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1,—_

LIBRARY

Michigan State
University

 

 

 

 

 

 

 

 

ABSTRACT
Two DIMENSIONALITY
IN MAGNETIC INTERPREEATION
By
Jeffrey J. Daniels

Although the two dimensional approximation is often
applied to three dimensional magnetic sources, rarely is
consideration given to the validity of this assumption.
For a body which can be considered a thin sheet, a finite
length correction factor is derived for vertical magnetic
intensity. This correction factor is also found to be
applicable to the total magnetic intensity for a thin sheet
at high magnetic latitudes.

Analysis of the error incurred by applying the two
dimensional approximation to vertical thick tabular sheets
covering a wide range of body parameters indicates that
(l) the error for N-S striking bodies is greater than for
E-w striking bodies, (2) as the L/W ratio increases, the
percentage error decreases. and (3) the error increases as
the depth extent increases.

Application of depth determination techniques to
tabular bodies of varying strike length indicates that in
general the results vary less than ten percent when L/W

is greater than six.

TWO DIMENSIONALITY
IN MAGNETIC INTERPRETATION

By
5""
Jeffrey J. Daniels

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Geology

1970

C, 95/32 :17
/- /:3“" ’I’/

ACKNOWLEDGMENTS

The author wishes to sincerely thank Dr. w. J. Hinze
for his guidance and interest during the preparation of this
study. Acknowledgment is also made to Dr. D. W. Merritt,
Dr. H. B. Stonehouse and Dr. J. W. Trow for their sugges-
tions and helpful criticism pertaining to this study.

Thanks is also expressed to Michigan State University
for the use of the CDC 3600 computer.

11

TABLE OF CONTENTS

LISTOFTABLES................
LISTOFPIGURES................
Chapter
I. INTRODUCTION.............
II. TWO DIMENSIONALITY OF THIN SHEETS. . .
III. TWO DIMENSIONALITY OF THICK SHEETS . .

IV. DEPTH DETERMINATION METHODS AND THEIR
RELATIONSHIP TO TWO DIMENSIONALITY .

V. CONCLUSION . . . . .
BIBLIOGRAPHY . . . . . . . .

111

32
38
1&0

LIST OF TABLES

Table Page

1. Midpoint error incurred using the thin
sheet assumption . . . . . . . . . . . . . . 6

2. Maximum total magnetic intensity of thick
two and three dimensional bodies . . . . . . l9

3. Percentage of stations greater than ten
percent in error for N-S striking bodies . . 2h

h. Percentage of stations greater than ten
percent in error for E-W striking bodies . . 25

5. The variable coefficient for half-width
depth determinations as L/h and H/h
18 vuied. . O O C C O C O O C O O C 0 0 C 0 3h

6. Comparison of depth determinations from

anomalies derived from finite length,
vertical tabular sources . . . . . . . . . . 36

iv

Figure

10.

11.

LIST OF FIGURES

Illustration Of Symbols Used In This Study . .
Variation Of NZ” /Zzo) With l/(x' + z“)!i . . .

Comparison Of Calculated Three Dimensional
To Corrected Two Dimensional Total Intensity
values For E-w Striking Thin Sheets. . . . .

Comparison Of Calculated Three Dimensional
To Corrected Two Dimensional Total Intensity
values For N-S Striking Thin Sheets. . . . .

Examples Of Two And Three Dimensional
Total Intensity Profiles . . . . . . . . . .

Examples Of Two And Three Dimensional
Total Intensity Profiles . . . . . . . . .

Percentage Error For The Maximum Total
Intensity For N-S Striking Bodies. . . . . .

Percentage Error For The Maximum Total
Intensity For E-W Striking Bodies. . . . . .

Percentage 0f Stations Greater Than Ten
Percent In Error versus The L/W Ratio
For N-S Striking BOdeSo o o o o o o o o o 0

Percentage 0f Stations Greater Than Ten
Percent In Error Versus The L/w Ratio
For E-w Striking Bodies. . . . . . . . . . .

Body Parameters For Example Of Application
Of TObIOB 3 And he 0 o o o o a o o e o o o c

Page

10

11

16

17

21

22

28

29

30

cmmmi
INTRODUCTION

A commonly made assumption in magnetic interpretation
is that the source of magnetic anomalies may be approxi-
mated by a theoretical body of infinite strike length. A
theoretical body whose strike length is infinite may be
considered a two dimensional body. This assumption is
utilized in curve matching interpretation techniques and
magnetic depth determination because of the resultant
simplification of magnetic theory. This simplification
expedites computations. which is important even in computer
calculations, and results in easily and rapidly applied
interpretational techniques. Theoretically this assumption
can only be applied to magnetic anomaly profiles which
would remain unaffected by an increase in the strike
length of the source. Practically, however, the two
dimensional assumption is applied with much less rigorous
criteria.

Several rules of thumb have been stated which provide
guidelines for the practical application of two dimensional
theory. Hutchinson (1958) proposed that the magnetic
contours must be parallel for a distance at least equal
to the length of the profile required for analysis and
should exceed the distance between the inflection points

1

2
of the profile. According to Koulomzine and others, (1970)
”when a dike has a length of more than ten times its width
or depth of the top (whichever is the greatest) and a
similar extent in depth. the conditions rather rapidly
tend towards the limiting case of an infinite dike."
McGrath and Hood (1970) also state that a body may be con-
sidered to have an infinite strike length if the strike
length of the body is equal to or greater than ten times
the depth to the top of the body. No theoretical proof and
little empirical Justification has been given for these
guidelines.

