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The geophysical analysis of magnetic anomalies Hall, Donald Herbert 1959

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Faculty of Graduate Studies P R O G R A M M E OF T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of DONALD HERBERT HALL B. Sc., University of Alberta M. A., University of Toronto IN ROOM 303, PHYSICS BUILDING THURSDAY, APRIL 30, 1959 at 3:00 P. M. COMMITTEE IN CHARGE Dr. R. D. JAMES: Chairman J. A. JACOBS R. D. RUSSELL G. L . PICKARD R. W. STEWART D. L. LTVESEY S. W. NASH V. J. OKULITCH J. F. DEUTSCH External Examiner: G. D. GARLAND University of Alberta THE GEOPHYSICAL ANALYSIS OF MAGNETIC ANOMALIES Abstract Traditional methods of interpretation of the results of magnetic surveys neglect effects due to permanent magnetization. Recent geomagnetic research on the remanent magnetization of rocks has shown this to be unjustified. Moreover techniques now being employed provide better measurements of magnetic field vari-ations than have ordinarily been available in the past. In order to take advantage of these new developments, equations for the magnetic field over a point dipole, a horizontal line of dipoles, a thin dipping sheet, a thick dipping sheet and a sloping step are derived in the cases when both the directions of measurement and polarization are arbitrary. It is found that these directions combine with other properties of the bodies to form parameters, which detemiine various fea-tures of the magnetic anomalies over the bodies. In terms of these combined parameters, it is possible to give expressions for the higher derivatives of the fields over these bodies, and to develop methods of determining the unknown parameters of the bodies when magnetic profiles over them are given. Further, it is shown that the field over four of these bodies treated can be obtained by successive differentiation of a single function. This fact is used in drawing charts for computing values of the fields and their derivatives at points along profiles over any of these bodies. Tables of the higher derivatives are given, as well as graphs showing the position of special points such as peaks and inflection points on the profiles for any direction of polarization and measurement. It is shown how these more general methods may be applied to the interpretation of aeromagnetic surveys, and examples are given, of their use in the analysis of magnetic survey data over the La-Plonge area, Saskatchewan, and Texada Island, British Columbia. In the latter area, the general question of what geo-logical information may be obtained from magnetic data is considered and a comparison is made of aeromagnetic anom-alies with structural data obtained from aerial photographs. PUBLICATIONS "Least Squares in Magnetic and Gravity Interpretation", Transactions American Geophysical Union,v. 39, pp.34-35 1958. GRADUATE STUDIES Field of Study: Physics Electromagnetic Theory .. R. Barrie Noise .in Physical Systems . . . . _ R. E . Burgess Advanced Electronics -. R. E . Burgess Advanced Geophysics J . A. Jacobs Other Studies: Differential Equations ... C. A. Swanson Probability . . . . .- S. W. Nash Computational Methods F. M. C. Goodspeed THE GEOPHYSICAL ANALYSIS OP MAGNETIC ANOMALIES by DONALD HERBERT HALL B.Sc, University of Alberta, 194-8 M.A., University of Toronto,1950 A THESIS SUBMITTED IN PARTIAL PULPILffiENT OP THE REQUIREMENTS POR THE DEGREE OP DOCTOR OP PHILOSOPHY i n the Department of Physics We accept t h i s t h e s i s as conforming to the required standard. THE UNIVERSITY OP BRITISH COLUMBIA April,1959 i THE GEOPHYSICAL ANALYSIS OP MAGNETIC ANOMALIES ABSTRACT Tr a d i t i o n a l methods of i n t e r p r e t a t i o n of the res u l t s of magnetic surveys neglect e f f e c t s due to ( permanent magnetization. Recent geomagnetic research on the remanent magnetization of rocks has shown t h i s to "be u n j u s t i f i e d . Moreover techniques now "being employed