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Columbia University in the City of New York 


LAMONT GEOLOGICAL OBSERVATORY 



PALISADES. NEW YORK 


\ 


MAGNETIC ANOMALIES 
CAUSED BY TWO-DIMENSIONAL STRUCTURE: 
THEIR COMPUTATION BY DIGITAL COMPUTERS 

AND 

THEIR INTERPRETATION 


by 

J. R. Heirtzler, G. Peter, M. Talwani and E. G. Zurflueh 


Technical Report No. 6 


CU'6'62 Nonr-Geology 


A 


October 1962 


V-v. 

















LAMONT GEOLOGICAL OBSERVATORY 
(Columbia University) 
Palisades, New York 


MAGNETIC ANOMALIES CAUSED BY TWO-DIMENSIONAL STRUCTURE: 
THEIR COMPUTATION BY DIGITAL COMPUTERS 

AND 

THEIR INTERPRETATION 

by 

J.R. Heirtzler, G. Peter, M. Taiwan! and E.G. Zurflueh 


Technical Report No. 6 
CU-6-62-Nonr - Geology 


October 19&2 









CONTENTS 


Page 

CHAPTER I Introduction 

by J.R. Heirtzler and E.G. Zurflueh 1-1 

CHAPTER II The Mathematical Expression for the 

Magnetic Anomaly over a Two-Dimensional 

Body of Polygonal Cross Section 

by M. Talwani and J.R. Heirtzler 2-1 

A* Derivation of the Basic Formulas 2-1 

B. Induced and Remanent Magnetization 

and the Total Intensity Anomaly 2-10 

CHAPTER III Computed Magnetic Total Intensity Curves 

over Two-Dimensional Bodies 

by G. Peter 3-1 

A. Description of the Model 3-1 

B. Discussion of the Computed Curves 3-2 

CHAPTER IV A Method of Interpretation for Anomalies 

of Total Magnetic Intensity caused by 
Two-Dimensional Bodies 

by E.G. Zurflueh 4“1 

A* General Remarks 4-1 

B. Mathematical Expressions 4"3 

C. Description of Method 4“10 

D. Results of Practical Applications 4"l5 

E. Curves and Graphs l\.-2b 

APPENDIX A Digital Computer Program for the 

Computation of Magnetic Anomalies over 

Two-Dimensional Polygonal Bodies 

by M, Talwani and J.R. Heirtzler A-l 

A. Framing the Problem A-l 

B. Flow Diagram A-4 

C. A Simple Example A-10 

ACKNOWLEDGEMENTS R-l 

REFERENCES R-l 

i 





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Chapter I 


Introduction 

J.R. Heirtzler and E.G. Zurflueh 

In a sense this is a "housekeeping” report in that 
it brings together, in a more or less orderly fashion, for our 
own use some of the analysis techniques that have been used by 
various persons at the Lamont Geological Observatory, It is 
felt that such material may be of interest to individuals in¬ 
itiating an analysis of the geomagnetic field intensity for 
purposes of determining earth structure. 

In the measurements of the geomagnetic field intensity 
in marine areas by Lamont it has seldom been possible to take 
measurements over a systematic grid and construct magnetic contour 
charts. Measurements usually consist of a magnetic profile 
along the track of a research vessel, these measurements being 
taken with a towed magnetometer that records the total intensity 
of the geomagnetic field. Over a period of time our ships and 
other ships and aircraft have made sufficient measurements in 
some areas to permit a general description of the magnetic a- 
nomalies even though track spacing and heading prohibit contour¬ 
ing, Frequently such a general description can substantiate or 
refute geophysical hypotheses and suggest guide-lines for future 
investigations. 

An attempt to deduce exact geophysical information 
from a limited magnetic profile is recognized as a highly un¬ 
certain undertaking, especially when control of the time 


1-1 


1-2 


variations is doubtful. Alternatively, analyzing a high densi¬ 
ty of profiles on a statistical basis may be of value if no 
errors in basic physics are committed and the limitations of 
the techniques are understood. 

The structures causing magnetic anomalies can fre¬ 
quently be treated as cylinderical, or two-dimensional, i.e. 
extending to plus and minus infinity in a direction parallel 
to a coordinate axis. Two-dimensional structure seems to be 
particularly common in certain marine areas, see for example 
(Vacquier, et.al., 1961 ) and (Heirtzler, et.al., 1962 ). It is 
instructive in such cases to determine some of the configu- 
ations of magnetic materials that can produce anomalies similar 
to those observed. An electronic digital computer program to 
calculate the anomaly that would be caused by an assumed two- 
dimensional structure has been used by this institution for 
several years. It was originally written as a program for the 
IBM 6£0 by M. Landisman of Lamont. It was written again for 
the IBM 1620, because of the convenience of that computer. It 
is the formulation of this second program that is given here 
but, naturally, both programs give the same answers. 

In order to give some crude feeling of how the various 
geometrical factors affect the total field anomalies a brief 
selection of anomalies for simple bodies is given in Chapter III. 
With the large number of geometrical parameters that are important 
in this calculation it is clearly impossible to give a complete 
handbook of anomalies with one parameter varying at a time. The 


1-3 


relative importance of the parameters is discussed in Chapters 
III and IV, 

In Chapter IV a close examination is made of the calcu¬ 
lated anomalies for those bodies with vertical sides, A working 
set of rules is presented to determine the configuration of such 
bodies from observed profiles. In that Chapter it is also 
shown how far mathematical theory can be carried without having 
to make very specialized assumptions about body configuration. 

In connection with the presentation of interpretational 
techniques in this paper, a brief listing of some of the pertinent 
literature is given here with short comments on the contents 
of each publication. 

Several authors give sets of model anomalies, usually 
accompanied by rules for depth determination and formulas for 
the particular model used. In this way Haalck (192?)* Gulatee 
(1938) and Heiland (194&) compute model anomalies of the hori¬ 
zontal and vertical components for different bodies. Nettleton 
( 1942 ) determines the vertical intensity anomaly for various 
geometric configurations. Henderson and Zietz (1948) treat the 
effect of point and line sources with respect to total intensity 
measurements. In 1958 the same authors published a paper on 
the use of magnetic doublets in the interpretation of total in¬ 
tensity anomalies. Vacquier, et.al. (1951) interpret total in¬ 
tensity maps with the help of three-dimensional (prismatic) 
models and computed second vertical derivatives. Zietz and 
Henderson (1954) use three-dimensional double layer model fields 


i-4 


for the analysis of total intensity data. Baranov (1955) 
standardizes his model calculations by reducing measurements 
for different inclinations to vertical field conditions. 

Smellie (1956) gives total intensity depth factors for point 
and line configurations of poles and dipoles. From general 
mathematical considerations Smith (1959 and 1961 ) derives in¬ 
equalities which can be used to estimate depth and intensity 
of magnetization for arbitary bodies. 

Before the advent of the airborne magnetometer the 
horizontal or vertical components were measured almost ex¬ 
clusively and the model anomalies were treated accordingly. 

Since about 1946 total intensity measurements are used in 
general and the newer literature usually refers to total in¬ 
tensity anomalies. According to potential theory it is possible 
to calculate any component from measurements of any other com¬ 
ponent if certain conditions are fulfilled (for example see 
symmetry relations on page 2-9)• A group of papers deal with 
the conversion of one component into another and with the re¬ 
lated problems of computing derivatives and upward and downward 
continuation. 

Skeels (1947) was the first to explicitly point out 
the general possibility of the above mentioned conversions by 
surface integration. Vestine (194^) and Vestine and Davids 
( 1945 ) also point out the importance of integral methods in mag¬ 
netic interpretation. Hughes and Pondrom (1947) describe the 
calculation of the vertical intensity out of total intensity 


1-5 


data, Skeels and Watson (1949) treat the problem of conversions 
in a more general way. Henderson and Zietz (1949a) give a 
method for the computation of second vertical derivatives and 
in another peper ( 1949 b) they analyze the upward continuation 
of the field. Peters (1949) treats the problems of upward and 
downward continuation and the calculation of derivatives. He 
also derives methods for the direct determination of basement 
structure. Henderson (i 960 ) relates the different operations 
of conversion to one basic formula and uses expressions suita¬ 
ble for use on electronic computers. 

Another group of papers is concerned with the inverse 
problem of the calculation of magnetic anomalies caused by 
bodies of any given shape. Gassmann (1951) describes a graphi¬ 
cal method of effecting the triple integration necessary for 
the solution of this problem. Baranov (1953) works out another 
graphical method with a different sequence of integrations. 
Henderson and Zietz (1957) simplify the computation of a total 
intensity anomaly by projecting a map of a given body in the 
direction of the earth*s field and using a modified Gassmann 
process for the integration. Talwani and Ewing (i 960 ) calculate 
the anomaly by dividing the body into parallel and polygonal 
laminae, similar to the treatment in this report. Their method 
is put in a form which can be used for digital computers. 