The purpose of this study is to investigate the
applicability of two dimensional theory to three dimen-
sional sources in magnetic interpretation. This problem
could be solved by determining correction factors from the
ratios of two dimensional to three dimensional formulas
of magnetic sources of the same transverse geometry. This
approach has been utilized by Nettleton (l9h0) in deter-
mining the correction factor for a horizontal line element
of finite length. Unfortunately. the mathematical expres-
sion obtained by performing this division is, in general,
no simpler than the three dimensional formulas. One
exception to this statement is the case of a vertical
magnetic anomaly from a thin sheet of infinite depth
extent. This case is investigated theoretically and
checked against calculations. In addition, the limits of

the application of the thin sheet formula are investigated.

3
The application of two dimensional theory to three

dimensional, thick. vertical tabular bodies at 75 degrees
magnetic inclination is studied for a range of widths
(ranging upward from the thin dike limit), lengths,
orientations and depth extents by comparing calculated
values over the positive portion of the anomaly. These
comparisons are used to draw general conclusions con-
cerning the conditions under which two dimensional theory
may be applied to three dimensional bodies.

The effect of finite strike length on depth determi-
nation methods are determined over a wide range of condi-
tions. The half-width and Peter's half-slope methods,
which are both based on the assumption of infinite strike
length, are tested as well as the straight slope method.

A diagram of the vertical tabular sheet investigated
in this study is shown in Figure l with the symbols used

in the formulas.

 

 

 

 

 

 

Floors I: lllustratten Of Symbols Used In This Study

CHAPTER II
TWO DIMENSIONALITY OF THIN SHEETS

In magnetic interpretation, anomalies arising from
long tabular sources with an apical width much less than
the depth to the top of the source are treated by the so
called "thin sheet" or "thin dike" formulas. The advantage
of these formulas over the more general thick sheet formulas
is their relative simplicity. As a result calculations are
less tedious and magnetic interpretative generalizations
are more easily made. One result of the simplified
formulas is that correction factors for the finite length
of a thin sheet can be easily determined for vertical
magnetic intensity over the center line of the sheets.

The equations used in this chapter assume that the
body used in the calculations may be considered a thin
sheet. In practice, Werner (1953) states that the thin
sheet is applicable when its apical width is less than
half the depth to the upper edge of the sheet (i.e., the
thin sheet is applicable for a vertical sheet when the h/W
ratio is greater than or equal to two). To verify this
rule, total magnetic intensity values were calculated
along a profile perpendicular to the strike of a two
dimensional body extending infinitely deep. using both
the thin sheet formula and the more general thick sheet

5

6

formula. The body can be considered as approximating a
thin sheet for widths whose total intensity values calcu-
lated by the thick and thin sheet formulas closely agree.

The results of comparing the thin and thick sheet
formulas for various h/W ratios are shown in Table 1.
This table shows that the error in the total magnetic
intensity over the midpoint of the sheet is less than or
equal to two percent when the h/W ratio is greater than or
equal to two. Even for values of h/W = l the error is less
than ten percent. Gay (1961) points out that thick dike
curves do not depart significantly from thin dike curves
until h/W is less than one. In all cases tested the mid-
point error is greater than the average error for the
positive portion of the profile and the values obtained
from the thick dike are greater than the thin dike values.

Table l. Midpoint error incurred by using the
thin sheet assumption.

strike a N-S strike a E-w

h/W midpoint % error h/W midpoint % error
10.0 +0.09 10.0 +0.09
5.0 +0.34 5.0 +0.3“
3-3 +0-75 3-3 +0-75
2.5 +1.31 2.5 +1.31
2.0 +2.01 2.0 +2.02
1.? +2.86 1.7 +2.86
1.“ +3.82 1.“ +3.82
1.3 +u.88 1.3 +5.13
1.1 +6.0b 1.1 +6.u3
1.0 +7.28 1.0 +7.85

For a two dimensional, thin, vertical sheet extending

to an infinite depth the mathematical expressions for the

7
vertical (Z) and horizontal (X) components as derived by

Werner (1953) are as follows:

M

for the two dimensional case (1) X“): -2W (x‘+ z‘)

-zM +xM
and (2) 2": -2w W

while for the three dimensional case along the center line:

1 (3M + 2M
(3) X: o: "NEH-+1“. z z)§ ( (1'4- 27$— M)

-2w1 (_zM + xM )
and (h) Z30 =(x‘+ 21+Ir72 (x’+ 2')

Correction factors for the finite length of thin vertical
sheets can be derived by dividing equation (3) by equation
(1) and equation (h) by equation (2). The results are:

 

i (1H.,+ 2M1) -
(5) fix” A») " (xM,,- 2m) (x‘+ z‘+ 1‘ )9 (

8 8 1
2:32.31, + z ) and (6) Fwy/Z...) = mi

where the field parameters are represented by the same
symbols as those given in Figure l. M‘ for the simple
induction case is the product of the magnetic suscepti-
bility. the total magnetic intensity and the cosine of the
inclination of the earth's magnetic field while M" equals
the product of the susceptibility. the total intensity and
the sine of the inclination.

Enuation (6) is the finite length correction factor
for the vertical magnetic intensity due to a thin vertical
sheet and is equal to Nettletons (l9n0) gravity correction

factor for a horizontal line element. The variation of

8
NZ” /Z“) as a function of the parameters x. z. and l is
shown in Figure 2. The values used in this figure are the
same as those given by Nettleton (l9h0) in his table of
F(l/P) and l/P values for the gravitational effect of a

horizontal line element, where Q =’(x‘+ z‘)§.

 

 

 

 

FVHKSJLO : a e

0.9.

