Geometric Factors of 

Bipole-Dipole Arrays

GEOLOGICAL SURVEY BULLETIN 1313-B

Y^~~ **;_ """ 'fed

Geometric Factors of 

Bipole-Dipole Arrays

By ADEL A. R. ZOHDY

NEW TECHNIQUES IN DIRECT-CURRENT 

RESISTIVITY EXPLORATION

GEOLOGICAL SURVEY BULLETIN 1313-B

Nomograms 9 curves, and tables simplify 

evaluation of geometric factors of 

azimuthal, perpendicular, and parallel 

bipole-dipole electrode arrangements for 

measuring electrical resistivities of 

the earth

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

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress catalog-card No. 73-607298

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

Washington, D.C. 20402 - Price 25 cents (paper cover)
CONTENTS

Page

Abstract __ ______________________________________________________ Bl

Introduction ______________________________________________________ 1

Geometric factor of the azimuthal arrangement________________________ 4

Special azimuthal arrangements-____________________________________ 9

The equatorial arrangement___________________________________ 9

Effective spacing of the equatorial arrangement-__________________ 12

L-shaped azimuthal arrangement-_______________________________ 14

Geometric factor of the parallel arrangement._________________________ 15

Special parallel arrangements______________________________________ 17

L-shaped parallel arrangement.__________________________________ 19

Polar and asymmetric Schlumberger arrangements.________________ 20

Geometric factor of the perpendicular arrangement-___________________ 22

L-shaped perpendicular arrangement-____________________________ 23

Conclusions-______________________________________________________ 25

References cited-__________________________________________________ 26

ILLUSTRATIONS

FIGURE 1. Diagram showing bipole-dipole and dipole-dipole arrays_____ B3

2. Diagram showing bipole-dipole azimuthal array ____________ 5

3. Nomogram for evaluating the factor A$ of the azimuthal array- 8

4. Nomogram for evaluating the factor A eCL of the equatorial

array ______________________________________________ 10

5. Nomogram for evaluating the factor A *e<1 of the equatorial

array ______________________________________________ 11

6. Diagram showing the L-shaped azimuthal array.___________ 15

7. Nomogram for evaluating the factors An, AXL and Avi of the

L-shaped azimuthal, parallel, and perpendicular arrays-__ 16

8. Diagram showing bipole-dipole parallel array. _____________ 17

9. Nomogram for evaluating the factor A x of the parallel array__ 18

10. Diagram showing the L-shaped parallel array._____________ 19

11. Diagram showing polar and asymmetric Schlumberger

arrays. _____________________________________________ 20

12. Nomogram for evaluating the factors ^. a sym and A v of the

asymmetric-Schlumberger and polar arrays-___________ 21

13. Diagram showing the bipole-dipole perpendicular array.___ 23

14. Nomogram for evaluating the factor A u of the perpendicular

array.______________________________________________ 24

15. Diagram showing the L-shaped perpendicular array._______ 25

III
IV CONTENTS

TABLES

Page 

TABLE 1. Geometric factor of the equatorial array for variable AB/2... B13

2. Geometric factor of the equatorial array for 45/2 = 2,000

feet_.______________________________________ 13

3. Geometric factor of the equatorial array for 4.5/2=4,000

feet___________________________________ 14
NEW TECHNIQUES IN DIRECT-CURRENT RESISTIVITY

EXPLORATION

GEOMETRIC FACTORS OF 

BIPOLE-DIPOLE ARRAYS

By ADEL A. R. ZOHDY

ABSTRACT

Formulas are derived for computing the geometric factors of the azimuthal, 

equatorial, parallel, polar, perpendicular, and L-shaped bipole-dipole electrode 

arrays. The derivations are based on the assumption that a component of the 

electric field along the azimuthal, the parallel, or the perpendicular direction is 

approximately measured. For each array, the geometric factor is expressed as the 

product of two factors, K* and A. The factor K* is simple to compute, being 

expressed as (R*/MN) for all the considered arrays where R is the bipole-dipole 

spacing and MN is the distance between the measuring electrodes. The expression 

for the factor A is complicated, and its evaluation is generally cumbersome. 

Nomograms, curves, and tables are given to facilitate the evaluation of the 

various A factors with sufficient accuracy for most practical purposes.

