Electrical Resistivity 

Sounding With an 

L- Shaped Array

GEOLOGICAL SURVEY BULLETIN 1313-C

Electrical Resistivity 

Sounding With an 

L- Shaped Array

By ADEL A. R. ZOHDY

NEW TECHNIQUES IN DIRECT-CURRENT 

RESISTIVITY EXPLORATION

GEOLOGICAL SURVEY BULLETIN 1313-C

Mathematical criteria and field procedures for collecting, processing, and 

interpreting electrical sounding data 

obtained by means of an L-shaped array

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HIGKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress catalog-card No. 74-608192

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Abstract................................................................................................... Cl

Introduction.............................................................................................. 1

Geometric factor of the yL array................................................................... 3

The yL sounding curve............................................................................... 6

Relation between pyL and ps. ....................................................................... 7

Collecting and processing of yL sounding data................................................ 15

Summary and conclusions........................................................................... 18

References cited........................................................................................ 19

ILLUSTRATIONS

Page

FIGURE 1. Diagram showing the L-shaped perpendicular array............................ C2

2. Graph showing comparison between azimuthal, perpendicular, and

radial dipole-dipole three-layer sounding curves.............................. 3

3. Nomogram for the evaluation of the geometric factor of the yL array..... 6 

4-7. Graphs showing:

4. Two-layer master set of yL-sounding curves............................ 8

5. Two-layer yL sounding curves.............................................. 10

6. Variation of the effective resistivity factor ps (AO)lpyL(AO) as a

function of the spacing ratios AO/BO and AO\AB. ................. 12

7. (Variation of ps (BO)/ps (AO) as a function of AO/AB for an error

in pyL (AO) of 2 percent and 5 percent................................ 14

8. Diagram showing the L-shaped parallel array (xL array)....................... 16

TABLE

Page

TABLE 1. Variation of K-> and K^ as a function of AO/AB. ................................... C5

III

NEW TECHNIQUES IN DIRECT-CURRENT RESISTIVITY 

EXPLORATION

ELECTRICAL RESISTIVITY SOUNDING WITH 

AN L-SHAPED ARRAY

By ADEL A. R. ZOHDY

ABSTRACT

The effect of the distant current electrode, B, in the (AMN, B) yL array (L-shaped 

bipole-dipole perpendicular array) on the value of apparent resistivity is analyzed. A 

formula is derived for computing the geometric factor of the yL array, and a nomogram is 

given to simplify computations. A second formula is derived to express the relation between 

the yL apparent resistivity, pyL (AO); the ratio of the yL electrode spacings, AO/AB; and 

the Schlumberger apparent resistivities, ps (AO) and ps(BO), where AO and BO are the 

distances from the current electrodes, A and B, to the center, O, of the potential dipole MN. 

This second formula is used for transforming Schlumberger sounding curves into yL 

sounding curves. It is also used for determining the limits of the ratio AO/AB, for which a 

yL sounding curve coincides with a Schlumberger curve. When the resistivity decreases 

with depth, the ratio AO/AB can be made as large as 0.3 or 0.4 without pyL (AO) differing 

from ps (AO) by more than 2 percent or 5 percent respectively. When the resistivity increases with depth, the maximum value of the ratio AO/AB should be =£ 0.15 or =£ 0.25 

for an error of ^2 percent or =£ 5 percent respectively. Mathematical and field procedures 

are described for transforming yL sounding curves into Schlumberger curves.

INTRODUCTION

The L-shaped perpendicular array (yL array) is a special electrode 

configuration of the bipole-dipole perpendicular array (Zohdy, 1970); 

the measuring dipole MN is placed along a line perpendicular to the 

current bipole AB at one of the current electrodes A or B (fig. 1). An 

electrical sounding is made by moving the dipole MN away from the 

nearest current electrode at logarithmically spaced intervals. At small 

electrode spacings AO, that is, when AO <£ AB, the yL array reduces to 

the pole-dipole array (or half-Schlumberger array), which is frequently 

used by some investigators for shallow exploration. However, at large 

electrode spacings (AO > AB), the yL array reduces to the perpendicular 

dipole-dipole array (Al'pin, 1950).

