ARTICLE IN PRESS

Computers & Geosciences 35 (2009) 1748–1751

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Computers & Geosciences

0098-30

doi:10.1

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journal homepage: www.elsevier.com/locate/cageo

Computer program to calculate resistivities and layer thicknesses fromSchlumberger soundings at the surface, at lake bottom and with twoelectrodes down in the subsurface$

Franz Kohlbeck a,�, Dana Mawlood b

a Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29, A1040 Vienna, Austriab Department of Civil Engineering, College of Engineering, University of Salahaddin, Erbil, Iraq

a r t i c l e i n f o

Article history:

Received 27 August 2007

Received in revised form

3 March 2009

Accepted 8 March 2009

Keywords:

DC-geoelectrics

Schlumberger

Computer program

Hydrology

Lake bottom

04/$ - see front matter & 2009 Elsevier Ltd. A

016/j.cageo.2009.03.001

e available from server at http://www.iamg.

esponding author. Tel.: +4315880112826; fa

ail addresses: fkohlbec@pop.tuwien.ac.at (F. K

awlood@gmx.at (D. Mawlood).

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A computer program is presented that calculates thicknesses and resistivities of a layered earth from

three different variations of Schlumberger arrangements:

(1) The usual Schlumberger method with 4 electrodes at the surface.

(2) Lake bottom arrangement: 4 electrodes at the bottom of waters like lakes, rivers or the sea.

(3) Two potential electrodes at the same depth in the ground and two electrodes at the surface.

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ohlb

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

One of the most popular methods for investigating theresistivity of the subsurface is DC sounding. The programpresented here written to evaluate Schlumberger soundings andtwo further, similar arrangements (see Fig. 1). Schlumbergersoundings have a very long tradition and are frequently usedtoday in spite of the growing importance of two-dimensionalmeasurements with multi-array arrangements. The reason is thatthe overall effort required to carry out multi-array measurementsis much higher than that for Schlumberger measurements and inmany cases the information obtained from Schlumbergermeasurements is sufficient.

The method can distinguish between layers with differentresistivities, where sufficiently high resistivity contrast exists. Ingroundwater exploration a popular example is the differentiationbetween aquitards, consisting of low resistivity layers with highclay content, and aquifers, consisting mainly of gravel and sandwith low clay content and high resistivity. There is no one-to-onerelation between the resistivities obtained from the measure-ments and the materials found in the subsurface; however, inmost cases a very confident model can be obtained.

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3 15880112892.

eck),

A great variety of computer programs have been published,e.g.: Johansen (1977); Koefoed (1979); Zohdy (1989); Bas-okur,(1990); Vedanti et al. (2005) and are commercially available e.g.(1�1D from Interpexs, IPI2Win from Geoscans, Res1d fromGeotomos). In most cases the source code is not available or theprograms do not meet the requirements for easy usage. Thecomputer program presented here offers an open source code forboth calculation of resistivities and user interface. The programcalculates layer thicknesses (or depth) and resistivities fromsurface measurements. The user must be aware of the fact that aunique solution is not possible. This is not a deficiency of theprogram, but a consequence of the theory of electrical soundings.Unique solutions may be obtained by combinations with othermethods and by assuming layer continuity.

The program calculates layer thicknesses and resistivitiesfrom three different variations of Schlumberger arrangements(Fig. 1):

(1)

The usual Schlumberger method with 4 electrodes at thesurface.

(2)

Lake bottom arrangement: 4 electrodes at the bottom ofwaters like lakes, rivers or the sea.

(3)

Two potential electrodes at the same depth in the ground andtwo electrodes on the surface.

The program gives a numerical error estimate for all parametersand allows for rapid judgment of the reliability of the results with

ARTICLE IN PRESS

A B

M N

zzj-1

zn-1

z1

z0

zj

z = 0h1

hn-1

hj

ρ1

ρj

ρn

notation convention

Fig. 2. Notations used in equations.

A BM Nsurface

lake bottom

two electrodes down

A BM N

A B

M N

Fig. 1. Three electrode arrangements calculated by program. Currents are

introduced as point sources at head of arrows.

F. Kohlbeck, D. Mawlood / Computers & Geosciences 35 (2009) 1748–1751 1749

different numbers of layers. Furthermore, data from several linescan be read and the calculated depth resistivity graphs can becompared in the final window. The final window may by copied tothe clipboard and pasted into other programs.