0.8‘ .
0.7 « ~
0.6 - .
0.5~ -
(14+ r
0. 3- »
0.2 ‘ »
O.l - ~

0.: 03 0T5 1T0 3T0 5.0 1/(x'tz‘)

Figure 2.‘ variation of F(Z,°/Z“,) with l/(x‘ + 2‘);

The finite length correction factor as shown in equa-
tion (6) is unaffected by variations in the orientation of
the body with respect to the magnetic field, or the width
of the body as long as the body can be considered a thin
sheet. The equation for the correction factor for hori-
zontal magnetic intensity (equation 5) is too complex to
be useful, thus a correction factor for the total magnetic
intensity which is in a simple, easily applicable form
does not exist. However. the correction factor, F(Z3,/Zfl,),
can be applied to the total magnetic intensity profile of
a thin sheet at high magnetic latitudes with a minimum of

error.

9

Figures 3 and h are comparisons of the total magnetic
intensity anomalies of thin sheets calculated directly
from the three dimensional equations and total intensity
two dimensional thin sheet values corrected by the finite
length correction factor for vertical magnetic intensity.
Figure 3 is for E-W striking vertical bodies with a depth
to the top of unity, extending to an infinite depth with
l/h 8 5 and l/h = 5000 respectively. The bodies were
assumed to have a magnetic susceptibility of 0.002 emu/cc
in an earth's magnetic field inclined 75 degrees from the
horizontal at an intensity of 50,000 gammas. In all of
the cases cited little error is incurred by applying the
finite correction factor to the two dimensional total
intensity. This error for N-S striking. vertical thin
sheets (Figure h) is less than 0.001 percent. The differ-
ence between the corrected and computed total intensity
values is greater for E-W striking bodies (Figure 3). but
the error for the central part of the anomaly (x a -2.5
to 2.0) is less than ten percent.

To apply the finite magnetic correction factor the
same procedure is used as Nettleton (1900) suggests for
the gravity correction factor of a horizontal line element,
except that it may only be applied on a perpendicular to
the center of the strike length of the body. First, the
two dimensional vertical magnetic intensity (or total if
applicable) is calculated. then the finite correction
factor is determined and multiplied by the two dimensional

value to obtain the three dimensional magnetic value.

10

 

strlhe 8 E-VI +qommos IN. = 5
eeleeleted three 8 ‘
dlmeeslorocl _o__
corrected two 0‘
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01
40«
30+
/ 20+
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s 1 Q ' ' rk— a v N

 

 

 

 

/ '01 .
sf - - - . \fi ' N
40 -e -e -4 -2 0 2 4 IO x/r.

Fleets 31 Ceetperlsen Ot Coleuleted Three Olmenslcnel To
Corrected Two Dlmeaslenel Tetel letenslty Values For

E-w Strlklng ledles

 

 

 

11

 

 

 

 

 

 

sniheSN-S
adcuiated three
dimensional -O—— V" ’5
corrected two
dimensional __,_.
/ ‘04) \
int/I - _ - - - 9H
-6 -4 -2 0 2 4 6 Kill
aarnaes
:i/h 35000
301
e 20‘
./ '0) \.
-i . .
«4:6 , - I , - -.\.—’LE
4 -4 -2 O 2 4 6 X/h

Figure 4: Comeeriscn Oi Calculated Three Dimensional

To Corrected Two Dimensional

For N-S Striking Bodies

Total intensity Values

12
This procedure was applied to thin sheets with N-S and E-W
orientations with an inclination of the earth's magnetic
field of 75 degrees. The results of applying the finite
length correction factor to the vertical magnetic intensity
of two dimensional sheets gave the vertical intensity of a
three dimensional sheet of the same length. Assuming that
the inclination of the earth's magnetic field is vertical,
this correction factor can be applied to points other than
those on a perpendicular to the center of the body. If the
lengths of the body on the two sides of the perpendicular
profile on which the values are calculated are 11 and 12
and the corresponding correction factors are F. (Zn /Z,_D )
and F‘ (Z30 /Z" ), then the two dimensional magnetic value
can be multiplied by (F. (Zn /Z“,) 4- Paula /Z“ )/2 to yield
the magnetic effect of a finite body. This procedure was
applied to points off the center line of a three dimensional
body and found to equal values calculated directly from
the three dimensional formulas.

In summary. a factor for correcting the vertical
magnetic intensity of a two dimensional, vertical thin
sheet of infinite depth extent for finite length is derived
and is shown to apply to all orientations and inclinations
of the earth's magnetic field. The correction factor which
is simple and easy to apply is equivalent to the finite
length correction factor for the gravitational effect of
a horizontal line element. The thin sheet formula is

applicable to h/W ratios of two or more with little error

13
and h/V ratios of greater than one with less than ten

percent error.

CHAPTER III
TWO DIMENSIONALITY OF THICK SHEETS

Magnetic anomalies arising from long tabular sources
whose width is greater than the depth to the top of the
body cannot in general be treated as thin sheets. As a
result mathematical expressions must be modified, thereby
losing the simplicity and generality that is characteristic
of the thin sheet formulas. In order to isolate generali-
zations concerning the two dimensionality of thick sheets
it is necessary to compute the magnetic anomalies of thick
sheets over a wide variety of body parameters.

The total magnetic intensity profiles of 126 bodies
were calculated assuming the intensity of the earth's
magnetic field equals 50,000 gammas and the magnetic
susceptibility of the body is 0.002 emu/cc. The depth to
the top of the vertical tabular bodies was set equal to
one. Since the three dimensional computation requires
numerical integration in small increments over the
vertical extent of the body it is necessary to use a finite
depth extent for all bodies. Body parameters were based
upon H/h values of 5, 10, and 20 and H/V ratios of l. 5,
and 10 which gives an h/W range of 0.05 to 2.0. This
range includes a thin dike of limited vertical extent and

10

15
yields an equal h/W case for H/h values of 5 and 10 and
10 and 20.

