Review Article Sensitivity and Resolution Capacity of

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2013 Article ID 608037 12 pageshttpdxdoiorg1011552013608037

Review ArticleSensitivity and Resolution Capacity of Electrode Configurations

Cyril Chibueze Okpoli

Department of Geology and Applied Geophysics Faculty of Science Adekunle Ajasin University PMB 001 Akungba-AkokoOndo State Nigeria

Correspondence should be addressed to Cyril Chibueze Okpoli okpolicyrilgmailcom

Received 10 June 2013 Revised 26 July 2013 Accepted 25 August 2013

Academic Editor Francesco Soldovieri

Copyright copy 2013 Cyril Chibueze Okpoli This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper reviews the geological conditions data density and acquisition geometry that have direct influence on the sensitivityand resolution capacity of several electrode configurations The parameters appreciate the effectiveness of automated multichannelsystem which has evolved several electrode arrays that are cost effective reduction in survey time high sensitivity and resolutioncapacity in 2D and 3D resistivity tomographiesThe arrays are pole-pole pole-dipole pole-bipole dipole-dipole Wenner Wenner-120573 120574 gradient midpoint-potential-referred Schlumberger square and Lee-partition arrays The gradient array and midpoint-potential-referred are well suited for multichannel surveying and gradient array images are comparable to dipole-dipole and pole-dipole 2D electrical resistivity surveys can produce out-of-plane anomaly of the subsurface which could be misleading in theinterpretation of subsurface features Hence a 3D interpretation model should give more accurate results because of the increasein the reliability of inversion images and complete elimination of spurious featuresTherefore the reduction of anomaly effects anddamping factor due to signal to noise ratio result in better spatial resolution image thus enhancing its usage in environmental andengineering research

1 Introduction

Most targets of environmental and engineering interest areat shallow depths Geophysical responses from near-surfacesources usually treated as noise in traditional geophysicalexploration surveys are often the targets of interest in envi-ronmental and engineering investigations The subsurfacegeology can be complex subtle andmultiscale such that spa-tial variations can change rapidly both laterally and verticallyThus a closely spaced grid of observation points is requiredfor their accurate characterization high spatial resolution ofthe anomaly and good target definition Survey design ingeoelectrical resistivity surveys must take into account thecapabilities of the data acquisition system heterogeneitiesof the subsurface electrical resistivities and the resolutionrequired Other factors to be considered are the area extentof the site to be investigated the cost of the survey and thetime required to complete the survey [1]

Geoelectrical surveys were introduced by ConradSchlumberger and Marcel Schlumberger in 1932 and havesubsequently been investigated by many researchers for nu-merous electrode configurations and subsurface structures

This concept has been used in the qualitative comparison ofthe effective responses of different electrode arrays and in thegeneral understanding of the subsurface origin of observedapparent resistivity [2] Early attempts assumed that thedepth of investigation solely depends on the depth of currentpenetration which is a function of the current electrodesseparation For example [3] presented computations of two-and three-layer earth models by considering only the currentelectrodes separation

The depth of investigation has been defined as the depthat which a thin horizontal layer makes the maximum contri-bution to the total observed signal at the surface [4]This def-inition was used by [5 6] to compute characteristic curves forvarious conventional electrode arrays over a homogeneousisotropic medium showing the response of a thin horizontallayer with varying depth The point where these ldquodepth ofinvestigation characteristicrdquo (DIC) curves attain a maximumdefines the depth of investigation of the electrode array Thetheoretical basis [6] supported by [7] for Roy and Apparaorsquosresearch work has been strongly criticized [8 9] However[10] demonstrated using a boundary value approach that therelationships developed by [5] are correct for a homogeneous

2 International Journal of Geophysics

earthmodelwhich provides a general case useful for electricalresistivity investigationsThe practical usefulness and validityof these relationships have been discussed by [11 12]

A more practical and useful value for the depth ofinvestigation may be provided by the median value of Royand Apparaorsquos normalized DIC curves and not the maximum[13 14] This median depth measure is referred to as theeffective depth of investigation It is the depth at which thetotal of one-half of the signal originates from above andthe other half from below In both definitions the depth ofinvestigation practically depends on the relative positions ofboth the current and potential electrodes and not on thedepth of current penetration only which is determined by thecurrent electrode separation Another factor that influencesthe depth of investigation is the measurability of the signalby the measuring equipment The ability of the equipmentto detect the amplitude of the signal and of the existingnoise the power specification of the equipment and itsability to filter noise all affect the depth of investigationThe depth of investigation also depends on the subsurfacelayering [14] a low-resistivity surface layer reduces the depthof investigation

In electrical resistivity field observations four electrodesare generally placed at arbitrary locations However a num-ber of electrode configurations have been used in recordingresistivity field data each suitable for a particular geologicalsituation with varying anomaly resolution The conven-tional arrays most commonly used include the conventionalWenner (Wenner-alpha) Schlumberger dipole-dipole pole-pole and pole-dipole arrays Other variants of the conven-tional Wenner array are Wenner-beta Wenner-gamma andWenner-Schlumberger arrays [15 16] and are illustrated inFigure 1Wenner-Schlumberger array is a hybrid between theWenner-alpha and Schlumberger arrays [17] It is a digitizedform of the Schlumberger array that allows it to be used onautomated multielectrode systems with constant electrodespacing The Wenner-beta array is a special case of dipole-dipole array with the 119899-factor equal to one In Wenner-gamma array the current and the potential electrodes areinterleaved

The apparent resistivity values observed by the differentarray types over the same structure can be very different Thechoice of a particular array depends on a number of factorswhich include the geological structures to be delineatedheterogeneities of the subsurface anomaly resolution of thearray sensitivity of the resistivity meter the backgroundnoise level and electromagnetic coupling Other factors tobe considered are the sensitivity of the array to vertical andlateral variations in the resistivity of the subsurface its depthof investigation and the horizontal data coverage and signalstrength of the array [18]

These measurements sets include every possible conven-tional and nonconventional electrode arrays Surveys withthis suite of electrode configurations result in comprehensivedata sets which would contain all resistivity subsurface infor-mation that the n-electrodes system is capable of gatheringHowever a large portion of these configurations have largegeometric factors and are capable of reducing the stabilityof the inversion of the observed data sets The works of [19]

assessed the imaging potential of data sets acquired withmodern multielectrode resistivity systems using syntheticand field examples They showed that comprehensive datasets recorded with large numbers of four-point electrodeconfigurations provide significantly more information thanthose with standard electrode arrays However the recordingof comprehensive data sets requires too many measurementsand is not cost effective in routine geoelectrical resistivitysurveys

An optimization procedure that utilizes a goodness func-tion that ranks the sensitivities of all electrode configurationscan be used to define suite of electrode configurationsthat yields images comparable in quality to comprehensiveimages [20ndash22] The goodness function includes weightingterms which counterbalance the high sensitivities of themodel relative to deeper parts and minimize the influenceof well resolved regions of the model based on the exper-imental design procedure Data generated with electrodeconfigurations that yields large amounts of new informationaccording to their high sensitivities and depth of influenceare incorporated into the successively increasing optimal datasets

Many electrode configurations such as pole-pole pole-dipole pole-bipole Wenner Schlumberger dipole-dipoleWenner-120573 and -120574 array have been used in electric explorationPole-bipole is a special case of pole-dipole with 119899-factor equalto one Wenner-beta is special case of dipole-dipole andWenner of Schlumberger as well as gradient array and mid-potential-referred measurement They have their advantagesand limitations in fieldwork but provide useful practicalapplicability for surface sounding profiling for 2D and 3Dtomographies that is pole-pole Schlumberger pole-dipoleand dipole-dipole [23 24] In practice it is always difficultto acquire data in pole-pole due to limited access to remoteelectrodes and sometimes a long potential reference electrodelayout may pick up noise The sensitivities and abilitiesof the several electrode configurations in relation to theirdifferences in spatial resolution tendency for artefacts toappear in images deviation from the true model resistivityand interpretable maximum depth are limitations usuallyencountered in fieldwork

