Bhattacharyya - Magnetic Anomalies Due to Prism Shaped Bodies With Arbitrary Polarization

GEOPHYSICS, VOL. XXIX, NO. 4 (AUGUST, 1964) PP. 517-531, 10 FIGS.

MAGNETIC ANOMALIES DUE TO PRISM-SHAPED

ARBITRARY POLARIZATIONS

B. K. BHATTACHARYYA*

BODIES WITH

A study is made of magnetic anomalies due to prism-shaped bodies with arbitrary polarization. Expressions of the total field and its first and second derivatives are derived on the assumption of uniform magnetization throughout the body. Formulas for all possible cases in connection with a rectangular prism with vertical sides can be obtained either directly from this paper or by simple extension of the formulas given here. Using the exact expressions given in this paper, the total field and its derivatives are evaluated conveniently and rapidly with the aid of a digital computer.

The effect of the dip angle and declination of the polarization vector on the size and shape of the magnetic anomaly is then studied for the case when the earth’s normal total field vector has a dip angle of 60” and declination of 0”. With an increase in the dip angle of the polarization vector, the negative anomaly occurring on the north of the causative body diminishes in magnitude, whereas the positive and second derivative anomalies increase to maximum values and then decrease. With an increase in declination, this latter trend is repeated with the positive anomaly but the negative and second-derivative anomalies decrease systematically. Both the positive and second- derivative anomalies become more and more symmetrical with respect to the prismatic body with increase in either the inclination or declination of the polarization vector.

INTRODUCTION

In the interpretation of aeromagnetic data, theoretical calculation of total field anomalies for various types of model sources has an important place. For this simple reason, there is no paucity of published papers outlining methods for calcu- lating the total-intensity field for different model sources, e.g., point pole, line of poles, point dipole, and line of dipoles (Henderson and Zietz, 1948; Smellie, 1956) and prism-shaped bodies (Vac- quier et al, 1951; Hughson, 1962). Of all these, the paper by Vacquier et al has probably been used most extensively because of the importance of block-type bodies in aeromagnetic interpretation. All the papers mentioned above have assumed the induction in the earth’s magnetic field to be responsible for the magnetization of the rock mass.

Experimental work on rock magnetism has made it abundantly clear that, contrary to the earlier belief, presence of permanent magnetization is often the rule, rather than the exception, in the rocks of the earth’s crust. Permanent m’agnetization associates itself with induced magnetization to orient the polarization vector of the rock mass in some arbitrary direction. The direction of this polarization vector influences appreci-

ably the size and shape of the associated magnetic anomaly. The necessity of a study of this influence is being increasingly felt. A few papers on this subject have already been published. In most of these papers the results for specific model sources have been presented, e.g., infinitely long dikes (Hutchison, 1958); point dipole and line of dipoles (Sutton and Mumme, 1937); a point dipole, a horizontal line of dipoles, a thin, dipping sheet, a thick, dipping sheet, and a sloping step (Hall, 1959).

The present paper treats only the case of prism- shaped bodies with arbitrary polarization. The effect of the dip angle and declination of the polarization vector on the size and shape of the magnetic anomaly has been studied.

An expression of the total field in closed form is derived on the assumption of uniform magnetization throughout the body. Without this assumption, analysis of this problem will be practically impossible because of our lack of knowledge of the probable distribution of magnetization in a given rock mass. Expressions in closed form for the first and second vertical derivatives of the total field are also obtained.

Though the obtained expressions look cumber- some, it is much faster to evaluate the total field

t Manuscript received by the Editor December 10, 1963.

* Geological Survey of Canada, Ottawa, Ontario, Canada.

517

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518 B. K. Bhattacharyya

5 z

FIG. 1. An elementary volume of the prism anti the coordinate system.

and its derivatives using these exact expressions rather than using the approximate numerical method as given by Vacquier et al (1951). In two hours with an IBM 1620 computer it is possible to obtain exact values of the total field and its first and second vertical derivatives at about 1,000 points in a horizontal plane above a par- ticular prism with known direction of polarization.