provide "better measurements of magnetic f i e l d v a riations than have o r d i n a r i l y heen available i n the past. In order to take advantage of these new developments, equations f o r the magnetic f i e l d over a point dipole, a horizontal l i n e of dipoles, a t h i n , dipping sheet, a thick, dipping sheet and a sloping step are derived i n the cases when both the directions of measurement and p o l a r i z a t i o n are a r b i t r a r y . It i s found that these directions combine with other properties of the bodies.to form parameters, which determine various features of the magnetic anomalies over the bodies. In terms of these combined parameters, i t i s possible to give expressions f o r the higher derivatives of the f i e l d s oyer these bodies, and to develop methods of determining the unknown parameters of the bodies when magnetic p r o f i l e s over them are given. Further, i t i s shown that the f i e l d over four of the bodies treated can be obtained by successive d i f f e r e n t i a t i o n of a single function. i i This fact i s used i n drawing charts f o r computing values of the' f i e l d s and t h e i r derivatives at points along p r o f i l e s over any of these bodies. Tables of the higher derivatives are given, as well as graphs showing the position of special points such, as peaks and i n f l e c t i o n points on the p r o f i l e s for any d i r e c t i o n of p o l a r i z a t i o n and measurement. It i s shown how these more general methods may be applied to the interpretation.of aeromagnetic surveys, and examples are given of t h e i r use i n the analysis of magnetic sur-vey data over the La-Plonge area, Saskatchewan, and Texada Island, B r i t i s h Columbia. In the l a t t e r area, the general question of what geological information may be obtained from magnetic data i s considered, and a comparison i s made of aeromagnetic anomalies with s t r u c t u r a l data obtained from a e r i a l photographs. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of Physics  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver $, Canada. D a t e April 6th, 1959 i i i TABLE OF CONTENTS Page 1 . INTRODUCTION 1 2. THE MAGNETIC FIELD OVER SOME ELEMENTARY BODIES 11 a. The point dipole......... 11 b. • The horizontal l i n e of dipoles... 17 c. The i n f i n i t e l i n e of dipoles.... 19 3. EXTENSION AND GENERALIZATION OP METHODS 19 a. Bodies obtainable by integration 21 ( i ) The t h i n , dipping semi-infinite sheet of dipoles 21 ( i i ) The thick, dipping polarized sheet 25 ( i i i ) The sloping step 26 (iv) Special cases 28 b. A system of equations of i n t e r p r e t -ation 29 4. METHODS OP SOLUTION 36 5. CALCULATION OP PARAMETERS OP BODIES 41 6. EXAMPLE OF CALCULATION OF PARAMETERS IN THE CASE OF AN INFINITE LINE OF DIPOLES 48 7. EXAMPLE IN THE CASE OF A THIN, DIPPING POLARIZED SHEET 50 i v Page 8. APPLICATION TO THE INTERPRETATION OP AEROMAGNETIC MAPS ,. . 54 9. THE GEOLOGICAL SIGNIFICANCE OP MAGNETIC DATA - AN INTERPRETATION OF AN AIRBORNE SURVEY OVER TEXADA ISLAND, B.C . 61 10. SUMMARY AND CONCLUSIONS 71 APPENDIX I - DERIVATIVES AND INTEGRALS OF SPECIAL FUNCTIONS 74 REFERENCES . ., 8u TABLES to follow page.* I. Derivatives of f i e l d values along traverse over a single dipole « » .... «... .»«•««..•«............16 5 1, Derivatives of F =octan~ x/h + 2 2 |3 l O g ^ ( X "rh ) . . 0 » o « . 0 0 . . . . . • » • . . « « . . . . f t«.»o.e.o.29 III'- Forms of equations f o r f i e l d s as s p e c i a l cases on page 30) IV Derivatives of F and t h e i r significance for various bodies ......