Author in [25] compared the resolution of crossholetomography with pole-pole pole-dipole and dipole-dipolearrays and he suggested that dipole-dipole surveying whenthe instrument is high is more suitable for resolving complexstructures than pole-pole array and pole-dipole may be agood compromise between resolution and signal strength

The reliability and resolution capacity of Wenner arrayas examined by [26 27] pointed out that Wenner datadensity is important and that the inverted model providesapproximate guide to the true geometry and true formationresistivity In order to get a high resolution and reliable imagemultielectrode cable with a fixed interelectrode spacing isusually utilized to capture maximum anomaly informationreasonable data coverage and high signal noise ratio Con-secutive usage of 119886 and 119899 in a theoretical complete data set ofarrays with low noise contamination is essential in obtaininghigh-resolution image but increases the fieldwork time

International Journal of Geophysics 3

a

C1 P1

(a)

aa naC2 C1 P1 P2

(b)

anaC1 P1 P2

(c)

C2C1 P1 P2aa na

(d)

aaC1 P1 P2

(e)

aa aC2 C1 P1 P2

(f)

aaaC2C1 P1 P2

(g)

middot middot middotP(1)1

P(1)2 P(2)

2

P(2)1 P(n)

1

P(n)2

C2C1

(h)

nana aC2C1 P1 P2

(i)

middot middot middot middot middot middotP(1)1 P(2)

1 P(2n)2

P(n+1)(n+2)(2n)1

P(1)(2) (n)2 C2C1

(j)

Figure 1 Sensitivity and resolution diagrams of electrode configurations (a) Pole-pole (b) dipole-dipole (c) pole-dipole (d) 120574-array (e)pole-bipole (f) Wenner-120573 (g) Wenner (h) gradient array (i) Schlumberger and (j) midpoint-potential-referred [15]

In all the electrode arrays larger spacing of 119886 and 119899gives deeper information about the subsurface while smallspacing of 119886 and 119899 probably offers a relatively good horizontalresolution for the shallower sections of the ground

The effectiveness of different electrode arrays in estima-tion of anomaly effects was developed by [28] which aresignificantly larger than background noise from the imagingperspective

The square array was originally developed as an alterna-tive to Wenner or Schlumberger arrays when a dipping sub-surface bedding or foliationwas present [29] Techniques foranalyzing directional-resistivity data provided by the square-array method have been developed [30] but the methodhas not been widely used Few case studies or interpretivemethods specifically applied to the square array are found inthe literature although commercial software is available forlayered earth interpretations

The square array is more sensitive to a given rock het-erogeneity and anisotropy than the more commonly usedSchlumberger and Wenner arrays Thus it is advantageousover equivalent survey using a Schlumberger orWenner arrayin that it requires about 65 percent less surface area

Field measurement in the square-array method is con-ducted in a manner similar to that for traditional collineararrays The location of a measurement is assigned to thecenter point of the square The array size (119860) is the length ofthe side of the square The array is expanded symmetricallyabout the centerpoint in increments of 119860(2)12 [29] so thatthe sounding can be interpreted as a function of depth

For each square three measurements are mademdashtwoperpendicular measurements (alpha a and beta b) and onediagonal measurement (gamma 119892) (Figure 2) The a and bmeasurements provide information on the directional varia-tion of the subsurface apparent resistivity (119903

119886) The azimuthal


A

B M

3

1

M

M

4

AA

2

B

B

N

N

N

5

6

7

8

Electrode configuration per square array

Starting orientation

Second orientation

about center point

Current electrode configuration for crossed square array

AB = current electrodesMN = potential electrodes

rotated 45 degrees

120572

120572

120572

120573

120573

120573

120574

120574

120574

120572998400

120572998400

120573998400

120573998400

120574998400

120574998400

1ndash35ndash7

1-2 = Pa3

5-6 = Pa4

1ndash4 = Pa1

5ndash8 = Pa1

Figure 2 Electrode configuration for square array

orientation of the 119886 and 119887 measurements is that of the lineconnecting the current electrodesThe 119892measurement servesas a check on the accuracy of the a and b measurements

In Figure 3 Lee-partition array is the same as theWennerarray except that an additional potential electrode 0 is placedat the centre of the array between the potential electrodes Mand N Measurements of the potential difference are madebetween 0 and M and between 0 and N The formula forcomputing the Lee-partitioning apparent resistivity is givenby

120588LP = 4120587119886Δ119881

119868 (1)

where Δ119881 is the potential difference between 0 and M or 0and NThis array 119868 has been used extensively in the past [31]

2 Sensitivity Function

Sensitivity is a reasonable quantity in electrical resistivitydata interpretation process It captures the changes in thepotential due to changes in resistivity of a cell volume Forinversion schemes the sensitivity provides a link betweenthe observed data and the model vector in terms of the

Jacobian matrix and is useful for the interpretation of fielddata employing forward modelling Thus the resolvabilityof the model parameters can be assessed using sensitivity[32] Sensitivity studies are usually carried out in conjunctionwith inversion techniques [33] depict the sensitivity as anexample for a single pole source at the surface for 1D and2D cases and show the distortion of the sensitivity for a 2Dprismatic body Frechet derivatives are deduce by [34] in thesense of single scattering theory and they compared theirevolved DC expression with the one for a 1D earth modeldescribed by [35] The presentation of a coarse 3D sensitivityplot is done by [36] for a surface dipole-dipole configurationusing two orthogonal sectors through 3D space A verticalsection of the dipole-dipole arrangement at the surface wasalso presented by [37] while [38] carried out sensitivitystudy using small 3D grids [39] has presented four waysin which the sensitivity of three-dimensional bodies can becomputedThese techniques include an analytical solution fora homogeneous half-space and three numerical approachesfor arbitrary resistivity structures which include DC forwardmodelling using a source at the transmitter and a fictitioussource at the receiver location finite difference approachsolving a set of partial differential equations defining thesensitivity problem and a two DC forward runs employingperturbation of a resistivity block to derive the sensitivity

The depth of investigation of an array can be assessedquantitatively by using sensitivity function which is givenby the Frechet derivative [33] The sensitivity function of anarray indicates the degree towhich the variations in resistivityof a section of the subsurface will influence the potentialmeasured by the array Higher values of the sensitivityfunction show greater influence of the subsurface regionon the measurements The sensitivity functions of severalelectrode configurations over homogeneous and layeredearth models are presented by [39] For a given source andreceiver configuration of electrodes the sensitivity function119878 is expressed as the inner product of the current densities119869119878and 119869

119877produced by a current source of strength 119868 at the

source and receiver positions respectively integrated over theperturbed volume 119889119881

119878 =ΔΦ

Δ120588=1119868intV

119869119878sdot 119869119877119889119881 (2)

3 Fundamental Theory of ElectricalResistivity Surveys

Electrical and electromagnetic (EM) methods are essentialtools in the field of applied geophysics for about a cen-tury particularly for shallow and near-surface investigationsThese geophysical prospecting methods are defined by theirfrequency of operation the origin of the source signals andthe manner by which the sources and receivers are coupledto the ground The signal frequencies range from a few hertz(Hz) in direct-current (DC) resistivity surveys up to severalgigahertz (GHz) in ground-penetrating radar (GPR) mea-surementsThemethods are generally governed byMaxwellrsquosequations of electromagnetism [40ndash43] Wave propagationdominates at high frequencies whereas diffusion is the


aa

aa

C1

C2

P1

P2

(a)