INTRODUCTION

Several electrode arrangements are known to geophysicists engaged 

in electrical prospecting. The preference of one arrangement over 

another is partially governed by the simplicity of the expression of 

the geometric factor, K. For example, one of the simplest geometric 

factor expressions is that of the well-known Wenner arrangement, 

l?w-=2ira, where a is the distance between the equally spaced electrodes A, M, N, and B (A and B are current electrodes; M and N are 

potential electrodes). In the practical AMTVB-Schlumberger array, 

the geometric factor, Ks, is more complex and is given by KS=TT 

[(AB/2) 2 -(MN/2Y]/MN, where AB>5MN. In practice, one or more 

tables of precalculated values for If s are used to facilitate the computation of the apparent resistivity, ps, from the well-known general 

formula

-J-,

Bl
B2 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

where AY is the measured voltage (between M and JV) and 7 is the 

electric current. For a given distance between current electrodes, AB, 

and a given current intensity, /, one measures a smaller potential 

difference, AY, when using the Schlumberger array than when using 

the Wenner array. This is mathematically expressed in an inverse 

relationship for the corresponding geometric factors. Thus more 

current is required when using the Schlumberger than when using the 

Wenner array to produce the same AY at the same AB. However, the 

sensitivity of the Schlumberger array for detecting a given layer, at a 

given depth and at a given AB, is greater than that of the Wenner 

array (Deppermann, 1954: Zohdy, 1964).

In recent years, the possibility of making electrical soundings using 

dipole-dipole arrays (Al'pin, 1950) has received considerable attention 

(Alekseev and others, 1957; Anderson and Keller, 1966; Cantwell and 

others, 1965; Cantwell and Orange, 1965; Berdichevskii, 1957; 

Berdichevskii and Zagarmistr, 1958; Jackson, 1966). The expressions 

for the geometric factors of the various dipole-dipole arrays are not 

particularly complex, but the magnitudes of these geometric factors 

are large and increase very rapidly as the spacing between the centers 

of the two dipoles is increased. The large values of these dipole-dipole 

geometric factors reflect the requirement for tremendous electric 

power and a high-sensitivity potential-measuring instrument. Using 

bipole-dipole arrays instead of dipole dipole arrays overcomes these 

prohibitive requirements. In a bipole-dipole array, the distance, R, 

between the centers, Q and 0, of the bipole and the dipole is not 

exceedingly larger than AB/2, that is, R<lQ(AB/2); and the length 

of the current bipole, AB, is large in comparison to that of the measuring dipole, MN (fig. 1). Geometric factors of bipole-dipole arrays 

are generally smaller in magnitude but more complex in form than 

those of corresponding ideal dipole-dipole arrays.

In general, the value of the geometric factor for any quadrupole 

array can be computed from the expression (Heiland, 1946)

p l-I-I+i) '

\n r2 rz n/

where rl} r2 , r3 , and r4 are the distances AM, AN, BM, and BN between 

the current electrodes, A and B, and the potential electrodes, M and 

N. In electrical sounding using bipole-dipole or dipole-dipole arrays, 

AM nearly equals AN and BM nearly equals BN; therefore the 

calculation of p from the above equation requires the measurement 

of AM, AN, BM, and BN with great accuracy. Furthermore, by 

expressing the distances AM, AN, BM, and BN in terms of quantities 

(AB, MN, R, and 6} that are readily measured in the field, the expres-
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B3

M M

B A Q B

FIGURE 1. Bipole-dipole (left) and dipole-dipole (right) arrays. A and B, current 

electrodes; Q, center of current bipole (left) or dipole (right); M and N, measuring 

electrodes; 0, center of measuring dipole; R, distance between centers of bipole 

and dipole; R, distance between center of dipole and current electrode.

sion of the geometric factor using the above equation becomes very 

complicated for nonlinear bipole-dipole arrays. To simplify the comutation of the geometric factor of bipole-dipole arrays, the apparent 

resistivity is expressed in terms of the electric field, E, instead of 

the potential difference, AF, and then the approximation E=AV/MN 

is used.

The use of bipole-dipole arrangements for making electrical soundings poses problems in the presentation and interpretation of the 

sounding data. Difficulties arise because of the introduction of additional variables not present in dipole-dipole, Schlumberger, or Wenner 

arrays. These variables are the azimuth angle, 6 and the finite length 

of the bipole, AB. The problems arising from the introduction of these 

variables are not mathematically formidable, at least for horizontally 

stratified media, where the sounding data may be processed in part by 

using "effective spacing factors," "effective resistivity factors," or 

other transformation procedures. The development of these types of
B4 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

analyses, however, is beyond the scope of this paper. With the exception of the equatorial array, the following discussion will be concerned 

only with the first fundamental problem, namely simplifying the 

computation of the geometric factors so that the magnitude of the 

resistivity can be evaluated easily.

The author thanks Donald Plouff and Paul G. Hoffman, of the 

U.S. Geological Survey, for writing the necessary programs for the 

CDC-3600 computer.

GEOMETRIC FACTOR OF THE AZIMUTHAL ARRANGEMENT

In the azimuthal arrangement, the perpendicularity of MN to R is 

maintained at all times, or <f>=ir/2 (fig. 1). An electrical sounding can 

be made by varying the distances R, AB, and (or) the angle 0. 1 The 

general formula for computing the apparent resistivity, using the 

azimuthal arrangement, is given by

A,B

n ,.