The ideal perpendicular dipole array (AB-^0 and MN~^ 0) has an 

intermediate probing depth (Keller, 1966) and an intermediate resolving 

power (Zagarmistr, 1957) between the ideal equatorial (or azimuthal) and 

the ideal polar (or radial) dipole-dipole arrays. For example, figure 2

Cl
C2 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

A/0 

0

AO

B

FIGURE 1. Diagram showing the L-shaped perpendicular array 

(yL array). A and B, current electrodes; M and N, potential 

electrodes; O, center of dipole MN; Q, center of bipole AB; AO, 

distance from current electrode to center of dipole.

shows that, for a given three-layer section of the H type (pi > p2, Pa °°), 

the ratios of electrode spacings necessary to register the effect of the 

electric basement (p3 = °°) are re:ry :rr l :1.5:2; but the ordinates of the 

apparent resistivities of the minimum points on the respective sounding 

curves are pe . > pv . > pr . Therefore, the presence of the second

r mm ^ w mm ^ min ' r

and the third layers is detected on the PB curve at shorter electrode 

spacings than on the py or pr curves, which demonstrates the greater 

probing depth of the equatorial (or azimuthal) array. However, the effect 

of the second layer is most emphasized on the pr curve, which demonstrates the greater resolving power of the polar (or axial) array. The 

perpendicular-dipole sounding curve occupies an intermediate position 

between the equatorial and the polar curves, which demonstrates its 

intermediate probing depth and resolving power.

Bipole-dipole arrays are used more often than dipole-dipole arrays, 

and in practice it is impossible to use a true pole-dipole array. It is 

important, therefore, to establish the ratios of the electrode spacings at 

which a bipole-dipole array approximates either a pole-dipole or a 

dipole-dipole array. It is also important to find methods for the interpretation of the data obtained when the dimensions of the electrode array 

do not approximate a pole-dipole or a dipole-dipole array.
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C3

100

FIGURE 2. Comparison between azimuthal (pe), perpendicular (pa), and radial (pr) dipoledipole three-layer sounding curves for an //-type section (p^/pi = l/a, ps/pi =°°, h2/hi = 3). 

p, true resistivity; p, apparent resistivity; r, dipole spacing; h, thickness of layer.

GEOMETRIC FACTOR OF THE yL ARRAY

The general expression for the geometric factor, K, of a quadripole 

array is derived from the following expression for the potential difference 

(Heiland, 1946):

1 1 1 \pl a)

where p is the true resistivity of the medium; AM, AN, BM, and BN are 

the distances between the electrodes A, M, N, and B; and /is the electric 

current. For the perpendicular yL array, the geometric factor K in 

equation 1 may be written as

K= 27T

MN

AMAN'

(2)

When the potential electrode spacing, MN, is small (in comparison 

to the distance from its center, 0, to the nearest current electrode, A, 

MN^Q.lAO), the expression for K is simplified by considering the 

electric field, E, rather than the potential difference, AF. For the bipole-
C4 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

dipole yL array, the magnitude of the net electric field Eyi B is expressed 

by

E*i B = Ei0 -El0 cosa, (3)

where E^0 and EJ0 are the magnitudes of the electric fields caused by 

the point electrodes A and B along the directions AO and BO, respectively, and cos a = AOIBO=\l VI + (AB/AO) 2 .

For a homogeneous and isotropic half space of resistivity p, the 

magnitudes of the electric fields E\0 and E|0 are

EB =

B0

and

Therefore, substituting equations 4 and 5 in equation 3 and rearranging, 

we get

-1 ]?A,B -?^> (6)

AO

where pyL(AO) is the true resistivity, p, of the homogeneous and isotropic 

semi-infinite space as measured by the yL array at the spacing AO. Thus, 

according to equation 6, the geometric factor for the ideal yL array

is

'

In practice, when the distance between the measuring electrodes 

M and A^ is finite but nevertheless is sufficiently small (in comparison to 

the distance from the nearest current electrode), the approximation

is applicable. Substituting approximation 8 in equation 7 we get

AO2 ( 1

1 AO
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C5

Therefore the geometric factor for the practical bipole-dipole yL array 

can be approximated by

AO2 { 1 I" 1

AO

Approximation 10 is simpler than equation 2, and it can be simplified 

further by expressing K as the product of two factors Ki and K2 so that

K = K1K2 , (11) 

where

K - Kl ~MN

and

f 1-

A nomogram for evaluating the factor K2 for 0.1 ^ AO/AB ^ 10 is shown 

in figure 3. If the distances AO and AB are measured in feet and the 

resistivity is evaluated in ohm-meters, then the curve denoted by K2 

(where K2 = 0.3048 K2) should be used. The values of K2 and K2 are given 

in table 1, and an example showing the use of the K2 curve is given below.