Surfacea a

C1 C2P1P2

P0

(12)a (12)a

(b)

Figure 3 Schematic diagrams of electrode configurations (a) Square array (b) Lee-partition

dominant physical mechanism of electromagnetic inductionat lower frequencies (in the quasistatic 16 approximation) Inthe direct-current (DC) frequency the diffusion term is zeroand the field is thus governed entirely by Poissonrsquos equation

Electrical and electromagnetic methods may either bepassive or active Passive methods use electromagnetic (EM)fields created by natural phenomena as source signals (egtelluric currents) while active methods employ signal gen-erators to generate the required input signals Sources andreceivers can be coupled to the ground through galvaniccontacts such as planted electrodes or through EM inductionsuch as coils of wireThese possibilities result in a greater vari-ety of field techniques than any other geophysical surveyingmethod where a single field of force or anomalous property(such as elasticity magnetism gravitation or radioactivity) isused

Electricalmethods of geophysical investigations are basedon the resistivity (or its inverse conductivity) contrasts ofsubsurface materials The electrical resistance 119877 of a materialis related to its physical dimension cross-sectional area119860 and length 119897 through the resistivity 120588 or its inverseconductivity by 120590 by

120588 =1

120590=

119877119860

119897 (3)

Low-frequency alternating current is employed as sourcesignals in the DC resistivity surveys in determining thesubsurface resistivity distributions Thus the magnetic prop-erties of the materials can be ignored [44 45] so thatMaxwellrsquos equations of electromagnetism reduced to

nabla sdot =1

1205760

119902 (4)

nabla times = 0 (5)

= minusnabla + Φ (6)

where is the electric field in Vm 119902 is the charge density inCm3 and 120576

0is the permittivity of free space (120576

0= 8854 times

10minus12 Fm ) These equations are applicable to continuousflow of direct-current however they can be used to representthe effects of alternating currents at low frequencies suchthat the displacement currents and induction effects arenegligible

4 Signal-to-Noise Ratio andElectromagnetic Coupling

Signal-to-noise ratio and electromagnetic coupling are otherfactors that influence the choice of an array If a collinearelectrode configuration is assumed the signal-to-noise ratioof an array would depend onwhether the potential electrodesare placed within or outside the current electrodes Thepotential difference and the signal strength would be higherfor potential electrodes placed within the current electrodesthan those placed outside the current electrodes The geo-metric factor of an array is indirectly related to its signalstrength it reflects the range of potential differences to beexpected for a particular electrode configuration Low valuesof geometric factor indicate that high potential differencewould be observed and vice versaThis implies that electrodeconfigurations with low geometric factor will yield highsignal-to-noise ratio Thus electrode configurations withlow geometric factor would be preferred if high signal-to-noise ratio is desired since a combination of high valuesof geometric factor with low voltage measurements may beoverwhelmed with noise

The frequencies of source signals in direct currentgeoelectrical resistivity surveys are usually very low Thisensures that electromagnetic effects are minimal Howevermost commercial resistivity measuring instruments employsquare waves or pulsed direct currents as source signals Thespectral analysis of such signals shows that they generate high


frequency harmonics which may result in electromagneticcoupling between the two dipoles (current electrodes andpotential electrodes) and the connecting cables Stability ofthemeasured response can be ensured by choosing long cycletimes of the input signal relative to the decay time Otherthings being equal electromagnetic coupling may be due tothe following

(i) increasing frequency of the input signal a slowlyvarying sinusoidal signal would generally suppresscoupling effects

(ii) the sensitivity of the electrode configuration to cou-pling effects

(iii) heterogeneities of the subsurface(iv) cultural effects arising from inductive coupling due

to power lines transmission lines pipelines buriedcables and so forth that may be close to the surveyline

5 Comparison of the ConventionalElectrode Arrays

The conventional Wenner and Schlumberger arrays arerelatively sensitive to vertical variations in the subsurfaceresistivity below the centre of the array but less sensitiveto horizontal variations in the subsurface resistivity Thearrays have a moderate depth of investigation and generallystrong signal strength which is inversely proportional to itsgeometric factor which is used in calculating the apparentresistivityThemajor limitation of these arrays is the relativelypoor horizontal coverage with increased electrode spacingThe Wenner array is preferred for surveys in a noisy sitebecause of its high signal strength however the array is lesssensitive to 3D structures

The dipole-dipole array is the most sensitive to resistivityvariations below the electrodes in each dipole pair and is verysensitive to horizontal variations but relatively insensitiveto vertical variations in the subsurface resistivities Thus itis the most preferred array for mapping vertical structureslike dykes and cavities Dipole-dipole array is howeververy poor in mapping horizontal structures such as sillssedimentary or horizontal layers Additionally it is the mostsensitive array to 3D structure among the common arrays[46] The depth of investigation of the array depends on boththe current electrode spacing and the distance between thetwo dipoles and is generally shallower than that of Wennerarray However dipole-dipole array has better horizontaldata coverage than Wenner array The major disadvantage ofthis array is the decrease in signal strength with increasingdistance between the dipole pair

The pole-dipole array is an asymmetrical array withasymmetrical apparent resistivity anomalies in the pseudo-sections over a symmetrical structure which could influencethe inversion model It has a relatively good horizontalcoverage and higher signal strength compared with dipole-dipole array It is much less sensitive to telluric noise than thepole-pole array Repeating measurements with the electrodesarranged in the reverse order can eliminate the asymmetrical

effect The combined measurements of the forward andreverse pole-dipole array would remove any bias in themodeldue to asymmetry [16] However this will increase the surveytime as the number of data points to be measured will bedoubled The signal strength of the pole-dipole array is lowerthan that of Wenner and Schlumberger arrays and is verysensitive to vertical structures

The pole-pole array consists of one current and onepotential electrode with the second current and potentialelectrodes at infinite distances Finding suitable locations forthese electrodes so as to satisfy this theoretical requirementis often difficult In addition to this limitation the pole-pole array is highly susceptible to large amount of telluricnoise capable of degrading the quality of the observeddata The array has the widest horizontal coverage and thedeepest depth of investigation but the poorest resolutionTheresolution of the pole-pole array is very poor as subsurfacestructures tend to be smeared out in the inversionmodel [16]If the electrode spacing is small and good horizontal coverageis desired the pole-pole array is a reasonable choice

6 Data Acquisition Instruments

Geoelectrical resistivity field data are acquired using earthresistivity meter commonly referred to as Terrameter Theequipment is portable light weight and relatively cost effec-tive when compared with other geophysical data acquisitionsystems A conventional setup of the earth resistivity meter(Figure 4) basically consists of the following The automateddata acquisition systems generally consist of a resistivityinstrument a relay unit (electrode selector) a portablecomputer (commonly a laptop) electrode cables variousconnectors and electrodes [24 47] Two ormore componentsof the multielectrode systems may be housed in the samebox making the systems more compact and portable Somemultielectrode systems employ intelligent switches at eachelectrode take-out instead of a central switching unit Theautomated multielectrode systems are either single channelor multichannel For any 119873-channel multielectrode systems119873 numbers of data can be recorded simultaneously therebyincreasing the data acquisition speed by a factor of 119873The multichannel systems are also capable of measuringfield data with a single channel if desired A number ofmultielectrode systems are commercially available for nearsurface investigations such as the SAS 4000 Lund (ABEM)Geo Tom (Geolog) Tomoplex (Campus Ltd) Super String R8(AGI Inc) and RESECS (DTM GmbH)

