Magnetic-Doublet Theory in the 

Analysis of Total-Intensity 

Anomalies

GEOLOGICAL SURVEY BULLETIN 1052-D

Magnetic-Doublet Theory in the 

Analysis of Total-Intensity 

Anomalies

By ROLAND G. HENDERSON and ISIDORE ZIETZ

EXPERIMENTAL AND THEORETICAL GEOPHYSICS

GEOLOGICAL SURVEY BULLETIN 1052-D

Discusses typical magnetic-doublet profiles and their use in the interpretation 

of aeromagnetic maps

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1958
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D. C. - Price 15 cents (paper cover)
CONTENTS

Page

Abstract__________________________________________._ _____ 159

Introduction. ________________________________________ _____ 159

Limitations of magnetic interpretations____________ 159

Fundamentals of depth estimation___________________ _ _ 160

Purpose and scope of investigation_________________ _ 160

Acknowledgments_______________________________________. 161

Theoretical doublet anomalies_______________________________ 161

Formulation of interpretation factors___________________________ 161

Analysis of anomalies________________________________________ 172

Other uses of computed anomalies____________________________ 173

Magnetic model experiments__-_______-____-_-___________-_-_______- 174

Tests on cylindrical models____________________________________ 174

Magnetic double-layer models____-___-_--_--_________-______-___ 176

Practical applications.____________________________________________ 179

Southeast Missouri__________________________________________ 179

Indiana____________________________________________________ 181

Ontario, Canada____________________________________________ 183

Adirondack Mountains, N. Y______--_-____-______-___-__-____ 184

Conclusions______________ _________________________-_-__________ 185

Literature cited_________________________________________________ 185

ILLUSTRATIONS

Page

FIGURE 27. Total-intensity anomaly of inclined doublet; coordinate

system and meaning of/SandS____________________ 162

28. Total-intensity anomalies of magnetic doublet of length,

Z=0.01 depth unit________________________ 163

29. Total-intensity anomalies of magnetic doublet of length,

Z=0.1 depth unit__________________________ 164

30. Total-intensity anomalies of magnetic doublet of length,

1 = 0.5 depth unit_________________________ 165

31. Total-intensity anomalies of magnetic doublet of length,

Z=1.0 depth unit__________________________ 165

32. Total-intensity anomalies of magnetic doublet of length,

1=1.5 depth units__________________________ 166

33. Total-intensity anomalies of magnetic doublet of length,

1=2 depth units__________________________ 167

34. Total-intensity anomalies of magnetic doublet of length,

1=5 depth units__________________________________ 168

35. Total-intensity anomalies of doublets showing effect of

increasing length, I____________________________________ 169

36. Total-intensity anomaly of inclined cylinder; obtained from

model experiment,__________________________________ 172

in
IV CONTENTS

Page 

^ST. Total-intensity anomaly of magnetized plate; obtained from

model experiment_______________________ 176

38. Anomalies of inclined cylinders showing effect of increasing

diameter_________________ ___ _________ 178

39. Total-intensity aeromagnetic map of part of Coldwater

quadrangle, Missouri.________ __ 180

40. Total-intensity aeromagnetic map of area near Bryant, Jay

County, Ind____________________________ 181

41. Total-intensity aeromagnetic map of T. 34 N., R. 3 W.,

La Porte County, Ind______________________ 182

42. Total-intensity aeromagnetic map of area near Marmora,

Ontario, Canada_________ ___ _________ 183

43. Total-intensity aeromagnetic map of northern part of Stark

quadrangle, New York__ 184

TABLES

Page 

TABLE 1. Comparison of numerical values for two different definitions

of ft________________________________ 166

2. Parameters of magnetic doublets for inclinations, I, and

length-to-depth ratios, lid________________________ 170

3. Comparison of calculated doublet lengths and depths with

those used in model experiments___________________ 175

4. Application of magnetic-doublet theory to field of horizontal slab___________________________ 177
EXPERIMENTAL AND THEORETICAL GEOPHYSICS

MAGNETIC-DOUBLET THEORY IN THE ANALYSIS OF 

TOTAL-INTENSITY ANOMALIES

By ROLAND G. HENDERSON and ISIDORE ZIETZ

ABSTRACT

Analysis of horizontal- or vertical-intensity magnetic anomalies in terms of 

equivalent magnetic doublets has been extensively used for rough estimates of 

depth and depth extent of disturbing rocks. In this investigation, factors have 

been theoretically determined by which appropriate total-intensity anomalies 

can also be analyzed in terms of their magnetic-doublet equivalents. The effectiveness and limitations of the method have been checked by application to 

anomalies derived from model experiments and to observed anomalies. For 

satisfactory results the effective radius of the disturbing body must be less than 

its depth of burial. The calculation of doublet length is not so reliable as the 

calculation of depth; however, the former is better at low magnetic latitudes.

INTRODUCTION

LIMITATIONS OF MAGNETIC INTERPRETATIONS

^ The results of aeromagnetic surveys are usually presented in the 

form of contour maps showing lines of equal total magnetic intensity. 

The anomalies of interest to exploration geophysicists are departures 

from smooth regularity, for they are indicative of inhomogeneities 

in the magnetism of the earth's crust. The relation of anomalies to 

subsurface geologic structure and to ore bodies is usually uncertain. 