Figure 5 shows the two and three dimensional profiles
through the maximum point of the anomaly for E-W and N-S
striking bodies with H/w = 1, L/W = 100, and h/W a 0.05.
and H/h = 20. For this body, as well as for most of the
other bodies tested, the sign of the difference between the
three and two dimensional total magnetic intensity remains
constant over the entire positive portion of the anomaly.
In addition, this is an example for which the error is
greater for the N-S case than for the E-W case. In both
instances the error is quite consistent over the entire
positive portion of the profile.

Figure 6 shows the two and three dimensional profiles
through the maximum point of the anomaly for E-W and N-S
striking bodies with H/h = 5. h/w = 2, H/w = 10 and L/W = a.
This body is one of the few examples in which there is a
crossover of the two and three dimensional profiles in the
positive portion of the anomaly. In this example, as well
as with the other cases in which a crossover occurs, the
crossover occurs near the zero amplitude portion of the
profile thereby having little effect upon the error
analysis. In considering the effect of a vertical, thin
sheet whose depth extent is theoretically infinite, it was
shown that the two dimensional anomaly is always greater
than the three dimensional anomaly. However, for vertical
thick sheets of finite depth extent the three dimensional
anomaly may be greater than the two dimensional anomaly,

16

two dimensional —0—
three dimensional _._
H/w= l, h/W8.05

H/h =20, L/W 8 mo

 

300.
°\strihe fE-W
200‘ \o
\O
lOO- 9\
O
v T I \.I N
W'sz-e'd' 4 8 m
. , .-——(r— 9“.
‘ gammas
400

 

300‘

2004

L00,

l-\\

 

 

O

\. strike 8 N- S

Figure52 Examples Of Two And Three Dimensional

Total intensity Pretiles

’17

two dimensional ___._ OJOOMMOO strike ‘E-W
three dimensional _,_ 0a

ti/hzs, h/w:2

H/w=l0,L/w=4 ' {/co\\o

 

 

'3 '2 -i 0 l E '3 X/h

7% \:
/.. \\

/ \
/ "\\

 

é X/h

Figure 62 Examples Of Two And Three Dimensional
Total intensity Profiles

18
as in Figure 5 or less than the two dimensional anomaly
as in Figure 6.

Table 2 gives the maximum two and three dimensional
total magnetic intensity and percentage difference for all
bodies computed. The sign of the difference (three dimen-
sional minus two dimensional) is positive in most cases.
Exceptions generally occur for low values of the L/W ratio
for bodies that approach the thin sheet approximation. The
graphs of Figures 7 and 8 illustrate the variation in the
percentage difference of the maximum amplitude as the
parameters are changed. It should be noted that for groups
of curves of constant H/h ratio, the variables are only L
and W. The main generalities to be derived from these
graphs are: (1) for a given H/W, H/h and h/W ratio the
shape of the percentage error curves derived from N-S
striking bodies correlates with the E-W curve (2) for a
given set of characteristic ratios the percentage differ-
ence is greater for a N-S striking body than for the
corresponding E-W striking body (3) as the L/W ratio is
increased the percentage difference between the maximum
total intensity of three dimensional and two dimensional
sources decreases and (0) negative percentage error values,
representing a situation in which the maximum two dimen-
sional total intensity is greater than the three dimen-
sional total intensity, occur only for L/W values less
than ten

Although comparing the maximum values does give some

idea as to the sign and magnitude of the difference between

19

Table 2. Maximum total magnetic intensity of
thick two and three dimensional bodies

N-S E-W N-S E-W N—S E-W
3-D 3-D 2-D 2-D zerrcr flhrror

H/h==5. H/Wal, h/Wz. 2

 

 

 

L/W=2 366.9 361.3 296.9 325.3 2a 11

u 36u.8 3b3.3 23 6

6 363.8 33u.8 23 3

8 363.3 331.0 22 2

10 363.1 329.1 22 1

100 307.6 325.3 a 0

1000 296.9 325.3 0 0

H/h=5, H/W=5. h/W=l

L/w=2 117.3 116.9 192.0 1a2.6 -17 .13

u 1u7.a 1&5.9 n 2

6 15a.9 151.9 9 7

s 156.5 152.5 10 7

10 156.8 151.7 10 6

100 156.0 1&3.R 10 0

1000 1&2.0 1u3.6 0 0
H/ha5, H/Wle, h/W=2

L/W=2 39.9 39.9 75.9 76.3 -h7 -h9

h 62.9 62.9 -17 -19

6 72.8 73.3 -u -u

8 78.8 77.9 a 2

10 s1.1 79.9 7 5

100 82.8 76.7 9 1

1000 77.0 76.3 1 0
H/h=10, H/W=l. h/W=.l

L/W=2 u71.1 u6o.u 353.3 u18.3 33 10

u u68.0 u37.2 32 5

6 u66.9 028.0 32 2

8 #66.5 uzh.1 32 1

10 u66.3 h22.1 32 1

100 386.9 u18.u 10 0

1000 353.3 u18.3 0 0

 

20

 

 

 

 