Vajk (1951) and Nettleton (1954) analyze the first 
step in magnetic anomaly interpretation, namely the separation 


1-6 


of anomalies from an assumed regional field. A summary of the 
different interpretation methods is given by Kohler (1958)* 

His work also contains a rather complete list of references 
which includes papers from the large Russian literature on this 
subject. Lastly, there are extensive, company classified tech¬ 
niques used by the geophysical exploration companies. 


Chapter II 


The Mathematical Expression for the Magnetic Anomaly 
over a Two-Dimensional Body of Polygonal Cross Section 

M* Talwani and J.R. Heirtzler 

A* Derivation of the Basic Formulas 

It is characteristic of geophysics texts to derive the 
expressions for the horizontal and vertical magnetic anomalies 
by analogy with the expressions for gravitational anomalies* In 
the present report the expressions for the horizontal, vertical, 
and total field anomalies are derived from basic magnetostatic 
theory* 

Consider the volume element Ax, located in 

an external magnetic field (Fig* 2-1). 



Fig. 2-1 


2-1 









2-2 


We assume that the volume element has a uniform mag¬ 
netization M. Let be the magnetic moment of the element* 
Then 

-j* * v 


The magnetic potential, at the origin, due to the volume 

element is 



A7 ^ x v- Af ^ y -f* % 


42* 


where 

p* = i X ^ j y -t i i : 


At the origin the potential of a rod of cross section 
and infinite in the y-direction is 


Jo. O'*** y**+ ^j"** ^ 


— J Ax A 2 / *. 7 

/ x 1 *-*'* / 


The vertical magnetic field strength, V, is 



3/* 

2 * 





ZX 2 ^ a - Mj(x '-Z x ) 

( + i L J 1 


( 2 - 1 ) 







2-3 


and the horizontal field strength, H, measured in the x direction 
is 


(V- -j 

a z J (2-2) 

The horizontal field strength measured in the y-direction is zero 
since y is not contained in the expression for the potential. If 
M is the induced magnetization this means, physically, that we 
have not included demagnetization effects. The above represent 
the magnetic anomalies caused by the rod, that is the regional 
field strength is not included. 

Consider the lamina shown in Fig. 2-2, infinite in the 
y-direction and infinite in the positive x-direction. 


M = - 


d_P 


= JL AX A*- 



Fig. 2-2 















2-4 


By the use of eqs (2-1) and (2-2) one gets 


\/ = 2 4%| M x zxa-H a (x*-eV 


Cx* 


Z A* I H a x i 

t x u -(. *•>- J 


(2-3) 


H - 2Alj H^Zx- fc 

'j ,Lx 


24 %| 'V -<- M-, fc I 

1 X'-H-i'- ) 


(2-4) 


The variables x and z in (2-3) and (2-4) represent the coordinates 
of the end of the lamina. If we had taken the lamina running in 
the opposite direction, as indicated in Fig. 2-3 
















2-5 


0 


T 

I 


t 


I 


" s/ZZZZZZZZZZZZZZZk 


Az 




Y 

* 


Pig. 2-3 


we would have obtained 


v = Z&i , 


“* f’l x 2 : +• M, X 


X*-h- 


H =• 


**"4- ^ 














2-6 


Notice that V and H bear the opposite signs to eqs (2-3) and 
(2-4)* Next consider the prism shown in Fig. 2-4, infinite in 
the y direction. 



Pig. 2-4 


The expressions for V and H at the origin can be obtained by 








2-7 


the integration with respect to z in eqs (2-3) and (2-4): 




(2-6) 


The integration is carried out from the small to the larger 0 
In eqs (2-3) and (2-4) the variables x and z represented the co 
ordinates of the end of the lamina. For the prism x and z are 
related by the equation for the line AD: 

X = Cx , + ~ (co-t <$>) -fc j 

If eqs (2-5) and (2-6) are integrated and the variables r and 
used instead of x and z one obtains: 







2-8 


* - - - *[". f«. - •. ) p s*-** P ^~°f (** / *i ) j 

M -i &,) *'—<#' - ce>j <t>tof(r L /r.)\l 

SJ ( 2 - 7 ) 


W - 2. j ~4>[K*f(e> ir *,)sJ- < t>-n,,<t>t of (, l / r .)'l 


*-M. 


I c®, - 0 Jm.s t + S ^pJL.f (n/r,)}J 


(2-8) 


Had it been assumed that the prism was oriented as in 
Pig, 2-5 the above would have been preceded by a minus sign. 


0 



w/e, 

IX, - 2 - ) 

HU 





Y 


Fig. 2-5 







2-9 


In Figs. (2-4) and (2-5) the prism extended infinitely 
in one of the x directions. It is clear that H and V for a body 
of finite x dimensions can be treated by assuming that it consists 
of two bodies (ref. Fig. 2-6). One must be careful with directions 
of angular measurements and with signs in working with these 
equations. 



Fig. 2-6 


The procedure indicated above can be extended to any 
number of prismatic or polygonal bodies each with many faces and 
each with its own magnetization. This derivation does not take 
into account nonuniform magnetization such as would exist near 
the corners of the bodies. In fact the actual geological bodies 
probably do not have distinct comers and nonuniform magneti¬ 
zation is believed to be unimportant. 

It is interesting to note the symmetrical relationships 
of the factors in eqs (2-7) and (2-8). That symmetry shows that 
vertical magnetization effects the horizontal anomaly to the same 
extent that the transverse magnetization effects the vertical 









t 


2-10 


anomaly and that the transverse magnetization effects the hori¬ 
zontal anomaly to the same extent that the vertical magnetization 
effects the vertical anomaly, 

B* Induced and Remanent Magnetization 
and the Total Intensity Anomaly 

If one wishes to utilize eqs (2-7) and (2-8) for the 

calculation of the V and H anomalies due to induced magnetization 

only he would write 

M = k P 

where k = magnetic susceptibility 

F = undisturbed regional total intensity vector 
and as illustrated in Pig, 2-7 
Mj = k F x = k P cos I sin s 
Mg = k P z = k P sin I 



Pig. 2-7 










2-11 


I = Inclination, positive if F is below horizontal 
(northern hemisphere) 

s = angle of strike, measured from horizontal pro¬ 
jection of P and either positive or negative y- 
axis. s must always be less than 180° (and thus 
sin s positive) since we will specify that the co ¬ 
ordinate axes are oriented to make F x positive . 
This orientation of the axes must be borne in mind 
for the correct Interpretation of results . 

If one wishes to utilize eqs (2-7) and (2-8) for the 
calculation of the V and H anomalies due to a body with remanent 
magnetization, and not to include induced anomalies, he would 
write 

M x = M rei n cos a sin b 

Mj; — sin a 

where 

a = inclination of remanent magnetization vector below 
horizontal 

b = strike angle of remanent magnetization vector 
measured between horizontal projection of M rem 
and positive or negative y axis, less than 180°. 

To apply the equations as written the axes must be 
oriented to make M x positive. 

If it is desired to use eqs (2-7) and (2-8) to get the 
V and H anomalies due to the total magnetization (induced and 



















2-12 


remanent) one should write 

M* = “tot cos * 3in p 

“y = “tot sin * 

where 

oc = inclination of total magnetization vector below hori¬ 
zontal 

p = strike angle of total magnetization vector measured 

between horizontal projection of M^ot and positive or 
negative y axis, less than 180° To apply the equations 
as written, the axes must be oriented to make positive. 
Regardless of the assumptions used to determine V and 
H, the anomaly in total intensity is obtained as follows: 

Let T be the anomaly in total intensity, then 

(P + T)2 = (F x + h)2 + (F y )2 + (f z + V)2 

Since there is no anomaly in the y direction (ref, p, 2-3)* 
Squaring, dropping terms that contain the square of V or H and 
remembering that 



Refering to Fig. 2-7 

T = H cos I sin s + V sin I * (2-9) 


2-13 


An electronic digital computer program to calculate 
V, H, and T is given in the Appendix. 

Eq (2-9) illustrates that, if the anomalies are small 
relative to P, then the anomaly in total intensity is the com¬ 
ponent of V and H in the Direction of P. See Fig. 2-8. 



Pig. 2-8 

The magnitude of T is composed of the projection of H 
(H cos I sin s) and of V (V sin I) in the direction of P. This 
geometrical picture is invalid if the anomaly is large (we 
dropped squared terms in the derivation of eq 2-9) and a more 
adequate vector diagram must be used. 