For this reason interpretations of magnetic anomalies are often 

qualitative in nature, the map being discussed in terms of "grain" 

of the anomalies and generalization about likely contrasts in magnetic 

properties of the basement rocks from which the anomalies probably 

arise. In general, broad magnetic gradients several tens of miles in 

linear section represent regional variations in magnetization of the 

crust; anomalies 10 or 20 miles in lateral extent are associated with 

more abrupt changes in the magnetic character of the basement rocks; 

and areally small, high-gradient anomalies are indicative of nearsurface concentrations of ferromagnetic materials.

159
160 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

f The quantitative interpretation of aeromagnetic maps is at once 

a complicated and challenging area of geophysical investigation. The 

complications stem largely from the presence of induced and permanent 

(sometimes called remanent) magnetism in the disturbing body, from 

lack of knowledge of the magnetic and geometric parameters of the 

body, and from an inherent ambiguity affecting interpretation of 

potential fields. The ambiguity arises from the fact that there are 

many distributions of magnetic material at various depths which can 

produce a given anomaly. Supplementary geological and geophysical 

information is therefore necessary for unequivocal solutions.

FUNDAMENTALS OF DEPTH ESTIMATION

Useful estimates of depths to disturbing bodies are often possible 

from magnetic anomalies despite these limitations. The relative 

positions of the maximums, minimums, and inflection points of an 

anomaly (the so-called shape factors) are functions of the depth and 

are affected only in a limited way by the remaining geometric parameters of the configuration of the body. These characteristic shape 

factors are unaffected by blockwise lateral changes in magnetic susceptibility of the basement or by the magnitude of the remanent 

magnetism when the latter is in the direction of the induced magnetism. Most methods of estimating depth are based on quasi-empirical 

facts deduced from studies of the shape characteristics of theoretical 

anomalies. Induction theory is used almost exclusively. It is 

assumed that the magnetization is the product of the susceptibility 

contrast and the normal field strength, and that the direction of 

magnetization is that of the normal field.

Since the advent of the airborne magnetometer, there have been 

several papers on methods of depth analysis. In the method outlined by Vacquier and others (1951), the shape characteristics of an 

observed field are compared with those of prismatic-model fields, the 

comparison being facilitated by the use of second derivatives of the 

anomaly which bring the shape characteristics into sharper focus. 

The method has been used successfully in depth analyses of many 

broad, large-amplitude anomalies (Zietz and Henderson, 1956; Henderson and Zietz, 1958).

PURPOSE AND SCOPE OF INVESTIGATION

Where the breadth of the anomaly is comparable to the depth of 

the source, the Vacquier method is less effective because the shape 

characteristics become more sensitive to parameters of the geometry 

other than the depth. Analyses of such anomalies can be made in 

terms of equivalent magnetic doublets. By equivalent magnetic 

doublet we mean two magnetic poles of equal strength and opposite
MAGNETIC-DOUBLET THEORY 161

sign, whose depths below the surface and mutual displacement in the 

direction of the earth's normal field are sufficient to account for the 

shape characteristics of the anomaly. In standard texts on geophysics, 

such as Heiland (1940), Jakosky (1950), and Nettleton (1940), magnetic doublets are discussed in relation to the anomaly in the vertical 

component (AZ) and the anomaly in the horizontal component (Afl), 

but no general treatment for depth calculations is given. Vestine and 

Davids (1945) presented an ingenious method for analyzing AZ in 

terms of magnetic doublets.

In airborne magnetometry, however, we are interested in the 

component of the total-intensity anomaly in the direction of the total 

field vector. It is a linear combination of AZ and AH and existing 

theory of the interpretation of doublets is not in general applicable. 

The work reported here was undertaken to establish a set of model 

anomaly curves for AT doublets and to deduce from them factors 

that are diagnostic of the doublet parameters. Major interest is 

centered in the calculation of depth to the upper (or nearer) pole; 

however, the more intractable problem of doublet length is also considered. There is some latitude in the choice of specific portions of 

an anomaly profile for formulating the factors. In every case the 

choice was dictated by considerations of usefulness in practical applications. The method has been tested by application to laboratory 

magnetic models, and its application to practical anomalies is demonstrated.

ACKNOWLEDGMENTS

We acknowledge our indebtedness to Mr. L. K. Alldredge, formerly 

of the Electricity and Magnetism Division of the Physics Research 

Department of the Naval Ordnance Laboratory, for the use of the 

magnetic testing equipment, and to Messrs. C. L. Parsons and K. E. 

Dornstreich, also of the Naval Ordnance Laboratory, for valuable 

technical assistance.

THEORETICAL DOUBLET ANOMALIES 

FORMULATION OF INTERPRETATION FACTORS

In this study it is tacitly assumed that the doublet is produced only 

by induction in the earth's magnetic field, and that the north dip of 

the doublet is equal to the magnetic inclination. A right-handed 

system of coordinates is adopted (fig. 27) with origin at 0, x positive 

to magnetic north, y positive to the east, and s positive vertically 

downwards. The components of the magnetic field, AX, AY, and 

AZ, are positive along the respective coordinate axes. The doublet of 

strength m and length I lies in the xz plane; the coordinates of the 

pole nearer the origin are (0, 0, d), and those of the remote pole are 

(I cos I, 0, d + 1 sin /).
162 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

AT/m cgs units 

T 0.9OO

FIOTIBE 27. Total-intensity anomaly of inclined doublet; coordinate system and meaning of 0 and S.