N-S E-W N—S E-W N-S E-W
3-D 3-D 2-D 2-D %error %error
H/heio, H/W=5, h/W=.5
L/W=2 257.7 256.6 259.3 265.1 -1 -3
9 282.1 278.9 9 5
6 286.2 279.2 10 5
8 286.7 277.3 11 5
10 286.7 275.2 11 9
100 266.3 265.3 3 0
1000 266.3 265.1 3 0
H/h-io, H/W=10, h/W=l
L/W=2 119.7 120.9 156.1 156.0 -23 -23
9 159.8 152.6 -1 -2
6 169.9 161.3 157.3 5 3
8 167.8 163.9 7 9
10 169.1 169.5 8 5
100 159.6 157.7 2 0
1000 159.6 157.3 2 0
H/h=20, H/w=1, h/W=.05
L/Wsz 558.7 595.0 383.2 999.9 31 9
9 555.2 519.8 31 9
6 965.7 510.5 17 2
8 965.3 506.5 17 1
10 96 .0 509.6 16 1
100 96 .6 500.8 16 0
1000 386.6 500.7 1 0
H/h=20, H/W=5. h/W=.25
L/W=2 906.5 909.7 377.8 399.1 8 1
9 923.0 916.5 12 9
6 929.9 919.5 12 9
8 925.0 911.5 12 3
10 929.8 908.8 12 2
100 399.8 398.1 9 0
1000 377-5 397.9 0 0
H/h=20, 8/8210, h/W=.5
L/w=2 259.2 258.1 275.9 280.5 -6 -8
9 288.5 289.2 5 1
6 295.1 288.3 7 3
8 297.1 288.6 8 3
10 297.9 288.0 8 3
100 282.5 279.1 3 o
1000 282.5 278.7 3 -1

21

Hill h/W W

L\ (a)
5 .2 I .. .

 

 

 

 

 

 

 

 

 

(e)
IO .5 5 1,
(1)
IO l IO - fi—
1 x 1 (g)
20.05 I r \
L (h)
20 .25 5 f.-. X -
(ll
20 .5 lo
‘10» ”L
SC'OIC: mi 5 z z 37. ..'. 75. W
-s-

Figure 72 Percentage Error For The Maximum Total Intensity
For N-S Striking Bodies

(a)

 

L>5 -
5 ' 5 }_74§ (bl

(c)

 

 

to .l l {—5.1 . (d)
Lfla

(e)

 

(f)

 

is)

 

 

L} .
20 .25 5 L“ . ‘1’
L

Scale: "

v V ‘7 f
,4. O 4 d I re roe leee

 

 

‘lw

Figure 8: Percentage Error (((30-20)/30)Xi00) For
The Maximum Total intensity For E-w Striking Bodies

(a)

 

5 ' 5 1'7“? (b)

(c)

 

 

l0 .i l Lha— , (d)
Lama

i0 .5 a (i)

 

(f)
to i l0 .--~-

 

2O .05 i

 

20 05 D 1%"-‘ . —- -(.'71

Scale: "

 

 

 

Figure 82 Percentage Error (((30-20)/30)Xl00) For
The ktaximum Total intensity For E-w Striking ladies

23

the three and two dimensional anomalies. it is of limited
value when considering the total profile length. This can
only be done by sampling and comparing values along the
entire profile. The percentage of profile points sampled
on the positive portion of the principal profile that are
greater than ten percent in error are shown in Tables 3
and h. The fractional error for a given profile point
equals the three dimensional total intensity minus the two
dimensional total intensity divided by the three dimensional
total intensity. The principal profile is oriented per-
pendicular to the strike of the body through the point of
maximum intensity to the nearest 0.1 depth unit. A
sampling interval of 0.2 depth unit was used for the
entire positive portion of the profiles obtained from E-W
striking bodies. resulting in a minimum of thirty stations.
For N-S striking bodies. a sample interval of 0.1 depth
unit was used from the center point to the zero magnetic
value. This procedure, although different from that used
for E-W bodies, resulted in approximately the same number
of stations. Checking this procedure against 0.2 depth
unit interval over the entire positive portion of the
profile produced a maximum difference in the results of
two percent. By using all of the positive profile points
rather than Just the maximum point, variations in the
shapes of profiles are also being utilized.

values for the parameters of thick bodies were chosen
to give maximum generality and applicability to the study.

The L/W ratios were chosen to give as broad of a range of

 

2*

 

 

 

 

 

 

 

 

H/w Miss - MW
:0 «7 :00 :00 62 74 BO 33 2 r 2
5 . :00 74 so 90 92 89 0 - :.
i ~ :00 :00 :00 :00 :00 29 3 .2
T T 8 3 :0 :50 :000 um:
i-l/hth
:0 T :00 65 s: so as 54 54 r :
s . cs as 92 95 94 50 53 l .5
l . :00 :00 :00 :00 :00 40 0 9 :
2 i : i 6 lb :00 :000 mu.
*‘H/hazo
:0 i re 90 92 93 94 97 87 . e
s < 90 9: 92 92 9: 73 40 . .25
: . :00 100 :00 :00 :00 :00 5 . .05
'l
2' i c 8 70 :00 :000 L18:

Table 32 Percentage Oi Stations

in Error For N-S Striking Bodies

Greater Than Ten Percent

 

25

 

 

 

 

 

 

 

 

n/w w/nzs rim
:0 T :00 9e 75 52 7: 22 2 2
s . 94 52 76 7e 77 :2 0 :
l < :00 60 40 25 24 0 0 - .2
2 7i 6 J :0 :00 :000 L18!
H/h8iO
:0 lOO 65 65 74 so :2 0 7 :
5 i 55 7s 65 s: 77 6 0 ~ .5
: 4 86 44 27 2: :9 0 0 - .:
2 4i 6 5 lb :00 :000 L/w
ti/h=20
i0 94 69 55 66 86 7: 70 f .5
5 60 or s: 79 77 62 62 - .25
: 86 43 3: 25 23 :6 :6 ~ .05
2 i 3' 8 :‘0 :00 7600 L/w

Table 4: Percentage
in Error

For

0t Stations Greater
E-W Striking Bodies

Than Ten Percent

26

values as possible. A variation in increments of L/W of
one depth unit were tested for several cases in the (0-10)
range and the results did not vary appreciably from those
using an increment equal to two units. Thus. results were
obtained for L/W ratios of 2 increments up to 10, and at
ratios of 100 and 1000. The small inconsistencies or
variations in the lower L/W ratios are probably due to the
edge effect caused by increasing the strike length. Ratios
of H/h and h/W were chosen to yield a practical range of
values.