Chapter III 


Computed Magnetic Total Intensity Curves 
over Two-Dimensional Bodies 


G. Peter 


A. Descr iption of the Model 


With the aid of the computer program (see Appendix) a 


few families of magnetic total intensity curves have been com¬ 
puted in order to show the changes in these curves, when the 
ambient magnetic field directions and the geometric properties 
of the model are varied. The different secs of curves have been 
obtained by varying one parameter at a time. Grouping the curves 
in this manner enables one to make a qualitative analysis of them 
The curves are discussed in section B, 



Pig. 3-1: Ground Plan and vertical 
section of assumed two-dimensional model 


x = line of measurement 
d = depth of burial 
D = vertical extent 


s = strike 
W = Width 


The model used in these calculations is a two-dimension 
al rectangular body, which has infinite length in a horizontal 


3-1 















3-2 


direction (see fig. 3-1). It has vertical sides, except for the 
group 6 curves, where the sides have ± 30 and ± 60 degree angles 
from the vertical. For all calculations the model has uniform 
isotropic induced magnetization of 0.001 emu, it is in the northern 
hemisphere where the field strength is £0000 gammas. The pro¬ 
jection of the magnetic north on the profiles points toward the 
right. 


The total intensity values are computed for a 


measurement (profile), which 
the axis of the body. 

The various curves 

1. Variable inclination 

2. Variable strike 

3. Variable depth 
4# Variable width 

5* Variable vertical extent 
6. Slanted bodies 


is horizontal and at right 

are grouped as follows: 

I (figure 3-2) 
s (figures 3-3 to 3-7) 
d (figures 3-8 to 3-12) 

W (figures 3-13 to 3-15) 
D (figures 3-1& to 3-18) 
(figures 3-19 to 3-22) 


line of 
angle to 


These curves are only a selection for a few varied 
parameters. Additional information can be found in the papers 
of Haalck (1927), Gulatee (1938) and Heiland (1946). 

B. Discussion of the Computed Curves 


The following is a short discussion of the general 
characteristics of the computed curves and of the effect of the 
various magnetic and model parameters on them. Some elementary 
characteristics are also mentioned. Conclusions are derived 
mathematically in Chapters II and IV. 



3-3 


1 . Variable inclination I. (Figure 3-2). 

The major features of the magnetic anomaly picture 
due to induced magnetization in the northern hemisphere con¬ 
sist of a maximum on the south, and a minimum on the north 
side of a structure. For a symmetrical model, if the strike 
s = 90 °, (the strike s is defined as the angle between the 
geological strike of the two-dimensional body and the magnetic 
north) in the case of I = 45the size of the maximum and the 
northern minimum are equal in amplitude, and the maximum and 
minimum are symmetrical about the center of the model. For 
smaller inclinations the northern minimum is the dominating 
feature. Figure 3-2 also shows that there is a symmetry about 
I = 45°; in other words for equal inclination differences from 
I = 45° the amplitudes of the maximum and minimum are equal, 
and their positions with the respect of the center of the body 
are equidistant. 

The above statements applied to a symmetrical model 
are true for the negative inclinations (southern hemisphere) 
except that the north and south sides of the picture are inter¬ 
changed. 

The above mentioned symmetry around I = 45 ° exists be¬ 
cause of the symmetrical model used in the calculation, and is 
due to the change in the relative importance of the horizontal 
and vertical components of the magnetic field at I = 45°* 

2. Variable strike s (Figures 3-3 to 3-7)* 

Figures 3-3 to 3-7 show the effect of the strike with 
various inclinations. It can be seen that with decreasing strike 





3-4 


angle the maximum of the total intensity curve becomes wider, it 
moves toward the center of the body, and its minimum toward the 
north decreases* In the s = 0° case there is no minimum (only a 
maximum with all the inclinations) and the maximum is symmetrical 
about the center of the body. Because of the type of the model, 
at 1=0° and s = 0° there will be no anomaly observed. The type 
of the model and the increasing importance of the horizontal com¬ 
ponent of the magnetic field toward low inclinations requires 
that the strikes have a greater effect on the low inclination 
curves. 


d (Figures 3-8 to 3-12) 
W (Figures 3-13 to 3-15) 


3, Variable depth 


4 » 


Variable width 


5* Variable vertical extent D (Figures 3-1& to 3-18) 
These three parameters of the model were individually 
changed, keeping all the other parameters constant, in order to 
see how the shape of the total intensity curve respond to these 
changes at various inclinations. 

The variable depth curves show wider and smaller anoma¬ 
lies with increasing depth. The variable width curves cannot be 
analyzed alone, because the shape of the curve depends on the W/d 
ratio. It can be deduced from these curves that when the W/d ^ 
1, the horizontal extension (A) of the anomaly curve (the hori¬ 
zontal extension at maximum value, or % 0 % minimum value at 
low inclinations) is quite insensitive to the width of the body. 
When W/d ^ 5* the change of A almost entirely depends on the 
width, and A approximately equals W. 


3-5 


The flattening of the maximum peak, or at low incli¬ 
nation of the minimum peak, is due to the effect of the lower 
edge of the model; that is, the effect of the vertical extent 
of the model D. On the curves near I = 45° there is a shallow¬ 
ing between the maximum and minimum due to this effect (see Pig* 
3-17) which can clearly be seen at smaller values of D. This 
flattening effect is evident when D < W. When the width is much 
greater than the vertical extent, the maximums or minimums of the 
anomaly curve will appear at the edges of the body, while over 
the center the curve will be shallow, and nearly horizontal. 

This effect can be seen also on the variable width curves at W = 

15 km, when the width of the model is equal to its vertical extent. 

It is evident that there is an altitude of observation 
above which the effect of the lower edge cannot be observed. 

6. Slanted bodies (Figures 3-19 to 3-22) 

The present calculations are over four different models. 
The tops are horizontal and symmetrical about the center of the 
coordinate system, the sides are 30 ° and 60 ° from the vertical 
toward the north, and on the other two models, toward the south. 

The lower edges are horizontal also, and they are at 15 km depth. 
The magnetic total intensity has been calculated for three incli¬ 
nations over these models, in order to show the change on these 
curves due to the asymmetry of the models. 

The general picture of the magnetic total intensity over 
these asymmetrical bodies is the same as for symmetrical bodies: 
there is a maximum to the south, and a minimum to the north of the 



3-6 


■ 


body (northern hemisphere), but now this maximum or minimum does 
not depend entirely on the inclination, but depends upon the 
shape of the body also. The various symmetries in the total in¬ 
tensity picture, which were mentioned at the group 1 calculations, 
are only valid for symmetrical models. The total intensity 
picture is different at all inclinations for an asymmetrical body. 

Because the mass is more distributed, the amplitudes 
are smaller; over the slopes of the model the anomaly becomes 
wider, and over the sharp tapering edge near the observation line 
(surface) the anomaly is sharp and pronounced. These effects can 
be seen by comparing the two oppositely slanted models: Pig, 3-19* 
Pig, 3-21, and Pig, 3-20, Fig. 3-22, All four figures show that 
the shape of the upper part of the model has a dominating effect 
on the anomaly curve, while the deep extension of the model only 
causes the widening of the maximums or minimums over the sloping 
side. 

Except for the symmetry relations, the same conclusions 
given in the previous sections can be drawn for these slanted 
bodies at various strike angles s, and the d, W, and D changes 
would also have similar effect as they had on the symmetrical 


models. 


3-7 
































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-9 


















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-10 













































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-11 
























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-13 


















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-14 




































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-1* 


























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-17 

























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-19 









































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-20 


































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-22 













































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-23 





































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-24 





































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-25 














































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-26 





















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































3-27 


































































































































































































































































































































































































































































































































































































































































































































































































































































































































































Chapter IV 


A Method of Interpretation for Anomalies 
of Total Magnetic Intensity caused by 
Two-Dimensional Bodies 

E.G. Zurflueh 


A. General Remarks 

There are numerous methods of interpretation for mag¬ 
netic data, each one having a certain range of application* 

The method of Egyed (1948) which refers specifically to two- 
dimensional bodies depends on the measurement of the hori¬ 
zontal and vertical components and can, therefore, not easily 
be used for total intensity data. 

Among the various existing procedures for the analysis 
of total intensity anomalies, the most widely used are probably 
the ones of Vacquier et.al. (1951)> Henderson and Zietz (1958) 
and different mathematical methods as described by Peters (1949)* 

The mathematical methods are complicated and can only 
be applied to the interpretation of accurate magnetic maps. 

The more graphical methods on the other hand have limitations 
imposed by the geometrical configuration of bodies to be ana¬ 
lyzed* The method of Vacquier et.al., for instance, becomes 
inaccurate when applied to bodies with a horizontal extent 
equal or smaller than its depth of burial. The method of 
Henderson and Zietz has been partly devised to overcome this 
difficulty and gives good results for narrow bodies, while it 
is, in turn, not applicable to wide bodies. Both methods re¬ 
late to prismatic or cylindrical bodies of more or less iso¬ 
metric horizontal cross section. 


4-1 



4-2 


Since many of the important geological structures causing 
magnetic anomalies are much more extended in one direction than 
in others, the computations of Chapter III are expanded to 
furnish a quantitative method of analysis for two-dimensional 
bodies. 