PRISM WITH INFINITE VERTICAL SIDES

Let us consider a prism-shaped body at a depth lz below the level of observation, the upper surface of which is a horizontal plane and the vertical sides infinitely extended. The polarization vector is taken to be at an angle 0 with the direction of the earth’s field. Since the magnitude of the anomalous field is generally small as compared to the earth’s total field, the direction of the re- sultant of the two field vectors will practically be

invariant over the area of observation and lies in the same direction as the earth’s field. In the case when a total field measurement will be made, the direction of measurement \vill, therefore, be the same as the earth’s field vector which is de- fined by the direction cosines, 1, m, and ~1.

Let N, p, and y be the coordinates of the volume element dcvdfldy in the prism (Figure 1). Let the polarization vector be characterized by the direction cosines L, M, and N. Then the field produced by the volume element dad/3dy is given by

(1)

where I, is the polarization, ds is an element of length in the direction of the polarization vector, and dt is an element in the direction of the total field of the earth. Let (x, y, z) be the coordinates of the point of observation. Then,

a a a a -=l-+m--++lz--, (2) at ax ay a2

a --=,;+A!t;+x;, (3 as .

and

Y2 = (a - X)” + (p - y)Z + (y - 2)‘. (1)

By utilizing equations (2), (3), and (4), (1) may be written as

dF = I, - * COS 0 + ; iiL(a - x)? + mM(P - y)" + n~T(y _ z)~ + 4cy - x)(/3 - y)

where

+ a13(a - X)(Y - 2) + CY~~(P - Y)(Y - z) I]~d&&, (3

CYST = Lm + Ml, aI3 = Ln + Sl and cy23 = MTZ + Sm. (6)

(5) may now be integrated with respect to y and the volume integral reduced to a surface integral:


Magnetic Anomalies, Prism-Shaped Bodies 519

nhere

cos % = 1L + mM + nS

cyl = O( - x, p1 = p - y, a2 = cyl? + PI”,

ro? = cY12 + p1* + (I2 - z)?.

We shall now take the plane of observation to be the z=O plane and integrate (7) with respect to LY,. The integrals appearing in (7) are all of standard type and so they pose no serious difficulty. Con- siderable algebraic and trigonometric manipulations are, however, needed to obtain

/‘(.I., y, 0) = I, s .mlvP1, (8)

\vhere

_$!(1 2) -“I;p’(l_“)

After the integration of (8) \vith respect to /3,, the expression of the total anomalous field may finally be written as

)

where LY,, flu are the upper limits and LYL, fin the loner limits of (~1 and pi, respectively. Let 10, Do, be the inclination and declination, respectively, of the earth’s field and I, D be the

inclination and declination of the polarization vector. In the case when the n: axis is directed north ward and when the direction of polarization is the same as that of the earth’s field, we have

I = I, = sin 6

I? = .Y = cos 6

m = M = 0,

where

6 = OIP - I.

Substitution of the above equations in (10) yields

F(.t_, Y, 0)

I, = [cos’* tan-l(s) - sin* G.tan-’ (_““‘_)



This is exactly the same case as treated by Vacquier et al (1951). When the s-axis is directed northward but the direction of polarization is different from that of the

earth’s field, we have

- 1L tan-’ (YlPl

w2 + rob + la? >

where, in the present case,

(~1~ = cos I cos lo sin D

al3 = cos 10 sin I + cos I sin 1” cos D,

and

CYZZ = cos I sin IO sin D.

VERTICAL DERIVATIVES OF THE TOTAL FIELD

In (10) if we substitute (/z--z) for 11, we obtain the expression of the anomalous field F(s, y, z). Since

a a _= -____ a2 c3(h - z)

and

a2 a? _=___, a22 a(12 - z)?

it is fairly easy to oblain first and second vertical derivatives of the anomalous field at (x, y, 0). After some algebraic and trigonometric manipulations, we obtain at z=O,

. ((CL” - 2hz)ro2 - a2fi,2{ + cxl? k - mM --~--~__ @&/?(3r”? - cq?)