The general expression for the anomaly at any point P : (a;, y, z) is 

given by 

___ (x l cos /) __

[(I cos I-x)*+y*+(d+l sin I-z)2]5"2 

x

l-z) ______ \sin/ 

(d+l sin /-s)2]3«/sm J [(I cos I-x)*+if+(d+l

where it has been assumed that the magnetic inclination is constant 

over the area. If y=z 0 and m=l, and all linear measures are 

expressed in terms of d 1, a profile along the x axis is given by
MAGNETIC-DOUBLET THEORY

(x l cos 7) x

[(I cos 7-a;) 2+(l+J sin 7)2]3/2 

1 1+Zsin/

3/2

[(I cos 7-a:) 2+ (l+l sin 7) 2]3'2

cos 7+ 

sin7

163

(1)

From formula (1), A!T(aO was computed for doublets with parameters 

Z/d=0.01, 0.10, 0.50, 1.00, 1.50, 2.00, 5.00; and inclinations, 7=0°,

30°, 45°, 60°, 75°, 90°. These are shown in figures 28 to 34.

Doublet length U0.01 

South pole at z=0, depth <Z=°1 

/'inclination of doublet and normal field

FIOUEE 28. Total-intensity anomalies of magnetic doublet of length, 1=0.01 depth unit.

From an empirical study of theoretical doublet-anomaly curves of 

this type, it is possible to determine factors, herein designated B and 

5, which are useful in estimating I and d from observed anomalies. 

The dimensionless factor 8 involves the ratio between certain horizontal distances subtended by the curve, and d is a horizontal distance, usually between half-maximum abscissas or between nearly 

inflectional points. In figure 27, |8=(a;/o.i aw)/(aw EO.S) and 

5=«'o^ £0.5 where y^^, x0.l} x0£) XQ^ are the abscissas of A!Tmax, 0.1

0.5 A!TmiU[, and 0.8 &TmM) respectively. Prime marks refer to 

the more northerly of two points.

461011 58 2
164 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

flT/m 

cgs units

Doublet length, £=0.1

South pole at z=0, depth, <Z = ^

/= inclination of doublet and normol field!

-L -O.IZO

FiGUEE 29. Total-intensity anomalies of magnetic doublet of length, 1=0.1 depth unit.

Practical considerations require that j8 be determined from portions 

of the curve that are easily identifiable and, at most, only slightly 

affected by nearby disturbances. Peripheral features should be 

avoided as far as possible. Moreover, /3 and 5 must be so chosen that 

there will be resolution between successive values of l/d. To satisfy 

these requirements, /3 and 5 are necessarily different for each inclination. The selection of abscissas is facilitated if the vertical scale of 

each of the family of AT(a?) curves is adjusted so that all maximums 

have the same amplitude, as is shown in figure 35 for 7=75°. If in 

this figure, /3 were chosen in two different ways, say ft = (x0m5 a?max)/ 

fonax s'o.s) and /32 = (z'o.i zmax)/(a;max ZO.B), the tabulation 

would be that shown in table 1.
MAGNETIC-DOUBLET THEORY 165

Doublet length, i =0.5 

South pole ot z=0, depth d» I 

-0200 / .inclination of doublet ond normal field

FIGURE 30. Total-intensity anomalies of magnetic doublet of length, 1=0.5 depth unit.

Doublet length,

South pole at x =0, depth, d =1

/ -inclination of doublet and normal field

FIGTJKE 31. Total-mtensity anomalies of magnetic doublet of length, 1=1.0 depth unit.
166 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

4T/m T0900 

cgs units I

I. rb.eoo

-azo° Doublet length f»l. 5

South pole at £=0, depthd=l 

/^inclination of doublet and normal field

FIGUEEJJ2. Total-intensity anomalies of magnetic doublet of length, 1=1.5 depth units.

TABLE 1. Comparison of numerical values for two different definitions of /8

Ud

ft ... ft...._-

0.01

1.24

2.52

0.10

1.23

2.68

0.50

1.22

2.79

1.00

1.17

3.00

1.50

1.14

3.19

2.00

1.12

3.21

5.00

1.08

3.59
MAGNETIC-DOUBLET THEORY 167

flT/m

cgs units /

Doublet length, 1-2

South pole ot x = 0, depth d«l "V

/ 'inclination of doublet and normol fiela!

FIGUBE 33. Total-intensity anomalies of magnetic doublet of length, J=2 depth units.

The incremental changes in ft are too small to permit us adequately 

to distinguish among the different terms l/d, but those of |82 are 

satisfactory for this purpose. If the choice were /3 = (xmto - aw)/ 

(aw »o.s)} the increments in /3 would be larger, but this gain would 

be offset by uncertainties in the location of the minimum. Similar 

studies were made of families of theoretical &T(x) profiles for all the 

remaining inclinations. Abscissas yielding satisfactory values of /3 

and 5 are given in table 2, together with magnitudes of maximum and 

minimum values of &T(x) and their respective coordinates.
AT/m 

cos un

FIGURE 34. Total-intensity anomalies of magnetic doublet of 
MAGNETIC-DOUBLET THEORY 169

.1
170 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

TABLE 2. Parameters of magnetic doublets for inclinations, I, and lengthJo-deplh

ratios, l/d

{The positions xt.i, xo.i, 29.3 ... indicate, respectively, abscissas of 10 percent, 20 percent, 30 percent . . . 