(1)

where

7>9= resistivity measured by the azimuthal arrangement, 

K e= geometric factor of the azimuthal arrangement,

AV^ = potential difference between the measuring electrodes M and N 

(placed around the point 0) due to the current electrodes A and 

B, and 

7= intensity of the electric current.

If MAT" is sufficiently small in comparison to AO and BO (AO>1Q 

MN, BO>10 MN), then the electric field component at the point 0, 

along the direction MN, is approximately measured. In other words,

dV

or

where

AF

, ,,, j/i,r-»7-\ MN d(MN)

Mtf E (2)

EMtf\o=Ee\ 0= azimuthal component of the electric field at the point 0, along 

the azimuthal direction, 9, of MN.

Let R and R be the distances from the center Q of AB and from a 

point electrode, respectively, to a given point on the earth's surface 

where the electric field is measured. The magnitude of the azimuthal 

component of the electric field, Ee , at the point 0 (fig. 2) is given by

?A,B

JMN ER. COS cos ]8, (3)

' In the ideal dipole-dipole azimuthal, radial, and perpendicular (but not the parallel) arrays, the resistivity measurements are independent of the azimuth angle B (Al'pin, 1950).
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B5

B

FIGURE 2. Bipole-dipole azimuthal array. A and B, current electrodes; Q, center 

of current bipole; M and N, measuring electrodes; 0, center of dipole; R, distance 

between centers of bipole and dipole; AB, distance between current electrodes; 

R, distance between current electrode and center of dipole; E, point-source 

component of electric field; Eg, azimuthal component of electric field.

where

Ei R\

and

radial components of the electric fields, along R\ and #2, at 

the point 0, due to the electrodes A and B, respectively,

a, (8 = the angles (E^ , MN) and (E^ , MN), respectively. 

From figure 2, the following relationships apply:

(AB/2) sin 6 COS a = ~ -=: >

(AB/2) sin 6

- -=*: j

R,

(4a) 

(4b)
B6 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

and

" Cos8. (4d)

For a homogeneous isotropic semi-infinite medium of resistivity p, 

the radial components of the electric fields due to the point electrodes 

A and B, at the point 0, are given by

E_ A =^2 (5a) 

and

E_ (5b)

2-irR

respectively. Hence, substituting equations 4a-d and 5a,b in equation 3 

and rearranging, we get

(6)

Substituting equation 6 in equation 2, solving for p, and rearranging, 

we get

p =

(7)

In practice, for any given electrode arrangement, the apparent 

resistivity, p, is calculated by using the formula derived for a homogeneous isotropic half space (whereas the actual measurements might 

be made over a heterogeneous generally anisotropic medium). Hence 

equation 7 is valid for computing the apparent resistivity by the 

azimuthal arrangement, and from equation 1 it follows that

(8)
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B7

The numerical evaluation of Ke from equation 8 is cumbersome. 

Therefore, let

Ke=K'eAe, " (9) 

where

K'-=

and

(ID

Values of the factor Ag , which is independent of the dipole length 

MN, were computed for 0.1< (AB/2R)< 1 and for 0°<0<90° and 

contoured as shown in figure 3. An example of the evaluation of the 

factor Ag , using the nomogram, follows:

Let R= 1,000 meters, AB/2=500 meters, MN=100 

meters, and 0=45°. To determine Kg, we first find K*g :

R2

 =104 meters. MN

The value of Ae as estimated from the nomogram (fig. 3), 

at the ordinate AB/2R=0.5 and the abscissa 0=45°, is 6.2. 

Therefore Ke=6.2X104 meters. (If R, AB/2, and MN are 

measured in feet, then the value of Ke must be multiplied by 

the conversion factor 0.3048 for the resistivity to be expressed 

in ohm-meters.)

In accordance with equation 11, the values of A^ approach infinity 

as the orientation of the measuring dipole MN tends to coincide with 

an equipotential line (as, for example, at 0=0°). This reflects the fact 

that as the potential difference between M and N becomes infinitely 

small, the geometric factor must become infinitely large in order that 

the value of p remain finite.

A nomogram for 0.01 < (AB/2R) < 0.3 and 60° <0< 90° was published 

by Berdichevskii (1957) and by Bordovskii (1958). The expression for 

Kg given by Berdichevskii is *

Ke=K$A$, 

where

and

sin 6

AB -1-3/2 r /AB\2 AB (13)
B8 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

40 50 

, IN DEGREES

FIGURE 3. Nomogram for evaluating the factor Ae of the azimuthal array. 

Values for Ae given on curves. AB, distance between current electrodes; R, 

distance between centers of bipole and dipole; 6, angle between current electrode, center of bipole, and center of dipole; MN, distance between measuring 

electrodes. (Ke =K*A e ; K* = R*/MN.)
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B9

Therefore the present nomogram for A8 is different from that of 

Berdichevskii not only in the range of AB/2R and 6, but also by a 

factor for 4R/AB in the value of A3 ; this further simplifies the evaluation of K*g (compare eq 10 and 12) and therefore of K6 .