Y03 r”B(?“? _ cyl?)?

- IL alhh

ro3(h2 + CU~)~ (CT12 + It2 + 2Y”?) + fL.J- -_zP_-

Y”(cQy31~ + ro?h2)

. 2 (ro2 + h2j2

a12fl12 + ro2h2 Y,,~ (14



It is easy to handle equations (13) and (14) with the help of a digital computer. When the x axis is directed northward and the polarization vector and the earth’s field vector are in

the same direction, (13) and (11) are reduced to the following equations:

- 2 sin 6 cos 6 Plk -I1 !%I

r&o’ I’ I - PI?) L[ SI’ (1.5)

and

Pl 2sin6cos6--------

a1 Pl h

r”s(r”* - pr?)? ( (a’ - 2h?)ro’ - a2pr?) - sin?6 - (or? + 19 + 2ro?)

ro3(lz” + al?)?

SOME GENERAL CONSIDERATIONS

Equation (10) may be conveniently utilized to obtain the total field for bodies of finite vertical extent. For example, if the prismatic body under consideration has its top at a depth h from the plane of observation and its bottom at a depth w, we have to determine fields due to two bodies of infinite vertical extent, one at a depth lz and the other w, and then subtract the latter from the former in order to evaluate the anomalous field for the given body. Applying the same technique to (13) and (14) we may also obtain the first and second vertical derivatives of the field.

In cases where none of the horizontal sides of the body are parallel to either the x- or y-axis but where the sides are perpendicular to each other, the problem can be solved by a rotation and simultaneous translation of the axes and then substituting the new values of R: and y in (lo), (13), and (14). For example, let the center of the prism be (x0, yO) and let the longer side of the body make an angle B0 with the n--axis. If we draw a.new coordinate system with the origin at the top of the center of the body and with x’ and y’ axis parallel to the sides of the body, the new (CC’, y’) coordinates of any point will correspond to the (x, y) coordinates in (to), (13), and (14). To refer the whole system to the (r-y) coordinate system, we have to use the following relations:

y’ = (y - y0) cos 00 - (.r - .r”) sin 00

z’ = (y - yo) sin 00 + (.r - 2”) cos BO.

Thus, the total field and its first and second vertical derivatives may be calculated with the help of (lo), (13), and (11) for a prism with rectangular sides situated anywhere at any angle with the horizontal axis, provided the top of the prism is kept horizontal.

CONTOUR MAPS OF THE TOTAL FIELD AND ITS

VERTICAL DERIVATIVES

In order to study the characteristics of the contour maps for the total field and its derivatives, only the prismatic body with infinite vertical dimension is considered in this paper. X11 the coordinates are measured in terms of the spacing which is taken as unity. The center of the prism is taken to be vertically below the origin of the coordinate system chosen. The top of the prism is outlined by dashed heavy lines in all the contour diagrams. The total field is shown by solid light lines and the second derivative by dashed light lines. Due to the necessity of making the diagrams less complicated and keeping the number of diagrams as small as possible, the first-derivative maps have not been presented at all. They are much smoother than the second-derivative maps. It will, however, not be out of place to



NORTH

1 \

0. i o*

IO I so0

_- _______---

,_-- ,

I

/’ .-

/’ ,’

I i

/’ :’ __A 25

FIG. 2. Contour diagram of the total field and its vertical second derivative when D=II,,=O’ and I=la=600. Second derivatives are shown by dashed lines. a~=Pr= -4; 1~=3.

emphasize here the importance of the first-derivative maps for various types of calculation in

,interpretation. For a rectangular prism when a,= -crU and

/3‘= -&, a study of the different terms in (10) reveals that the coefficients of IL, mM, and nN are even functions of both .X and y. The coefficient of aI3 is even in y but odd in .Y, whereas the coeff-

cient of cyZ3 is even in N but odd in y. The remain- ing term, which is the coefficient of cyr?, is an odd function of both x and y. Thus, it can be shown with the aid of (lo), (13), and (14) that the total field and its vertical derivatives are symmetrical with respect to the y-axis when the polarization vector and the earth’s field vector are perpendicular to the y-axis, i.e., m= M=O. This is also the



case when the x-axis is directed northward and the magnetization of the rock mass is due to induction alone. This explains why all the contour maps published by \-acquier et al (1951) are symmetrical with respect to the y-axis. It should, however, be emphasized that a contour map is, in general, not likely to be symmetrical with respect to any axis whatsoever.