In field surveys the cables are rolled out along the surveyline(s) as predesigned by the field crew and the electrodesare connected to the electrode take-outs which are usuallynumbered on the cables Two to four cables may be usedtogether in a given survey depending on the acquisition sys-tem and the electrode configuration usedThe electrode take-out number should increase in the direction of increasingcoordinate number The acquisition systems automaticallycheck the electrode contacts and scan through a predefinedmeasurement protocol The extension of the survey line canbe achieved through roll-along technique (Figure 5) in which


Current meterBattery

Volt-meter

Electrodes

Soil and gravel

Path of electric current

Bedrock

Figure 4 A schematic earth resistivity meter and field observation [48]

part of the layout is shifted along the survey so as to makenewmeasurements in areas not already coveredThe systemsallow automatic updating of the coordinates in both 119909- and119910-axes if roll-along technique is employed

7 Types of Geoelectrical Resistivity Surveys

Geoelectrical resistivity surveys for delineating subsurfaceresistivity contrast may be classified as follows

(i) vertical electrical sounding or electric drilling

(ii) geoelectrical resistivity profiling (constant separationtraversing)

(iii) two-dimensional (2D) geoelectrical resistivity imag-ing

(iv) three-dimensional (3D) geoelectrical resistivity imag-ing

71 Vertical Electrical Sounding (VES) In vertical electricalsounding (VES) or ldquoelectric drillingrdquo [50] the centre pointof the array remains fixed (fixed potential electrodes) but thecurrent electrode separation is progressively increased TheSchlumberger array is mostly used for VES surveys becauseof its logistic simplicity however other configurations suchas Wenner array with equal electrode spacing can be usedThe vertical variation in the subsurface electrical resistivityat the point of investigation is obtained The subsurface isassumed to consist of horizontally stratified layers in whichthe resistivity varies only with depth but not in the horizontal(or lateral) direction Thus the model of interpretation ofVES is one-dimensional (1D) and is inherently insensitive tolateral variations in the subsurface resistivity which may leadto significant changes in apparent resistivity values Thesechanges are often misinterpreted as changes in resistivitywith depth However useful results have been obtained for

geological situations such as depth to bedrock where the 1Dmodel is approximately true

The observed apparent resistivities are usually plottedagainst half-current electrode spacing on a double loga-rithm graphs The resulting curves are then matched againsttheoretical model curves of typical subsurface resistivitystructures commonly referred to as master curves usually2-layer curves The presence of a turning point in thecurves indicates a boundary (an interface) between twolayers The shape of the curves depends on the resistivitycontrast between two layers Sequences of several layers canbe interpreted using auxiliary point diagrams A variety ofalgorithms for numerical computation of apparent resistivityversus depth were developed in the 1960s and 1970s [50 51]This was followed by the development of forward modellingand inversion schemes that allowed the resistivity of layeredmodels (1D) to be obtained on computers [52ndash54] Withthe computer inversion schemes which are based on linearinverse theory the covariance matrix of the inverse problemcan be analysed

For a horizontally stratified earth model which is homo-geneous and isotopic the potential and apparent resistivitydistribution can easily be computed using recurrence formu-las and linear filtering techniques

72 Geoelectrical Resistivity Profiling Constant separationtraversing (CST) or profiling method is the second approachemployed in classical resistivity surveys where the electrodeseparation remains fixed both in dimension and in orienta-tion The entire array is progressively moved along a straightline usually in the direction perpendicular to the geologicstrike This gives lateral variations in the subsurface resis-tivity at an approximate constant depth and is incapable ofdetecting vertical variations in the subsurface resistivity Dataobtained from profiling are mainly interpreted qualitativelyThe Wenner configuration is best suited for this approachdue to the equidistant spacing between the electrodes For


11

1

1

11

Station 3Station 2Station 1

First spread

Electrodeaddresses

Short setupLong setup

Cable

Second spreadThird spread

Etc

2

2

3

3

4

4

2 3 423 4

112121

414131

Figure 5 Roll-along technique using four electrode cables [49]

special problems such as the determination of a steeplydipping structure or the precise determination of a faultthe dipole-dipole array is preferred Generally the gradientarray could be a good choice in electrical resistivity profilingbecause only the potential electrodes need to be moved[55] Usually a completely homogeneous and isotropic earthmedium of uniform resistivity is assumed For a continuouscurrent flowing in an isotropic and homogeneous mediumthe current density 119869 is related to the electric field throughOhmrsquos law

119869 = 120590 (7)

The electric field vector can be represented as the gradientof the electric scalar potential

= minusnablaΦ (8)

Thus the current density 119869 becomes

119869 = minus120590nablaΦ (9)

By combining (4) and (6) we obtain the fundamental Poissonequation for electrostatic fields given by

nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)

The equation of continuity for a point in 3D space and time 119905defined by the Dirac delta function is given as

nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)

The current sources in a typical electrical resistivity surveyare usually point sources Thus the current and the currentdensity over a volume element Δ119881 around a current source 119868located at (119909 119910 119911) are given by the relation [56] as

nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)

where 120575 is the Dirac delta function Hence the potentialdistribution in the ground due to a point current source is

nabla sdot 119869 = [120590 (119909 119910 119911) nablaΦ (119909 119910 119911)]

= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)

This partial differential equation which is a self-adjointstrongly connected and nonseparable elliptic equation ofsecond order [57] gives the subsurface potential distributionin an isotropic nonuniform3Dmediumdue to a point currentsource Numerous techniques have been developed to solvethis problem that is to determine the potential distributionthat would be observed over a given subsurface structureThepotentialΦ(119909 119910 119911) and the normal component of the currentdensity 120590120575Φ120575119899 are continuous across the boundary betweentwo media of different resistivities but the current lines arerefracted in accordance with the boundary conditions

8 Potential Distribution due to Point Sourcein a Homogeneous Half-Space

All resistivity methods employ an artificial source of currentinjected into the subsurface through point electrodes and theresulting potential difference is measured at other electrodepositions in the neighbourhood of the current flow Fora semi-infinite conducting layer of uniform resistivity (acompletely homogeneous and isotropic medium) boundedby the ground surface as shown in Figure 6 a current ofstrength +119868 is injected at a point 119862

1into the ground surface

This current will flow away radially from the point of enteringand its distribution will be uniform over a hemisphericalshell of an underground region (Figure 7) of resistivity 120588 Thepotential varies inversely with the distance from the currentsource The currents flow and the equipotential surfaces areperpendicular At a distance 119903 of a point in the medium fromthe point source the surface area of the hemispherical shell is21205871199032 so that the current density 119869 becomes 11986821205871199032


Source Sink

rsource rsink

P

Figure 6 A diagram showing a homogenous and isotropicmedium

Thus the potential for the homogeneous half-space is

Φ (119903) =119875119868

2120587119903 (i)

In practice two current electrodes the current source +119868and the sink minus119868 (Figure 6) are often used The potentialdistribution is symmetrical about the vertical placed at themidpoint between the two current electrodes The potentialat an arbitrary point from a given pair of current electrodesby applying (i) is obtained as

Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862

2are the respective distances from the first

(source) and second (sink) current electrode to the arbitrarypoint

9 Apparent Resistivity and Geometric Factor

Usually the potential difference between two points is mea-sured The injecting (current) electrodes could be used intheory to measure the potential difference But the influ-ence of the resistances between the subsurface and currentelectrodes is not precisely known [58] Thus two potentialelectrodes are dedicated to detect the response signal If 119875