It is believed that the method can close a certain gap in 
the range of graphical methods and it is also considered desirable 
to work out a simple and uniform method which can be used for 
bodies of small as well as large width compared to depth of burial. 

The method has been devised for simplicity of procedure. 

In order to minimize the influence of adjacent anomalies only 
distances near the main extreme of the anomaly have been used for 
the depth determinations. As a general rule it is only feasible 
to use differences between abscissas of certain points on the 
curve since all measures involving the total intensity values 
are dependent on the susceptibility of the body. Prom various 
distances that have been considered, the distance h between points 
of half maximum (or minimum) value has been selected as the main 
depth measure (see Pig. 4“2)* The computations have shown that 
this distance (hereafter called half-width) depends mainly on 
depth and width and is comparatively little influenced by other 
factors. 

The distance a has been chosen so that the ratio b = h/a 
gives a clear distinction between different values of W/d. In 
order to satisfy this condition two different distances a had to 
be used for middle and for low and high magnetic latitudes. Por 
inclinations near 0° or ± 90 ° there is a choice between both 


4-3 


values of a and depending on the circumstances one or the other 
may give better results. 

For the practical applications it is useful to keep thr 
limitations of a method in mind. For this purpose the basic 
assumptions that have been made are briefly summarized h^ e: 

1* two-dimensional body. 

2. vertical sides (this assumption implies that the 
method is, in general, to be applied to anomalies 
which are caused by susceptibility contrast and not 
by topography). 

3* vertical extent larger than depth of burial. 

4. uniform magnetization. 

5* no remanent magnetization (if there is a remanent 

magnetization parallel to the present earth's field, 
the method can be used but susceptibility estimates 
will yield too high values). 

B. Mathematical Expressions 

The derivation of a few mathematical expressions for the 
anomalies to be analyzed will be helpful in evaluating the 
properties of the computed curves. 

1. Formula for the anomaly curve 

Assuming induced magnetization only, we obtain the following 
expression for the total intensity anomaly produced by a two- 
dimensional body with vertical sides and rectangular cross section 
(see Fig. 4~l) by using equations (2-7)> (2-8), (2-9) and the 
expressions on page 2-10: 



oc/ > 


(4-2) 


at which 

K = 




and 


X U F* C -L g~^s — X) 

1 k F LX >^Wx ^ 




+ a 


i 

i 




i 



= depth of burial 
= width 

= vertical extent 
= field point (on 
the x-axis) 

= origin (note that 
the origin used in 
this chapter is located 
on the body*s axis of 
symetry and not at the 
field points as in 
chapter II.) 


section of assumed 


body and coordinate system. 















4-5 


Using distances instead of angles and setting d = d + D for 
simplification we obtain the following expressions for A and 


B: 


A = 


B ' 


X -*• 


sV 


—— -J' — — .<4/rz4-U' 

i I x j 1 

A VO**, : A 


o X~ v i 


w- *+• 


t7 A - £ 




/ w . ‘<V \ 1 ; 

* ' V * I) J - 


^ ^ ^ ^ J 


— ' -> 


(4-3) 




r 'vc:,. 


Equation (ip—ij.) can also be written as: 


^ i 

1 f . ^-j ! 

v x. yj 


J 


( 4 - 4 ) 


B 


• } 


+ ( 


x.« 


4. x . ,x .. f T c i^ i- . / '?* V/M/ vyA\ 

* _** v ^ ^_f* - J > ' - \ A - ^ / V r ^ v p j -A- p. j 

X , x. / r ^ .1 jj X'» ,,, i/' :i u-. . , 




W-; 

£ y 
( 4 - 4 *) 


Equation (4-1) gives the resolution of the total intensity anomaly 
into an even and an odd term as will be shown here: 

The function A(x) represents the even term since replacing 
x by -x in equation (4-3) does not change the expression. B(x) 
can be written as B(x) = log f(x), where f(x) is the square root 
of the fraction in equation (ip—4 a ) • If B is odd it must satisfy 
the condition log f(x) = -log f(-x), and therefore, f(x) = l/f(-x). 
That this condition is fulfilled can easily be seen from equation 

( 4 - 4 *)• 


2, Extremes positions 

In order to evaluate the maxima and minima positions we 
take the derivative of formula (4-1) 

T» = K-jA* + K 2 B* 








4-6 


Taking the derivatives of A and B from equations (4-3) 
and (4-4) we obtain: 

d d ch d 


A(v)= - z 


/ W\< »x / •-c -a \ vv\<* jx , 

J +( v ^x) ^ + d + d + 

(4-5) 


BU) - » 


Y v 


y/ 

x* 




. w 

Y -v* — 

_ _ 4 — [ 

ti W x 

A +^ V T j 


Y - 


vY 


V X 


V/ 

* - r 

( \.v 


V / < t - \/4f . 

d d tV < --/(4-6) 

A_ 


In a more concise notation the derivative of the total intensity 
anomaly can then be written as: 


- ' X -I 


. r / rr, iz' f tc^t- 

i 1 <0 - 


(4-7) 


whe re 


£>, & -.x s i' * 

c 1 - 7 (^(d.D^-D.DJ),) 
■=<* ?• , (* r(^r)(D 1 i) 4 i> J 'qD 1 Dj 


-r 1 

Expression (4-7) is of the form \ C<J *s *' 


D^ J 

rev ) 

p ^) 

r ix) 


where P^8)( x ) and p(5)( x ) are polynomials of the eighth and fifth 
degree respectively. 

Setting T* (x) = 0, p(8) ( x ) contributes the two solutions x = ± «jo 
( where the anomaly curve approaches the x - axis asymptotically), 
and in the following we will only consider the equation: 


p(5)) x ) = K X E* + K 2 P* = 0 


(4-8) 














4-7 


Substituting for D x through w© obtain the following ex¬ 
pressions for E* and F»: 


= - 2WD/ •+ WD(4- <1 d (4-9) 

+ -+3jw' 1 d-}w 4 d> 



For the general case it is not possible to obtain an exact so¬ 
lution for equation (4-8)* However, since E* is an odd function 
of the fifth degree and F* an even function of the fourth degree, 
we can solve the problem for cases where either K-j_ or K 2 is equal 
to zero* 

a* Kg = 0• 

According to equations (4-2) this condition is satisfied 
for I = 0° or ± 90° and for s = 0°, Solving the equation E* (x) = 0 
we get the following extremes positions: 



This condition can also be written as tan I = ± sin s 
(from (4-2). The relation can only be satisfied for + 45 < ^T?-45°« 
Two extreme values are: I = 45 °» 3 = 90° and I = 0°, s = 0°. 





4-8 


For the latter case both K]_ and K 2 are equal to zero and no a- 
nomaly will be observed. 

From the equation F*(x) = 0 we obtain the extremes positions: 



3• Summary 

Formulas have been derived for anomaly curves over two- 
dimensional, rectangular and vertical sided bodies and for the 
positions of the extremes of the curves for special cases of 
inclination and strike. It is not possible to obtain exact so¬ 
lutions for some other quantities like the half-width that are 
used in the method of analysis presented here. 

However, the formulas give information about the general 
properties of the anomaly curves and they also allow conclusions 
about the relationship between the different parameters involved. 
Many of the relations mentioned here are illustrated in a quali¬ 
tative way by the curves shown in Chapter III of this report. 

The following main conclusions can be pointed out: 

The parameters k and F are merely multiplying factors and 
have no influence on the shape of the anomaly curve (see equations 
(4-1) and (4-2)). 

The angular parameters I and s determine the relative im¬ 
portance of the even and odd terms of the curve. From equations 
(4-2) it follows that for positive and negative inclinations of 






4-9 


equal magnitude the anomaly curves are symmetric with respect to 
the z-axis, if the other parameters are held constant. This is 
true if the origin of the coordinate system is taken over the 
center of the body, and the profiles are drawn in the standard 
way with the northern end to the right. 

It can be shown that for s =90°, constant body parameters 
and for inclinations that differ from 45° by equal and opposite 
amounts, there is a central symmetry with respect to the origin. 
Mathematically this can be formulated: 

I = 45° ± T(x, oO = -T(-x, - oi) for s = 90 ° 

The two symmetry relations (see also page 2-9) reduce the 
amount of computation necessary for the method of interpretation. 
Since we can account for different strikes by applying a cor¬ 
rection to the values computed for s = 90 ° (see page 4-43)> we 
can restrict ourselves to computing curves for inclinations be¬ 
tween 0° and + 4£°. 

The first solution of equation (4-12) gives the distance 
of the point with maximum value from the origin which coincides 
with a zero position of the anomaly curve for the cases where 
= 0, This is the distance a which is used for the interpre¬ 
tation at certain inclinations (see Fig. lj.-2). The formula 
shows that for great width it is approximately equal to half the 
body’s width and that, therefore, the main extremes are located 
over the edges of the body. 