of the maximum value, A Tm»x, of a given profile. Prime marks refer to more northerly abscissas. Upper 

pole is at x =0, d=l. Pole strength, m=l]

l/d ATmaz ATmto Xmti Zmin 0 s

Inclination 0°

0.01

. 10

.50

1.00

1.50

2.00

0.002

.020

.098

. 180

.235

.274

-0.010

-.098

-.452

-.706

-.766

-.707

-1.20

-1.20

-1.00

-.90

-.80

-.80

0.01

.05

.25

.50

.75

1.00

3.16

3.20

2.91

2.80

2.46

2.14

0.78

.78

.83

1.02

1.28

1.67

Inclination 30°

5=z'o.i-a;o.iJ

0.01

.10

.50

1.00

1.50

2.00

5.00

0.008

.076

.290

.440

.516

.564

.649

-0.008

-.070

-.252

-.347

-.332

-.306

-.185

-0.60

-.58

-.53

-.46

-.43

-.40

-.35

0.35

.43

.60

.82

1.02

1.15

1.32

1.27

1.31

1.40

1.45

1.61

1.67

1.72

2.67

2.77

2.95

3.16

3.39

3.45

3.90

Inclination 45°

0.01

.10

.50

1.00

1.50

2.00

5.00

0.013

. Ill

.392

.440

.638

.690

.780

-0.005

-.046

-. 154

-. 192

-. 192

-. 178

-. 104

-0.40

-.35

-.35

-.35

-.35

-.35

-.35

0.58

.65

.85

1. 10

1.30

1.50

1.80

1.60

1.69

2.00

2.38

2.66

2.94

3.77

1.71

1.84

1.94

2. 16

2.30

2.40

2.66
MAGNETIC-DOUBLET THEORY 171

TABLE 2. Parameters of magnetic doublets for inclinations, I, and length-to-depth

ratios, lid Continued

lid ATm« ATmin Zmu Xmin ft s

Inclination 60°

1/9= (io.i0.01

0.10

0.50

1.00

1.50

2.00

5.00

0.017

.142

.480

.662

.750

.801

.882

-0. 003

-.026

-.081

-.098

-.096

-.089

-.050

-0.24

-.26

-.22

-.20

-.19

-.19

-.18

0.85

1.00

1.20

1.40

1.68

1.80

2.50

1.65

2.12

2.14

2.32

2.42

2.58

2.79

1.05

1.06

1.19

1.29

1.36

1.40

1.48

Inclination 75°

0.01

.10

.50

1.00

1.50

2.00

5.00

0.018

. 163

.540

.730

.817

.865

.950

-0.001

-.011

-.033

-.040

-.038

-.043

-.021

-0. 12

-.13

-. 11

-.10

-.10

-.09

-.09

1.20

1.40

1.60

2.00

2.30

2.50

3.50

2.52

2.68

2.79

3.00

3.19

3.21

3.59

1.01

1.05

1. 19

1.30

1.37

1.41

1.50

Inclination 90°

[|S=z'o.05 Z0.5. 8=20.5]

0.01

.10

.50

1.00

1.50

2.00

5.00

0.020

.174

.556

.750

.840

.889

.938

0.000

-.003

-.019

-.011

-.011

-.010

-.005

0

0

0

0

0

0

0

1.90

2.00

2.50

3.00

3.50

3.50

5.70

2.22

2.25

2.25

2.38

2.46

2.57

2.91

0.51

.52

.60

.64

.68

.70

.75
172 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

ANALYSIS OF ANOMALIES

To determine the magnetic-doublet equivalent of a suitable observed 

anomaly, a north-south profile is drawn through the maximum at any 

convenient scale. The zero datum line is drawn by inspection and is 

based on the mean undisturbed magnetic level. The magnetic inclination for the area is determined from isoclinal charts published by 

the U. S. Coast and Geodetic Survey (Deel and Howe, 1948) or from 

tables by Vestine and others (1947). The abscissas to be used in 

computing j8 are taken from table 2, for the inclination nearest to that 

observed. If the computed value of /3 falls within the range of those 

listed in the table, there is reason to believe that a doublet representation by the method described here is possible. Table 2 is entered for 

this value of /3 and a quantity lid and a depth factor 5 are determined 

by interpolation. The depth, d, to the near pole is calculated by 

dividing the corresponding horizontal distance on the observed profile 

by 5. The doublet length is then obtained from the product of d 

and l/d.

The calculation is best explained by an example. Consider the 

total-intensity profile over the inclined cylinder shown in figure 36, 

which was obtained from experiments on models. As the inclination

Abscissas indicated are used in 

determining equivalent doublet

= 10 I inches

FIGURE 36.- Total-intensity anomaly of inclined cylinder; obtained from model experiment.
MAGNETIC-DOUBLET THEORY 173

is 75°, the abscissas to be used in computing /3 and 5 are ormax, a?Yi> 

#o.5> s'0.5 and #0.8, as indicated by table 2. The maximum value of AT 

is 35071 therefore the x coordinates for 357, 1757, 280y, and 3507 are 

plotted as shown. Then f} (x'o.i xm&^)l(xm&x XQ.&) =3.24. Entering table 2 for 0=3.24 we obtain by interpolation an l/d of 2.2 and a 5 

of 1.42 inches. As x'0 .5 BO.5=7.30 inches on the profile, the depth is

dc=(a;V5 _ar0 .5)/5==(7.30)/(1.42)=5.1 inches. The length is le= 

(l/d)dc= (2.2) (5.1) =11.2 inches. The depth and length are known to 

be d0=5.Q5 inches and Z 0=10.1 inches from model experiments. The 

results are unusually good even for a carefully controlled laboratory 

experiment.

