IEEE Transactions on Power Apparatus and Systems, Vol. PAS-103, No. 2, February 1984

EARTH RESISTIVITY MEASUREMENT INTERPRETATION TECHNIQUES

F. Dawalibi, Senior Member*Safe Engineering Services Ltd.12201 letellier, Montreal, Canada.

ABSTRACT

Earth resistivity measurement interpretation techniquesdeveloped as part of a major EPRI research project ontransmission line grounding are described and discussed. Theinterpretation techniques include graphical curve matchingand an advanced computer program (RESIST). The curvematching technique requires a set of theoretical Master Curveswith which a field curve can be compared directly. ProgramRESIST is based on the analytical methods used in a moreelaborate computer program which has been in operation forseveral years. For an electrode spacing of 2.5 meters orgreater, satisfactory agreement is obtained between measuredand computed results using both interpretation techniques.

1.0 INTRODUCTION

This paper describes earth resistivity measurementinterpretation techniques developed as part of a major researchproject on transmission line grounding. The Final Report [1lincludes a comprehensive description of advanced theoriesand techniques pertaining to the analysis, design andmeasurement of transmission line grounding systems with a

particular emphasis on safety and mitigation techniques toimprove safety around exposed structures. Several analyticalmethods described in the report are new or were not availablepreviously in the open literature. This is the case of theresistivity interpretation method described in the appendixof this paper. The research project was sponsored by theElectric Power Research Institute (EPRI).

The design of a power system requires that normal andabnormal conditions be considered in order to correctlydetermine the design requirements and characteristics of theinstalled power equipment. Of the abnormal conditions whichcan occur on a power network the two most frequent are:

D Lightning strokes

o Phase to ground faults

* F. Dawalibi is the principal author of the EPRI reportreferenced in this paper . C. Blattner served as an

industry advisor on the EPRI Task Force.

83 SM 456-1 A paper recommended and approvedby the IEEE Transmission and Distribution Committee

of the IEEE Power Engineering Society for presenta-tion at the IEEE/PES 1983 Summer Meeting,Los Angeles, California, July 17-22, 1983. Manu-

script submitted January 10, 1983; made available

lor printing May 13, 1983.

C. J. Blattner, Senior Member*Niagara iMohawk Power Corp.300 Erie Blvd., Syracuse, New York.

In both cases, the overhead network and the earth path,including buried metallic conductors such as counterpoisesand ground electrodes, are part of the circuit in which thesurge or fault current circulates. Generally, an analysis ofthese abnormal conditions is based on a reasonably accuraterepresentation of the overhead circuit. The earth path,however, is usually modelled as a perfect conductor or in avery simplified form. This seldom leads to realistic results.The apparent inconsistency of these engineering approachescan be explained by the mathematical difficulties involvedin the solution of three dimensional current flow in earth.Often, the wide variations observed in the characteristics ofearth, generally described as a semi-infinite nonhomogeneousmedia, are used as justification for not pursuing detailedmodelling of the earth path for fault currents.

Thirty years ago, the lack of suitable high-speed digitalcomputers was a serious obstacle to accurate modelling ofthe earth. Now, there are no computational limitations tothe development of an accurate model of the structure ofthe earth. Recently published analytical works on powersystem grounding describe accurate, computer based,computational techniques for the design of grounding systems.

Large variations in earth resistivity need not be an obstacleto the development of detailed earth structure models.Relatively simple equivalent earth models can effectively beused to accurately predict transmission line groundingperformance as evidenced by the field measurements describedin reference [1]. Finally, the earth structure at any particularsite can be accurately determined by a suitable selection ofthe method and test equipment.

1.1 DISCUSSION OF MODELLING PROBLEM

The development of a mathematical model to represent theelectrical properties of earth can be a formidable task becauseof the widely nonuniform characteristics of earth.

Fortunately for transmission line grounding purposes, the earthcan be reasonably approximated by a two-layered soilstructure. This soil structure is characterized by the layerresistivities P1, P2 and the upper layer thickness h. Thelower layer is considered infinite. In some cases the thicknessof the upper layer is large enough so as the earth modelmay be considered fairly uniform.

The variables P1, P2 and h are generally determined byinterpreting the apparent resistivity values measured usingthe Wenner (or four probe) array.