With respect to the body parameters d, W and D it can be 
said that, as test computations have shown, changes in D in 


4-io 


general affect mainly the amplitude of the anomaly, while they 
have less influence on the location of the extremes and of the 
zero positions. Furthermore, for physical reasons it is clear 
that if D is greater than a certain value, say greater than d 
or W, an increase in D will only slightly affect the anomaly 
curve because the added magnetized mass is too distant from the 
plane of measurement. 

C. Description of Method 

The quantitative analysis of total intensity anomalies de¬ 
scribed below is based on theoretical curves which have been com¬ 
puted for rectangular and vertical sided two-dimensional bodies 
(see Fig. 4“l)* Sets of curves have been obtained for fixed ratios 
of width to depth by varying deoth and width proportionally (Figs. 
4-11 to 4-26). In addition the curves for variable strike of 
Chapter III (Figs. 3-3 to 3-7) are used. The analysis can be 
carried out in the following steps: 

1) The inclination I of the region is determined by some 
means (isogonic maps, tables or measurements)• 

2) The approximate strike s of the body causing an a- 
nomaly is determined from the magnetic or geological 
map or from other data available. 

If no information of this kind is at hand a rough 
estimate of s can sometimes be obtained by comparing 
the measured anomaly with the curves with variable 
strike in Chapter III. 

If a magnetic map is to be interpreted, a profile is 


3) 



4-ii 


drawn through the center of an anomaly at right angles 
to its strike. 

If only profiles have been measured, the profile 
has to be projected on the direction perpendicular to 
the strike. For the practical application it is suf¬ 
ficient to multiply the distances h and a (see below) 
by the sine of the angle between profile and strike. 

4) The profile then is compared with the computed curves 
and a curve that most closely resembles the actual one 
is chosen, while keeping in mind the changes in shape 
produced by the strike. 

Since often the exact regional field is not known, 
the main purpose of this comparison is to determine a 
zero line for the anomaly. 

5) From the measured anomaly profile the distances h and 
a and the ratio b are determined from the main extreme 
of the curve (see Fig. 4“2)« It is essential that the 
part of the anomaly which lies between the main maximum 
and main minimum be used for the determination of a. 

h = x* -x . (half-width) 

0.^ 0.5 

a = x n - -x_ or x -x 

0.5 0 m 0 

according to the inclination 
(see nomograms A), 
b = h/a 

Fig. 4-2: Total intensity anomaly and explanation 



of h, a and b. 





4-12 


6) If the strike s differs substantially from 90° a cor¬ 
rection has to be applied to h and b by using tables 
4-1 to 4“4 (page 4“£5)* 

7) With the corrected values of h and b enter nomogram A 
for the nearest inclination, (Figs, 4-28 to 4-37)* 

Read a ratio of W/d for the body, 

8) Nomogram A also gives an estimate of the depth of burial 
d, A more accurate depth estimate can be obtained by 
using nomogram B with h and the previously determined 
value of W/d, The width W of the body can then be cal¬ 
culated from d and W/d, 

9) For inclinations near the middle of the l£° intervals, 
of the computed curves an interpolation is necessary. 

For this purpose a W/d - ratio is determined from nomogram 
A for the two adjacent inclinations. Then the average of 
the two W/d - ratios is taken and a depth d determined 
from nomogram B for both inclinations. The average of 
these two values gives the depth, 

10) The susceptibility k of the rock body can be calculated 
from the maximum value of the anomaly. The model a- 
nomalies have been computed for k = 10“^ emu and a total 
field strength of £ 0*000 gammas. After having corrected 
for the total intensity of the earth’s field in the area, 
we obtain the susceptibility in electromagnetic CGS-units 
by the following relationship: 


4-13 


k = 10*3 T max/T*max 

where T max = maximum value of measured anomaly 
T*max = maximum value of computed anomaly 
It has to be borne in mind, however, that the vertical 
extent D and the strike s have a strong influence on 
the amplitude of the anomaly, and therefore, affect the 
susceptibility measurements* 

The interpretation procedure can be clarified by stating an ex¬ 
ample. The profile In Fig. 4-3 has been computed for an incli¬ 
nation of 26 ° and the true body parameters are indicated in the 
figure* 

The half-width h taken from the profile is 6*5 km, tne 
distance a = Xj^-Xq is 4*2 km and the ratio b, therefore, equals 
1*55* Correcting for the strike according to table 4-1 we add 
l\% to the half-width and obtain h = 6.8 km. Similarly the cor¬ 
rected value of b is 1*44* 

Entering nomogram A on page 4“£l with the adjusted values 
of h and b, we estimate a W/d - ratio of 3*3* With this and 
with h nomogram B on page 4-5>2 gives us the estimated depth of 
the body which is approximately 2 km. 

In order to complete the interpretation it is necessary 
to determine the location of the body*s center. This can be 
done in a more or less qualitative fashion as follows: 




4—14 




















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-15 


According to eq. (4-11) for I = 0° or ± 90° and for s 
= 90 ° the anomaly is symmetric with respect to a vertical 
line through the center of the body and the main extreme is 
located over the center* 

For the cases where the anomaly curve represents an 
odd function (in the standard coordinate system, see eq* (4- 
12)* The maximum and minimum are located approximately over 
the edges of the body, while its center is given by the zero 
position between them* 

For other inclinations and strikes the center of the 
body can be found between the main extreme and zero positions 
by interpolation according to the computed curves* 

D. Results of Practical Applications 
1* Analyzed Cases 

Some anomalies on Canadian aeromagnetic maps have been 
interpreted with the method described above. From geological 
maps it can be inferred that the chosen anomalies are caused 
by rocks exposed on the surface* The depths determined, 
therefore, can be directly compared with the flight altitudes* 
Since no susceptibility values measured from rock samples 
were immediately available, no susceptibilities have been cal¬ 
culated for the examples. 

Fig. 4-4 4-^0 show parts of the aeromagnetic maps 

with the analyzed anomalies. The chosen profiles are indicated 





by the lines AB and the distances h and a are also shown in 
the figures. The zero level used is given by the point 0 
on the profiles. The inclination and the strike for each 
case are indicated in parenthesis in the accompanying text. 

a. Port Hawkesbury, N.S. (Pig. 4-4) 

The ratio of W/d was found to be approximately 7 and 
the depth determined is 290 m (I = 73s = 57°) • The flight 
altitude is about 1000 feet or 305>m above ground. The geo¬ 
logical map shows an outcrop of Pre-Carboniferous granitic 
rocks surrounded by Mississippian sediments at the location 
of the anomaly. 

b. Deskenatlata Lake South, NW Territories 

For the anomaly near the Taltson River (Pig. 4"*5) a 
W/d of 8 and a depth of 290m were obtained (I = 81°, s = 

22 °). 

The anomaly A^ in Fig. 4-6 yielded a W/d ratio of 
4 and a depth of 225 m (I = 8l°, s = 13 °), the profile A 2 B 2 
a W/d of 2.5 and a depth of 320m (I = 8l°, s = 10°). 

The flight altitude for all three cases is 1000 feet 
or 305m* The average of the three depths determined is 278 m 
which differs less than 10$ from the average flight altitude. 

Geologically the area is composed of Archaean and Pro¬ 
terozoic acid rocks. 

Since 75° is the nearest inclination and because the 




4-17 


graphs for I = 75>° are more reliable for small depths than 
the ones for I = 90 °, no interpolation was attempted and only 
the graphs for I = 7£° were used for these cases. 

c. Mack Township, Ontario (Pig. 4*7) 

With a W/d ratio of 2 we estimate the depth to be 170m 
(I = 76 °, s = 90 °)• The average flight altitude is 500 feet 
or 15>3ia above ground. The rocks of the area are Huronian 
quartzites with quartz diabase and quartz norite intrusions. 

d. Gladstone Township, Ontario (Pig. 4“®) 

The estimated depth is 130m and the W/d ratio 1 (I «s 
76°, s = 55°)• The geological map indicates Huronian sedi¬ 
ments of the Gowganda Formation with intrusives of the same 
type as found in Mack Township. 

e. Bright Township, Ontario 

The profile in Pig. 4~9 gives a W/d of 1.3 and a depth 
of 120m (I = 7^°, s ~ 90°). The results for Fig. 4**10 are: 
W/d = l/2 or smaller and d = 170m (I = 7&°, s = 90°) • 

Pre-Huronian granite gneiss and related plutonic rocks 
are the rock formations encountered near the anomalies. 