OTHER USES OF COMPUTED ANOMALIES

Although figures 28 to 34 were computed primarily for determining 

the 0 and 5 factors, they can be used for qualitative interpretations. 

For example, they provide information on the general shape of totalintensity anomalies to be expected from different rock masses magnetized by induction. When the horizontal dimensions of the mass 

are less than the depth to its top, a single doublet can be used to represent it. When the mass has a width less than the depth, but is extensive in the plane of the magnetic meridian, the anomalies of several 

doublets alined in parallel in the meridian can be combined to give an 

equivalent anomaly. This simple addition of two or more anomalies 

is permissible by the superposition law of potential theory.

The following generalizations based on figures 28 to 35 are helpful 

in interpretations:

1. The anomaly of the vertical doublet is symmetric with respect 

to the axis.

2. For inclined doublets, the maximum of an anomaly is always 

south of the upper pole, and the amount of the south shift increases 

with decrease in the inclination.

3. For a given depth and doublet length, a decrease in the inclination diminishes the amplitude of the maximum and increases that of 

the minimum. The effect on the steepest gradients is practically 

negligible, and the north slopes of the profile remain parallel for all 

inclinations. The latter fact suggests that fitting a slope is meaningless, because the diagnostic characteristics of the anomaly are involved 

in the south slope and minimum as well.

4. If other parameters are equal, an increase in doublet length 

decreases the steepest gradients of the anomaly, as can be seen explicitly in figure 25.

5. For an inclination 7=0, a double minimum develops when I 

is about three times the depth, which indicates resolution of the 

effects of the two poles.
174 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

6. For I very large, the theory reduces to that of a pointjpole, 

which we have already treated (Henderson and Zietz, 1948).

MAGNETIC MODEL, EXPERIMENTS

To investigate the method of study when applied to fields produced 

by physically realizable bodies, a series of experiments on magnetic 

models was carried out. Facilities of the Naval Ordnance Laboratory 

at White Oak, Md., were made available for these and other tests. A 

detailed account of the experiments is contained in a preliminary 

report by Zietz and Henderson (1956).

The experimental set up is an elaborate modification of that used 

by Alldredge and Dichtel (1949) in their interpretation of the Bikini 

magnetic data. Three large mutually perpendicular Helmholtz-type 

coils on 30-foot frames are individually energized by an adjustable 

d-c voltage supply. In this way, the magnitude and direction of the 

uniform field of the earth can be simulated over a relatively large 

volume about the center of the system. A model is placed at the 

center and the anomaly field produced by the induced magnetism of 

the model is measured by means of a flux-gate magnetic detector of 

fixed orientation on a tower having powered vertical positioning. 

The detector tower is mounted on a carriage which moves in an eastwest or a north-south direction. Continuous registration of data is 

accomplished with a null-type recorder. Once the model has been 

set up and the detector oriented in the direction of the simulated field, 

all operations are conveniently effected from control panels mounted 

in a console. The accuracy of the magnetometer system is ±87.

The models were prepared from a mixture of gypsum plaster and 

powdered magnetite in the ratio of 2:1 by volume. For this study, 

the models were primarily cylinders; however, a few double-layer 

models 0.5 inch thick were also used. All remanent magnetism in 

the models was carefully removed by electrical "shaking" in a zero 

ambient field. The cylinders and the detector were alined codirectionally with the field at the center of the Helmholtz system. The applied 

total field was invariably 53,400y.

TESTS ON CYLINDRICAL MODELS

A profile of a typical AT anomaly over a cylinder is shown in figure 

36 together with data for computation. Similar profiles for 5 cylinders 

in fields of inclinations 7=0°, 75°, and 90° were also analyzed and 

the results are given in table 3. The measured length of the cylinder 

and depth to the center of the base nearer the origin are in columns 

headed "10" and "d0" and the corresponding computed lengths and
MAGNETIC-JXHJBLET THEQKY 175

depths are in columns "I" and "df." In general the calculated depths 

are in better agreement with those measured than are the calculated 

lengths.

TABLE 3. Comparison of calculated doublet lengths and depths with those used in

model experiments

[Dip of cylinders equal to inclination of impressed field. Linear dimensions in inches]

IflfiHTlfl»

tion

0°

30°

75°

90°

Cylinder 

No.

I

II

III

IV

V

I

II

III

IV

V

I

III

IV

V

I

II

III

V

VI

Diameter

2. 15

2. 15

2. 15

1.00

1.00

405

4.05

1.60

1.60

1.50

1.50

2. 15

2.15

1.00

4.05

1.60

1.50

2.15

2. 15

4.05

1.60

1.60

1.50

2.15

1.00

4.05

4.05

1.50

5.85

Observed values

I.

5.00

5.00

5.00

5. 10

5. 10

10.00

10.00

10. 10

10.10

18. 00

18.00

5.00

5.00

5.10

10.00

10. 10

18.00

5.00

5.00

10.00

10.10

10.10

18.00

5.00

5. 10

10.10

10.10

18.00

9.90

d.