Unlike most engineering problems, interpretation of earthresistivity measurements is an "inverse" problem; i.e., fromthe electrical response to impressed current at specificlocations on the earth surface, the electrical properties ofthe conducting media (earth) are to be determined. In contrast,conventional electrostatic problems determine the electricalresponse or the excitation current sources, based on theknown properties of the conducting material. These are known

0018-9510/84/0002-0374$01.00 ( 1984 IEEE

374

375

as the Laplace and Diriclet problems. Obviously, the "inverse"problem, where the physical constants of the material areunknown, presents more difficulties than those problems wherethe physical constants of the material are known functionsof position.

Moreover, the number of parameters required to represent amodel of the earth structure is usually so great that it isdifficult to choose initial values to these parameters andhave a computer algorithm converge to an acceptable solutionwithin a practical time frame. Consequently, the selectionof initial values becomes a fundamental task in theinterpretation process.

Success or failure in this important initial assessment isgenerally dependent on the experience of the engineer andthe knowledge of earth electrical properties available to theengineer responsible for the interpretation of themeasurements.

There is one further problem with the "inverse" solution ofresistivity measurements. It is not always possible to obtaina unique solution to a data interpretation problem.Because of inaccuracies in the measurements (usually 5%with classical geoelectric instruments), several models ofearth structure can be found to give satisfactory agreementwith the measured results. These models will usually differin the characteristics of the deep soil layers.

The above discussions are not presented to discourage thepower system engineer from performing a scientificinterpretation of resistivity measurements, but rather to makehim aware that this task requires careful preparation,investigation, and engineering judgement. The difficultiesmentioned previously, while imposing a considerable challengeto the geologist, have significantly smaller impact on theelectrical engineer. Firstly, the existence of multiple solutionsto the substratum structure is of little consequence indetermining the response of ground electrodes, particularlythose of the transmission line tower grounds. Secondly, atwo-layer earth model, is generally sufficient for modellingsubstation and transmission line tower grounding systems.Finally, there are numerous charts, algorithms, and simpleengineering visual estimation techniques, which can beused to determine an equivalent two-layer earth-model withreasonable accuracy.

1.2 RESISTIVITY MEASUREMENT

Because of the wide variations in the structure and propertiesof earth materials, there are numerous methods and techniquesfor determining the structure of earth. A complete descriptionof all of these methods is beyond the scope of this paper.However, information on a variety of these methods may befound within [1].

In power system engineering applications, the Wenner methodL2] is used almost exclusively for resistivity measurements.It is our opinion that this method, when used with suitabletest equipment, will provide sufficient data for an accurateearth structure model for the analysis of power systemgrounding installations. Because of the relative simplicity ofresults interpretation and widespread availability of testequipment supporting the use of this method, it isrecommended as an effective and suitable standard testprocedure.

The Wenner four-electrode arrangement is shown in Figure1.1. Four electrodes are driven in the earth alonq a straightline. The electrodes are uniformly spaced and the burial depthof the electrodes is usually less than 10 per cent of thespacing between two adjacent electrodes. Thus, each electrodewill appear as a point with respect to the distances involvedin the measurement.

FOUR TERMINALTEST SET

POTENT ALPROBES

CURRENTPROBE

Figure 1.1 Wenner Arrangement

1.3 POINT SOURCE ELECTRODE IN A TWO-LAYER EARTH

There are several mathematical approaches which have beenused to calculate potentials in a layered soil structure [3-10].The method which is adopted here is to search for a particularsolution to Laplace's equation which satisfies the boundaryconditions of the problem.

One of the simplest and most important eprth structuremodels is the two-layer earth model. It can be shown thatthe potential in earth can be always expressed as the sumof a normal potential (uniform soil), and a disturbing potential,which accounts for the deep layers of soil. Therefore, thetwo-layer model can be used as the simplest equivalent earthstructure for interpreting practical resistivity measurements,as shown in Figure 1.2.

h zP1 M(x,y Z)

Figure 1.2 The two-Layer Earth Model

The apparent resistivity, Pa, as measured using theWenner method is derived for the two-layer earth conditionin Chapter 4 of LlJ and is expressed as:

p = Pi 1 + 4kn [ ) 2 2

1

[4+(2nh/a)21]2 JJ

(i - 1)

where,

Pa = the apparent resistivity as measured using theWenner Method.

a = separation distance between current and potentialprobes.