Tables 3 and b show that in general the error for N-S
striking bodies is greater than for E-W striking bodies.
This is due to the greater positive magnetic effect of the
E-W margin of N-S striking bodies in contrast to the effect
of changes in the length of E-W margins of E-W striking
bodies.

Tables 3 and 9 also indicate that in most cases, for
both E-W and N-S striking bodies. an increase in the MN
ratio decreases the error. This is to be expected because
as the strike length increases, the body approaches the
infinite or two-dimensional case.

Figures 9 and 10 are plots of the percentage of
stations over the positive portion of the profile that
are greater than ten percent in error versus the L/W
ratio. This gives a good basis for comparing Tables 3 and
9 with the maximum values given in Figures 7 and 8. For
N-S and E-W striking bodies, Figures 7 and 8 show almost

identical curves when the h/W ratios are equal (eg.,

27
Figure 7(b) and Figure 7(f) for N-S striking bodies).
Although such similarities do exist in Figures 9 and 10,
they are not as pronounced as those in Figures 7 and 8
(eg., Figure 8(a) and 8(1) versus Figure 10(e) and 10(1)
for E-W striking bodies). Anomalous values in Figures 7
and 8 and Figures 9 and 10 correspond to the occurrence of
negative percentage differences for the maximum values
in Table 2. Figures 7 and 8 and Figures 9 and 10 show a
decrease in percentage error as L/W is increased and a
greater percentage error for N-S striking bodies as com-
pared to E—W striking bodies. In addition, these figures
show that the error increases as the depth extent increases.
Comparison of Figures 7 and 8 with Figures 9 and 10 show
that care must be used in using the maximum anomaly value
as a guide to the error in magnitude of the entire
positive portion of the profile used for analysis.

The following example serves to show how the graphs
can be used to predict the amount of error that would be
incurred by applying the two dimensional approximation to
a particular situation. Given a body shown in Figure 11
and normalizing to the depth to the top of the body.
yields an h/W = 1. H/W = 5, H/h = 5, and LN = 5.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Whit/WWW
[ (a)
5 .2 : g “g- i
i
5 l 5 ,4 :7: a -
5 z '0 4% ‘fft‘ i :—
' .
1 (d)
to .i l - -c a
‘ /-\ 1.. (c)
'0 .5 5 , ; : t 2
I W* m
'0 ' ’0 _: :71; i “t
i \ (g)
20 .05 I A t::‘ L f
20.25 5 s s: 1 s i:
f \
(l)
20 .5 i0 v .e.: .1
$e'“.
“i
. 2
Scale. 5 f 3 ”5 .3. .r.. to

Figure 9: Percentage 0t Stations Greater Than Ten Percent
in Error Versus The L/w Ratio For ti-S Striking Bodies

29

 

 

Mil/WWW
{ \ (a)
5 2 I A T?::+ \-
i (b)
5 i 5 I -1

 

 

L (c)
5 2 '0 - : :::. -
L¥¥ (6)
l0 .l i f ,5- s a
' \\ (.)
i0 .5 5 . -t:f 4
t ' (r)
iO i l0 - if: 1

 

 

 

 

 

 

 

 

 

 

20 .05 ' : ;'f 3 1
LA in)
20 .25 5 t :':, , ‘L
W
L (i)
20 .5 IO - v r,-- c i
“r g
1
O
Q
Scale: " , --.- . -
I 400‘ I0. noel,“

Figure i0: Percentage Of Stations Greater Than 'Ten Percent
in Error Versus The L/ltl Ratio For E-VI Striking Bodies

29

 

 

 

M
‘i
1
fl
0
l
in
p
A
Q
U

l0 .i

l \\ (.)
l0 .5 5 r -¢:.A :

L . (f)
IO i no A“: A

 

 

 

 

 

 

 

 

20 .05 l - ,f c L
4 A m
20.25 5 t :v;, - _5_
W
L
20 .5 IO - , “fl 1 t
M

Q [I I:l
ee
0
Scale : "

Figure i0: Percentage 0t Stations Greater Than Ten Percent
in Error Versus The L/W Ratio For E-W Striking Bodies

V

"'f V
i 4000 re. “00%

30

(not to scale)

 

5000'

l

  
 

Figure 11. Example figure for the application of
graphs in Tables 3 and h.

Table 3 is chosen for consideration because the H/W and
h/w values correspond closely to the parameters of the
example. The graph shows that in this case about 80
percent of the profile points would be expected to be
greater than 10 percent in error. In this situation it
would be advisable to use a three dimensional analysis on
the body.

In summary, the error due to applying the two dimen-
sional approximation to model a magnetic anomaly caused by
a thick vertical prismatic body of finite depth extent
decreases as the L/W ratio increases. In addition. with
few exceptions. for the thick vertical sheets tested, a
greater percentage of error occurs when applying the two
dimensional approximation to a magnetic anomaly caused by
a N-S striking body than for an E-W striking body. The

difference between magnetic values derived from two and

31
three dimensional bodies remains rather constant with in-

creasing L/W for large values of H/W’and H/h.