In addition to the source code, the executable program forWindowss operating system, together with examples, is alsoavailable.

2. Theoretical background

The calculation of layer resistivities and thicknesses frommeasurements is denoted as ‘inversion’ in the following context.The calculations start with an initial subsurface model consistingof horizontal layers of given thickness and resistivity (see Fig. 2 fornotations). Currents I are induced at the current electrodes (A, B)and the potential difference DV is measured at the potentialelectrodes (M, N). With the known positions of the current andpotential electrodes, the resistances DV/I can be calculated for allAB/2 distances of electrode positions. Now the model parametersmay be changed in order to minimize the sum of the squareddifferences between measured and calculated values.

The potentials of the model are calculated using the formulasderived by Schulz (1985), for electrodes placed arbitrarily within alayered model.

The potential at location ~r caused by one current electrode atlocation ~ramay be expressed as

Vð~r;~raÞ ¼I

2p

Z 10

f ðz; za; lÞJoðlRaÞdl (1)

with

Ra ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xaÞ

2þ ðy� yaÞ

2q

(2)

and J0(lRa), the Bessel function of order zero. (x,y,z) and (xa,ya,za)are the components of~rand~ra, respectively, with z being positivedownwards.

One recognizes that the horizontal components (x,y)of ~r in Eq. (1) are only contained within the Bessel-functionand that the vertical component z is contained solelywithin f(z,za,l). When the potential at ~r ¼~r1 is known, thepotential at ~r1 þ D~r with sufficiently small D~r may be approxi-mated as:

Vð~r1 þ D~r;~raÞ ¼ Vð~r1;~raÞ þ@Vð~r;~raÞ

@x

� �~r¼~r1

Dx

þ@Vð~r;~raÞ

@y

� �~r¼~r1

Dyþ@Vð~r;~raÞ

@z

� �~r¼~r1

Dz

with (Dx,Dy,Dz) the components of D~r.The potential difference DVa between two adjacent points

(Dx5Ra) located at the same z-level with y constant (Dy ¼ 0,Dz ¼ 0) is as follows:

DVa ¼ Vð~r;~raÞ � Vð~r þ D~r;~raÞ

¼IDx

2p �Z 1

0f ðz; za; lÞJ1ðlRaÞldl

with J1ðl~raÞ the 1st order Bessel function. Ra Eq. (2) simplifies to:

Ra ¼ x� xa and � ¼ 1 case x4xa

Ra ¼ xa � x and � ¼ �1 case xoxa

The potential difference DV for a positive source at ~ra and anegative source of the same strength (as necessary for a currentintroduced into the earth) located at ~rbwith

~ra ¼

xa

ya

za

0B@

1CA; ~rb ¼

�xa

ya

za

0B@

1CA

follows as:

DV ¼ 2IDx

2p

Z 10

f ðz; za;lÞJ1ðlRaÞldl (3)

This formula applies to all three electrode arrangements of thecomputer program described.

Placing the current electrode at the top layer (za ¼ 0), theexpression for f(z,za,l) from Schulz (1985) may be modified andexpressed as (see Fig. 2 for notations and definitions of the layerindex):

f ðz;0; lÞ ¼Yj�1

i¼1

�Ti þ ri

�Tiþ1 þ ri

!" #1

2ð �Tj þ rjÞe

�lz

� 1þ e�2lðzj�zÞ�Tjþ1 � rj

�Tjþ1 þ rj

" #jX2

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F. Kohlbeck, D. Mawlood / Computers & Geosciences 35 (2009) 1748–17511750

�TiðlÞ is defined by the following recursion formula:

�Tn ¼ rn

�Ti ¼ ri

�Tiþ1 þ riai

ri þ�Tiþ1ai

1pion

ai ¼ tanh lhið Þ

with n being the number of the deepest layer and tanh(x) thehyperbolic tangent.

This formula applies to the general case of two electrodes atthe surface and two electrodes at the same level down.

For Schlumberger measurements we may use this formulawith z ¼ 0.