PI = surface layer resistivity of thickness h

P2 = second layer resistivity which extends to an infinitedepth.

k = reflection coefficient = (P2 - P1) / (P2 + P1)

376

1.4 POINT SOURCE ELECTRODE IN EARTH WITH INCLINEDLAYERS

If the boundary separating two regions of earth with differentresistivities is not horizontal, but inclined at an angle to thesurface, an exact mathematical solution for the potentialfunction is arduous to obtain. The potential function is obtainedthrough a double integration process on two dummy variables,of a complicated function containing hyperbolic sines andBessel functions of the second type [3,9,10].

When the angle of dip is 90ga simple solution can be obtained.Figure 1.3 illustrates this case. The case where a verticalinterface between two dissirnilar earth resistivity areas existswill be referred to as a vertical fault structure. In practice,this effect can be the result of a number of differentmechanisms, included but not limited to geological faulting.

Figure 1.4 Vertical Fault (Top View)

a - SIDE-VIEW

V2 I

- TOP-V IEW

Figure 1.3 Vertical Fault

The apparent resistivity, Pa,Wenner method, for the verticalwithin [1] and is expressed as:

as measured usingfault condition, is de

Pi K( 1-K)P = 11+ k2 + 1a (1-k) L [4(sinw+h/a)2+COS2J]2

K(1+K )

[4 (s nub-h/a) 2cos2~]21[4(s inw-h/a)2+cos2]2 Iwhere,

Pa = the apparent resistivity as measured by the WennerMethod.

a = separation distance between current and potentialprobes.

P1 = resistivity of region on one side of vertical faultline.

P2 = resistivity of region on opposite side of verticalfault line.

h = distance from center of array direction and theline of fault (See Figure 1.4).

ci = angle of array direction to line of fault (SeeFigure 1.4).

Ithe

1.5 INTERPRETATION OF THE MEASUJREMENTS

The simplest interpretation problem is the case where themeasured apparent resistivities, Pa, vary minimally around anaverage value P. This indicates that earth at the measurementsite is reasonably uniform and has a resistivity P.

Observed resistivity variations can be attributed to smalllocal discontinuities, which may be neglected, or toinaccuracies in the measurements due a number of factorssuch as stray currents in earth or inadequate sensitivity ofthe measuring equipment.

Unfortunatley, such cases rarely occur in practice. In mostcases, apparent resistivity, plotted as a function of theelectrode spacing, shows large variations with probe spacing.This indicates that the earth is nonuniform.

Drived In general, apparent resistivity curves change smoothly anddo not exhibit sudden changes. When the latter occurs, it isa clear indication that the array has just crossed a verticalfault or a local discontinuity close to earth surface. Themagnitude of the jump is an indication of the differencebetween the resistivities of the two adjacent earth materials.The presence of buried pipes or other metallic structuresclose to the surface is also a typical cause of sudden changes

(1 -2) in apparent earth resistivity, as shown in Figure 1.5.

L/)

cLa-

P I P E SPAC NG a

Figure 1.5 Presence of Buried Metallic Structures

P 2 /10"-C2Line of Fault

d2.A r L

- -I- x

th I0w m

d It P I

z

The method used for interpreting the measurements can begrouped into two simplified categories:

o Empirical interpretation

o Analytical interpretation

Analytical interpretation is, in theory, independent of theperson conducting the interpretation. In contrast, the resultsof an empirical interpretation are significantly influenced bythe background and experience of the interpreter.

It is preferable to use a combination of both approaches formaximum accuracy and a minimum of uncertainty. Forexample, when analytical methods indicate that two or moreearth models are reasonable, the most realistic choice canbe determined from empirical considerations or visualinspection of the curves. In any case, it should be emphasizedthat experience is of paramount importance in theinterpretation process.

Empirical methods are based on experience gained throughnumerous measurement and interpretation exercises. Thus,such methods can be described as statistical in nature.Essentially, it is observed that the shape of an apparentresistivity curve is closely related to the earth structure andits characteristics at the site. Therefore, certainproperties of the measured curve are used to deduce theresistivity and thickness of the earth layers. Although theremay be inherent inaccuracies in some of these methods, theyare of less consequence to the design engineer than they areto the geologist. Empirical methods may be useful for onsite interpretations and serve as a good starting point formore rigorous methods. A complete discussion of empiricalmethods is included in the referenced EPRI report, and willnot be repeated in this paper.