4.90

6.80

a 50

3.70

5.75

9.00

1420

6.45

11.50

5.85

7.35

3.75

5.65

3.15

5.00

5.70

492

5.00

6.65

5.00

5.05

10.10

450

450

460

5.00

10.00

400

5.05

OAO.

1.02

.74

.59

1.38

.89

1. 11

.70

1.57

.88

3.08

2.45

1.33

.88

1.62

2.00

1.77

3.66

1.00

.75

2.00

2.00

1.00

400

1. 11

1. 11

2.02

1.01

450

1.96

Computed values

CAO.

1.3

1.0

1. 1

1.8

1.8

1.5

1.2

1.8

1.8

3.8

2.9

1.2

1. 1

1.2

1.2

.8

3.5

.9

1.4

2.6

2.2

1. 1

5.7

1.8

1.5

2.6

1.5

5.0

1.2

I,

5.5

5.8

7.7

5.4

7.7

11.2

148

10.8

15. 1

17.1

21.5

5.2

7.4

42

9.5

6.4

20.7

49

9.1

146

11. 1

11. 1

25.6

7.9

6.6

146

15.0

21.5

7.8

<z.

42

5.8

7.0

3.0

44

7.5

12.3

6.2

8.4

45

7.4

43

7.0

3.5

7.9

7.7

5.9

5.3

6.4

5.6

5. 1

10.0

4 5

45

44

5.6

10.0

43

6.5

Comparisons of columns (l/<f) 0 and (l/d) e also afford some idea of 

the accuracy involved. Calculations of doublet length are more 

reliable for low magnetic inclinations than for high inclinations. In 

low inclinations the more remote source is nearer the surface and can 

therefore have a greater expression in the data.
176 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Uncertainties in the location of apparent poles of a cylindrical body 

impose limitations on the extent to which comparisons of observed 

and computed values of I and d can be made. Theoretically, at least, 

it is expected that dc will be greater than d0 and le will be less than 10, 

because the apparent poles must lie within the body. In this particular investigation, the errors developed in the process rendered 

indiscernible any such systematic trends.

For a given cylinder the errors in computed lengths and depths 

increase as Ifd decreases, which is to be expected. The depth dc in 

table 3 was computed with a mean error of 14 percent; the le was 

computed with a mean error of 29 percent.

MAGNETIC DOUBLE-LAYER MODELS

The method was applied to anomaly fields measured over models 

consisting of plaster-magnetite horizontal slabs 0.5 inch thick and of 

different rectangular dimensions. The 5 inches X 5 inches model at 

a depth of 5 inches gave fair results, as shown in table 4. Here, the 

doublet length, I, is logically the slant distance in inches (see inset, 

fig. 37) measured along the direction of the field between the upper 

and lower faces, and is given by I0=5(cos I for 05=7^0.032*- and 

Z0=0.5 inch sin / for 0.0327r^/^7r/2. The computed 10, although not 

entirely satisfactory for determining the thickness, does in a general 

way decrease with increasing I and gives the order of magnitude. The 

results suggest that the calculation of depth to the top surface is more

I inch 

1 > Depth scale

All linear dimensions in inches 

Susceptibility K * 0.06 cgs unit

= 53,400 V

South North

Abscissas indicated are used in 

determining equivalent doublet

J~ 200 

FIGUEE 37. Total-intensity anomaly of magnetized plate; obtained from model experiment.
MAGNETIC-DOUBLET THEORY 177

accurate at low magnetic inclinations. When applied to doublelayer models of larger horizontal dimensions, for example, 0.5 inch 

X 10 inches X 10 inches, the method failed, because the /3 values 

fell outside the range of table 2. For models 0.5 inch thick but of 

plan less than 5 inches X 5 inches, the method would very likely be 

successful.

TABLE 4. Application of magnetiodoublet theory to field of horizontal slab

[Fields obtained from laboratory measurements on body of dimensions 0.5 in. X 5 in. X 5 in. at a depth of 

5 in. Linear dimensions in inches]

Inclination l

0°

30°

45°

60°

75°

Model

5.00

1.00

.85

.58

.52

0/<0.

1.00

.20

. 17

. 11

. 10

did).

1.12

.60

.56

. 10

.20

I,

5.5

3.3

3.2

.6

1.2

d.

4.9

5.0

5.8

6.3

6.2

 Observations over the center line at 1=90° were not made.

Some idea of the limits of the size of bodies that can be analyzed 

in this manner can be gained from the following expression for the 

AZ anomaly on the axis of a vertical cylinder in a vertical field:

where Z0 is the impressed vertical intensity field, p the radius, d the 

depth to the upper base, and K the susceptibility. If the expression 

(2) is expanded in a binomial series the result is

For p<cd only the first term is significant

(4)

which is valid for 

and (3) may be written

(cylinder) AZ(0)« TrKZoP2 [l/cP- l[(d+l) 2].

The expression (4) has the same geometrical form as the formula for 

a vertical doublet in a vertical field:

AZ(0) =m [l/d?- (5)

Although (4) and (5) are meaningful only at the origin, and although 

I for the cylinder is not identical with I for the doublet, the comparison nevertheless suggests that the mean radius of a body must be 

less than its depth of burial if it is to be successfully represented as a 

magnetic doublet. The experiment above on the 5 inches X 5 inches 

body at a depth of 5 inches, with an effective radius p=5/V^ confirms
178 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Oiom«ter 

A Doublet (inches)

6 ft--*-- Cylinder 1.6

C o o Cylinder 4.0

O o o Cylinder 10.0

A and 0 computed

6 andC obtained from model

experiment 

(inch 

i i Scale

MAXIMUMS HAVE BEEN ADJUSTED TO DOUBLET WITH ' T m -0.6S5 egsunrt

FinuEK 38. Anomalies of inclined cylinders showing effect of increasing diameter.

this result. In the 10 inches X 10 inches body at the same depth, 

the effective radius is p= 10/Vx and failure is to be expected.