TABLE 1. Variation ofKz and K% as a function of AO/AB

AOIAB K2 K!, AO/AB K2 K^

0.1

.125

.166

.25

.333

.5

.714

1

6.289

6.295

6.311

6.474

6.488

6.900

7.818

9.72

1.917

1.919

1.923

1.943

1.977

2.103

2.382

2.963

1.25

1.666

2.5

3.333

4.0

5.0

7.14

10

11.99

17.00

31.48

51.81

72.28

109.9

218.9

424.1

3.652

5.182

9.595

15.79

22.03

33.52

66.74

129.3

Example.-Let AB = 500 feet, /f0 = 600 feet, and MN=80 feet 

(MN= Q.133AO). Therefore, Ki = 36 X 10 4 /80 = 4.5 X 103 feet; for 

AO/AB = 600/500= 1.2, the value of K2 as determined from the nomogram 

is about 3.5. Therefore the value of the geometric factor is KVL=KiK2 ^ 

1.58X104 meters. The exact value of KVL according to equation 2 is
C6 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

1.564 X 104 meters, which indicates that the error resulting from the use 

of the graph in figure 3 is only 1.02 percent. In this example, if 

0.1 AO (MN^6Q feet), the error would be less than 1 percent.

1000

100

10

MN

2£X0.3048

0.01 0.1 1

AQ_ 

AB

10 100

FIGURE 3. Nomogram for the evaluation of the geometric factor of the yL array. K, geometric factor; AO, distance from current electrode to center of dipole; AB, distance 

between current electrodes; MN, distance between potential electrodes.

THE yL SOUNDING CURVE

For a horizontally stratified, laterally homogeneous and isotropic 

medium, the value of the apparent resistivity measured by means of a 

quadripole array depends not only on the geoelectric parameters of the 

layers (their thicknesses and resistivities) and on the spacings of the 

electrode array, but also on the geometry of the array. The dependency 

on the spacing distances is indicated by considering the rearranged 

form of equation 1 which (for the computation of the apparent resistivity, 

p,) is written as

- 2 rj _ i _ i . i

P [AM AN BM AN I

(14)

In equation 14 the value of p depends on four variable distances. 

Consequently, several electrode arrays (Wenner, Schlumberger, and ideal
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C7

dipole-dipole arrays) were invented to reduce the dependency of p from 

four distance variables to one distance variable, which in turn reduces 

the number of theoretical curves necessary for the interpretation of 

field data.

In yL soundings (M/V-*0), p yL depends on the distances AO and BO 

during the transition from the pole-dipole array (AO < BO) to the perpendicular dipole-dipole array (AO ~ BO). The solution to this problem 

for horizontally stratified, laterally homogeneous media is achieved by 

one of the following methods:

1. The spacing AO is made much smaller than BO at all times 

(AO ^0.1 BO), and the sounding curves are interpreted by means of 

the albums of theoretical curves for the Schlumberger array (Compagnie Generale de Geophysique, 1963; Orellana and Mooney, 

1966). However, the condition that AO is always less than or equal 

to 0.1 BO cannot always be fulfilled in practical applications except 

for very shallow depths of exploration.

2. The ratio AO/BO is held constant for all the sounding measurements. 

This is done by moving the electrodes MN as well as the current 

electrode B after each measurement. Then the interpretation of the 

sounding data is made by calculating special albums of theoretical 

curves for the designated constant value of AO/BO. A two-layer set 

of yL sounding curves is shown in figure 4 for the interpretation 

of data obtained with AO/BO = V2 (or AOJAB = 1). The calculation 

of this set was made by using the album of theoretical curves of 

the Compagnie Generale de Geophysique and by using a formula 

given in the following section. (See eq 18 and 19.)

RELATION BETWEEN pyL AND p,

The pattern of the electric field at the surface of a horizontally stratified, laterally homogeneous and isotropic half space, caused by a bipole 

AB (placed at the surface of the ground), is the same as the pattern obtained over an unstratified homogeneous and isotropic half space. 