Analytical methods, as termed herein, follow a constantmethodology, i.e:

Step 1The measured results are examined and preliminaryinterpretation is performed, typically based on the empiricalmethods described previously.

Step 2One or several possible earth models are proposed.

Step 3The measured results are compared with those calculatedfrom the proposed imodels.

Step 4The most suitable model is retained. If more that one modelis suitable, these are considered to be equivalent.

Step 5The selected model is optimized. Often the opimizationprocess is based on engineering judgment. Sometimes onemay choose to conduct additional surveys in order to checkthe validity of some assumptions or to eliminate uncertainties.However, in power system design this step is generally omitted.

Two analytical methods of earth resistivity measurementinterpretation were developed for the EPRI project andare described in the following sections. First, the use ofprecalculated curves is described. Next, a computer program(RESIST) based on the method of steepest-descent is described.

The word 'analytical" can be misleading as it is ofteninterpreted as meaning "accurate" or "rigorous". Earthresistivity measurements are rarely accurate to within 1%,even when sophisticated equipment is used. Usually,careful measurements with conventional equipment areaccurate to within about 5%. Careless measurements,inexperience or poor equipment can lead to measured resultssignificantly different from the real values. Even

377

measurements taken by an experienced crew under the bestof conditions, will never give a perfect match with analyticalresults computed from the optimum earth model derived fromthe measurement data.

2.0 LOGARITHMIC CURVE MATCHING

The apparent resistivity functions (equations 1-2 and 1-3)may be written in terms of the dimensionless ratios K, Pa'P1and h/a:

Horizontal Two-Layer Earth00

pa/p = 1 + 4 Kn[ [ + 4n2(h/a) 2]

- 4 + 4n2(h/a)2 2]j

Vertical Fault

Pa"/ (1 K) [ 1 + K2 + K(1-K) [4(sinw + h/a) 2

(2-1 )

+ cos 2w ] - K(1+K) [(2sinw-h/a)2+cos2j1

If one of the above dimensionless apparent resistivity functionsPo = Pa/ Pl is plotted in logarithmic coordinates, i.e., ln(Pa/ PIF(ln(a/h)) the coordinates, of a point on the resistivity curvewill be:

y = lnpa - lnp

x = Ina - lnh

on the y axis

on the x axis

(2-3)

(2-4)If we now assume that a number of apparent resistivityreference curves, designated as Master curves, are plotted,for various reflection coefficients K, then:

y = lnp

x° = lnaAssuming that Pl = 1 Q-m and h = 1 m.

(2-5)

(2-6)

Resistivity curves derived for two-layer earth structureswhere P1 and/or h are not equal to unity, will be shiftedby -ln(P1) vertically and -ln(h) horizontally with respect tothe corresponding Master curves. The shapes of the curvesare thus preserved. This property of the apparent resistivitycurve when plotted in logarithmic coordinates, is the basisfor the logarithmic curve matching method.

Thus, a field curve can be compared directly with a setof theoretical Master curves, through a series of appropriatetranslations of the field curve plotted on a transparent paper.If a satisfactory match is found between the field and a

theoretical curve, then the real earth reflection factor K, isequal to that of the computed curve.

This method requires that a set of precalculated Masterreference curves be available to the interpreter. Such curvescan be easily determined based on Equation 2-1 for horizontallayers and Equation 2-2 for a vertical fault. Figure 2.1 is a

set of Master curves for the case of a two-layer earth. Whena field curve falls between two curves, the corect valuescan be interpolated. If more precision is required, additionalcurves for closer K values should be constructed.

Figure 2.2 is a set of Master curves applicable to verticalfault when the direction of the traverse is at 00 angle withthe line of fault. It should be noted that additional charts(different traverse angles) are required for a complete setof master charts. The construction of such charts isstraigthforward with a programmable calculator. Referencecharts for 30, 60 and 900 angles are provided in Volume 2if El].

n to -,).

378

Figure 2.1 Horizontal Two-Layer Earth Master Chart

3.0 COMPUTER PROGRAM RESIST

The resistivity measurement interpretation process isconsiderably simplified when program RESIST is used. Detailedinformation on the use of this program is included in Chapters3, 4 and 10 and in Appendix B of [1].