To investigate further the effect of the radius on doublet representation, the anomalies of 3 coaxial cylinders 10 inches long in a 

field of inclination 75° were adjusted so that the ATmax of each was 

equal to that of a doublet of the same length, as shown in figure 38. 

For all, the depth to the center of the top is d and the length 2d. 

The radii of cylinders B, C, and D are p=0.16d, QAd, and d. The 

anomalies of B and C were obtained from model experiments. The 

doublet anomaly was computed from formula (1). The anomaly for 

cylinder D was computed numerically by means of a modification
MAGNETIC-DOUBLET THEORY 179

(Henderson, R. G., and Zietz, Isidore, 1958) of Gassman's threedimensional integration process (Gassman 1951). The latter was 

checked by application to cylinder C whose anomaly it reproduced 

faithfully. In figure 38, it is quite clear that the shape of the anomaly 

does not alter radically from that of a doublet as the radius increases 

to 0.4 depth. When the radius equals the depth, the shape of the 

anomaly bears little resemblance to that of a doublet; therefore the 

doublet-analysis method cannot be used. It also appears from 

figure 38 that the possibility of determining the radius of a cylinder 

from a dipolelike anomaly is very remote. A rigorous investigation 

of this matter is not possible because of the lack of a closed form for 

calculating the AT anomaly of a cylinder.

PRACTICAL APPLICATIONS

Aeromagnetic observations over geologic bodies of known depth, 

dimensions, and magnetic properties were not available for testing. 

The method was therefore applied to the surveys in areas for which 

we have some knowledge, although fragmentary, of the depth to 

crystalline magnetic rocks. If the rocks are exposed, the problem is 

simply to determine from magnetic data the altitude of the surveying 

airplane above the surface. In other areas where the crystalline 

rocks are buried, the "known" depths can be inferred from drilling 

data or from geologic information. Although calculations of I were 

made, the results are not included here because depth-extent data 

for purposes of comparison were nonexistent.

The anomalies to which we have applied this method are shown in 

figures 39 to 43. The line A-A' ha each figure indicates the profile 

across the anomaly, and B-Bf indicates the horizontal distance used 

with the appropriate 5 in the depth calculation. The values of /8 are 

not shown for these applications.

SOUTHEAST MISSOURI

On the U. S. Geological Survey aeromagnetic map of the Coldwater 

quadrangle, in southeastern Missouri (fig. 39), there are several 

anomalies over exposed felsite, particularly where the felsite is in 

contact with granite. Calculations of the depth of the source were 

made on the closed magnetic high about 1 mile west of Cedar Mountain in the Coldwater quadrangle and a depth of 870 feet was obtained. 

As the elevation of the airplane above the surface was 800 feet, the 

calculated value is regarded as satisfactory.
180 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Flown 1000 feet above surface

I MILE

Contour interval 50 gammas

ff-ff' HALF-MAXIMUM DISTANCE ON PROFILE A-A'

. Total-intensity aeromagnetic map of part of Coldwater quadrangle, Missouri.
MAGNETIC-DOUBLET THEORY 

INDIANA

181

The anomaly near Bryant in T. 24 N., R. 14 E., in Jay County, 

Ind. (fig. 40), was analyzed by the doublet method and a depth to 

magnetic rocks of 5,400 feet below the aircraft was found. On the 

"basis of a well in sec. 9, T. 24 N., R. 13 E., Precambrian igneous rocks 

in the area are estimated to be at a depth of 3,350 feet, or 4,350 feet 

below the level of observations.

R.I4E.

Flown 1000 feet above surface

Contour interval 10 gammas 

B-B'lS HALF-MAXIMUM DISTANCE ON PROFILE A-A'

FIOTJKE 40. Total-Intensity aeromagnetic map of area near Bryant, Jay County, Ind.
182 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

In La Porte County, Ind. (fig. 41), information on depth to igneous 

rocks is scanty, as the nearest wells are about 50 miles away. Geologists estimate the depth to the Precambrian is 6,200 feet at the location of the anomaly 6 miles northeast of La Crosse, in T. 34 N., 

K. 3 W. The depth calculated by doublet analysis is 6,900 feet below 

the airplane.

R.3W.

Flown 1000 feet above surface 

i I MILE

Contour interval 10 gammas

ff-ff'lS HALF-MAXIMUM DISTANCE ON PROFILE A-A' 

FIOTTBE 41. Total-intensity aeromagnetio map of T. 34 N., R. 3 W., La Porte County, Ind.
MAGNETIC-DOUBLET THEORY 

ONTARIO, CANADA

183

A prominent magnetic anomaly about 1 mile southeast of Marmora, 

Ontario, appears on the Campbellsford aeromagnetic map (Canada 

Geological Survey [1950]). Drilling information indicates there are 

90 to 150 feet of limestone of Paleozoic age overlying iron-bearing 

Precambrian rocks in the area. The disturbing rocks are known to 

contain magnetite, A calculation on the northeasterly profile (A-A1 

in fig. 42) gave a depth of 850 feet below the airplane in comparison 

with probable depths of 600 to 650 feet.

