain large scale (generally low frequency) magnetic fields and the terrestrial currentsystems are induced by these fields. The terms ‘magnetotelluric’ and ‘telluric’ are generallyused to designate these fields and currents respectively.

The existence of natural large scale earth currents was first established by Barlow in 1847in the course of studies on the first British telegraph system. Long term records of telluriccurrents were made at Greenwhich, Paris and Berlin in the late nineteenth century; nowadaysthey are recorded at various observatories around the world. The source of these currents hasbeen fairly located outside the Earth. Periodic and transient fluctuations can be correlated withdiurnal variations in the Earth’s magnetic field caused by solar emission, aurora and so forth.These activities have a direct influence on currents in the ionosphere; it is known that thetelluric currents are induced in the Earth by ionospheric currents. The inductive mechanismis an electromagnetic field propagated with slight attenuation over large distances in the spacebetween the ionosphere and Earth’s surface somewhat in the manner of a guided wave betweenparallel conducting plates. That is to say, it proceeds by bouncing back and forth betweenthese boundaries and hence has a large vertical component. At large distances from the sourcethis is a plane wave of variable frequency (from about 10−5 Hz up to the audio range at least).Obviously these magnetotelluric (MT) fields can penetrate the Earth’s surface to produce thetelluric currents ([14, 16] etc).


However, a method such as the resistivity method in which currents are introduced intothe Earth, is called an artificial method. Electrical prospecting methods are being employedto an increasing extent in engineering geology, where resistivity measurements are used forfinding the depth to bedrock and also in geothermal exploration. All resistivity techniques, ingeneral, require the measurements of apparent resistivity.

2.1. Methodology

In our attempt to measure the conductivity of the Earth, we employ the Wenner andSchlumberger electrode arrays [12]. Although there are commercially available specialpurpose four-electrode instruments, which should simplify the experimental method, ouraim is to present the theory with some sophistication and to show how it applies in the contextof instrumentation, commonly available in physics laboratory, from our experimentation andcalculations by physical excavations of a section of the profile site.

2.2. The general four-electrode method

Consider an arrangement consisting of a pair of current electrodes and a pair of potentialelectrodes (figure 3(a)). The current electrodes A and B act as source and sink, respectively.At the detection electrode C the potential due to the source A is +ρI/(2πrAC), while thepotential due to the sink B is −ρI/(2πrCB). The combined potential at C is

φC = ρI

2π

(1

rAC− 1

rCB

)(3)

Similarly, the resultant potential at D is

φD = ρI

2π

(1

rAD− 1

rDB

). (4)

The potential difference measured by a voltmeter connected between C and D is

φ = ρI

2π

{(1

rAC− 1

rCB

)−

(1

rAD− 1

rDB

)}. (5)

All quantities in this equation can be measured at the ground surface except the resistivity,which is given by [12]

ρ = 2πφ

I

1{(1

rAC− 1

rCB

) − (1

rAD− 1

rDB

)} . (6)

This resistivity is known as apparent resistivity, which is equivalent to the true resistivityonly when the latter is uniform throughout the subsurface. Otherwise it must be looked uponas the most convenient way to represent the resistivity in the subsurface on the basis of surfacemeasurements. If the electrodes are laid out along a line and their separations are increasedin a systematic manner, the change in the apparent resistivity (as defined in equation (6)) withelectrode spacing makes it possible to determine the variation of resistivity with depth withinlimits of precision that depend on the subsurface layering configuration [4].

2.3. Wenner arrangement

In the Wenner arrangement (figure 3(b)), each potential electrode is separated from the adjacentcurrent electrode by a distance a which is one third the separation of the current electrode(figure 3(b)).


I V

A C D B

rAC rCB

rAD rDB

I

V a a a

A C D B

I L V

a A C D B

(a)

(b)

(c)

Figure 3. (a) General four-electrode configuration for resistivity measurement, consisting of a pairof current electrodes (A, B) and a pair of potential electrodes (C, D). (b) The Wenner array. (c) TheSchlumberger array—a remains constant for some time as L is increased during the measurement.However if a � 1

5 L then a increases systematically. The separation of the current electrodes is L,and the separation of the potential electrodes is a.

Comparing this electrode arrangement with equation (6), we have rAC = a, rAD = 2a andrDB = a.