The average value of all depth estimates for the ex¬ 
amined anomalies in Ontario (which are relatively close to¬ 
gether) is l48m* which comes very close to the average flight 
altitude of 15 > 3 nu 





4-18 


2. Conclusions 

The test cases have shown that the method gives reliable 
depth estimates for anomalies that are reasonably elongated 
in shape. F or carefully chosen anomalies the results often 
are accurate to within 10$, although only very small depths 
of burial have been considered. For greater depths it is to 
be expected that the method will give even better results, 
because deviations from the basic assumptions (uniform mag¬ 
netization etc.) will be less important for those cases. 

For this method, like for others, it is of advantage 
to choose isolated and simple anomalies. As a general cri¬ 
terion it can be said that good results can be expected, if 
the shape of the anomaly profile is in its major parts com¬ 
parable to the theoretical curves. 

For anomalies which are nearly isometric in the map 
projection, the depth estimates obtained with this method 
tend to become too small, because in this case the bodies 
considered are not two-dimensional. However, if an anomaly 
of more or less elliptic shape is about three times as long 
as broad, the method can be applied successfully. 

It was not possible to check the W/d ratios determined 
for the actual anomalies considered above. But measurements 
on computed curves have shown that they are somewhat less re¬ 
liable than the depth estimates. 


4-19 



Ctaminond Island 


Morrison ■ 


Mclnnes Pt 


Head Ba 

Cove . 


Ounphey 
Head s' 


McIntosh 


Ballam 


Dundee 


Rear Black 
/ River 


Balmoral 


Buchanan 


FIG. 4-4= PORT HAWKESBURY, N S 
MAP 238G 
























































































' *200 


4—20 





FIG. 4—5= DESKENATLATA LAKE SOUTH, NW TERRITORIES 


MAP 107 G 














































4-22 



FIG. 4-7' MACK TOWNSHIP, ONTARIO 





























































4-23 



FIG. 4-8= GLADSTONE TOWNSHIP, ONTARIO 


2 











































































4—24 



FIG. 4-9 s 


BRIGHT TOWNSHIP* ONTARIO 


















































4-25 





FIG. 4-10* 


BRIGHT TOWNSHIP, ONTARIO 










































4-26 


E. Curves and Graphs 
1. Computed curves 

The bodies for which the anomaly curves have been com¬ 
puted are of the kind described in paragraph B and the co¬ 
ordinate system used is essentially the same as shown in Pig 
4-1• As abscissa the ratio of the horizontal distance x to 
the depth of burial d for each curve has been plotted in 
order to facilitate the graphical representation. The verti 
cal extent D is the same for all bodies, namely 15 km. The 
depth of burial d in km is indicated on each curve. In all 
cases the assumed total intensity P of the undisturbed field 
is 50*000 gammas and the susceptibility k is 10“3 emu. 

For width to depth ratios smaller than one the curves 

practically do not change their character except that their 

amplitude is multiplied by a factor proportional to W/d. 

That is the curves for W/d = l/2 e.g. are approximately the 

same as the curves for w/d = 1 multiplied by the factor of 

l/2. Therefore, only curves for W/d ^ 1 are shown here. 



FIGURE 4-11 



**8 1=0° S s 90° 

d 


4-27 


















































































































































































































































































































































































































































































































































FIGURE 4-12 



~ » 4 1*0° S*90° 


4-28 






































































































































































































































































































































































































































































































































































FIGURE 4-13 





^ r 2 1=0° S s 90° 


4-29 












































































































































































































































































































































































































































































FIGURE 4-14 



^ = I 1 = 0° S=90° 


4-30 


















































































































































































































































































































































































































































































































































































FIGURE 4-15 





4-31 




































































































































































































































































































































































































































































































































































































































































FIGURE 4-16 



^=4 1=15° S=90° 


4-32 







































































































































































































































































































































































































































































































































































































































FIGURE 4-17 





4-33 






































































































































































































































































































































































































































































































































































FIGURE 4-18 



4-34 


































































































































































































































































































































































































































































FIGURE 4-19 



^7*8 1 = 30° S=90° 

a 


4-35 


















































































































































































































































































































































































































































































FIGURE 4-20 



4-36 













































































































































































































































































































































































































































































































































FIGURE 4-21 





1 = 30* 


S=90* 


4-37 
























































































































































































































































































































































































































































































































































































































FIGURE 4-22 



= I 1 = 30° S=90° 


4-38 


























































































































































































































































































































































































































































































































































FIGURE 4-23 



4-39 

































































































































































































































































































































































































































































































































































FIGURE 4-24 



4-40 



















































































































































































































































































































































































































































































































































































FIGURE 4-25 





4-41 

































































































































































































































































































































































































































































































































































































































































































FIGURE 4-26 



w 

d 


1=45° 


S=90° 


4-42 








































































































































































































































































































































































































































































































































































2. Nomograms and Tables 

The values of the tables and the nomograms have been 
obtained by measuring the corresponding distances on the com¬ 
puted curves. 

a. Nomograms 

Nomogram A for each inclination shows two families of 
curves. Both of them give the half-width h as a function of 
the ratio b. For the solid curves the W/d ratio is held 
constant, while the dashed curves are plotted for constant 
depths of burial. 

On nomogram B the half-width is plotted as a function 
of depth for constant W/d ratios. 

b. Tables 

As equations (4-1) and (4-2) show, the strike influences 
anomaly curves for different body parameters in a different 
way. Corrections for the strike for different values of depth 
and width are given in tables 4“1 to 4“4* ^ h and ^ b are 

the percentages that have to be added to the measured values 
of h and b in order to obtain the values which would have been 
measured for a strike of 90 °* 

For inclinations of less than 4£° and strikes near 0° the 
positive part of the anomalies becomes larger than the negative 
one, and in these cases it is more convenient to measure the 
distances h and a from the maximum instead of using the minimum 
(see figure 4“27)♦ 








4-44 















4-45 


h(km) 



I 


>b 


0 


2 



































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-46 


h(km) 






























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-47 


h(km) 

A 



































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































9 

e 

7 

6 

5 

A 

3 

2 

fC 

9 

0 

7 

6 

5 

A 

3 

2 

9 

e 

7 

6 

5 

4 

3 

2 

\i 


4-48 


m) 


i=o° a + 90° 


NOMOGRAM B 



7 S 9 10 


5 6 7 B 9 100 


d (km) 


FIG. 4-31 







































































































































































































































































































































































































































































































































































































































































































































































































































4-49 


h (km) 





























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-50 


h(km) 


I = ± 15° & ± 75° 


NOMOGRAM B 



FIG. 4-37 































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-51 


h(km) 


















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-52 


h (km) 
a 


| 5 - 30° a 1 60 ' 


NOMOGRAM B 



FIG. 4-35 





























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-53 


h(km) 






























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































4-54 


NOMOGRAM 


h(km) 



I = ±45° 


B 



> d(km) 


FIG. 4-37 


m ^ 






































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































k-H 


Table 4“1 

Correction for the Strike: W/d=3* d=2, D=l3 



s 

Ah(%) 

Ab($) 

Ah($) 

AbW 

i=i5° 

6o° 

+ 1.8 

+ 2.8 




45° 

+ 4.1 

+12.4 




30° 

+10.0 

+26.9 




15° 

+14.4 

+23.4 

+11.4 

+14.4 


0° 



- 4.2 

+13.9 

1=30° 

6o° 

+ 3.1 

- 3.3 




45° 

+ 4.4 

-10.9 

- 1.1 

-36.1 


30° 

- 1.3 

-33.4 

+ 4.4 

-11.7 


15° 



- 3.4 

+ 3.3 


0° 



-14 -7 

- 7.2 

i=43° 

6o° 

+ 3.3 

+13.4 




45° 

0.0 

+21.6 




30° 

- 4.0 

+33.7 




15° 

-13.1 

+33.4 




0° 

-l6.5 

+19.7 



i=6o® 

6o° 

- 2.7 

+ 2.2 




45° 

- 3.8 

+ 2.1 




30° 

- 8.3 

+ 0.9 




15° 

-14.1 

- 3.7 




0° 

-14.8 

- 7.9 



i=75° 

45° 

- 1.3 

-10.4 




15° 

- 3.1 

+ 2.9 




0° 

- 4.8 

+19.7 












4-56 


Table 4"2 

Correction for the Strike: W/d=5, d=0 # 5> D=l5 



s 

Ah(%) 

Ab (%) 

Ah(^) 