CHAPTER IV

DEPTH DETERMINATION METHODS AND THEIR RELATIONSHIP
TO TWO DIMENSIONALITY

Depth determination methods are commonly dependent
upon the assumption of two dimensionality. therefore errors
are anticipated in the determinations as the length of the
source of the anomaly decreases. The relationship of this
error to the length of tabular bodies has been the subject
of little discussion in the literature. However, McGrath
and Hood (1970) show for a self-adjusting curve fitting
procedure that the percentage error in the depth determi-
nation of a thin dike falls off as a function of L/h. the
error being less than one percent when L/h is greater than
20.

In an effort to determine the relationship between
depth determinations and the length of vertical sheets,
Peters' half-slope method, the straight slope method and
the half-width.method were applied to profiles drawn from
the total magnetic intensity values used for the graphs
in Chapter III. Maintaining a consistent scale for all
profiles, the estimated error for the determinations is
approximately I 0.05 depth units. Consistent with prac-
tical application. the north or steep slope side of the

anomaly was used for determinations for the case of an

32

33
E-w striking body. For Paters' method (19h9) the depth
to the top of the body equals the product of a variable
coefficient which is dependent on the h/W factor and the
horizontal distance between points of tangency of lines
that have one-half the maximum slope. In this study the
primary interest is in relative changes in the depth,
therefore, the variable coefficient was arbitrarily set at
one unit. With this method, as well as the other methods»
used in this study, absolute values have no validity
because no attempt has been made to modify the depth deter-
mination techniques for the limited vertical extent of the
sources, the variable h/W ratio, and the effect of trans-
verse magnetization. In addition, the half-width and
Peters' slope methods are based upon the use of vertical
intensity values in the depth determinations. Peters'
method is based upon induction theory. The assumptions
made in arriving at this method are that: (l) the
anomalous mass is in the shape of an infinitely long slab
with vertical sides which has a horizontal top and extends
to an infinite depth and (2) it is uniformly magnetized in
the vertical direction.

For the straight slope rule, the depth to the body
equals the product of the horizontal distance between the
lower and upper points of the apparent straight segment
of the flank of the curve and a variable coefficient which
is assumed to be one in this study. There is no theoretical
basis for this method although vacquier and others (1951)
achieved excellent results with the technique.

3h

With the half-width rule the depth to the top of the
body equals the product of a variable coefficient and the
distance equal to one-half the width between the flanks
of the profile at an amplitude of one-half the maximum
intensity. The method assumes a two dimensional body while
strike and depth extent is considered infinite. vertical
magnetization is also assumed. Table 5 is given by
Parasnis (1962) for variations in the half-width coeffi-
cient used on vertical intensity anomalies for a variety

of finite bodies.

H/h
L/h 1 2 u a.
0.0 1.99 1.5a 1.37 1.31
1.91 1.u5 1.26 1.18
u.0 1.88 1.u3 1.21 1.08
8.0 2.02 1.u8 1.20 1.03
0° 2.06 1.53 1.26 1.00

Table 5. The variable coefficient for half-width
depth determinations as L/h and H/h is
varied. (after Parasnis, 1962)

The results of applying the depth determination tech-
niques to the profiles used for the graphs in Chapter III
are shown in Table 6. As the length of the anomaly source
changes, the amplitude and width of the anomalies vary in
relatively constant proportions. Thus the depth deter-
minations, which are largely based on anomaly gradients,
remain nearly the same. For Peters' method the depth
determined when L/W = 2 is generally lower than when
L/W = 1000. values for an L/w greater than or equal to 6

35

are relatively constant and have a maximum deviation
(with one exception) of not greater than ten percent.

Depth determinations using the half-width method are
erratic for L/W = 2. There is generally an increase and
then a decrease in values for an L/W ratio between 6 and
1000. The variation of depths in this range is generally
not more than ten percent. As anticipated. comparison of
Table 6 with Table 5 (Parasnis' correction factor table)
shows that with a variation in the strike length of the
body an inverse correlation exists between the variable
coefficient and changes in the uncorrected depth determined
by the half-width method.

As with the other methods the depths determined by
the straight slope procedure for L/W a 2 are erratic.
'Although little absolute variation exists for values for
an L/W ratio between 6 and 1000. the percentage deviation
for a given H/h and h/V ratio is, in most cases, greater
than ten percent.

From this study it is concluded that within the range
of conditions studied there is no definite correlation
between the L/W ratio and the results of the depth deter-
mined. Generally. however, the depth determinations based
on anomalies derived from bodies with an L/W ratio greater
than 6 are within ten percent of the depth obtained from
bodies with an L/w = 1000. For the bodies used in this
study Peters' half-slope and the half-width method seem
to show a more consistent variation as the parameters are

varied, than does the straight slope method.