However, the intensity of the electric field at any given point on the 

horizontally stratified medium is different from the intensity of the 

field at the same point on the unstratified half space. Therefore, equation 

4 for the ideal yL array (MN >0) can be applied for a horizontally 

stratified, laterally homogeneous medium, provided that the electric 

field EA and EB are expressed in terms of the apparent resistivities 

ps(AO) and ps (BO) of a half-Schlumberger array. Therefore, using 

equations 3, 4, and 5 we get:

_ Ps (AO)I 

A0 ZrrAO* '
C8 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

100

0.004

1000

FIGURE 4. Two-layer master set of yL sounding curves for AO = AB. Numbers on curves 

designate values of pzlpi- A and B, current electrodes; O, center of dipole; p, true 

resistivity; p, apparent resistivity; AO, distance from current electrode to center of 

dipole; h, thickness of layer.
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C9

= Ps (BO)I 

B0 2>rrB0 2 '

and

Substituting equation 17 in equation 6, and rearranging we get (Al'pin, 

1956):

(18)

or in terms of AO and AB,

i

(19)

Equations 18 and 19 are used to transform theoretical Schlumberger 

curves (Compagnie Generale de Geophysique, 1963; Orellana and 

Mooney, 1966) into yL sounding curves. However, because theoretical 

Schlumberger curves are expressed in terms of the dimensionless 

quantities AO/hi and ps/pi (where pi and hi are the unit resistivity and 

unit thickness of the first layer), equation 19 may be written in the 

dimensionless form

1

r Aoihj i 3

L V(AO/hi) 2 +(AB/hi) 2 \

ps(AO/hi) ( AO/hi \ 3 ps (£0//M)1

V Pi J

»'

Pi

The calculated values of the reduced resistivity p VL(AO/hi)/pi may 

be plotted as a function of the reduced spacing AO/hi, but the resulting 

yL sounding curve will also depend on the value of AB/hi. To illustrate 

this dependency, a set of yL sounding curves was computed for a twolayer earth model in which the second layer is a perfect insulator (p2=0°).
CIO TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

100

100 200

FIGURE 5. Two-layer yL sounding curves obtained over a buried insulator (p 2 = 00) for 

variable AO/AB. p, true resistivity; p, apparent resistivity; AO, distance from current 

electrode to center of dipole; A and B, current electrodes; M and N, potential electrodes; 

O, center of dipole; hi, thickness of first layer. VES, half-Schlumberger, or pole-dipole 

curve; DYS, dipole-dipole perpendicular curve.

The set shown in figure 5 is composed of five curves corresponding to 

ABlhi-* oo (half-Schlumberger, pole-dipole, or VES curve), AB/hi = 5Q, 

10, 5, and AB/hi > 0 (dipole-dipole perpendicular, or DYS, curve).

Considering the set of curves shown in figure 5, the following conclusions are drawn for yL sounding curves obtained over a two-layer 

structure with p2 = °°, AB = constant, and AO = variable:

1. An infinite number of theoretical yL sounding curves is obtained as 

a result of the infinite number of values that the ratio AB/hi may 

attain. A single curve, however, is obtained if AO/AB= constant.

2. A yL sounding curve (AB = constant) coincides with the pole-dipole 

(Schlumberger) curve as AB\h\ > °°, and coincides with the ideal 

dipole-dipole perpendicular sounding (DYS) curve as AB/hi ^O.

3. The general form of a two-layer yL sounding curve (p 2 = °°), obtained 

with ABIhi = constant, resembles a three-layer Schlumberger curve 

of the A type (Kalenov, 1957; Keller and Frischknecht, 1966) or the 

sounding curve obtained by a symmetric AMNB Schlumberger 

array over a two-layer medium with a buried vertical step fault 

(Faradzhev, 1958a, b; Kunetz, 1966).
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY Gil

4. In order to obtain a sufficiently long segment of a two-layer yL sounding curve (AB constant) that can be interpreted by means of a 

two-layer Schlumberger curve, the distance between the current 

electrodes must be at least 20 times the depth to the buried insulator 

(AB/hi 5= 20). Larger AB/H ratios generally would be required for 

multilayer earth models, where H is the total depth to the electric 

basement, because of the possible effect of pseudoanisotropy 

(Zohdy, 1965) owing to the presence of resistive layers within the 

section.

5. The two-layer yL sounding curve (p2 = °°) deviates from the corresponding VES curve at AO/AB ~ 0.2 and merges with the corresponding DYS curve at AO/AB ~ 4. Therefore, if the ratio AO/AB ^ 

0.2, then the data can be interpreted using theoretical VES curves. 

This, however, does not necessarily mean that the obtained curve 

segment (at AO/AB ^ 0.2) would be of sufficient length to justify 

a dependable interpretation (as, for example, when AB\h\ ^ 10).