Because RESIST requires a minimum of data from the user,it can be used by engineers with very little exposure topower system grounding. The input data consists of the probespacings and apparent resistances (or resistivities)measured using the Wenner method. RESIST automaticallyselects and calculates all the other data needed toproceed with the final computations.

For example, RESIST must select initial values for the upperlayer thickness and layer resistivities before starting itsiterative search algorithm. The initial values retained areappropriate for relatively small grounding systems such asthose employed in transmission line structures.

This program determines an equivalent two-layer earth modelfrom the measured apparent resistivity data. The resistivityvalues must have been measured using the equally-spacedfour probe or Wenner method. The equivalent earth model ischaracterized by the thickness of the first layer and by theresistivity values of the upper and lower layers of soil.

RESIST was developed for use in an interactive mode. In thismode, the user is prompted to provide the required inputdata and answers in free-format.

The program can also be readily operated in batch mode. Inthis mode, the user must organize the data in a file priorto running the program. The batch mode is particularlyadvantageous when running several cases with smalldifferences in the input data.

a a PA)o h h° -to -

/10'VERTICAL FAULT/

-.1

-.5

-.7

-.9 . -

-.99

.. .... . . . . . . . . . .~~~~~~~~~~~~1.0_2 _5 _1 .5 _1

Figure 2.2 Vertical Fault Master Chart

Dr

The following method was used as the basis for the computerprogram RESIST, developed by SES. This method was selectedfor the following reasons:

* A more elaborate version of the method has beenoperational for several years and has beenextensively tested in practical cases. It has provedto be a very stable, reliable method which requiresa minimum amount of input data. Initial valuesfor the presumed earth model are not requiredfrom the user. Therefore, it can be used byengineers with limited experience in resistivityinterpretation.

* The method is easy to implement.

* The program is particularly suited to transmissionline tower grounding design where very largespacings between the electrodes of an array arenot necessary.

This program is based on the method of steepest-descent.

The method of steepest-descent is most readily vizualizedby an analysis of the two-variable function p(x,y), illustratedin Figure 3.1. The analytical development of this method isshown in the Appendix.

The gradient of this function is calculated at an initial pointMo defined by xo, yo. The values of x and y are then selectedso that the function decreases along the direction definedby the gradient vector. The process is repeated until thefunction along the initial direction starts to increase. Theprocess will stop when all possible directions of the gradientindicate that the present (x,y), coordinates corresponds to aminimum of the function (zero gradient).

This process will normally converge to a minimum of thefunction. However, there is no guarantee that the rninimumobtained will be the only one nor that it is the minimum ofthe minima. Experience shows that when a secondary minimumis obtained, the initial starting point was most likely withinthe zone of influence of this minimum. In this case anotherpair of initial x and y values should be selected and theprocess started again.

M (x ,y)MINIMUM

Figure 3.1 The Method of Steepest-Descent

4.0 A TYPICAL EXAMPLE

379

Table 4.1 clearly shows that the soil at the site is notuniform. Therefore, an equivalent two-layer earth model mustbe determined to improve the accuracy of the groundingperformance calculation.

The resistivity values of Table 4.1 were used as input datato computer program RESIST as shown in Figure 4.2.

......... etc.

YOU MAY ENTER1- THE APPARENT MEASURED RESISTANCE (V/I), OR2- THE APPARENT RESISTIVITY VALUES

APPARENT RESISTANCE ?

ENTER YOUR FIELD DATA RESULTS IN SEQUENCE.ONE LINE FOR EACH MEASUREMENT AT A GIVEN SPACING.TERMINATE WITH END AT THE BEGINNING OF A NEW LINE.EXAMPLE SPACING (M OR FT), MEASURED VALUE (OHMS OR OHMS-METER)

5.0,433.0?2.5y320?5.0,245X7.5,18? 10.0,162X 12.5,168? 15.0,152? end

......... etc.

Figure 4.2 Program RESIST Input Data

The computer printout, shown in Figure 4.3, shows that theequivalent earth resistivity structure determined by RESISTcorresponds to a 2.56 m (8.4 feet) thick first layer with aresistivity of 383 Q-m underlain by a second layer with alower resistivity value of 147.7 Q-m.