Substituting these in equation (6), the apparent resistivity equation becomes

ρ = 2πφ

Ia. (7)

Therefore

φ = ρI

2πa. (8)


Figure 4. Two-layered Earth model in which the depth increases downward (along the+Z direction).

2.4. Schlumberger arrangement

Then from figure 3(c) (Schlumberger array) and equation (6)

rAC = rDB = (L − a)

2(9)

and

rAD = rCB = (L + a)

2. (10)

Substituting into equation (6), we get

ρ = π

4

φ

I

(L2 − a2

a

). (11)

The resistivity in equation (7) is called the apparent resistivity. The phenomenon ofapparent resistivity was necessary because in nature the Earth’s crust is neither homogeneousnor isotropic

3. Stratification of the Earth

We now introduce stratification into the theory (figure 4) so that we assume homogeneitywithin the strata. By adding this, we have introduced some degree of inhomogeneity. Becauseof this stratification, the potential will no longer be spherically symmetrical about point P.Hence a solution for ∇2φ = 0 must be arrived at, which must be made to satisfy someboundary conditions.

In the Cartesian coordinates, Laplace’s equation of potential can be written as

∇2φ = ∂2φ

∂x2+

∂2φ

∂y2+

∂2φ

∂z2= 0. (12)

If we take the z-axis to be vertical and positive in the downward sense, by separating Laplace’sequation into the cylindrical coordinate system (r, θ, z) with P as the origin (note that the polarangle φ is usually used where we have used θ ; this is done in order that angle φ may not beconfused with the potential φ), Laplace’s equation [9] becomes

∇2φ = ∂2φ

∂r2+

1

r

∂φ

∂r+

1

r2

∂2φ

∂θ2+

∂2φ

∂z2= 0, (13)


suppressing the polar angle, due to azimuthal symmetry (i.e. there is no variation of thepotential along the polar direction [3]), Laplace’s equation becomes

∇2φ = ∂2φ

∂r2+

1

r

∂φ

∂r+

∂2φ

∂z2= 0. (14)

By separating the variables

φ(r, z) = R(r)Z(z), (15)

where R(r) is a function only of r and Z(z) is a function of z. Substituting equation (15) backinto equation (14) gives two second-order ordinary differential equations

d2R

dr2+

1

r

dR

dr+ λ2R = 0 (16)

and

d2Z

dz2− λ2Z = 0, (17)

where λ is known as the separation constant. Equation (16) is the zeroth-order Bessel equation.This equation has a solution

R(λr) = aλJo(λr), (18)

where Jo(λr) is the regular zeroth order Bessel function [10] and aλ is an arbitrary constant.In addition, there is a linear independent Bessel (irregular) function, the solution of whichdiverges (unphysically) in the limit λr → 0 where the function was suppressed. However,equation (17) has a solution

Z(z) = Aλ e−λr + Bλ e+λr , (19)

where Aλ and Bλ are arbitrary constants. Both R(r) and Z(z) are regular and well behavedinside each distinct medium. They represent the contribution to the surface charge of z = h,where h is the depth of the first layer. By superposition, we include in the potential the irregularpart from the point charge of the electrodes. Hence for 0 < z < L

φ(r, z)0<z<L = ρ1I

2πR+

ρ1I

2π

∫ ∞

λ

(Aλ e−λz + Bλ e+λz)Jo(λr) dλ, (20)

which leads to (see appendix A)ρa

ρ1= [2G(a, k, h) − G(2a, k, h)]. (21)

The apparent resistivity ρa is given as

ρa = 2πa�φ

I. (22)

Depth determination is done for a two layer Earth model by fitting the formulae (21)and (22) to observe values of the apparent ground resistivity measured at various electrodespacings. The fitting process is greatly facilitated by preparing in advance a set of curves, onlogarithmic paper, of the ratio ρa

ρ1plotted against the dimensionless inter-electrode spacing a/h

for different values of K (where K = ρ1−ρ2

ρ1+ρ2). The complete set of theoretical curves for the

Wenner electrode array for one, two or three horizontal layers have been published by Mooneyand Wetzal [13]. The curve fitting is then carried out simply by plotting the apparent electrodedistance on a sheet of a similar paper and matching these values to one of the field profilesin the direction of ρa

ρ1or a/h (see appendix B (figure 6)). Such displacements, when applied

to logarithmic quantities, merely fix the scale ratios. Then when a suitable match has been


Figure 5. The experimental set-up.

found, h is given by the inter-electrode spacing which falls at a/h = 1 and ρ1, by the limitingvalue of apparent resistivity as a → 0. However, there is modern software for analysing theresistivity data. The trademarks of the ones used in the analysis of the data obtained by thefabricated equipment are ‘Offix’ [17] and ‘Resist’ [18]. Virtually all modern software relieson inverse modelling and iteration of the subsurface parameters based on the initial value ofρ1 obtainable from direct measurements of the subsurface or from the curve matching. Thetwo types of software rely on this technique to generate model results.