6. For different values of AB/hi, the general form of the yL sounding 

curve remains unchanged from where it deviates from the rectilinear right branch of the Schlumberger curve (S-line) to where it 

merges with the rectilinear rising branch of perpendicular dipoledipole curve.

The mathematical proof for conclusion 6 is as follows. Equation 18 is 

rewritten as

BO 

but for ps values on the S-line,

(BO) 1 AO 

am BO Sps (AO) '

where S is the total longitudinal conductance (Kalenov, 1957; Keller and 

Frischknecht, 1966). Therefore, on the S-line,

BO

pvL (AO) ,iAO

(BO

Equation 20 indicates that the ratio ps (AO)/pyL(AO), which is an index 

of the deviation of the yL sounding curve from the Schlumberger 

sounding curve, along the S-line, is a function only of the ratio AOjBO.
C12 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

AO AO 

BO' AB

FIGURE 6. Variation of the effective resistivity factor ps (AO)/pyL(AO) as a function of 

the spacing ratios AO/BO and AO/AB. p, apparent resistivity; AO and BO, distances 

between current electrodes and center of dipole; AB, distance between current 

electrodes; M and N, potential electrodes.

The ratio pyL(AO)lps (AO) may be called the "effective resistivity 

factor." In equation 20 when BO > AO (pole-dipole array), ps (AO)/ 

pyL(AO) = 1, and when BO ~ AO (perpendicular dipole-dipole array), 

ps (AO}lpyL (AO} = 3/2; this is proved by applying I'Hospital's rule 

(Wylie, 1951) to equation 20.

In figure 6, the variation of the effective resistivity factor ps (AO}l 

pyL(AO} as a function of AO/BO is shown as the curve having open 

circles, and its variation as a function of AO/AB is shown as the curve 

having solid circles. These curves are useful in constructing the S-line 

of the Schlumberger curve from data obtained by the yL array, provided 

that the effect of a high-resistivity electric basement is detected on the 

yL sounding curve. The procedure for this construction is as follows: 

(1) Calculate the value of AO/AB, (2) determine from the solid-circle 

curve in figure 6 the value of the ordinate ps (AO}lpyL (AO}, (3) multiply 

the measured value of pVL(AO] by the ratio ps (AO)lpyL(AO) to determine 

ps (AO) and (4) plot the value of ps (AO) at the spacing value AO.

It should be emphasized that the transformation procedure using the 

effective resistivity factor ps (AO}lpvL (AO} is strictly applicable to 

apparent resistivity values that correspond to the detection of a highresistivity electric basement (pbedrock ~* °°) and that it does not 

accurately transform all the data on the yL sounding curve into a 

Schlumberger sounding curve. This limitation also applies to the transformation of sounding data using the so-called effective spacing factor as 

denned by KeUer (1966).
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY CIS

As mentioned earlier, when the ratio AO/AB ^ 0.2, the two-layer yL 

sounding curve for p2 -* °° does not depart considerably from the poledipole (half Schlumberger) curve. When p% ^ oo, the limiting value of the 

ratio AO/AB, for which a pole-dipole curve is obtained, depends on the 

geoelectric parameters of the section. For example, it is obvious that, 

when sounding measurements are made over a homogeneous and 

isotropic half space, the true resistivity of the medium is determined 

correctly irrespective of the value of AO/AB (provided that the exact 

value of the geometric factor is used in the calculation). Therefore, to 

study the effect of the distant current electrode B and the error introduced in the results when that effect is neglected, we write equation 18 

in the following form:

- (A0\ - ~ (Af}\ \ l (AOIBOy ps(BO) PvL (AO) - ps (AO) (AO\*~l-(AOIBOyp.(AO)

(BO)

Adding 1 and subtracting 1 in the term in the brackets and rearranging, 

we get

^

ps (AO) 1 - (AO/BO) 3 L ps(AO) y

The second term in the preceding equation is the disturbing factor that 

makes the ratio pvL(AO}lps (AO) different from unity. Therefore, by 

setting this second term to be equal to or less than 0.05 or 0.02, respectively, or by setting

l-(AOIBO)>\ l -jMffir ± °X* <21 > 

and

" /D^ "" (21')

1 - (AO/BO) 3

we determine the limiting conditions for the values of pyi^AO) to be 

equal tops (AO) with an error of ±5 percent, or±2 percent, respectively.