1

and 1198752are the potential electrodes the potential difference

between 1198751and 119875

2is shown in Figure 3

10 Modelling and Inversion Techniques

Thenumericalmodelling and inversion techniques employedin different geological conditions usually encountered ingeophysical application for environmental hydrogeologi-cal and engineering investigations in tropical regions arenumerically studied A net of parallel and perpendicular 2Dmodel structures was approximated from these 3D syntheticmodels Apparent resistivity values were then calculatedover the resulting 2D synthetic models for the followingarrays Wenner-alpha Wenner-beta Wenner-Schlumbergerdipole-dipole pole-dipole and pole-pole arrays The appar-ent resistivity values were calculated for different minimumelectrode spacings and interline spacings over the samemodels In addition the apparent resistivity values that wouldbe observed over the syntheticmodels if rectangular or squaregrids of electrodes are used for the survey were computedfor the different arrays except for Wenner-beta which is

basically a dipole-dipole array with a dipole separation equalto the dipole length using the different minimum electrodeseparations considered

It is usually recommended that apparent resistivity valuesfor the parallel and perpendicular sets of 2D profiles over thesynthetic model for each of the arrays under considerationshould be collated into 3D data sets More so the electrodearrays should be paired such that the apparent resistivityof all parallel 2D profiles must be computed with the samearray and those in a perpendicular direction are computedwith the second array that makes the pair Inversions of thecollated 3D data sets as well as the 3D data sets calculatedusing RES3DMOD were usually carried out using a three-dimensional inversion code (RES3DINV) that allows thevariation of model resistivity values in all directions Thesmoothness constrained and robust inversion methods areused separately on the same data sets so as to evaluate theirrelative effectiveness in 3D inversion Among the variouselectrode arrays considered the dipole-dipole and pole-dipole arrays generally yield higher anomaly effects whileWenner-Schlumberger array produced moderate anomalyeffect on the synthetic models In both synthetic modelspole-pole array yields the least anomaly effects

The normalized average sensitivity model values ofdipole-dipole and pole-dipole are much higher than those ofother arrays andWenner-Schlumberger array yields amoder-ate model sensitivity values This indicates that dipole-dipoleand pole-dipole arrays are more sensitive to 3D features andthus would yield better resolution of 3D inversion modelsTheWenner-Schlumberger array may be a good compromisebetween the high sensitivity dipole-dipole and pole-dipolearrays and the low sensitivity pole-pole Wenner-alpha andWenner-beta arrays in terms of inversion model sensitivityand resolution The sensitivity and resolution of the arrays to3D features were found to generally improve for orthogonalpaired arrays

11 Sensitivity and Resolution

In geoelectrical resistivity imaging the imaging capabilities ofdifferent electrode configurations are different when appliedto a particular geologic structure These differences in imag-ing capability are often reflected in the spatial resolution ofthe arrays the tendency of the array to produce artefacts inthe resulting model inversion images deviations from thetrue resistivity model values and the maximum depth ofinvestigations attained by the arrays To obtain reliable andhigh-resolution images in geoelectrical resistivity imagingthe electrode configuration used in themeasurements shouldideally give data with maximum anomaly information rea-sonable data coverage and high-signal to noise ratio Intheory a complete data set of an array with minimum noisecontamination is required to obtain such a high-resolutionimage But acquiring such large number of data pointswould significantly increase the time required for the fieldmeasurements even when automatic data acquisition systemsare used A large number of data points could also make itdifficult for the inversion to attain a good data misfit and


Surface

EquipotentialsCurrent flow

C2C1

Uniform resistivity 120588

Figure 7 Current from a point source and the resulting equipotential distribution

probably produce more artefacts in the inversion images dueto the unknown characteristics of the noise contamination inthe data

Generally the overall sensitivity of the image planedecreases rapidly with depth indicating a significant loss ofresolution with depth

The average model sensitivities observed in dipole-dipoleand pole-dipole arrays are much higher than those in theother arrays investigated This implies that dipole-dipole andpole-dipole arrays using combination of 2D profiles aremore sensitive to 3D features than any of the conventionalarrays investigated The observed high model sensitivity val-ues for dipole-dipole and pole-dipole arrays are in agreementwith the high anomaly effects calculated for these arraysIn general pole-pole and Wenner-beta arrays showed theleast sensitivity in all the inversion models tested This ispartly due to the fact that these arrays have the highest depthof investigation under similar conditions with other arraysand sensitivity generally decreases with depth The Wenner-Schlumberger array shows moderate sensitivity values andthus may be a good compromise between the more sensitivedipole-dipole and pole-pole arrays and the less sensitivearrays for 3D survey

The sensitivity pattern of an electrode array is an impor-tant factor in the determination of its imaging capabilityEach apparent resistivity measurement usually made with aunique set of four electrodes can be viewed as a weightedsum of the distributed electrical resistivity within a volumesampled by the set of four electrodes The best-fit of theelectrical resistivity distribution can be determined by simul-taneous inversion of large number of apparent resistivitymeasurements made with overlapping sampled volumesThesampled volume and the manner in which the distributedelectrical resistivities are weighted depend on the locationsand the configuration of the electrodes used for the measure-ments The spatial sensitivity distribution of each set of four-electrodes used for the measurements accumulates to definethe spatial sensitivity of the entire survey

The sensitivity of a set of four electrodes can be definedas the change in potential measured with the potential elec-trodes because of given change in the subsurface conductivity(or resistivity) in a small area located at (119909 119910) An equivalentform can be written in terms of the apparent resistivity Thesensitivity of the set of four electrodes can be calculated for

many locations throughout the subsurface to produce a mapof spatial sensitivity The sensitivity provides information onthe section of the subsurface with the greatest effect on themeasured apparent resistivity

12 Factors Affecting Sensitivityand Resolution

The following are some of the factors that could influencethe sensitivity and resolution of 3D inversion in geoelectricalresistivity imaging

(i) Data Density and Lateral Coverage The sensitivity ofgeoelectrical resistivity increases with increasing data densityrelative to the grid size of the electrodesThe higher the lateraldata coverage of an electrode array the greater the modelsensitivity values that would be obtained in the inversion

(ii) Depth of PenetrationThesensitivity of electrical resistivityof the subsurface for a given electrode configuration generallydecreases with depth of investigation Thus electrode con-figurations with greater depth of penetration are largely lesssensitive than those with less depth of penetration for thesame data level

(iii) Damping Factor and Noise ContaminationThe choice ofthe damping parameters for a particular data set also affectsthe overall sensitivity of the resultingmodel resistivity valuesAppropriate damping factor that would yield the optimumsensitivity can be selected through trial and error methodor with experience based on the knowledge of the geologyNoisy data generally require a higher damping factor thanless noisy data Some arrays are more sensitive to noise thanothers the sensitivity of the arrays to noise contaminationscan significantly affect the sensitivity of the inversionmodels

(iv) Inter-line Spacing and Electrodes Grid Size In the caseof 3D geoelectrical resistivity imaging using orthogonal orparallel 2D profiles the interline spacing relative to theminimum electrode separation can significantly affect theoverall sensitivity of the resulting inversionmodels In theorythe interline spacing should be the same with the minimumelectrode separation this may not be practical considering


the volume of work and time required for field data measure-ments Electrode grid sizes with regular or uniform electrodespacing in both directions are found to yield higher sensitivityvalues than corresponding grid sizes with irregular electrodespacing

(v) Geological Conditions The nature of the subsurfacegeology geometry of subsurface features and the electricalproperties (or resistivity contrast) of the anomalous bodycan also significantly influence the overall sensitivity of theinversion models