4. Apparatus

The industrial equipment for this work is called the Terrameter. It is an instrument that,although it measures the supplied current and potential difference across the two potentialprobes, has now undergone a lot of changes. It now has a logic unit which converts themeasured current and potential difference into resistance. Basically, any geoelectric equipmentconsists of two parts: the ‘power unit’ and the ‘measuring unit’. A simple instrument wasfabricated adapting the experimental set-up of Avants et al [1] using commonly availablephysics laboratory devices (figure 5). The choice of various components is from personalinitiative and this may also be duplicated in any standard physics laboratory. The experimentalset-up was designed and constructed and this is used for the experiment (figure 5).

The power supply unit was constructed to have the voltage range of 0–25 V so that wecan vary the current. The signal generator was constructed using operational amplifiers.

All the operational amplifiers used are the LM-741 range with a maximum voltage input25 V. The typical output impedance for the LM-741 is 75 �. The sine/square wave generatorwas used as our generator. We choose our components so that the signal can be adjustedbetween frequency 25 and 650 Hz. However our output is taken from the sine wave Voutput

and not the square wave output. This is so because a square wave is made up of a fundamentalsquare wave and its infinite harmonics. The output of the fundamental square wave reducescontinuously when other sine waves (harmonics) are added.

The lock-in amplifier (in figure 5) is used to detect a phase difference between the outputsignal from the ground and the input signal to the ground. This was constructed using two10 k� resistors connected in input and forward bias through a 741 operational amplifier


Figure 6. The interpretation of a two-layered apparent resistivity graph by comparison with a setof master-curves [8].

operating in a differential amplifier mode. The lock-in amplifier also consists of a phaseshifter and a CMOS HEF 4066 switch (any FET, i.e. field effect transistor, switch will do). Thephase shifter balances by adjusting the difference in phase using a capacitor and an adjustableresistor. The various units are coupled together as shown in figure 5 and short aluminium rodswere used as the current and potential electrodes so as to avoid corrosion of electrode.

5. Results and discussions

We used current in the neighbourhood of 20 µA. To produce signal as large as possible, it isnecessary to maximize the current. Only a coherent direct current source will align with thetheory. Aluminium electrodes are used since such electrodes reduce corrosion. To eliminatenoise, frequencies greater than 50 Hz but less than 100 Hz were used. All wires were raisedabove the Earth’s surface to prevent spurious current generation by wires. The Schlumbergerand Wenner configurations were used for the experimentation with the values a taken between0.05 and 17.00 m with a sampling interval of 0.5 m and 1.0 m for the potential electrodesas the case may be. Also in accordance with the theory, the aluminium poles were pushedslightly but firmly into the earth. The Schlumberger configuration was used for two profilesVES-1 and VES-2 (VES—vertical electrical sounding) while the Wenner was used for onlyone profile, i.e. VES-3. The results obtained were automatically analysed using commerciallyavailable software known as ‘Offix’ [17] and ‘Resist’ [18]. The results are shown in table 1and graphically in figure 7.

The results obtained from the field using the fabricated equipment via Schlumberger andWenner electrode arrangements were analysed quantitatively by the partial curve-matchingmethod. The model obtained from the partial curve matching now served as the initial modelfor the computer-assisted technique based on the inverse filter coefficient [2] and the resultsshown in table 1. Each of the field data curves was analysed quantitatively to obtain respectivegeoelectric layers, thickness, resistivity values as well the lithology (table 1).