At lake bottom one obtains with z1 ¼ h1:

f ðz1; z1; lÞ ¼

�T1 � a1r1

1� a21

a1a1

r1r2

r1 þ r2

a1 ¼ 1

8>>><>>>:

9>>>=>>>;

The integral of Eq. (3) is carried out by convolution with a linearfilter containing 141 coefficients. The filter method was firstintroduced to DC-geoelectrics by Ghosh (1971). The calculation ofthe model parameters from the measurements starts with Eq. (3).The measured quantity is DV/I, which is multiplied by a weightingfactor K to give the apparent resistivity ra:

ra ¼2pDV

IK ¼ 2DxK

Z 10

f ðz; za; lÞJ1ðlRaÞldl (4)

with K ¼ (Ra2�a2)/4aERa

2/4, Ra the horizontal distance from thecentre of the arrangement to the current electrodes and a ¼ Dx/2.

ra approaches the true resistivity of a uniform half-space forSchlumberger measurements only. The designation for ra,apparent resistivity, is not correct in the case of lake bottommeasurements because ra will not approach the true resistivity inthat case.

Taking a set of measurements with different values ofRa ¼ Ra,m. Eq. (4) with K substituted may be written as

ra;m ¼ R2a

Z 10

f ðz; za; lÞJ1ðlRa;mÞldl m ¼ 1;M (5)

This equation contains the known values ra,m, Ra,m, z, za and the(2N�1) unknown layer parameters (rj, (j ¼ 1,N); hj, (j ¼ 1, N�1))with N the number of layers and M the number of measurements.All unknowns are contained within f(z,za,l).

The inversion algorithm uses an iterative Levenberg–Marquardt algorithm with singular value decomposition de-scribed in Johansen (1977). The algorithm minimizes the sum ofthe squared differences between measured and calculated valuesra,m on a 20 points/decade grid of AB/2 values on log scale. Themeasured data are interpolated to the 20 points/decade grid to fitthis requirement.

3. Programming considerations

The program was written in Fortran95 using the Laheys

compiler with graphical interface to Windows by the Winter-acters programming package.

The inversion algorithm was originally programmed byK. Arnason (personal communication, 1994). A 141 point filterwas introduced by A. Bas-okur (personal communication 1997).The algorithm was further modified by the authors of this paper.The first-order partial derivatives for the parameters p, necessaryfor the Marquardt algorithm, are computed by calculating thefunction difference as Dr/Dp instead of qr/qp. A test showed thatthe accuracy of this approximation is sufficient. The computation

time is about the same as that for the analytical solution and theprogramming code is much simpler.

Eq. (5) allows to first calculate f(z,za,l) with Bessel transformand then evaluate the unknown parameters from f(z,za,l)(Bas-okur, 1990). This method is called the ‘direct method’.However, it turns out that the direct method is not stable incases where the approximation of horizontal layering is not goodenough. For that reason the direct method is not used in thepresented computer program.

It is well known that – apart from one- and two-layer cases –clear cut calculations of layer-thicknesses and resistivities frommeasured soundings are not possible in most cases. A wide rangeof variations in layer thickness and resistivity will give the samesounding curve within the limits of data spread. This phenomen-on is called the principle of equivalence in geoelectrics. Furtherinformation from other investigations must be included to resolvethese uncertainties. The described program calculates the stan-dard errors for layers and thicknesses as random errors from thevariance matrix during inversion. Biased error distributions orsystematic errors originating, for example, from non-horizontallayering are not considered. The error estimates are veryimportant because they show the possible range of equivalentmodels.

The program allows for fixing layer thicknesses and/orresistivities. The number of layers can be given or can be selectedautomatically; likewise, a starting model for the iteration processcan be selected manually or automatically. The comparison of theparameters with their error ranges allows for a quick estimate ofrealistic models.

One of the main advantages of the presented program is thepossibility to adapt geological models to the measurements veryeasily. One can fix layers and/or resistivities during the approx-imation process and obtains error estimates for the variable layersand resistivities. High uncertainties for a specific layer thicknessor resistivity indicate that other models are as likely as the currentone.

3.1. Program usage

After calling up the program, a main menu, a data dialog, amodel dialog and a checkbox with the title ‘‘ready for calculation’’appear (Fig. 3). From the main menu, the electrodes submenushould be selected first to start either surface, lake bottom or the‘two electrodes down-two electrodes up’ electrode configuration.In case nothing is selected, a standard Schlumberger array isassumed.