Flown 500 feet 

above sea ievel

Canada Geological 

Survey

I MILE

Contour intervals 25, 100, 200, 500, and 2,000 gammas 

B-B'\$ HALF-MAXIMUM DISTANCE ON PROFILE ATotal-intensity aeromagnetic map of area near Marmora, Ontario, Canada.
184 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

ADIRONDACK MOUNTAINS, N. Y.

Analysis of the isolated magnetic anomaly in the north-central part 

of the Stark quadrangle in the Adirondack Mountains, (fig. 43) indicated a depth of 1,050 feet for the source. The airplane was 1,000 

feet above the ground. The rocks in the area have been mapped as 

hornblende granite with a few layers of metasediment. It is very 

likely that the disturbing rocks are near the surface.

Flown 1000 feet 

above surface I MILE

J

Contour interval 100 gammas 

B-B'\S HALF-MAXIMUM DISTANCE ON PROFILE A~A'

FIOTJKE 43. Total-intensity aeromagnetio map of northern part of Stark Quadrangle, New York;
MAGNETIC-DOUBLET THEORY 185 

CONCLUSIONS

The application of magnetic-doublet theory has been shown to be 

reasonably effective for determinations of the depth of cylindrical 

laboratory models oriented in the direction of the earth's magnetic 

field. The determination of doublet length is less reliable. In general the method yields better results for bodies in low magnetic inclinations. The method cannot be expected to give satisfactory values 

in high magnetic inclinations when the mean radius of a horizontal 

section exceeds the depth of burial. The most practical use of the 

method is in depth determinations. The anomaly curves of figures 

28 to 34 are also useful qualitatively in obtaining some idea of the 

anomalies to be expected at various inclinations and the relative 

displacement of maximums.

LITERATURE CITED

Alldredge, L. R., and Dichtel, W. J., 1949, Interpretation of Bikini magnetic

data: Am. Geophys. Union Trans., v. 30, p. 831-385. 

Balsley, J. R., and others, 1954, Total aeromagnetic intensity and geologic map

of Stark, Childwold, and part of Russell quadrangles, New York: U. S.

Geol. Survey Geophys. Inv. Map GP 117, scale 1 inch=about 1 mile [1955]. 

Canada Geological Survey [1950], Aeromagnetic map, Campbellford sheet

[Ontario] No. 315: Canada Geol. Survey Geophysics Paper 13, scale 1 inch=sl

mile. 

Deel, S. A., and Howe, H. H., 1948, United States magnetic tables and magnetic

charts for 1945: U. S. Coast and Geod. Survey Serial 667, 137 p. 

Dempsey, W. J., and Duffner, R. T., 1949, Total intensity aeromagnetic map of

Coldwater quadrangle [Madison and Wayne Counties] Missouri: U. S. Geol.

Survey Geophys. Inv. Map, scale 1 inch=^ mile. 

Dempsey, W. J., Henderson, J. R., and Duffner, R. T., 1949, Total intensity

aeromagnetic map of La Porte County, Indiana: U. S. Geol. Survey Geophys.

Inv. Map, scale 1 ineh=l mile. 

Gassman, Fritz, 1951, Graphical evaluation of the anomalies of gravity and of

the magnetic field, caused by three-dimensional bodies: 3d World Petroleum

Gong., The Hague 1951, Proc., sec., 1, p. 613-621. 

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

1013 p. 

Henderson, J. R., and Meuschke, J. L., 1951, Total intensity aeromagnetic map

of Jay County, Indiana: U. S. Geol. Survey Geophys. Inv. Map GP 86,

scale 1 inch=l mile. 

Henderson, J. R., Jr., and Zietz, Isidore, 1958, Interpretation of an aeromagnetic

survey in Indiana: U. S. Geol. Survey Prof. Paper 316-B [in press], 

Henderson, R. G., and Zietz, Isidore, 1948, Analysis of total magnetic intensity

anomalies produced by point and line sources: Geophysics, v. 13, p. 428-436. 

 1957, Graphical calculation of total-intensity anomalies of threedimensional bodies: Geophysics, v. 22, p. 887-904.
186 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Jakosky, J. J., 1950, Exploration geophysics: Los Angeles, Calif., Trija Publishing Co., 1195 p.

Nettleton, L. L., 1940, Geophysical prospecting for oil: New York, McGraw-Hill 

Book Co., 444 p.

Vacquier, Victor, Steenland, N. C., Henderson, R. G., and Zietz, Isidore, 1951, 

Interpretation of aeromagnetic maps: Geol. Soc. America Mem. 47, 151 p.

Vestine, E. H., and Davids, Norman, 1945, Analysis and interpretation of geomagnetic anomalies: Terrestrial Magnetism and Atmospheric Electricity, v. 

50, p. 1-36.

Vestine, E. H., Laport, Lucile, Lange, Isabelle, Cooper, Caroline, and Hendrix, 

W. C., 1947, Description of the earth's main magnetic field and its secular 

change, 1905-1945: Carnegie Inst. Washington Pub. 578, 532 p.

Zietz, Isidore, and Henderson, R. G., 1956, A preliminary report on model studies 

of magnetic anomalies of three-dimensional bodies: Geophysics, v. 21, p. 

794-814.

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