SPECIAL AZIMUTHAL ARRANGEMENTS

Two special azimuthal arrangements the equatorial and the 

L-shaped are of practical interest.

THE EQUATORIAL ARRANGEMENT

The equatorial arrangement has been used rather extensively in the 

Soviet Union and has proved to be an effective tool in deep resistivity 

exploration (Berdichevskii and Zagarmistr, 1958; Yakoubovskii and 

Lyakhov, 1964; Zohdy, 1966; Zohdy and Jackson, 1966). In this special arrangement the angle 6 is always equal to 90°. Furthermore, the 

perpendicularity of MN to R (<£=7r/2), a condition of the azimuthal 

arrangement, is also maintained. During the sounding process, the 

dipole MN is moved away from the bipole AB along the perpendicular 

bisector of AB. This perpendicular bisector may be regarded as an 

equator to the poles at .A and B; hence the name "equatorial arrangement."

Under these conditions equation 7 reduces to

R* *

According to equation 14, the geometric factor Ke<l may be expressed 

as follows

 ft- e q == " e <t"- e q >

where

R*

and

-

The value of Ae(l for 0.1<(AB/2K)<10 can be determined easily 

from the curve in figure 4. The curve A^=j(ABj2R} exhibits a minimum &tAB/2R=Q.7Q71 ; which is easily determined by differentiating 

equation 15, setting the result equal to zero, and solving for AB/2R. 

The existence of a minimum value for^lea does not, however, mean that 

the geometric factor Ke(l= (R2/MN)Aeq attains a minimum value at 

AB/2.R 0.71. In fact, the factor Ke<l increases continuously, as 

expected, as R increases.
BIO TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

CQQ;

_ -R=pR

FIGURE 4. Nomogram for evaluating the factor A efl of the equatorial array 

(K* = R2/MN). AB, distance Jaetween current electrodes; R, distance between 

centers of bipole and dipole; R, distance between current electrode and center 

of dipole (effective spacing); MN, distance between measuring electrodes; K,

geometric factor; p, effective spacing factor f 2? = -i/l + ( T7p~) )

Equation 14 is a special case of equation 7 which in turn is valid 

only provided MN<O.IAO (see eq 2) . However, the general expression 

for the resistivity, using the equatorial arrangement, may simply be 

given by

AM AN AFAg (lfl) ( }

or

AN-AM I

_ AM AN

-* AN -AM

For the purpose of computation, equation 17 may be written as

, = TrR

r,.(AB MN\~ 

V. 1 + \2R~2R ) .

1 +

AB )']

I (AB MN\*_ 

where in equation 17

(AB^_MN\ 

\2R 2R)

(18)

and
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS Bll

Equation 18, in contrast to equation 14, does not imply any restrictions on the length of MN with respect to AO and BO. However, if a 

component of the electric field is to be approximately measured, then 

for AB(2R<1, MN/2B<Q.l; and for 1 < (AB/2R) < 10, 0.1<(MN/

In order to simplify computations, equation 18 can also be expressed as the product of two factors,

where 

and

K::=R

/- /AB_MA/V~ " (AB ,MN\*~ 

V _ \2B 2R ) _ _ + \2R ^ 2R ) _

I (AB MN\*_ I (AB\ MN\

~\l * "T" I Q r> i" Q ?p J "\/ * i" V o E> / O E> i

Y \ ^/v z/v / V \ ^/t / ^/t' /

(19)

(20)

The variation of the factor A*e(l as a function of 0.1< (AB/2R) < 1.5 

for 0.01< (MN/2R) <0.1 is shown in the nomogram in figure 5.

Interpolation between the various curves for MN/2R can be made by 

using a logarithmic scale (rotated through 180°) of the same modulus 

as the one on which the curves are plotted. For example, let .5=1,000

FIGURE 5. Nomogram for evaluating the factors ^CQ of the equatorial array 

(K** = R).AB, distance between current electrodes; R, distance between centers 

of bipole and dipole; K, geometric factor; MN, distance between measuring 

electrodes.
B12 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

meters, AB= 1,000 meters, and MA7=34 meters; then AB/2R=Q.5, 

MN/2R=0.017, A*^257, and j£eQ =tf^* = 2.57X105 meters. Berdichevskii (1954) and Berdichevskii and Petrovskii (1956) presented 

another nomogram to assist in the computation of Ke(l . Their corresponding expression for K** is given by

P3 K** = ri lv i(\s

A ° AB MN U '

which is not as simple as the one given in equation 19.