COMPUTATION RESULTS

TOP LAYER RESISTIVITY 383.4982BOTTOM LAYER RESISTIVITY 147,6571REFLECTION FACTOR - 4440TOP LAYER THICKNESS 2.5626

OHMS-METEROHMS-METER(P.U.)METERS

Figure 4.3 Computation Results

A comparison between the actual measured resistivities andthe calculated apparent resistivities based on the twolayer earth structure determined by RESIST is shown in thecomputer printout in Figure 4.4.

The apparent soil resistivity at the site of a power systemgrounding installation was measured using the Wenner method.The results of the measurements are given in Table 4.1.

PROBE APPARENTSPACING (i) RESISTIVITY (Q-m)

2.5 320

5.0 2457.5 182

10.0 162

12.5 168

15.0 152

SPACING CALCULATED APPARENT MEASURED APPARENT DISCREPANCY(METERS) RESISTIVITY(OHMS-M) RESISTIVITY(DHMS-M) (PERCENT)

2.j005.0007,50010.00012.50015,000

3271.4372233.8807187.4388168.0307159.5344155 3505

320.0000245.0000182 .0000162,0000168.0000152.0000

2.32-4,542.993.72

-5,04220i- iO

Figure 4.4 Comparison of Resistivities

Table 4.1 Apparent Soil Resistivity

380

An almost identical two-layer earth model can be determinedusing the logarithmic curve matching technique described inSection 2. The field resistivity curve is plotted on a transparentlogarithmic graph and is then compared directly to the setof theoretical Master curves provided in Section 2 (Figure2.1). The comparison process, illustrated in Figure 4.5, consistsof obtaining a satisfactory match between the measured andtheoretical curves through a series of appropriate horizontaland vertical translations of the transparent graph sheet. Whenthis is accomplished, the thickness of the upper layer is thevalue of the abscissa on the vertical line passing through theMaster chart abscissa corresponding to a/h=1. From Figure4.5 it is found that the upper layer thickness is approximately2.5 m (8.2 feet). Similarly, the upper soil resistivity is thevalue of the ordinate on the horizontal line passing throughthe Master chart ordinate corresponding to Pa/P1 = 1. Fiqure4.5 shows that the upper soil resistivity is about 390 Q-m.Finally, this figure indicates that the measured apparentresistivity curve corresponds to a reflection factor in therange of -0.4 to -0.45. This leads to a lower resistivity valuebetween 148 and 167 Q-m.

A comparison of the two-layer earth models obtained bycomputer program RESIST and the curve matching techniqueis shown in Table 4.2. The results demonstrate that the resultsare almost identical.

Top Layer Resistivity

Bottom Layer Resistivity

CONCLUSIONS

Advanced earth resistivity measurement interpretationtechniques have been developed and demonstrated. The newtechniques include:

a- Logarithmic curve matching utilizing precalculatedMaster curves with which a field curve can be compareddirectly.

b- An advanced computer program RESIST which isbased on the method of steepest-descent. The programis particularly suited to transmission line tower groundingdesi gn.

The interpretation techniques demonstrate that largevariations in earth resistivity measurements need not be anobstacle to the development of detailed earth structure models.For power system grounding purposes, the earth can bereasonably approximated by a two-layer soil structure. Foran electrode spacing of 2.5 meters or greater, satisfactoryagreement has been demonstrated between measured andcomputed results using both techniques.ACKNOWLEDGEMENTS

The authors wish to aknowledge the financial support of ther7I--:-M-_n .- -^ h T FF F-for k;" ro r Oh -niof-felectrical t-ower researcn instLtute Tor rInis researcnt i rojeCL.

RESIST CURVES Mr. J. Dunlap, the EPRI Project Manager, and the Advisory-------- ---------- Task Force members; Messrs. T. E. Bethke, A. C. Pfitzer,

G. B. Niles, and R. S. Baishiki, are also acknowleged for383 390 their assistance and guidance on this EPRI project.

147.7 148-167

Top Layer Thickness 2.56 m 2.5 m

Table 4.2 Comparison of Interpretation Results

P.P

Figure 4.5 Logarithmic Curve matching Technique

1- F. Dawalibi, "Transmission Line Grounding", EPRI ResearchProject 1494-1, Final Report EL 2699, October 1982.