For a 5-percent error we rearrange equation 20 and, in terms of AO 

and AB, write it in the forms

v l ^^ - / /*/"»\ -1- -1- V L ps (AO) J

and

( '
C14 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

100

over horizontal media 

(ps (BO) and pJAO) 

on S-line)

0.01

FIGURE 7.-Variation of ps (BO)/ps (AO) as a function of AO/AB for an error in pyL (AO) of 

2 percent and 5 percent. Hatched area results in errors mpyL(AO) of less than 2 percent; 

stippled area, in errors in pVL(AO) of less than 5 percent, p, apparent resistivity; AO and 

fiO, distances between current electrodes and center of dipole; AB, distance between 

current electrodes; S-line, rectilinear branch of a VES curve rising at an angle of 45° 

with the abscissa axis.

Similar expressions are found for evaluating a 2-percent error using 

equation 21'. The variation of ps (BO)lps (AO) as a function ofAO/AB for 

errors in pyL(AO)lps (AO) of ± 5 percent and ± 2 percent are shown in 

figure 7. A curve for the maximum values of ps (BO)lps (AO), which are
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY CIS

obtained when both ps (BO) and ps (AO] fall on the S-line, is also shown 

in figure 7. The curves in figure 7 indicate that, whenps (BO) < ps (AO) 

(steeply descending branch on a Schlumberger curve), the ratio AO/BO 

may be made as large as approximately 0.3 without introducing an error 

greater than 2 percent, or it may be made as large as approximately 0.4 

without introducing an error greater than 5 percent. For a moderately 

descending branch the ratio AO/BO can be made larger than 0.3, or 

0.4, respectively. When ps (BO) ^> ps (AO), the ratio of AO/AB should 

be made equal to or smaller than 0.15 for a 2-percent error or less, or it 

should be made equal to or smaller than 0.26 for a 5-percent error or less. 

In practice, the exact value of ps (BO) is not known beforehand, but 

if the general form of the sounding curve in the area can be anticipated 

from other data, then the proper "safe" ratio AO/AB can be estimated 

from the curves in figure 7. A similar analysis for horizontal profiling 

using the pole-dipole and the polar dipole-dipole array was made by 

Russian investigators (Blokh, 1957 and 1962; Blokh and Shemyakin, 

1959), who seemed to disregard the fact that this type of analysis applies 

only to horizontally stratified media and not to media with inclined or 

vertical contacts (unless the electric fields created by images placed 

across the contact are considered).

COLLECTING AND PROCESSING OF yL SOUNDING DATA

Several methods were outlined by Al'pin (1956) for the transformation 

of electrical sounding data obtained by the so-called asymmetric or 

angular soundings. The method for processing yL sounding data will 

closely follow Al'pin's derivations.

Equation 18 can be rearranged in the form

[

/ AO\Z~\ / /tn\3 1 ~ (w) \ PVL(AO] ~ + (Jd) ps(BO} - (18 ' }

The values of AO/BO and pVL(AO] are measured in the field, but in 

order to calculate the Schlumberger apparent resistivity ps (AO} from 

18' the value of the second Schlumberger resistivity ps(BO) at the spacing 

BO must be determined, especially when AO/AB is large. The value of 

ps (BO} cannot be determined independently from ps (AO) unless the 

effect of the current electrode A is eliminated. Therefore, in the field, 

by orienting the potential line MN at right angles to the line AO, we 

eliminate the contribution of the current electrode A, and we make a 

measurement of the apparent resistivity pxi with the L-shaped parallel 

bipole-dipole array (xL array).
C16 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

FIGURE 8. The L-shaped parallel array (xL array). A and B, current electrodes; M and N, 

potential electrodes; O, center of dipole; EB , electric field at O caused by point source at 

B; E%, horizontal component of the electric field EB; E£, electric field at O caused by 

point source at A.

Considering figure 8, the horizontal component of the electric field, 

%, is related to the total field EB (caused by the point source at B) by

or

= EB sin a, 

 ~~BO (24)

For a homogeneous and isotropic semi-infinite space, the true resistivity 

p is calculated from

EB 

p = 2TTB02 (25)

In the xL array, the horizontal component £%, rather than the field EB , 

is measured. Therefore, substituting equation 24 in equation 25 we get

P = ^-T5--f (26)
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C17

For a horizontally stratified, laterally homogeneous medium the 

apparent resistivity is substituted for the true resistivity, and we therefore 

write equation 26 as

BO3 EB

(27)

where pXL is the apparent resistivity measured by means of the xL array 

and is calculated by measuring the electric field

(28) ( }

T 

x MN 

Therefore, substituting approximation 28 in equation 27 we get

^ 27T BO3 kV*IN 

pxL ~ MN AB I ' 

or

(AO* + AB*)^ &V* N 

pxL ~ 27T ABMN ~T- (29)

The exact value of pxi may also be calculated from the exact but more 

complicated expression

+ A02

AJ'fc..