Three locations were chosen for this experiment in the premises of the Department ofPhysics, University of Ibadan, Ibadan, Nigeria. We carried out two vertical electrical soundingusing a Schlumberger electrode array and one vertical electrical sounding using a Wenner


C Schlumberger and Wenner CurvesSchlumberger-VES02

Schlumberger-VES01

1000

100

10

0

AP

PA

R. R

ES

IST

IVIT

Y O

HM

-M

Dep

th [

m x

100

]

0.0 0.1 1.0 10.0 100.0

SPACING [M]

3

2

1

0

RESISTIVITY OHM-M1 10 100 1000 10

4

Figure 7. Schlumberger and Wenner curves of the field data.

array. The results show the presence of four geoelectric layers. At VES-1, the geoelectriclayer was composed of topsoil, sandy clay, sandstone and clayey sand with resistivity values83.6 � m and 104.8 � m respectively. The total depth of penetration at this location is 6.1 mwith topsoil of 0.2 m thickness, while the sandy clay and sandstone layers have thicknesses of0.5 m and 5.4 m respectively.

At VES-2, three geoelectric layers were delineated, a topsoil of resistivity 13.3 � m andthickness 0.4 m, a second geoelectric layer consisting of shale/clay with resistivity 38.4 � mand thickness 0.7 m, which is underlain by a sandstone with resistivity 3162.1 � m. Becausethe main goal is to ascertain the accuracy of the fabricated equipment, the length of current


Figure 8. Geoelectric section of field data correlated with geoelectric section of physicalexcavation.

Table 1. Restitivity, thickness and lithology of various geoelectric layers.

Resistivity Thickness DepthElectrode array (� m) (m) (m) Lithology

VES 01 8.2 0.2 0.2 Top soil (lateritic)Schlumberger 179.9 0.5 0.7 Sandy clay

2813.6 5.4 6.1 Dry sandstone104.8 – – Clayey sand

Schlumberger 13.3 0.4 0.4 TopsoilVES 02 38.4 0.7 1.1 Shale/clay

3162.1 – – Dry sandstone

Wenner 1.0 0.12 0.12 TopsoilVES 03 0.5 0.684 0.84 Topsoil

32.0 – – Shale/Clay

electrode traverse was fixed between 15.0 m and 18.0 m and this has made the depth ofpenetration of the Schlumberger array between 1.1 m and 6.1 m.

At VES-3 where the electrode arrangement was Wenner, a maximum of three geoelectriclayers were delineated with the first and second geoelectric layers taken together as topsoilwith resistivity of 1.0 � m and 0.5 � m and a thin thickness of 0.12 m and 0.68 m. A thirdgeoelectric layer comprising shale/clay was delineated with resistivity 32.0 � m. It wasobserved that the Schlumberger array gave greater depth of penetration than the Wenner array,which is in agreement with the existing literature [12, 15, 16].

These final result were correlated with the physical excavation of subsurface within thestudy area and this gave good agreement with the observed soils of the excavated area.Figure 8 shows the geoelectric section of the field data when correlated with the geoelectricsection of the physical excavation.


6. Conclusion

It can be deduced on the basis of our findings that the combination of Wenner andSchlumberger configurations will go a long way in discriminating between resistivity ofdifferent layers. Because of poor depth of penetration, the use of the Wenner configurationexposes shallow inhomogeneous layers, while the Schlumberger configuration sees such layersas homogeneous. From the geoelectric section of both the field data and physical excavation,we can conclude that the fabricated equipment has been able to demonstrate some measure ofaccuracy when compared to the expensive common resistivity meters. This research has alsosucceeded in amplifying the theory by experimental set-up, measuring the resistivity of theEarth.

Acknowledgments

We are grateful to G A Adebayo of the Department of Physics, University of Agriculture,Abeokuta, Nigeria, who helped in the conversion of the figure files to the postscript files andthe Head of Department of Physics, University of Ibadan, Ibadan, Nigeria for kind permissionto publish the work and the reviewers whose remarks greatly improved the quality of the paper.JAO is grateful to the University of Agriculture, Abeokuta, for providing him with a sabbaticalappointment and the necessary logistics during the preparation of the paper.