2- F. Wenner, "A Method of measuring Resistivity", NationalBureau of Standards, Scientific Paper 12, NO. S-258, 1916,p. 499.

3- G. F. Tagg, "Earth Resistances", George Newnes Ltd.,London 1964 (book).

4- E. D. Sunde, "Earth Conduction Effects inTransmission Systems", Dover Publications, New York, 1968(book).

5- F. Dawalibi, D. Mukhedkar, "Influence of Ground Rodson CGrounding Grids", IEEE Transactions on PAS, Vol. PAS-98,No. 6, November/December 1979, pp. 2089-2098.

6- S. Stefanesco, C. & M. Schlumberger, "Sur la DistributionElectrique Potentielle Autour d'une Prise de Terre Ponctuelledans un Terrain a Couches Horizontales Homogenes etIsotropes", Journal de Physique et Radium, Vol. 1, Serie VII,No. 4, 1930, pp. 132-140.

7- M. Muskat, "Potential Distribution About an Electrodeon the Surface of the Earth", Physics, Vol. 4, NO. 4, April1933, pp. 129-147.

8- H. M. Mooney, E. Orellana, H. Pickett, L. Tornheim, "AResistivity Computation Method for Layered EarthModels", Geophysics, Vol. XXXI, No. 1, February 1966, pp.

192-203.

9- K. Maeda, "Apparent Resistivity for Dipping Beds,Geophysics, Vol. XX, No. 1, January 1955, pp, 123-139.

10- L. S. Palmer, "Examples of Geolectric Surveys", IEEJournal, Vol. 106, part A, June 1959, pp. 231-244.

R EERENCES

381

APPENDIX

Let PO(aj), j = 1, n be the series of apparent resistivity valuesas measured at a given site by the Wenner method for ndifferent inter-electrode spacings aj.

Let P (aj), j = 1, n be the calculated apparent resistivityvalues, based on a two-layer earth model at the same spacingsaj used during the measurements.

The interpretation task consists of finding the most suitableearth model for which the difference between the set ofmeasured and calculated values, according to certain criteria,is a minimum. In theory any criterion can be used (e.g., sumof the absolute value of the differences). In practice, theclassical least-square criterion is preferred.

Let + ( P1, K, h) be the square error function defined as:

n ~p°(aj) _p (aj ) l2

')(Pj, K, h) = E[- )P(a ) J (1)

The best fit is obtained when p is minimum. The values ofPi, K, h which lead to this minimum are determined by thesteepest-descent algorithm.

The gradient vector is defined as:

V= Xap1 ah aK

Each component of the potential vector is determined fromEquation 1. Thus:

apJnn2 a)P (2a)

or

F-T p1 +2[a) 12]At = a~ + IT

pil K~ rLh J(4)

The sought for minimum is obtained when AW = 0 or practicallywhen:

IA < E£ (5)

where E is the desired accuracy.

The main steps in the steepest-descent algorithm aretherefore:

1- Estimate initial values of P1, K and h (i.e., PlKO, h°)

2- Calculate a suitable value of T

3- Determine AP1,AK and Ah

4- Estimate a new starting point:

Pi 0) = (i-i) + AP

K(i) K(1-i) + AK

h( = h i-) + Ah

5- Calculate Ail and compare it with a:

a- if Ai < a , the fit is completed.

b- if IAyI > E , continue the process at step 2(or step 3 if T is maintained constant).

aK 1 p° aKn

0 ah

(2b) In order to calculate Aip from Equation 4, the partialderivatives of P must be known. These are determined fromEquation 5 where the partial derivatives of the theoreticaltwo-layer earth apparent resistivity function are obtained

(2c) from Equation 4-46 of the EPRI report [1]. These calculationslead to:

Assume now that AP1, AK, Ah are small stepwise changesalong the gradient represented as:

Ap1 = -T -

DP,

AK = -T -

00

-= I + 4Upl n=l

Kn( - n(I - K2)/2K) (A:

-- a ]

(3a)

ap 16plh(3b) -=

ah a2

Ah = -T (3c)

Where T is a positive value expressed in p.u. of V), suitablyselected to generate a smooth search for the m-ninimum. Theabove changes cause a small variation A in the error function11:

AWl= al APi + a* AK + a'p Ahapi 3K a~~h

4 nKnl (A-:n=l

- B)

where

A = 1 + (2nh/a) 2

B = A + 3

(6)

(7)

(8)

(9)

ap

ah

Kn A-' B-'

382

Discussion

Eldon J. Rogers, (Bonneville Power Adm., Vancouver, WA): It wouldappear the author's computer technique for resolving Wenner test earthresistivity into two-component earth would start with parameters deter-mined from the logarithmic curve matching. I have used Roman's loga-rithmic curve fitting method for two-layer earth, described by theauthors, to analyze earth resistivity data measured for the substationgrid sit (Refer to [3], Ch. 3, pp. 72). Generally, Wenner resistivity sur-vey data fall into several categories: The data are obviously due to two-layer earth and logarithmic comparison is easily made; the data appearto be two-layer but logarithmic fit is inaccurate; or, it is evident the dataare 3- or 4-layer earth. As most of the substation data fall in the last twocategories, the author's computer technique could be useful to deter-mine their two-layer equivalent. Even for towers, earth resistivity varia-tions may require more than one Wenner test location to adequatelydescribe the earth volume.One important aspect of resistivity survey is determining the earth's

resistivity near the surface. The earth's resistivity near the surface canbe calculated from the measured resistance (Rn) of each Wenner testprobe (Pn = YY L Rn/(In8L/D- 1) ). For a typical test probe length(L) of 0.3m, earth resistivity samples are obtained of an earth cylinder0.5m in depth and 0.6m diameter. The composite earth surface layerresistivity is found by the parallel combination of all sample volumes

00

[Pi = n/I (1/Pn) ).I

When the composite Wenner test data for a grounding electrode sitediffer significantly from the two-layer earth model (for example, datafrom three-and four-layer earth), will the author's RESIST programdetermine the best two-layer fit? How are the p.u. value of V, "r", andthe desired accuracy "e" Eq.(5) selected? It would appear their selec-tion depends on how well the data fit the theoretical model. Would theauthor please discuss? What are the theoretical considerations whichjustify the assumptions used to form Eqs. 3a, 3b, 3c shown in the Ap-pendix? There appear to be several typo errors in the Appendix:

Eq. (4): First term has incorrect numerator.Eq. (7): Should have a negative sign in front of 16.

a2, in the denominatorn2, between summation symbol and kn

Eq. (8): Should have P1 between 4 and summation symbol

In his closure will the author include equations for the Vertical Faultcase similar to Eqs. (6), (7) and (8) of the Appendix. Will his computertechnicque be adaptable to the case of the rod electrode penetratingtwo-layer earth?The authors are to be congratulated for reviewing the logarithmic

curve matching technique and developing a computer program to fit thetwo-layer earth model to the earth resistivity survey.

Manuscript received August 2, 1983.

F. Dawalibi and C. J. Blattner: We thank Mr. Rogers for his discussionand pertinent comments.The initial parameters required by the steepest-descent technique

could be determined from the logarithmic curve matching method. Mr.Rogers points out rightfully that our technique is particularly useful todetermine an equivalent two-layer model for complex soil structures.When soil is practically a two-layer configuration, then the logarithmiccurve matching method provides accurate solutions.Program RESIST is designed to give the best two-layer fit based on

the Wenner test values.There is no selection of an initial value of the gradient vector V. The

components of this vector are defined by Eq. 2 of the paper. Guidancefor the selection of the accuracy is included in [1] (see p. B-3).

Eq. 4 has a correct numerator, (it was corrected after the review bythe Technical Committee).We are grateful to Mr. Rogers for reporting the typo errors in Eq. 7

and 8. It seems that these mistypes appeared a few years ago when thefirst author was involved in the revision of IEEE Guide 81. Unfortu-nately, the mistypes were reintroduced in this paper and also in Ref. 1.Fortunately however, the code in program RESIST is correct.The case of vertical layers requires that the direction of the traverse

and the location of each probe of the array be known relative to the ver-tical fault plane (interface betweeen layers). This introduces severalpossibilities with 3 equations similar to Eqs. 6, 7 and 8. Although wehave not yet derived all these new equations., we believe that it is arelatively easy task (see p. 4-26 of [1]).The rod electrode penetrating two-layer earth requires major

modifications to the computer algorithm.

Manuscript received September 26, 1983.