 (30)

+AO* I

When MN^Q.lAO, the value of pxi as determined from equation 30 

differs from its value as determined from approximation 29 by less tK~n 1 

percent.

Substituting the measured value of pxi for ps (BO) in equation 18', 

and having measured p yi and substituting it in equation 18' also, the 

value of ps (AO) is calculated. The Schlumberger curve ps (AO) f(AO) 

is thus obtained by making two apparent resistivity measurements, 

pyL and pxL. Furthermore, the value of p XL(BO] =ps (BO) (used in the 

calculation of p s (AO) ) can be plotted as part of the Schlumberger curve 

at the spacing BO. Therefore, in effect, the two measurements paL(AO) 

and PXL(BO) are used to obtain two Schlumberger resistivities, ps (AO] 

(by calculation, using eq. 18' on p. CIS) and ps (BO) =pxL (BO) (by 

observation).

If we utilize the above procedure in practice, we realize that, as the
CIS TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

spacing AO is increased, the spacing BO= vAO^+AB2 is increased at 

a much smaller rate, especially for 0.2 < AO/AB < 2. For example, if 

AB= 10 length units, then for AO/AB<Q.2, puL (AO) ^ ps (AO), and 

there is no need to measure pxL (BO) ~ ps (BO). Then at A0 = 2. 5 length 

units, BO= 10.29 length units, and a measurement of pXL(BO) = ps (BO) 

is made to calculate ps (AO) (from eq. 18'), to plot ps (AO) at ,40 = 2.5 

length units, and to plot pxi(BO) at ,40=10.29 length units. Then, 

as measurements of pVL(AO) are made at ,40 = 3, 4, 5, 6, and 8 length 

units (which corresponds to an increasing logarithmic increment of 

1.2 to 1.3), the value of BO ranges from 10.29 length units (at ,40 = 2.5) 

to 12.8 length units (at AO 8). This means that as AO changes through 

a logarithmic interval of 8/3=2.66, BO changes through a logarithmic 

interval of about 1.24 only. Therefore, it is sufficient to make only 

two measurements of pXL(BO), one at AO/AB = 0.25 and the other at 

AO/AB = 0.8. The values of pxL (BO] for 0.25 < AO/AB < 0.8 that are 

necessary for the calculation of ps (AO) at ,40 = 3, 4, 5, and 6 length 

units are easily obtained by interpolation between the values of pxi(BO} 

at 50=10.29 and pxL (BO) at 50=12.8, which would be sufficiently 

close if the data are obtained over a horizontally homogeneous medium. 

The following AO/AB values are recommended for making measurements of pxL(BO): AO/AB = 0.25, 0.8, 1.2, 1.6, and 2.0. For values of 

AOjAB 3= 2 the azimuth angle 6= /_AQO, where Q is the center ofAB, is 

greater than or equal to 76°. It is difficult to measure p VL(AO) at 0 > 75° 

because KyL -^ °° as 0-» 90°. Furthermore, for AO/AB ^ 2.5, AO = BO, 

and it is of little use to measure ps (AO] and ps (BO) when the spacings 

,40 and BO are separated by a very small logarithmic interval. Therefore, 

for AOIAB 3= 2.5, it is sufficient to measure pj-L(BO) only and to plot it 

at the electrode spacing value of BO= VAO2 + AB2 .

SUMMARY AND CONCLUSIONS

Electrical soundings using L-shaped electrode arrays can be made 

either with the ratio of the electrode spacings AO/AB held constant or 

with AO/AB held variable and AB held constant. When the ratio AO/AB is 

held constant, the sounding curves are interpreted using sets of theoretical curves that are based on the given constant value of AO/AB. 