In Figure 2 is shown the contour map of the total field and its vertical second derivative when D = Do = 0” and Z = lo = 60”. This serves as a rcfer- ence and at the same time an ideal map.

The total field and its vertical second deriva- ti1.e are drawn in Figures 3, 4, 5, and 6 for D = O’,

Z=Z0=600, and for four values of DC, e.g.,

Do=@, 90”, 135”, and 180’. With the help of these diagrams it is possible to study the variation of the characteristics of the total field and its vertical second derivative with the change in declination of the polarization vector with respect to that of the earth’s field vector.

The positive anomaly oi the total field is at a maximum distance away from the center of the prism when D= Do (Figure 2). This anomaly shifts toward the center and the negative anomaly away from the center as (DO-D) increases. Ho\v- ever, a second negative anomaly, small in magnitude, appears at the south end of the map and

FIG. 3. Contour diagram of the total field and its vertical second derivative when 11=0°, I= Ia=60”, and &=45’. Second derivatives are shown by dashed lines. ar=/3~= -4; h=3.



FIG. 4. Contour diagram of the total field and its vertical second derivative when ZI=O”, I=Zo=600, and &=90’. Second derivatives are sholvn by dashed lines. ac=fi,= -4; 1~=3.

gradually moves toward the center with increase in Do, When Do= 180”, the contours become practically symmetrical with respect to the center of the prism and look as if the prism has been taken to the magnetic pole and the polarization is due to induction alone. The amplitude of the positive anomaly decreases gradually with increase in (Do-D), reaches a minimum, and then begins to increase.

The second-derivative contours are also initially very unsymmetrical with respect to both x and y axes and look more and more symmetrical as Do increases. The magnitude of the second derivative, however, decreases systematically with in-

crease in Do.

For studying the variation in the characteristics of the total field and its vertical second derivative with change in the inclination Ia of the polarization vector, four contour maps are drawn in Figures 7, 8, 9, and 10 by setting I= 60”, D = 0” and for values of Z0 from 0’ to 135” at steps of 45”. When Zo=O’, i.e., the polarization vector is horizontal, the negative high of the anomaly occurs at the north of the positive high, and the former is of higher amplitude than the latter. The zero contours of the second derivative nowhere follow the outline of the body when lo = 0”. ,4s lo increases, the negative high diminishes and the



NORTH

h 0, = 139

I, : boo -

FIG. 5. Contour diagram of the total field and its vertical second derivative when D=O”, Z=Z,=60”, and DO= 135”. Second derivatives are shown by dashed lines. al=fi~= -4; h=3.

positive high begins to increase, reaches a maximum, and then starts to decrease. The same thing applies for the second derivative of the total field. Both of them become more and more symmetrical with respect to the body of the prism with increase in lo.

In all the contour diagrams presented in this paper, the prism has been taken to be rectangular with equal sides. A few diagrams drawn for un-

equal sides of the prism show that, according to whether a side of the prism increases or decreases in length, the shape of the anomaly bulges out or shrinks in the direction of the side. Besides this, there is no other relationship observed between the physical dimensions of the prism and the shape of the magnetic anomaly. Hence, these diagrams have not been included in this paper.

(Text continued on page 531)


B. K. Bhattacharyya

1 I, : 600 _- ____--_

</--- .’

,’ ,’

/’ ,.r- , /’ ----\ ---------\

, /’

,’ I’ ,

;I;;: ,/I 1 I, I, I i 1

I

1, 1, \_ ------- .______- \+_--____- _ ’

______-____#

/ / -________----

: ‘\ -.05

\\ ‘.