AbW 

1=15° 

45° 

+2,0 

- 1.5 




15° 

+3.2 

-12.7 

+4.9 

+ 2.8 


0° 



-1.2 

+330 

1=30° 

45° 

+4.2 

-4.5 

-6.5 

-44*o 


15° 



-0.8 

+10.7 


0° 



+5*4 

+58.0 

i=45° 

45° 

+6.4 

+34-0 




15° 

-1.6 

+57.0 




0° 

-3.1 

+123 



i=6o° 

45° 

+1.7 

+12.6 




15° 

-3.2 

+ 18.6 




0° 

-4.3 

+60.5 



i=75° 

45° 

-1.5 

+ 2.0 




15° 

-1.9 

+72.4 




0° 

-2.1 

+172 




4-57 


Table 4“3 

Correction for the Strike: W/d=l, d=2, D=l5 



s 

At W 

i=*i5° 

•6o° 

- 2.6 


45° 

- 4.8 


0 

O 

-12.6 


15° 

0° 

-31.3 

1=30° 

0 

0 

sO 

- 4-3 


45“ 

-12.8 


30° 

-28.5 


15° 


o° 


x=45° 

60° 

+ 7.4 


45° 

+16.2 


O 

O 

+30.2 


H 

vn 

0 

+44*o 


0® 

+52.0 

i=6o° 

45° 

+ 11.5 


15° 

+20.0 


0® 

+21.0 

1=75* 

45° 

+ 1.9 


15° 

+ 3.0 


0° 

+ 5*o 


-14.8 

-31.0 

-42.1 


b(#) 

-72.8 

-30.0 

-71.0 


+ 7.2 

+185 

- 5.8 

-19.3 

-30.8 

-37.9 

-37.4 

-14-3 

-26.0 


+ 8.8 

+21.6 


+20.1 

+190 


+15.0 

+24*4 

+55.0 

+114 

+306 

+32.2 

+99.4 

+184 

+ 24.5 

+91.7 

+154 










4-58 


Table 4*4 



Correction 

for the Strike 

>: W/d=l, 

d=0.5, D=i5 



s 

& h (%) 

&b {%) 

& h (%) 


1=15° 

45° 

- $.0 

-37.0 




15° 

-35.6 

-81.7 

-29.2 

-75.8 


0° 



+ 4*6 

+400 

1=30° 

45° 

-14.0 

-23.6 

-34.2 

-41.7 


15° 



+10.6 

+19.6 


0° 



+25.0 

+440 

i=45° 

45° 

+19.2 

+23.8 




15° 

+58*0 

+188 




0° 

+60.7 

+730 



i=6o° 

45° 

+12.5 

+23.6 




15° 

+22.7 

+160 




0° 

+23.8 

+396 



i=75° 

45° 

+ 1.8 

+39.4 




15° 

+ 5*6 

+190 




0° 

+ 6.5 

+360 




Appendix 


A Digital Computer Program 
for the Computation of Magnetic Anomalies 
over Two-Dimensional Polygonal Bodies 

M. Talwani and J.R. Heirtzler 


1* Framing the Problem 

Rewrite eqs (2-7) and (2-8) for a prism of Fig. (2-ij.) expressing 
<P in terms of distance: 


x , - **. = x = - K, 


J I “^X * "^11 ~ “ ^1/ 


5 Ls%r\ ^ IT i / 


c. & 


z/ */*• ) 


S = 


IT. 




6 cf > <- c>S < p > — 


■^z/ * r. 


*?, + * 


= *>z. **/ 






t = 


z/ 


* */\ 


/z. 


’ / 2 . 




A-l 























A-2 


This yields 


V = Z (PI, o' - M t P') 


(A-l) 


H = t (n t p‘ *■ H t d') 


with 


(A-2) 





*■ X 



4 U 


— —- JU. (r„r ,J 


(A-3) 



K/fc 



•f- x 


/v 








sCo$ (r L /n ) 


(A-l*) 


By refering back to page 2-7 , it is seen that integrations leading 
to this result were carried out in a counter-clockwise direction, 
that is B moving c.c.w. during the integration. For a computer 
program there is an advantage in using a different labeling of 
body points (see Fig. A-l): 











The previous calculations can be utilized provided we replace 
*9 ^7 *1 » by \r, , Q, by , and O t by €>, # These changes 
give the following equations: 


V ~ Z (A7 y Q - /t 1i p ) 


(A-5) 


H - Z (PT A F -h <Q ) 


(A-6) 


p = 


it 


*1, 


i*t - ) + 


*it *n. 


toy (r u /tr { J 


(A-7) 


■3./ 


'it 


O - (6>,-6>J - - *£*$(*. Jr,) (A-8) 

**/t 4i/ + */t 

For a given field point (point of observation or origin location) 
the computer computes eqs (A-5), (A-6), and the total intensity 
anomaly (eq 2-9) for each face of the body using eqs (A-7) and 
(A-8) as internal subroutines# It sums the anomalies for each 
consecutive face and then proceeds to do the same for other bodie 
present# After the various sums are printed for a given field 
point it goes to the next field point, etc# 

The program has been written specifically for bodies 


with induced magnetization only, but can also.be used without 
modification for bodies having remanent magnetization or bodies 
with both remanent and induced magnetization# 






For computing convenience we can write (A-5> and (A-6) 


above as: 


V - fc<-oi x'/iU* s'JQ - 

L (A-9) 

u =r z-kFfc^i s'Jf + z';q.~\ 

L (A-10) 

For bodies with induced magnetization only, k represents the 
susceptibility and F the magnitude of the total undisturbed field, 
I* =1# s’ = s as in Fig. 2-7* For bodies with remanent or mixed 
magnetization kF has, of course, no significance; it is merely 
retained for convenience so that one can use the same program as 
for bodies with induced magnetization. In practice kF is made 
to represent the magnitude of M rem for bodies with remanent mag¬ 
netization by giving k the dummy value of 1 and F the value for 
M rem , Also I 1 = a, s = b as mentioned on p. 2-11, Similarly for 
a body with mixed magnetization k is again given the dummy value 
of 1 and F the value of M tot . I* = o**, s* =p as mentioned on 
p, 2-12, 

We note that the expression (2-9) for the evaluation of 
T, the total anomaly, remains unchanged whether the magnetization 
is induced, remanent, or mixed, 

2, Flow Diagram 

To assist in the understanding of the flow diagram and 
the subsequent program the following list of symbols is defined: 



A-5 


Table A-l 
Program Symbols 


Symbol 

DIP* 

DIED* 

D» 

DD* 

F* 


C* 

EXX, ZEE* 


JTOT* 


CONS* 


FO* 

DELX* 

THETA, THETB 


Meaning 

Inclination or Dip, I, in degrees 

Modified Inclination or Dip, I* 
in degrees 

Strike, s, in degrees 

Modified Strike, s', in degrees 

Total Magnetic Field intensity, 
in gammas 

Magnetic susceptibility, k, in e*m.u. 

Coordinates of body comers, entered 
in program in clockwise sequence in¬ 
cluding first twice 

Total number of body points including 
first twice 

Constant value of z of field points, 
zero for shipboard surveys, negative 
for aeromagnetic surveys taking sea 
level as z = 0 

x value of first field point 
Increment In x value of field points 
Angle to corner measured from x-axis. 


OMEGA 

K 

KTOT* 

LNO* 


Angular span of face from field 
point, 

Subscript for consecutive field 
points 

Total number of field points 
Subscript for body number 










LNOT* 

Total number of bodies 

H 

Horizontal anomaly for single face 
of body at field point, in gammas 

V 

Vertical anomaly for single face of 
body at field point, in gammas 

PSUM 

The sum of P*s for the various 
faces of the body 

Q3 UM 

The sum of Q*s for the various 
faces of the body 

HSUM, VSUM 

Horizontal and Vertical anomalies, 
in gammas, at field point for 
given body 

HASUM, VASUM 

Horizontal and vertical anomalies, 
in gammas, at field point for all 
bodies 

TSUM 

Anomaly in total intensity, in 
gammas, at field point for given body 

TASUM 

Anomaly in total intensity, in gammas 
at field point for all bodies 


•a-Data which must b© entered into computer from typewriter. 
D and DIP are used in the calculation of V and H* 


DD and DIPD are used to calculate TSUM and TASUM. 



A-7 























































A-8 


C FORTRAN PROGRAM FOR IBM 1620 

DIMENSION FX(47) ,FZ (ij.7) ,VASUM(47) ,HASUM(47) 
DIMENSION VSUM(47),HSUIvl(47) ,EXX(30),ZEE(30) 
ij.00 ACCEPT,DD,DIPD,KTOT.LNOT,CONS 

402 sdd=sin(. 0174533 *dd) 

CDIPD=COS(.0174§33*DIFD) 

SDIPD=SIN(.0174§33*DIPD) 

601 ACCEPT,FO,DELX 

603 DO 604 K=1,KT0T 
RK=K 

FX(K)=(FO-DELX)+DELX*RK 

FZ(K)=CONS 

VASUM(K)=0. 

604 hasum(k)=o. 

IO 5 ACCEPT,LNO,C,F,JTOT 
ACCEPT,D,DIP 

410 sd=sin(.0174533*d) 

CDIP=COS (.0174533*DII>) 

SDIP=SIN(.0174533*®IP) 

DO 11 J=l,JTOT 

411 ACCEPT,EXX(J),ZEE(J) 

11 CONTINUE 

DO 36 K=1,KT0T 
PSUM=0. 