Such sets are easily derived from the available albums of Schlumberger 

theoretical curves. When AO/AB is variable and AB is held constant, 

yL sounding data are almost identical to Schlumberger sounding data 

for most resistivity contrasts, provided that AO/AB < 0.2. When 

0.2 =S AOIAB ^ 2, the yL sounding data must be complemented by 

measurements with the parallel L-shaped array (xL array) in order to 

calculate the equivalent Schlumberger apparent resistivity. Only three 

or four resistivity measurements, made by the xL array, are necessary
ELECTRICAL RESISTIVITY SOUNDING, L-SHAPED ARRAY C19

for the entire range of 0.2 =S AO\AB ^ 2, and forAO/AB > 2 it is sufficient 

to measure the value of pXL(BO), which is equal to the Schlumberger 

resistivity ps (BO).

Electrical sounding with an L-shaped array with AO variable and AB 

held constant is very useful in obtaining subsurface information at 

great depths in areas where road networks make it difficult to make all 

the required symmetric AMNB Schlumberger soundings. Moreover, for 

each setup of the current electrodes A and 5, the potential electrodes 

can be moved along four lines perpendicular to the line AB at the electrodes A and B. Consequently a great amount of detailed study can be 

made with a minimum movement of the current electrodes.

REFERENCES CITED

Al'pin, L. M., 1950, [The theory of dipole sounding]: Moscow, Gostoptekhizdat, 88 p. 

(in Russian); English translation in Keller, G. V., 1966, Dipole methods for measuring 

earth conductivity, New York, Consultants Bur., p. 1 60.

 1956, [The asymmetrical (angular) sounding]: Prikladnaya Geofizika, v. 14, p.

65 96 (in Russian). 

Blokh, I. M., 1957, [Dipole electroprofiling]: Moscow, Gosgeoltekhizdat, 190 p. (in Russian).

 1962, [The resistivity method of electroprofiling]: Moscow, Gosgeoltekhizdat, 238 p. 

(in Russian).

Blokh, I. M., and Shemyakin, E. A., 1959, [Construction of dipole and three electrode 

electroprofiling charts from investigations of straight-line unsymmetric BAMN systems]: Akad. Nauk SSSR Izv. Ser. Geof. 70.6, p. 872-879 (in Russian); English 

translation by J. J. Loferski, 1960, Bulletin (Izvestiya), no. 6, p. 614-619.

Compagnie Generate de Geophysique, 1963, Master curves for electrical soundings: 

The Hague, European Assoc. Explor. Geophysicists.

Faradzhev, A. S., 1958a, [Model studies of electrical soundings over a screen and a step]: 

Razved. i Promyslovaya Geofizika, v. 23, p. 38-44 (in Russian).

 1958b, [A study of the effect of non-horizontal boundaries on the results of electrical soundings]: Prikladnaya Geofizika, v. 19, p. 109-128 (in Russian); English 

translation by Ivan Mittin, 1966, in U.S. Geol. Survey library, Denver, 36 p.

Heiland, C. A., 1940, Geophysical exploration: New York, Prentice-Hall, Inc., 1013 p.

Kalenov, E. N., 1957, [Interpretation of vertical electrical sounding curves]: Moscow, 

Gostoptekhizdat, 471 p. (in Russian).

Keller, G. V., 1966, Dipole method for deep resistivity studies: Geophysics, v. 31, p. 

1088-1104.

Keller, G. V., and Frischknecht, F. C., 1966, Electrical methods in geophysical prospecting: 

Pergamon Press, 517 p.

Kunetz, Geza, 1966, Principles of direct current resistivity prospecting: Berlin-Nikolassee, 

Gebriider Borntraeger, 103 p.

Orellana, Ernesto, and Mooney, H. M., 1966, Master tables and curves for vertical electrical sounding over layered structures: Madrid, Interciencia, 34 p., 125 tables, and 

supplement of theoretical curves. (In English and Spanish.)

Wylie, C. R., Jr., 1951, Advanced engineering mathematics [1st ed.]: New York, McGrawHillBook Co., Inc.,696 p.

Zagarmistr, A. M., 1957, [Utilization of the increased resolving power of dipole-axial 

soundings in investigating a "type-H" section]: Prikladnaya Geofizika, v. 16, p.
C20 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

130-144 (in Russian); English translation by Ivan Mittin, 1966, in U.S. Geol. Survey

library, Denver, 21 p. 

Zohdy, A. A. R., 1965, The auxiliary point method of electrical sounding interpretation and

its relationship to the Dar Zarrouk parameters: Geophysics, v. 30, no. 4, p. 644-660. 

 1970, Geometric factors of bipole-dipole arrays: U.S. Geol. Survey Bull. 1313-B,

26 p.

U.S. GOVERNMENT PRINTING OFFICE : 1970 OL 389-744