Appendix A. Derivation of apparent resistivity equations

By superposition, we include in the potential the irregular part from the point charge of theelectrodes. Hence for 0 < z < L

φ(r, z)0<z<L = ρ1I

2πR+

ρ1I

2π

∫ ∞

λ

(Aλ e−λz + Bλ e+λz)Jo(λr) dλ, (A.1)

the constant aλ has been absorbed in Aλ and Bλ. In matching boundary, it is most convenientto use the mathematical identity called Lipschitz integral identity [19]

1

R= 1√

(r2 + z2)=

∫ ∞

λ=0Jo(λr) e−λz dλ, (A.2)

which is valid for z > 0. Hence equation (23) can be written as

φ(r, z)0<z<L = ρ1I

2π

∫ ∞

λ

[(1 + Aλ e−λz) + Bλ e+λz]Jo(λr) dλ. (A.3)

For z � h

φ(r, z)h<z = ρ1I

2π

∫ ∞

λ

[(1 + Cλ e−λz)]Jo(λr) dλ. (A.4)

In order to solve Laplace’s (or Poisson’s) equation uniquely, appropriate boundaryconditions must be applied. The first boundary condition to be satisfied is

∂φ1

∂z= 0 at z = 0, (A.5)

which implies that at the interface between the conductor (i.e. layer 1) and insulator (i.e. airabove the ground, figure 4.) there is no accumulation of charges (or normal current flowbetween an insulator and conductor is zero). The second and third conditions to be satisfiedare related to potentials and the potential gradient at the interface of two different conductors.


The potential in layer 1 is φ1 and that in the second layer is φ2. At the interface where z = h

(figure 5) the two potentials must be equal i.e.

φ1 = φ2 at z = h (A.6)

and for the potential gradient at the interface of two different conductors

1

ρ1

∂φ1

∂z= 1

ρ2

∂φ2

∂zat z = h. (A.7)

Applying condition (A.5) in equation (A.3) yields

Aλ = Bλ. (A.8)

Applying conditions (A.6) and (A.7) in equation (A.3) yields

(1 + Aλ) e−λh + Bλ e+λh = Cλ e−λh (A.9)

and

− 1

ρ1[(1 + Aλ) e−λh − Bλ e+λh] = − 1

ρ2(1 + Cλ) e−λh, (A.10)

respectively.Using equation (A.8) in equation (A.9) gives

(1 + Aλ) e−λh + Aλ e+λh = Cλ e−λh, (A.11)

which leads to

Cλ = e−λh + Aλ(e−λh + eλh)

e−λh. (A.12)

From equation (A.10) we obtain (using equation (A.8))

ρ2[(1 + Aλ) e−λh − Aλ e+λh] = ρ1(1 + Cλ) e−λh. (A.13)

Therefore

Cλ = ρ2

ρ1[(1 + Aλ) − Aλ e+2λh] − 1. (A.14)

Let us define

K = ρ2 − ρ1

ρ2 + ρ1. (A.15)

Equating equation (A.12) to equation (A.14) and using equation (A.15), we obtain

Aλ = Bλ = K e−2λh

1 + K e−2λh. (A.16)

K obeys the inequality −1 � K � 1. Substituting equation (A.16) in equation (A.3), weget

φ(r, 0) = ρ1I

2πrG(r,K, h) (A.17)

where

G(r,K, h) = 1 + 2Kr

∫ ∞

λ=0

(e−2λh

1 − K e−2λh

)Jo(λr) dλ. (A.18)

For the four equally spaced electrodes and with two layer Earth, the potential differencebetween the two middle electrodes follows from the superposition application of


equation (A.17) written conveniently as [10]

�φ = ρaI

2πa= ρ1I

2πa[2G(a,K, h) − G(2a,K, h)]. (A.19)

Henceρa

ρ1= [2G(a,K, h) − G(2a,K, h)]. (A.20)

Apparent resistivity ρa is given as

ρa = 2πa�φ

I. (A.21)

Appendix B. Curve matching

Field data can be compared with graphs (master curves) representing the calculated effectof the layered models. A once important but now little-used technique is known as curvematching. Figure 6 shows an interpretation using a set of master curves for vertical electricalsounding with a Wenner spread over two horizontal layers. The master curves are preparedin dimensionless form for a number of values of the reflection coefficient K by dividing thecalculated apparent resistivity ρa by the upper layer resistivity ρ1 (the latter derived from thefield curve at electrode spacing approaching zero), and by dividing the electrode spacing a bythe layer thickness z1. The curves are plotted on logarithmic paper with the same modulusas the master curves. They are then shifted over the master curves keeping the coordinateaxes parallel, until a reasonable match is obtained with one of the master curves or with aninterpolated curve. The point at which ρa

ρ1= a

z= 1 on the master sheet gives the true value of

ρ1 and z1 on the relevant axes, ρ2 is obtained from the K value of the best fitting curve [11].

References

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measuring the electrical resistivity of the earth using a fabricated resistivity meter

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