EFFECTIVE SPACING OF THE EQUATORIAL ARRANGEMENT

If an equatorial array is placed at the surface of a horizontally 

stratified laterally homogeneous and isotropic half space, then because 

of the symmetry of the arrangement

AFfrl AF£y 

7 ~ J / '

Therefore, in accordance with equation 16

AM AN AF£*_ , 2n 

peq -21r AN=AM -J PAMN- W>

In other words, by using the equatorial arrangement at a spacing

-4#Y ^ * A f n ), one measures the same magnitude ol ap- K /

parent resistivity as by using a three-electrode AMN (half-Schlumberger) array at a spacing AO. The distance AO R of the equatorial 

array is referred to as the effective spacing because, by plotting the 

apparent resistivity as a function of AO R, one can use the available 

albums of theoretical Schlumberger sounding curves for horizontally 

stratified media to interpret equatorial sounding curves. The Schlumberger theoretical curves cannot be used, however, to interpret the 

results of any other array (including dipole-dipole ones, except for 

the azimuthal dipole-dipole) even when the resistivity is plotted as a 

function of the effective spacing as defined by Al'pin (1950) or by 

Keller (1966). The effective spacings of other arrays only translate 

the sounding curve such that the inverse slope of the terminal righthand rising branch (indicating the detection of a highly resistive 

basement) would be equal to the total longitudinal conductance of 

the section above the basement. _

In the equatorial arrangement, the effective spacing, R, is related 

to the spacing R by

R=pR, 

where
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B13

Variation of the effective spacing factor, p, as function of (AB/2R) 

is given in figure 4 together with the variation of the factor Ae(l .

At the present time, it is customary in the United States and 

Canada to use the foot as a unit of distance and depth and the ohmmeter as a unit of resistivity. To facilitate the fieldwork with the 

equatorial arrangement, in tables 1, 2, and 3 the parameters R, 

AB/2, MN/2, and R are in feet, and K^ is in meters (the current / 

is to be measured in milliamps and AY in millivolts). In this way 

when the current / is made numerically equal to the appropriate 

value of Ke(l , the AV, measured in millivolts, will be numerically 

equal to the resistivity, in ohm-meters.

TABLE 1. Geometric factor of the equatorial array for variable AB/0 

[R, AB/2, MN/2, and R, in feet; KeQ , in meters. Modified from Berdichevskii and Petrovskii, 1956]

AB/2 MN/2 -X103

200

400

400

600

800

800

1,200

1,600

2,000

2,000

3,000

3,000

4,500

6,000

8,000

10,000

12,000

200

200

500

200

200

500

500

500

500

1,500

500

1,500

1,500

1,500

1,500

1,500

1,500

20

25

25

50

50

50

50

150

200

200

200

200

300

300

400

400

400

283

447

640

632

825

943

1,300

1,676

2,061

2,500

3,041

3,340

4,743

6,184

8,139

10, 112

12, 091

2.665

8.518

10. 026

12.11

26.8

16.02

41.79

30.19

42.09

24.89

134. 75

59.95

112.9

251.0

428.8

821.5

1,405

1.332

4.259

5.013

6.055

13.4

8.01

20.89

15,10

21.04

12.44

67.37

29.97

56.45

125.5

214.4

410. 75

702.5

0.8883

2.839

3.342

4.036

8.933

5.340

13.93

10.06

14.03

8.296

44.92

19.98

37.63

83.66

142.9

273.8

468.3

0. 6662

2.129

2.506

3.027

6.700

4.005

10.45

7.547

10.52

6.222

33.69

14.99

28.22

62.75

107.2

205.4

351.2

0.533

1.704

2.005

2.422

5.36

3.204

8.358

6.038

8.418

4.978

26.95

11.99

22.58

50.20

85.76

164.3

281.0

0.3807

1.217

1.432

1.730

3.829

2.289

5.970

4.313

6.013

3.556

19.25

8.564

16.13

35.86

61.26

117.4

200.7

TABLE 2. Geometric factor of the equa torial array for AB/0= 0,000

feet

[R, MN/2, and 5, in feet; K^, in meters]

2,000

4,000

6,000

8,000

10, 000

12. 000

13. 000

14. 000

16, 000

20, 000

MAT/2

200

250

500

500

500

600

600

600

800

1,000

2,830

4,470

6,320

8, 250

10, 198

12. 165

13. 153

14. 142

16, 124

20, 099

26. 65

85. 18

121. 1

269. 809

509. 507

720. 873

910. 593

1, 131. 441

1, 259. 031

1, 951. 020
B14 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

TABLE 3. Geometric factor of the equatorial array for 

4,000 feet

[R, in feet; value in parentheses, in miles. MN/2 and R, in feet. Ke ^, in meters]

MN/2 R .KeqXlO"

2, 640 (0.5)

3, 960 (0. 75)

5, 280 (1)

6, 600 (1. 25)

7, 920 (1. 5)

10, 560 (2)

10, 560 (2)

13, 200 (2.5)

15, 840 (3)

18, 480 (3.5)

21, 120 (4)

21, 120 (4)

26, 400 (5)

31, 680 (6)