The sensitivity values of electrode configurations provideinformation on the section of the subsurface with the greatesteffect on the measured apparent resistivity values In geo-electrical resistivity imaging the sensitivity is a very impor-tant and indicative property in the interpretation processGenerally the spatial sensitivity patterns for homogeneousenvironments are good approximations for moderate con-ductivityresistivity contrasts not exceeding 1 10 if the sourceis located within conductive material [39]

13 Conclusion

Pole-dipole and pole-bipole have moderate anomaly effectsand relatively low signal to noise ratio and both can yieldbetter spatial resolution images than those of pole-poleSchlumberger and Wenner Increasing potential electrodesenhances spatial resolution and better signal to noise ratioand thus reducing noise contamination One remote elec-trode accessibility of surveys complicates data quality byreciprocal measurements since the noise from the remotepotential electrode can be higher Wenner and Schlumbergerhave similarity and resemblance in electric field measure-ments due to their imaging abilities with their main strengthin depth determination in comparison to other electrodesHowever Wenner has poorer spatial resolution than thatof pole-dipole pole-bipole dipole-dipole and Schlumbergerarrays while Schlumberger with reduction of the signal tonoise ratio may offer improved imaging resolution side byside with Wenner but may pick up noise than the normalarray when assessing the data quality using reciprocal mea-surements Dipole-dipole has relatively high anomaly effectsand risk of noise contamination but often produces low signalto noise ratio in surveys when compared with Wenner-betaWenner and gamma arrays Gamma array has advantage oflow noise contamination of imaging survey in all arrays butthe signal-to-noise ratio and anomaly effects are consistentlyhigh and spatial resolution of the imaging is not as good asthe other arrays Gradient array has good spatial resolutionwhile midpoint-potential-referred has higher signal-to-noiseratio with lower spatial resolution in the image Midpointpotential referred has an option for multichannel recordingarray due to its lower noise sensitivity The square-array ismore sensitive to a given rock heterogeneity and anisotropythan the more commonly used Schlumberger and Wennerarrays

References

[1] P A Ahzegbobor Acquisition geometry and inversion of 3Dgeoelectrical resistivity imaging data for environmental andengineering investigations [PhD thesis] Physics DepartmentCovenant University Ota Nigeria 2010

[2] R D Barker ldquoSignal contribution sections and their use inresistivity studiesrdquo Geophysical Journal vol 59 no 1 pp 123ndash129 1979

[3] M Muskat and H H Evinger ldquoCurrent penetration in directcurrent prospectingrdquo Geophysics vol 6 pp 397ndash427 1941

[4] H M Evjen ldquoDepth factor and resolving power of electricalmeasurementsrdquo Geophysics vol 3 pp 78ndash85 1938

[5] A Roy andAApparao ldquoDepth of investigation in direct currentmethodsrdquo Geophysics vol 36 pp 943ndash959 1971

[6] A Roy ldquoDepth of investigation in Wenner three-electrode anddipole-dipole DC resistivity methodsrdquo Geophysical Prospectingvol 20 no 2 pp 329ndash340 1972

[7] O Koefoed ldquoDiscussion on ldquoDepth of investigation in directcurrent methodsrdquo by A Roy and A Apparaordquo Geophysics vol37 pp 703ndash704 1972

[8] D S Parasnis ldquoMore comments on ldquoA theorem for Dc regimesand some of its consequencesrdquo by A RoyrdquoGeophysical Prospect-ing vol 32 pp 139ndash141 1984

[9] S C Guerreiro ldquoComment on ldquoA theorem for direct currentregimes and some of its consequencesrdquo by A Roy and somerelated papers and commentsrdquo Geophysical Prospecting vol 31no 1 pp 192ndash195 1983

[10] B Banerjee and B P Pal ldquoA simplemethod for determination ofdepth of investigation characteristics in resistivity prospectingrdquoExploration Geophysics vol 17 pp 93ndash95 1986

[11] M N Nabighian and C L Elliot ldquoNegative induced polariza-tion effects from layered mediardquo Geophysics vol 41 no 6 pp1236ndash1255 1976

[12] K K Roy andHM Elliot ldquoSome observations regarding depthof exploration in DC electrical methodsrdquo Geoexploration vol19 no 1 pp 1ndash13 1981

[13] L S Edwards ldquoA modified pseudosection for resistivity andinduced polarizationrdquo Geophysics vol 42 no 5 pp 1020ndash10361977

[14] R D Barker ldquoDepth of investigation of collinear symmetricalfour-electrode arraysrdquo Geophysics vol 54 no 8 pp 1031ndash10371989

[15] T Dahlin and B Zhou ldquoA numerical comparison of 2D resistiv-ity imaging with 10 electrode arraysrdquo Geophysical Prospectingvol 52 no 5 pp 379ndash398 2004

[16] M H Loke ldquoTutorial 2D and 3D electrical imaging surveysrdquo2004 httpwwwgeotomosoftcom

[17] O Pazdirek andV Blaha ldquoExamples of resistivity imaging usingME-100 resistivity field acquisition systemrdquo in Proceedings ofthe EAGE 58th Conference and Technical Exhibition ExtendedAbstracts Amsterdam The Netherlands 1996

[18] M H Loke ldquoElectrical Imaging surveys for environmental andengineering studies a practical guide to 2D and 3D surveysrdquo2001 httpwwwgeotomosoftcom

[19] P Stummer HMaurer and A G Green ldquoExperimental designelectrical resistivity data sets that provide optimum subsurfaceinformationrdquo Geophysics vol 69 no 1 pp 120ndash139 2004

[20] P B Wilkinson P I Meldrum J E Chambers O Kuras andR D Ogilvy ldquoImproved strategies for the automatic selection ofoptimized sets of electrical resistivity tomographymeasurement


configurationsrdquo Geophysical Journal International vol 167 no3 pp 1119ndash1126 2006

[21] A Furman T P A Ferre and G L Heath ldquoSpatial focusingof electrical resistivity surveys considering geologic and hydro-logic layeringrdquo Geophysics vol 72 no 2 pp F65ndashF73 2007

[22] T Hennig A Weller and M Moller ldquoObject orientatedfocussing of geoelectrical multielectrode measurementsrdquo Jour-nal of Applied Geophysics vol 65 no 2 pp 57ndash64 2008

[23] J Chambers R Ogilvy P Meldrum and J Nissen ldquo3Dresistivity imaging of buried oil- and tar-contaminated wastedepositsrdquo European Journal of Environmental and EngineeringGeophysics vol 4 no 1 pp 3ndash14 1999

[24] T Dahlin On the Automation of 2D Resistivity Surveying forEngineering and Environmental Applications Lind University1993

[25] Y Sasaki ldquoResolution of resistivity tomography inferred fromnumerical simulationrdquo Geophysical Prospecting vol 40 no 4pp 453ndash463 1992

[26] T Dahlin andM H Loke ldquoResolution of 2DWenner resistivityimaging as assessed by numerical modellingrdquo Journal of AppliedGeophysics vol 38 no 4 pp 237ndash249 1998

[27] A I Olayinka and U Yaramanci ldquoAssessment of the reliabil-ity of 2D inversion of apparent resistivity datardquo GeophysicalProspecting vol 48 no 2 pp 293ndash316 2000

[28] H Militzer R Rosler and W Losch ldquoTheoretical and exper-imental investigations for cavity research with geoelectricalresistivity methodsrdquo Geophysical Prospecting vol 27 no 3 pp640ndash652 1979

[29] G M Habberjam ldquoApparent resistivity anisotropy and strikemeasurementsrdquo Geophysical Prospecting vol 23 no 2 pp 211ndash247 1975