Data can be input either directly by typing into the data dialogspreadsheet, or by copying the data via clipboard from anotherspreadsheet like Excels, or by calling a data file from the filemenu.

The next step is to create a starting model with the modeldialog. The starting model can be typed into the spreadsheeteither directly, or selected automatically. When ‘automatic’ isselected, the layers are created with exponentially increasingthicknesses on going downward. This selection is in accordance tothe well-known fact that the resolution of DC soundingsdiminishes approx. exponentially (Johansen, 1977). The thick-nesses are fixed in the following inversion process if the automaticswitch is not turned off afterwards. When the number of layers isnot given, following Zohdy (1989), M�1 layers (M ¼ number ofmeasurements) are created. This is the most stable inversion andshould be selected first when no further information on thesubsurface is provided. For every layer, the thickness orthe resistivity may be fixed or kept free during the inversionprocess.

ARTICLE IN PRESS

Fig. 3. Screen with dialogs and graphics. Highlighted graphic may be exported or plotted with different formats.

F. Kohlbeck, D. Mawlood / Computers & Geosciences 35 (2009) 1748–1751 1751

The checkbox shows if all three steps have been selected.Thereafter, the inversion starts. The first frame shows thesounding curve of the measured apparent resistivities and thestarting model. After finishing the inversion process, the finaldepth model together with the sounding curve is shown inanother frame. Layer thicknesses and resistivities are drawn intothe model dialog and written onto an output file. The final framemay be copied to the clipboard and inserted into other reports, orplotted onto a file using the plot option on the main menu.

Two output files are created: ‘*.mod’ and ‘*.res’. The asteriskrepresents the name of the data input file without extension. Thesetwo files will rarely be used. ‘*.mod’ contains the calculatedresistivities and thicknesses and may be used as an input file forthe starting model in further calculations. ‘*.res’ contains detailedinformation on the calculations: The eigenvalues, the eigenvectors,the correlation matrix, and the calculated resistivities. The user mayread the paper by Inman (1975), in addition to the paper byJohansen (1977), for a deeper understanding of the singular valuedecomposition and the ridge regression method used within theprogram. An asterisk to the right of a parameter marks that theparameter has been kept constant during the inversion process.

Two further files, ‘testf’ and ‘TEMPI’ are created as scratch filesand deleted when the program is closed. Programmers maychange the output to be permanent. ‘testf’ contains programminginformation in order to find errors or to control the program flow.This file is of interest for programmers only. The content of thisprogram will be changed to the needs of the programmer. ‘TEMPI’contains output information for the final graph.

Acknowledgements

The Authors are grateful to Knutur Arnason (National EnergyAuthority of Iceland, Geothermal division) for providing the

programming code of non-linear least-square inversion ofSchlumberger soundings and to Ahmet T. Bas-okur (AnkaraUniversity Fen Fak. Jeofizik Muh. B.) for providing improvementsto this code.

Appendix A. Supplementary material

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.cageo.2009.03.001.

References

Bas-okur, A.T., 1990. Microcomputer program for the direct interpre-tation of resistivity sounding data. Computers & Geosciences 16 (4),587–601.

Ghosh, D.P., 1971. The application of linear filter theory to the direct interpretationof geoelectrical resistivity sounding measurements. Geophysical Prospecting22, 176–180.

Inman, S.R., 1975. Resistivity inversion with ridge regression. Geophysics 40,798–817.

Johansen, H.K., 1977. A man/computer interpretation system for resistivitysoundings over horizontally stratified earth. Geophysical Prospecting 25,667–691.

Koefoed, O., 1979. Resistivity Sounding Measurements, Ser. Methods in Geochem-istry and Geophysics 14A, Geosounding Principles 1. Elsevier, Amsterdam-Oxford-New York, 276 pp.

Schulz, R., 1985. The method of integral equation in the direct current resistivitymethod and its accuracy. Journal of Geophysics 56, 192–200.

Vedanti, N., Srivastava, R., Sagode, J., Dimri, V.P., 2005. An efficient 1D Occam’sinversion algorithm using analytically computed first- and second-orderderivatives for DC resistivity soundings. Computers & Geosciences 31 (3),319–328.

Zohdy, A.A.R., 1989. A new method for the automatic interpre-tation of Schlumberger and Wenner sounding curves. Geophysics 54 (2),245–253.