36

Table 6. Comparison of depth determinations from
anomalies derived from finite length,
vertical tabular sources

H/h=5, H/W=1. h/WsZ

N-S N-S N-S E-W E-W E-W
éwidth Peters' straight iwidth Peters' straight
slope slope

 

 

 

 

L/sz 2.52 1.72 .u5 2.87 1.50 .uo
6 2.52 1.80 .35 2.90 1.u3 .uo
10 2.53 1.85 .u5 2.77 1.50 .uo
100 2.h2 1.85 .u5 2.77 1.50 .uo
1000 2.37 1.75 .u5 2.77 1.50 .uo
H/haS, H/wzs. h/w=1
L/Wa2 .9u 1.17 .35 .85 1.10 .25
6 1J03 1.30 .27 .85 1.05 .25
10 1.05 1.30 .25 .77 1.25 .us
100 1.02 1.30 .25 .80 1.15 .35
1000 .95 1.27 .25 .80 1.15 .30
H/hBS . EN: 10, h/W22
L/w-z .80 .99 .25 .72 .95 .35
6 .92 1.13 .25 .7u .90 .25
10 .9u 1.18 .25 .79 .95 .35
100 .9h 1.18 .25 .79 .95 .25
1000 .89 1.18 , .30 .67 .97 .50
H/hzlo, H/w=1, h/w=.1
L/wuz 9.95 1.70 .50 6.80
6 n.95 1.95 .70 6.35
10 9.97 1.95 .50 6.u0
100 n.85 1.90 .50 6.35
1000 n.77 1.85 .50 6.36
H/hsio, H/w=5. h/W=.5
L/waz 1.30 1.23 .30 1.25 1.55 .35
6 1.37 1.65 .35 1.25 1.60 .n5
10 1. 6 1.55 .30 1 25 1.55 .30
100 1.32 1.55 .35 1.22 1.55 .35
1000 1.30 1.50 .35 1.22 1.55 .35

 

37

 

 

 

N-S N-S N-S E-W E-w E-w
Qwidth Peters' straight iwidth Peters' straight
slope slope
H/halo. H/wtlo, h/Wsl
L/wez .95 1.05 .35 .85 1.15 .35
6 1.07 1.30 .30 .85 1.20 .h5
10 1.08 1.25 .35 .85 1.10 .ho
100 1.02 1.20 .30 .85 1.10 .h5
1000 1.02 1.25 .35 .85 1.20 .35
H/hazo, H/w=1, h/W=.05
L/w-Z 9.97 1.10 .75 1h.75
6 9.88 2.10 .65 13.80
10 9.87 2.00 .h5 13.90
100 9.90 1.90 .ho 13.90
1000 9.72 1.90 .30 1h.00
H/hszo, H/W25, h/w=.25
L/waz 2.20 1.80 .95 2.u0 1.70 .60
6 2.25 1.85 .h5 2.h0 1.67 .uo
10 2.30 1.95 .70 2.h0 1.80 .uo
100 2.2h 2.00 .90 2.30 1.80 .ho
1000 2.15 1.95 .90 2.h0 1.80 .hO
H/hnzo, H/Velo, h/Wh.5
L/th 1.37 .95 .25 1.27 1.55 .60
6 l.h5 1.55 .h5 1.32 1.60 .60
10 1.h8 1.65 .ho 1.30 1.65 .55
100 1.h2 1.65 .ho 1.28 1.60 .55
1000 l.h2 1.60 .ho 1.27 1.60 .50

 

 

CHAPTER V
CONCLUSION

In conclusion. it has been found:

(1) The two dimensional approximation must be used
with great caution in the interpretation of magnetic
anomalies of thin sheets and thick sheets of finite depth
extent.

(2) The magnetic anomaly of finite length thin sheets
is less than the two dimensional anomaly. However, this is
not true for thick sheets. In fact, the maximum magnetic
anomaly of finite length thick sheets is generally greater
than the maximum value of two dimensional bodies except at
L/W ratios of less than 10.

(3) For a thin sheet. a finite strike length correc-
tion factor is derived for vertical magnetic intensity.
This correction factor may be applied with minimal error
to total magnetic intensity at high magnetic latitudes.

(R) No correction factor is available for thick
sheets but studies of these bodies for finite depth extent
over a wide range of conditions shows that the error due to
applying the two dimensional approximation decreases as the
L/W ratio increases and increases as the depth extent in-
creases. In addition. with few exceptions, for the thick

vertical sheets tested, a greater percentage of error

38

39
occurs when applying the two dimensional approximation to
a magnetic anomaly caused by a N-S striking body than for
a E-W striking body.
(5) Application of depth determination techniques to
tabular bodies of varying strike length indicated that in
general the results vary less than ten percent when L/W

is greater than 6.

BIBLIOGRAPHY

Hutchison. R. D., 1958. Magnetic Analysis by Logarithmic
Curves: Geophysics. v. 23. no. u. p. 7&8-769.

Koulomzine. T.. Lamontagne. Y. and Nadeau. A.. 1970.
New Methods for the Direct Interpretation of Magnetic
Anomalies Caused by Inclined Dikes of Infinite Dike:
Geophysics. v. 35. no. 5. p. 812-130.

McGrath. P. K.. and Hood. P. J.. 1970. The Dipping Dike
Case: A Computer Curve-Matching Method of Magnetic
Interpretation: Geophysics. v. 35. no. 5. p. 831-8h8.

Parasnis. D. S.. 1966. Mining Geophysics (v.) 3 of Methods
in Geochemistry and Geophysics: Amsterdam and New
York. Elsevier Publishing 00.. p. 356.

Peters. L. J.. 19h9. The Direct Approach to Magnetic Inter-
pretation and its Practical Application: Geophysics.
Ve 1h. nae 3, pe 290-320e

vacquier. Victor. Steenland. N. 0.. Henderson. R. G., and
Zietz. Isidore. 1951. Interpretation of Aeromagnetic
Maps: G. S. A. Memoir #7, p. 151.

Werner. Sture. 1953. Interpretation of Magnetic Anomalies

at Sheetlike Bodies: Sveriges Geologiska Undersokning.
Arsbok #3 (l9h9) no. 6. Stockholm.

#0