200

200

300

400

400

400

600

600

1,000

1, 000

1,500

2,000

2, 000

2, 000

4,793

5,628

6, 624

7,717

8,873

11, 292

11, 292

13, 793

16, 337

18, 908

21, 494

21, 494

26, 701

31, 932

65.85

106. 75

116. 11

137. 85

209. 44

431. 52

288. 21

524. 73

524. 57

812. 25

798. 01

531. 92

1, 147. 35

1, 959. 69

L-SHAPED AZIMUTHAL ARRANGEMENT

In this second special azimuthal arrangement, the perpendicularity 

of MN to R is maintained as usual while the center 0 of the dipole 

MN is shifted along a line perpendicular to AB, at one of the current 

electrodes (fig. 6). Thus, the system acquires an L-shaped configuration, where

AB

cos e=-2R

and

rin^i-j

Hence from equation 8, the geometric factor Keil may be written in 

the form

where

and

(23)

(24)

The variation of ABL as a function of the ratio (fig. , 6) is

shown in figure 7 (together with the curves for AxL and AvL), where 

y is the distance along the perpendicular to one of the current 

electrodes.
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B15

FIGURE 6. The L-shaped azimuthal array. A. and B, current electrodes; Q, 

center of bipole; M and N, measuring electrodes; 0, center of dipole; R, 

distance between center of bipole and dipole; y, distance along the perpendicular between current electrode and center of dipole.

GEOMETRIC FACTOR OF THE PARALLEL ARRANGEMENT

In the parallel arrangement, the potential dipole MN is always 

kept parallel to the current bipole AB (fig. 8). At 0=90° the parallel 

arrangement reduces to the equatorial, and at 0=0° it reduces to the 

asymmetric-Schlumberger arrangement (at R<^AB/2) or to the polar 

bipole-dipole configuration (at R^>AB/2).

Following the same procedures discussed for the azimuthal arrangement, it can be easily shown that the geometric factor of the parallel

arrangement is

KX=KIAX, (25) 

where

K=ws (26)
B16 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

h/U

FIGURE 7. Nomogram for evaluating the factors An,, A xi,, and A VL of the Lshaped azimuthal, parallel, and perpendicular arrays, respectively, y, distance 

along the perpendicular between current electrode and center of dipole; AB, 

distance between current electrodes; R, distance between centers of bipole and 

dipole; MN, distance between measuring electrodes. (Kn. K*Aei.', KXi K*A xLl;

and

fiMH) IT

The expression for Ax becomes negative if cos d^>AB/2R and the 

second term is larger than the first. These negative values of Ax result 

from a reversal in the direction of the electric field component Ex with 

respect to MN (fig. 8), which leads to the calculation of a negative 

resistivity.

A nomogram for the evaluation of the factor Ax is shown in figure 

9. The value of Ax becomes infinite along a given contour which 

represents the locus of points (AB/2R, 0) where the dipole MN is 

tangent to an equipotential line. Mathematically, the values of Ax 

to the right of this contour are positive and those to the left are 

negative. The values of Ax tend to + °° when the infinity contour is 

approached from the right and to °° when it is approached from 

the left.
GEOMETRIC FACTORS OF BIPOLE-DIPQLE ARRAYS B17

FIGURE 8. Bipole-dipole parallel array. A and B, current electrodes; Q, center 

of bipole; M and N, measuring electrodes; 0, center of dipole; E, point-source 

component of the electric field; EXt parallel component of the electric field.

In contrast to the dipole-dipole parallel array, where the absolute 

value of the geometric factor becomes infinite only at 0=54°44' 

(Al'pin, 1950), in the bipole-dipole array the absolute value of the 

geometric factor increases indefinitely at different values of the 

angle 6 depending on the value of AB/2R. At AB/2R<0.1 the bipoledipole becomes an approximate dipole-dipole, and the geometric 

factor tends to ± » at 0^54°44' (or at arctan 6^-^/2).

The procedure of evaluating Ax from the given nomogram (fig. 9) 

is identical with that of the azimuthal arrangement. The part of 

this nomogram for AB/2R^>1 is especially valuable in computing the 

geometric factor Kx for measurements made with the "rectangle of 

resistivity" method (Breusse and Astier, 1961).

SPECIAL PARALLEL ARRANGEMENTS

The parallel arrangement coincides with the azimuthal arrangement 

at 0=90°, and the general expression for Ax (eq 27) reduces to that 

of the equatorial configuration (eq 15). The L-shaped parallel arrangement may be of interest in exploration, but the polar and asymmetricSchlumberger configurations are more widely used. In the following, 

each of these special arrays is briefly discussed.
B18 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

O.I

10 20 30 40 50 60 

9, IN DEGREES

70

FIGURE 9. Nomogram for evaluating the factor A x of the parallel array. Values 

for A x given on curves. AB, distance between current electrodes; R, distance 

between centers of bipole and dipole; 0, angle between current electrode, 

center of bipole, and center of dipole; MN, distance between measuring 

electrodes. (KX =K*AX ; K* = R*/MN.)
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B19