QSUM=0. 

X1=EXX(1)-FX (K) 

Z1=ZEE(1)-FZ(K) 

RSQI=0CL**2+ZHh*2 

if (xi) no, 140,180 

110 IF(Z1)120,130,130 

120 THETA=ATN(Z 1 /X 1 )- 3.1415927 

GO TO 200 

130 THETA=ATN(Zl/Xl)+3.1415927 
go to aoo 

140 IF(Zl)l 50 ,l 60,170 
150 THBTA=-1.5707963 
GO TO 200 
160 THETA=0•0 
GO TO 200 

170 THETApI.5707963 
GO TO 200 

180 THETA=*ATN (Zl/Xl) 

200 J=2 

201 X2=EXX(J)-FX(K) 

Z2=ZEE(J)-FZ(K) 

RS Q2»X2*-*2+Z2*~*2 
IF(X2)210,240,280 
210 IF(Z2)220,230,230 
220 THETB^ATN(Z2/X2)-3.1415927 
GO TO 300 

230 THETB=ATN(Z2A2)+3.1415927 
GO TO 300 


A-9 


2k0 IF(Z2)250,260,270 

250 thetb=-i .5707963 
GO TO 300 
260 THETB= 0,0 


GO TO 300 

270 THETB=1 # 5707 963 
GO TO 300 

280 THETB=^TN(Z2/X2) 

300 IP(Z1-Z2)320,31,320 
31 P=0. 

Q=0. 


GO TO 32 

320 OMEGA=THETA-THETB 

IP(OMEGA)3201,3202,3202 
32021P(OMEGA-3.1415927 ) 330,330,340 
3201IF(OMEGA+3•1415927)34°* 330, 330 
330 THETD=OMEGA 
GO TO 370 

340 IF(OMEGA)350,360,360 
350 THETD=0MEGA+6.2831853 
GO TO 370 

360 thetd=omega- 6,2831853 

370 XL2=X1-X2 
Z21=Z2-Z1 


XSQ=X12**2 

ZSQfZ21**2 

XZ=Z21*X12 

GL=0 ,5'^LOG (RSQ2/RSQ1) 

P=( (ZSQ/(XSQ+ZSQ) )*THETD)+((XZ/(XSQ+ZSQ) )**GL) 
Q= (THETEHt( XZ/ (XSQ+ZSQ)))-(GL*(ZSQ/(XSQ+ZSQ)) ) 

32 IP(SENSESWITCH 3)33#34 

33 H=2.*C-*F#( (CDIP*SD*P) + (SDIP**Q)) 

V=2.*CttF*( (CD I P-ttSD*«-Q) - (SDI P-«-P)) 

PRINT, XI, X2,Z1,Z2,XSQ,ZSQ,XZ 
PRINT , RSQ1, RSQ2, THETA , THETB 
PRINT,K, J, THETD,GL, P, Q,H,V 

34 PSUM=PSUM+P 
QSUM=QSUM+Q 
X1=X2 


Z1=Z2 

RSQ1=RSQ2 

THETA=THETB 


J=J+1 

JR=J-1 

IP(JR-JTOT)201,202,202 

202 HSUM(K)=&,*CttF*( (CDIP*SD*PSUM)+(SDIP*QSUM)) 
VSUM(K)=2.*C*F*( (CDIP#SD*-QSUM) -(SDIP-&PSUM)) 
HASUM(K)=HSUM(K)+HASUM(K) 

VASUM (K) =VSUM(K) +VASUM (K) 

36 CONTINUE 

PRINT,LNO,C,F,D,DIP 
DO 37 K=1,KT0T 







A-10 


TS UM= (HSUM (K) *CD I PD*SDD ) + (VSUM(K)-b-SDIPD) 

37 PRINT,K,FX(K),HSUM(K) ,VSUM(K) ,TSUM 
IF(LNO-LNOT )105,38,105 
38 K=1 

3 81 TASUlfc (HASUM (K) *CDI PD*SDD) + ( V AS UM (K) *SD I PD) 
39 PRINT,K,FX(K),HASUM(K),VASUM(K),TASUM 
K=K+1 
KR=K-1 

IF(KR-KTOT) 381,391,391 
391 GO TO 400 

END 


A-ll 


3* A Simple Example 

Suppose we wish to determine the induced anomalies of 
three field points for an assumed body shape as given in Pig* A-2. 



Pig. A-2 


Suppose the axis of the body makes an angle of 60° with magnetic 
north, the inclination is 10 °, the ambient magnetic field strength 
is 50,000 gammas, the susceptibility is 0.001 emu. 

We have JTOT = 4, LNO = 1, LNOT = 1, CONS = 5., KTOT = 3, 

P0 = 5., DELX = 3. 

Information in the following form must be supplied to 
the machine by typewriter 

60. 10. 3 15. 

5. 3. 

1 .001 

60. 10. 

8 . 8 . 

8 . 6 . 

11 . 6 . 

8 . 8 . 


50000 . 4 












A-12 


With senseswitch 3 off numerical results will be printed as 
follows: 


(LNO) 

(c) 

(p) 

(D) 

(DIP) 

1 

1.0000000E-03 

50000.000 

60.000000 

10.000000 

(K) 

FX(K) 

HSUM(K) 

VSUM(K) 

TSUM(K) 

1 

5.0000000 

11.142248 

8.2548^30 

10.936315 

2 

8.0000000 

-25.503102 

53.009137 

-12.545855 

3 

11.000000 

-12.179321 

-43.110189 

-17.873370 


A-X3 


4# Console Operation for Lamont IBM 1620 


Switches: 


Clear Memory: 


Load Tape: 


Load Data: 


Senswitch 3 ON for detailed print out (used 
for trouble shooting only) 

Other senseswitches OPP 
Program switches OPF 
Other switches ON 

Reset, Insert, 160001000000, Release, Start, 
(after a few seconds) Instant Stop, 

Mount object tape, set tape console switch 
to Reel, Reset, Insert, 3^0000000300, Re¬ 
lease, Start, If machine fails to read, re¬ 
peat previous instructions. If functioning 
properly machine will pause 2/3 way through 
reading to think but will proceed to end of 
tape when it will type "Load Data”, 

Set typewriter tabs at approximately 


16 28 45 58 70. 

Reset, Insert, 4907500* Release, Start 
Numbers must be typed in upper case. 

Minus signs and decimal points in lower case. 
Zero in either case. 

A space must be typed between successive data 
within a line of data, 

A record mark after the last datum of a line 
is optional. 




A decimal point is required after floating- 


Typing Error: 


Stop Output: 


Check Stops: 


point data. 

A decimal point is optional after fixed- 
point data. 

If incorrect data has been entered in machine 
Reset, Insert, 4907500# Release, Start, re¬ 
enter all data. 

If incorrect data has been typed but not 
entered into machine Senswitch 4 ON, Release, 
Start, Senswitch 4 OFF, repeat line of data. 
Depress Stop. Neatness can be achieved by 
depressing Stop while carriage of typewriter 
is in return motion following typing of last 
line of desired data. If machine is not 
stopped it will automatically return to the 
stage where it is again ready to accept all 
data. 

If a check stop occurs during data entry it 
may be due to a parity error in that part of 
the memory related to data storage: Reset, 
Insert, 4907500, Release, Start, re-enter 
all data. 

If a check stop occurs at any other time or 
if the above procedure fails clear memory, 
re-load tape, re-enter all data. 


R-l 


ACKNOWLEDGEMENTS 

The figures of Chapters III and IV of this report 
were drawn by Jay Maxon and David Wolfe, 

The aeromagnetic maps of Chapter IV-D were reproduced 
with the permission of the Geological Survey of Canada, 

Financial support for this work was provided by the 
Office of Naval Research, Contract Nonr 266(48) and the National 
Science Foundation, Grant 4l5l* 

Reproduction of this document in whole, or in part, 
is permitted for any purpose of the United States Government, 


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List of maps used 

Geological Survey of Canada (1922): Portions of the districts 

of Algoma, Sudbury and Timiskaming, Ontario, map 155A. 
Geological Survey of Canada (1945): Geological map of the 
Dominion of Canada, map 820A. 

Geological Survey of Canada (1949 )• Geological map of the 
Maritime Provinces, map 910 A. 

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P 

Aeromagnetic series, sheet 11 map 238 G. 

Geological Survey of Canada (195&): Deskenatlata Lake South, 


NW Territories. Aeromagnetic series, sheet 85 


15 ’ 


map 107G. 

Province of Ontario, Department of Mines (1955): Aeromagnetic 
map. Bright Township, Distr. of Algoma. 

Province of Ontario, Department of Mines (1955): Aeromagnetic 
map, Gladstone Township, Distr. of Algoma. 

Province of Ontario, Department of Mines (1955): Aeromagnetic 


map. Mack Township, Distr. of Algoma