[30] G M Habberjam ldquoThe effects of anisotropy on square arrayresistivity measurementsrdquo Geophysical Prospecting vol 20 no2 pp 249ndash266 1972

[31] R GVanNostrand andK L Cook ldquoInterpretation of resistivitydatardquo U S Geological Survey Professional 499 pages 1966httpbooksgooglecom

[32] K Spitzer and H-J Kumpel ldquo3D FD resistivity modelling andsensitivity analyses applied to a highly resistive phonolitic bodyrdquoGeophysical Prospecting vol 45 no 6 pp 963ndash982 1997

[33] P R McGillivray and D W Oldenburg ldquoMethods for calcu-lating Frechet derivatives and sensitivities for the non-linearinverse problem a comparative studyrdquo Geophysical Prospectingvol 38 no 5 pp 499ndash524 1990

[34] D E Boerner and G F West ldquoFrechet derivatives and singlescattering theoryrdquoGeophysical Journal International vol 98 no2 pp 385ndash390 1989

[35] D W Oldenburg ldquoThe interpretation of direct current mea-surementsrdquo Geophysics vol 43 no 3 pp 610ndash625 1978

[36] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

[37] M Noel and B X Biwen Xu ldquoArchaeological investigation byelectrical resistivity tomography a preliminary studyrdquoGeophys-ical Journal International vol 107 no 1 pp 95ndash102 1991

[38] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[39] K Spitzer ldquoThe three-dimensional DC sensitivity for surfaceand subsurface sourcesrdquo Geophysical Journal International vol134 no 3 pp 736ndash746 1998

[40] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941

[41] R F HarrintonHarmonic Electromagnetic Fields Electrical andElectronic Engineering Series McGraw-Hill New York NYUSA 1961

[42] F S Grant and G F West Interpretation Theory in AppliedGeophysics McGraw-Hill New York NY USA 1965

[43] S H Ward and G W Hohmann ldquoElectromagnetic theoryfor geophysical applicationsrdquo in Electromagnetic Methods inApplied Geophysics Investigations in Geophysical Series MN Nabighian Ed vol 1 pp 131ndash312 Society of ExplorationGeophysicists Tulsa Okla USA 1987

[44] G V Keller and F C Frischknecht Methods in GeophysicalProspecting Pergamo Press 1966

[45] WMTelford L P Geldart R E Sheriff andDAKeysAppliedGeophysics Cambridge University Press Lodon UK 1976

[46] T Dahlin and M H Loke ldquoQuasi-3D resistivity imaging-mapping of three-dimensional structures using two-dimensional DC resistivity techniquesrdquo in Proceedings of the3rd Meeting of the Environmental and Engineering GeophysicalSociety pp 143ndash146 1997

[47] D H Griffiths J Turnbull and A I Olayinka ldquoTwo-dimensional resistivity mapping with a complex controlledarrayrdquo First Break vol 8 no 4 pp 121ndash129 1990

[48] E S Robinson and C Coruh Exploration Geophysics JohnWiley amp Sons New York NY USA 1988

[49] R A V Overmeeren and I L Ritsema ldquoContinuous verticalelectrical soundingrdquo First Break vol 6 no 10 pp 313ndash324 1988

[50] O Koefoed Geosounding Principles 1 Resistivity SoundingMeasurements Elsevier Science Amsterdam The Netherlands1979

[51] H M Mooney E Orellana H Pickett and L TornheimldquoA resistivity computation method for layered earth modelsrdquoGeophysics vol 31 pp 192ndash203 1966

[52] D P Ghosh ldquoThe application of linear filter theory to thedirect interpretation of geoelectrical resistivity sounding mea-surementsrdquo Geophysical Prospecting vol 19 pp 192ndash217 1971

[53] J R Inman J Ryu and S H Ward ldquoResistivity inversionrdquoGeophysics vol 38 no 6 pp 1088ndash1108 1973

[54] A A R Zohdy ldquoA newmethod for the automatic interpretationof Schlumberger andWenner sounding curvesrdquoGeophysics vol54 no 2 pp 245ndash253 1989

[55] P Furness ldquoGradient array profiles over thin resistive veinsrdquoGeophysical Prospecting vol 41 no 1 pp 113ndash130 1993

[56] A Dey andH FMorrison ldquoResistivitymodelling for arbitrarilyshaped three- dimensional structurerdquoGeophysics vol 44 no 4pp 753ndash780 1979

[57] R S VargaMatrix Iteration Analysis Prentice-Hall EnglewoodCliffs NJ USA 1962

[58] K S Cheng S J Simske D Isaacson J C Newell and D GGisser ldquoErrors due to measuring voltage on current-carryingelectrodes in electric current computed tomographyrdquo IEEETransactions on Biomedical Engineering vol 37 no 1 pp 60ndash65 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

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EarthquakesJournal of


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Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

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OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


earthmodelwhich provides a general case useful for electricalresistivity investigationsThe practical usefulness and validityof these relationships have been discussed by [11 12]

A more practical and useful value for the depth ofinvestigation may be provided by the median value of Royand Apparaorsquos normalized DIC curves and not the maximum[13 14] This median depth measure is referred to as theeffective depth of investigation It is the depth at which thetotal of one-half of the signal originates from above andthe other half from below In both definitions the depth ofinvestigation practically depends on the relative positions ofboth the current and potential electrodes and not on thedepth of current penetration only which is determined by thecurrent electrode separation Another factor that influencesthe depth of investigation is the measurability of the signalby the measuring equipment The ability of the equipmentto detect the amplitude of the signal and of the existingnoise the power specification of the equipment and itsability to filter noise all affect the depth of investigationThe depth of investigation also depends on the subsurfacelayering [14] a low-resistivity surface layer reduces the depthof investigation

In electrical resistivity field observations four electrodesare generally placed at arbitrary locations However a num-ber of electrode configurations have been used in recordingresistivity field data each suitable for a particular geologicalsituation with varying anomaly resolution The conven-tional arrays most commonly used include the conventionalWenner (Wenner-alpha) Schlumberger dipole-dipole pole-pole and pole-dipole arrays Other variants of the conven-tional Wenner array are Wenner-beta Wenner-gamma andWenner-Schlumberger arrays [15 16] and are illustrated inFigure 1Wenner-Schlumberger array is a hybrid between theWenner-alpha and Schlumberger arrays [17] It is a digitizedform of the Schlumberger array that allows it to be used onautomated multielectrode systems with constant electrodespacing The Wenner-beta array is a special case of dipole-dipole array with the 119899-factor equal to one In Wenner-gamma array the current and the potential electrodes areinterleaved

The apparent resistivity values observed by the differentarray types over the same structure can be very different Thechoice of a particular array depends on a number of factorswhich include the geological structures to be delineatedheterogeneities of the subsurface anomaly resolution of thearray sensitivity of the resistivity meter the backgroundnoise level and electromagnetic coupling Other factors tobe considered are the sensitivity of the array to vertical andlateral variations in the resistivity of the subsurface its depthof investigation and the horizontal data coverage and signalstrength of the array [18]

These measurements sets include every possible conven-tional and nonconventional electrode arrays Surveys withthis suite of electrode configurations result in comprehensivedata sets which would contain all resistivity subsurface infor-mation that the n-electrodes system is capable of gatheringHowever a large portion of these configurations have largegeometric factors and are capable of reducing the stabilityof the inversion of the observed data sets The works of [19]

assessed the imaging potential of data sets acquired withmodern multielectrode resistivity systems using syntheticand field examples They showed that comprehensive datasets recorded with large numbers of four-point electrodeconfigurations provide significantly more information thanthose with standard electrode arrays However the recordingof comprehensive data sets requires too many measurementsand is not cost effective in routine geoelectrical resistivitysurveys