L-SHAPED PARALLEL ARRANGEMENT

In this array, the parallelism of MN to AB is maintained as the 

point of measurement, 0, is moved along a line perpendicular to AB 

at one of the current electrodes (fig. 10). Therefore, cos d=AB/2R 

and the source at .A does not contribute to the electric field component 

along the direction of MN. Using equations 25, 26, and 27, the 

geometric factor Kxtl can be written as

where 

and

K __ jr*

._ R2 X ~MN

13/2

1/2

(28)

(29)

AB\*

1 + 3-

3/2

(30)

FIGURE 10. The L-shaped parallel array. A and B, current electrodes; Q, 

center bipole; M and N, measuring electrodes; 0, center of dipole; R, distance between centers of bipole and dipole; y, distance along the perpendicular between current electrode and center of dipole.
B20 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

where y is the perpendicular distance AO (fig. 10). The factor AxLl 

may be easily evaluated by using the curve of AxLl=f( . ^ . ) shown 

in figure 7.

POLAR AND ASYMMETRIC-SCHLUMBERGER ARRANGEMENTS

In these arrangements, the angle 6 is always equal to zero, and 

hence the measuring dipole is always in line with the current bipole 

(fig 11). Consequently, cos 0=1, and the expression for the geometric 

factor Kx (eq 25) is reduced to

*»-asym,p "- ^ asym.pi

where

P2 JT* ~MN ll

and

or

f MS )rr

-aaym.p _^_( AB . 1 V^Ai ^ V

The equation has four roots given by

A _t

A 0 B MON

' R '

R

FIGURE 11. Polar (upper) and asymmetric-Schlumberger (lower) arrays. A and 

B, current electrodes; Q, center of bipole; M and N, measuring electrodes; 

0, center of dipole; R, distance between centers of bipole and dipole.
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS

for (AB/2R)>1, and

B21

for (AB/2R)<1.

(34)

Equation 33 is used for computing the factor Aasym of the asymmetric Schlumberger arrangement (AB/2E>1). The positive root is 

for the AMNB arrangement, and the negative root is for the ANMB 

arrangement. Similarly equation 34 is used for computing the factor 

Ap for the polar arrangement (AB/2R<O), using the negative root for 

ABMN and the positive root for ABNM. In practice, the absolute 

values of equations 33 and 34 can be used (fig. 12) to avoid the calculation of a negative resistivity, but the distinction between the role 

of each equation must be emphasized for calculating the correct 

geometric factors.

The expressions for Ap and A&aym both vanish at AB(2R=1, and it is 

of interest to determine the geometric factors Kv and -K"aSym at ABf2R=

Asymmetric-Schlumberger array

Polar 

s

array.

\

Figure 12. Nomogram for evaluating the factors A &aym and A p of the asymmetric-Schlumberger and polar arrays, respectively (K* = RZ/MN). AB, 

distance between current electrodes; R, distance between centers of bipole and 

dipole; MN, distance between measuring electrodes.
B22 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

1. Using the condition BO>10MN, where B is the nearest current 

electrode to the center 0 of MN, it can easily be shown that for the 

polar arrangement

or

Therefore, if, for example, BO 1QQMN, then

R " ~"~0-£) <£)

AtAB(2R= 1, -Kp=Jj. Therefore, using L'Hopital's rule

« p -«^ 2R/ 

AB P AS d 

2.R ~* 2fl ~*

Similarly, by using the same type of analysis it can also be shown 

that KMym Q at AB/2R=l. Therefore both Kp and Kasym become 

identically zero at AB/2.K=1. The general geometric factor Kx of 

the parallel arrangement may, however, become infinite if the angle 

07*0 and AB/2R-+1. (See fig. 9.)

GEOMETRIC FACTOR OF THE PERPENDICULAR 

ARRANGEMENT

In the perpendicular arrangement the direction of MN is always 

maintained at right angles to AB (fig. 13). The analysis for evaluating 

the geometric factor Ky is similar to that of the azimuthal configuration. Hence it can be shown that

(35)

where

/?2 K>m (36)

and
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B23

FIGURE 13. Bipole-dipole perpendicular array. A and B, current electrodes; 

Q, center of bipole; M and N, measuring electrodes; 0, center of dipole; 

E, point-source component of the electric field; Ey , perpendicular component of the bipole electric field.

The nomogram for the evaluation of the factor Ay is given in 

figure 14. From equation 37, Ay becomes infinite at 0=0 and ir/2, which 

indicates that at these angles, MN is alined along an equipotential 

line. The perpendicular arrangement is most useful in the L-shaped 

configuration.

L-SHAPED PERPENDICULAR ARRANGEMENT

The perpendicularity of MN with respect to AB is maintained, and 

the measurements are carried out along a line perpendicular to one 

of the current electrodes (fig. 15). Therefore

cos 9=

and

AB 

2R

sin 9-

^B\2

(38)

(39)

Therefore, using equation 37, the geometric factor Ky can be written

(40)

(41)

as 

where

R2

MN
B24 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

40 50 60 

0, IN DEGREES

70 80 90

FIGURE 14. Nomogram for evaluating the factor A v of the perpendicular array. 