‘.._ -._

‘\

‘.._ . .

_--__

-_-____._els

I .I

FIG. 6. Contour diagram of the total field and its vertical second derivative when U=O”, I =10=600, and Do= 180”. Second derivatives are shown by dashed lines. CE,=~~= -4; h=3.


Magnetic Anomalies, Prism-Shaped Bodies

NORTH

527

I “i____ --______..+

,’ .I

,,’ i ,,_________~_- ---lx

\.._ -. -.. -..___.-.-l l’

FIG. 7. Contour diagram of the total field and its vertical second derivative when II = II0 = O”, I = 60”, and 10=O”. Second derivatives are sholvn by dashed lines. aI=ljl= -4; l/=3.



NORTH

: 1 ,I ,-

,’ ---Y \\Y\‘ 0

FIG. 8. Contour diagram of the total field and its vertical second derivative when D= Do= 0”, I = 60”, and I0=45”. Second derivatives are shown by dashed lines. oli=pi= -4; h=3.



‘\ ‘,

‘. ---___________

\ .25

FIG. 9. Contour diagram of the total field and its vertical second derivative when D = Do =O”, I = 60”, and Z0=90’. Second derivatives are shown by dashed lines. al=Pl= -4; h=3.



NORTN

I 0. : 0”

I, = 13S0

FIG. IO. Contour diagram of the total field and its vertical second derivative when ZI=D,=O~, 1=60”,

IO= 135”. Second derivatives are shown by dashed lines. ~l!=p!= -4; 11~3.


Magnetic Anomalies, Prism-Shaped Bodies

SUMMARY AND CONCLUSION

In this paper have been presented the results

of an investigation on the anomalies produced by

prism-shaped bodies with the total polarization

vector arbitrarily oriented with respect to the

earth’s field vector. Formulas of the total field

and its first and second vertical derivatives for

all possible cases in connection with a prism can

be obtained either directly from this paper or by

simple extension of the formulas given here.

The diagrams for some simple cases show how

pronounced the influence of permanent magnetiza-

tion may be on the total field and its vertical

second-derivative contours. Asymmetry in con-

tours is, in general, a rule in the presence of

remanence. The zero contours of the second deriv-

ative do not normally, as is often assumed, out-

line the body, and the maximum may occur at

quite some distance from the center of the body.

The problem of interpretation of anomalies may

thus become exceedingly difficult in cases where

magnetization due to induction plays an insig-

531

nificant role compared to that due to remanence.

Attempts to simplify this problem look to be

futile. Some special problems may call for very

simple answers, but, unfortunately, they are only

a few of the various possibilities.

REFERENCES

Hall, D. H., 1959, Direction of polarization determined from magnetic anomalies: Jour. Geoph. Research, v. 64, p. 19-L-1959.

Henderson, R. G., and Zietz, I., 1948, ;\nalysis of total magnetic intensity anomalies produced by point and line sources: Geophysics, v. 13, p. 428.-436.

Hughson, J. ‘I., 1964, The calculation of total-intensity magnetic anomalies for certain bodies by digital computer: Geophysics, v. 29, p. 54-66.

Hutchison, R. D., 1958! Magnetic analysis by logarith- mic curves: Geophysics, v. 23, p. 749-769.

Smellie, D. VV., 1956, Elementary apmosimations in aeromagnetic interpretation: geophysics v. 21, p. 1021-1040.

Sutton, D. J., and Mumme, \V. G., 19.57, The effect of remanent magnetization on aeromagnetic interpretation: ;\ustralian Jour. of Physics, v. 10, IL 547-557.

Vacquier, Victor, Steenland, N. C., Henderson, R. G., and Zretz, I., 1951, Intermetatron of aeromagnetic maps: Geol. Sot. .\merica Mem. 47, 151 p.


Bhattacharyya - Magnetic Anomalies Due to Prism Shaped Bodies With Arbitrary Polarization

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