An optimization procedure that utilizes a goodness func-tion that ranks the sensitivities of all electrode configurationscan be used to define suite of electrode configurationsthat yields images comparable in quality to comprehensiveimages [20ndash22] The goodness function includes weightingterms which counterbalance the high sensitivities of themodel relative to deeper parts and minimize the influenceof well resolved regions of the model based on the exper-imental design procedure Data generated with electrodeconfigurations that yields large amounts of new informationaccording to their high sensitivities and depth of influenceare incorporated into the successively increasing optimal datasets

Many electrode configurations such as pole-pole pole-dipole pole-bipole Wenner Schlumberger dipole-dipoleWenner-120573 and -120574 array have been used in electric explorationPole-bipole is a special case of pole-dipole with 119899-factor equalto one Wenner-beta is special case of dipole-dipole andWenner of Schlumberger as well as gradient array and mid-potential-referred measurement They have their advantagesand limitations in fieldwork but provide useful practicalapplicability for surface sounding profiling for 2D and 3Dtomographies that is pole-pole Schlumberger pole-dipoleand dipole-dipole [23 24] In practice it is always difficultto acquire data in pole-pole due to limited access to remoteelectrodes and sometimes a long potential reference electrodelayout may pick up noise The sensitivities and abilitiesof the several electrode configurations in relation to theirdifferences in spatial resolution tendency for artefacts toappear in images deviation from the true model resistivityand interpretable maximum depth are limitations usuallyencountered in fieldwork

Author in [25] compared the resolution of crossholetomography with pole-pole pole-dipole and dipole-dipolearrays and he suggested that dipole-dipole surveying whenthe instrument is high is more suitable for resolving complexstructures than pole-pole array and pole-dipole may be agood compromise between resolution and signal strength

The reliability and resolution capacity of Wenner arrayas examined by [26 27] pointed out that Wenner datadensity is important and that the inverted model providesapproximate guide to the true geometry and true formationresistivity In order to get a high resolution and reliable imagemultielectrode cable with a fixed interelectrode spacing isusually utilized to capture maximum anomaly informationreasonable data coverage and high signal noise ratio Con-secutive usage of 119886 and 119899 in a theoretical complete data set ofarrays with low noise contamination is essential in obtaininghigh-resolution image but increases the fieldwork time


a

C1 P1

(a)

aa naC2 C1 P1 P2

(b)

anaC1 P1 P2

(c)

C2C1 P1 P2aa na

(d)

aaC1 P1 P2

(e)

aa aC2 C1 P1 P2

(f)

aaaC2C1 P1 P2

(g)


P(1)2 P(2)

2

P(2)1 P(n)

1

P(n)2

C2C1

(h)

nana aC2C1 P1 P2

(i)


1 P(2n)2

P(n+1)(n+2)(2n)1

P(1)(2) (n)2 C2C1

(j)










A

B M

3

1

M

M

4

AA

2

B

B

N

N

N

5

6

7

8



Second orientation

about center point



rotated 45 degrees

120572

120572

120572

120573

120573

120573

120574

120574

120574

120572998400

120572998400

120573998400

120573998400

120574998400

120574998400

1ndash35ndash7

1-2 = Pa3

5-6 = Pa4

1ndash4 = Pa1

5ndash8 = Pa1




120588LP = 4120587119886Δ119881

119868 (1)








119878 =ΔΦ

Δ120588=1119868intV

119869119878sdot 119869119877119889119881 (2)




aa

aa

C1

C2

P1

P2

(a)

Surfacea a

C1 C2P1P2

P0

(12)a (12)a

(b)





120588 =1

120590=

119877119860

119897 (3)


nabla sdot =1

1205760

119902 (4)

nabla times = 0 (5)




0= 8854 times






















Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


a

C1 P1

(a)

aa naC2 C1 P1 P2

(b)

anaC1 P1 P2

(c)

C2C1 P1 P2aa na

(d)

aaC1 P1 P2

(e)

aa aC2 C1 P1 P2

(f)

aaaC2C1 P1 P2

(g)


P(1)2 P(2)

2

P(2)1 P(n)

1

P(n)2

C2C1

(h)

nana aC2C1 P1 P2

(i)


1 P(2n)2

P(n+1)(n+2)(2n)1

P(1)(2) (n)2 C2C1

(j)










A

B M

3

1

M

M

4

AA

2

B

B

N

N

N

5

6

7

8



Second orientation

about center point



rotated 45 degrees

120572

120572

120572

120573

120573

120573

120574

120574

120574

120572998400

120572998400

120573998400

120573998400

120574998400

120574998400

1ndash35ndash7

1-2 = Pa3

5-6 = Pa4

1ndash4 = Pa1

5ndash8 = Pa1




120588LP = 4120587119886Δ119881

119868 (1)








119878 =ΔΦ

Δ120588=1119868intV

119869119878sdot 119869119877119889119881 (2)




aa

aa

C1

C2

P1

P2

(a)

Surfacea a

C1 C2P1P2

P0

(12)a (12)a

(b)





120588 =1

120590=

119877119860

119897 (3)


nabla sdot =1

1205760

119902 (4)

nabla times = 0 (5)




0= 8854 times






















Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


A

B M

3

1

M

M

4

AA

2

B

B

N

N

N

5

6

7

8



Second orientation

about center point



rotated 45 degrees

120572

120572

120572

120573

120573

120573

120574

120574

120574

120572998400

120572998400

120573998400

120573998400

120574998400

120574998400

1ndash35ndash7

1-2 = Pa3

5-6 = Pa4

1ndash4 = Pa1

5ndash8 = Pa1




120588LP = 4120587119886Δ119881

119868 (1)








119878 =ΔΦ

Δ120588=1119868intV

119869119878sdot 119869119877119889119881 (2)




aa

aa

C1

C2

P1

P2

(a)

Surfacea a

C1 C2P1P2

P0

(12)a (12)a

(b)





120588 =1

120590=

119877119860

119897 (3)


nabla sdot =1

1205760

119902 (4)

nabla times = 0 (5)




0= 8854 times






















Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


aa

aa

C1

C2

P1

P2

(a)

Surfacea a

C1 C2P1P2

P0

(12)a (12)a

(b)





120588 =1

120590=

119877119860

119897 (3)


nabla sdot =1

1205760

119902 (4)

nabla times = 0 (5)




0= 8854 times






















Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


















Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



Volt-meter

Electrodes

Soil and gravel


Bedrock















11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


11

1

1

11


First spread

Electrodeaddresses


Cable


Etc

2

2

3

3

4

4

2 3 423 4

112121

414131



119869 = 120590 (7)


= minusnablaΦ (8)


119869 = minus120590nablaΦ (9)


nabla2Φ(119909 119910 119911) = minus1

1205760

119902 (119909 119910 119911) (10)


nabla sdot 119869 (119909 119910 119911 119905) = minus120575

120575119905119902 (119909 119910 119911 119905) 120575 (119909) 120575 (119910) 120575 (119911) (11)


nabla sdot 119869 = (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904) (12)



= (119868

Δ119881) 120575 (119909 minus 119909

119904) 120575 (119910 minus 119910

119904) 120575 (119911 minus 119911

119904)

(13)







Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Source Sink

rsource rsink

P



Φ (119903) =119875119868

2120587119903 (i)


Φ (119903) =119875119868

2120587(

11199031198621

minus11199031198622

) (ii)

where 1199031198621and 119903119862





1












Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Surface


C2C1



















13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





13 Conclusion


References






































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Documents

Review Article Sensitivity and Resolution Capacity of