Values for A v given on curves. AB, distance between current electrodes; R, 

distance between centers of bipole and dipole; 9, angle between current electrode, center of bipole, and center of dipole. (K U K* A v ; K* = RZ/MN).

and

27T
GEOMETRIC FACTORS OF BIPOLE-DIPOLE ARRAYS B25

N3 n

T. 0

R

0 B

FIGURE 15. The L-shaped perpendicular array. A and B, current electrodes; 

Q, center of bipole; M and N, measuring electrodes; 0, center of dipole; R, 

distance between centers of bipole and dipole; y, distance along perpendicular between current electrode and center of dipole.

Substituting

in equation 42, the value of AyLl may be plotted as a function of

(fig. 15) for

( A Jt/9 )

as snown m

CONCLUSIONS

Formulas for the geometric factors of bipole-dipole arrays are fairly 

complex, making routine computations very tedious. Most of the 

computations must be carried out to five significant figures for the 

final results to be of practical value. Consequently the computations 

should be made by an electronic computer. The results of such computations, presented here in the form of nomograms, reduce the necessary calculations considerably. The values of the various geometric 

factors, determined with the aid of the given nomograms, are generally
B26 TECHNIQUES IN DIRECT-CURRENT RESISTIVITY EXPLORATION

sufficiently accurate for evaluating field resistivity data, using any of 

the considered electrode configurations.

REFERENCES CITED

Alekseev, A. M., Berdichevskii, M. N., and Zagarmistr, A. M., 1957, [The use of 

new methods of electrical exploration in Siberia]: Prikladnaya Geofizika, 

no. 18, p. 103-127 (in Russian); English translation in Applied Geophysics, 

U.S.S.R., 1962, Pergamon Press, p. 196-222.

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.

Anderson, L. A., and Keller, G. V., 1966, Experimental deep resistivity probes in 

the central and eastern United States: Geophysics, v. 31, p. 1105-1122.

Berdichevskii, M. N., 1954, [Nomograms for determining the geometric factor 

for an equatorial array]: Razved. i Promyslovaya Geofizika, v. 10 (in 

Russian).

 1957, [The method of curved electrical probes]: Prikladnaya Geofizika, 

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U.S.S.R., 1962, Pergamon Press, p. 223-240.

Berdichevskii, M. N., and Petrovskii, A. D., 1956, [Procedures of bilateral equatorial soundings]: Prikladnaya Geofizika, no. 14, p. 97-114 (in Russian).

Berdichevskii, M. N., and Zagarmistr, A. M., 1958, [Methods of interpreting dipole 

resistivity soundings]: Prikladnaya Geofizika, no. 19, p. 57-108 (in Russian); 

English translation in Keller, G. V., 1966, Dipole methods for measuring 

earth conductivity, New York, Consultants Bur., p. 79-113.

Bordovskii, V. P., 1958, [The determination of the coefficients of dipole arrangements in the case of curvilinear soundings]: Razved. i Promyslovaya 

Geofizika, v. 24, p. 24-27 (in Russian).

Breusse, J. J., and Astier, J. L., 1961, Etude des diapirs en Alsace et BadenWurtemberg par la methode du rectangle de resistivite: Geophys. Prosp., v. 9, 

p. 444-458.

Cantwell, T., Nelson, P., Webb, J., and Orange, A. S., 1965, Deep resistivity 

measurements in the Pacific Northwest: Jour. Geophys. Research, v. 70, 

p. 1931-1937.

Cantwell, T., and Orange, A. S., 1965, Further deep resistivity measurements in 

the Pacific Northwest: Jour. Geophys. Research, v. 70, p. 4068-4072.

Deppermann, K., 1954, Die Abhangigkeit des scheinbaren Widerstandes vom 

Sondenabstand bei der Vierpunkt-Methode: Geophys. Prosp., v. 2, p. 

262-273.

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

Jackson, D. B., 1966, Deep resistivity probes in the southwestern United States: 

Geophysics, v. 31, p. 1123-1144.

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

p. 1088-1104.

Yakoubovskii, U. V., and Lyakhov, L. L., 1964, Elektrorazvedka [Electrical 

prospecting]: Moscow, Izdatel'stvo "NEDRA" (publisher) [Mineral resources], 414 p.

Zohdy, A. A. R., 1964, Earth resistivity and seismic refraction investigations in 

Santa Clara County, California: Stanford Univ. Ph. D. dissert. 132 p.

 1966, Geoelectrical exploration for ground water in the southwestern

United States [abs.]: Geophysics, v. 31, p. 1216. 

Zohdy, A. A. R., and Jackson, D. B., 1966, Application of resistivity soundings

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