(on page 31) V Derivatives of f i e l d values along traverse over i n f i n i t e l i n e VI Derivatives of f i e l d values along traverse i over i n f i n i t e l i n e of dipoles - reduced form....31 VII Derivatives o f . f i e l d values along traverse. over thin,, dipping sheet .31 VIII Derivatives of f i e l d values along traverse over t h i n , dipping.sheet - reduced form ........31 IX Derivatives of f i e l d values along traverse over thick, dipping dike, or sloping step. ..... .3.1 v i PLATES to follow page; I - Values of the function f 33 II - Values of the function f^ 34 III - P r i n c i p a l values of n0 when derivatives of the function P equal zero.............. 39 IV - Direction cosines of p o l a r i z a t i o n vector fo r "bodies where M = 0 ................... 44 V - S h i f t of peak and i n f l e c t i o n points, t h i n dipping sheet, and l i n e of dipoles ....... 45 VI - Separation of peak and i n f l e c t i o n points.. 45 VII - P r o f i l e s over horizontal l i n e of dipoles and, t h i n , dipping sheet 48 VIII - Aeromagnetic contours and p r o f i l e , La-Plonge, Saskatchewan. 54 IX - Topographic and aeromagnetic p r o f i l e s near Comet Mountain, Texada Island,B.C... 67 X - Aeromagnetic p r o f i l e over f a u l t zone east of Pocohontas Mountain, Texada Island, B.C., and topographic p r o f i l e over the southwest side of the mountain... 70 MAPS Magnetic trends Structural trends Magnetic d i v i s i o n s on the basis of f i r s t order anomalies. Magnetic d i v i s i o n s on the basis of a l l anomalies Proposed s t r u c t u r a l displacements v i i i ACKNOWLEDGMENT The author wishes to express his thanks to Professor J.A. Jacobs, University of B r i t i s h Columbia, for his c r i t i c a l examination and discussion of the work at a l l stages. The o r i g i n a l records and data f o r the aeromagnetic surveys used were supplied.by Dr. Hartley Sargent of the B r i t i s h Columbia Department of Mines and by Dr. L. W. Morley of the Geological Survey of Canada. F i n a n c i a l support to the author was given during the course of the work by the Geological Survey of Canada and the Canadian Kodak Co. Ltd. 1 1 * INTRODUCTION Research on the physics of rocks and minerals, and on the int e r p r e t a t i o n of geophysical surveys, has established that the d i s t r i b u t i o n of values of physical properties within any geol-i c a l unit may, under favourable conditions, be interpreted to give information on the d i s t r i b u t i o n of rock type and structure, on.the d i s t r i b u t i o n of elements, ind on the geological h i s t o r y of the u n i t . Prom laboratory measurements of the magnetic s u s c e p t i b i l i t y of core samples, f o r example, the p o s i t i o n of various units i n the geological section may be determined proding, 1952J. Measurements of the magnetic s u s c e p t i b i l i t y of rock samples may be used to delineate zones of varying concentrat-'i ion of magnetite i n igneous rocks, ^Hawes_, 1952J and measures of the remanent magnetization have been used to deduce.the presence of-stress, as f o r example, i n the past h i s t o r y of a geological unit |Kalashnikov,1952J. A l t e r n a t i v e l y , values of physical properties may be sampled by geophysical surveys made over the formations i n question. The l a t t e r methods have the advantage of widespread and rapid coverage, p a r t i c u l a r l y when a i r surveys are used, and the advantage of better sampling which comes from measurements on the formation i n place. It i s t h i s second method, using geophysical surveys to provide the geological information, which i s the subject of study i n the present t h e s i s . Magnetic surveys share with the other members of the so-c a l -led "force - f i e l d " group of methods of geophysics p3rant, 194-8, P.561] the advantage of speed of operation, and lend themselves well to airborne operations. Furthermore, the magnetic properties of rocks have been subject to considerable study i n 2 present-day geophysics, with a view to c l a r i f y i n g t h e i r r e l a t i o n s h i p to geological conditions. The magnetic properties of rocks have long been known to be sensitive indicators of the p a r t i c u l a r geological h i s t o r y of the rocks ^Davis,1935j . Por. these reasons, the geological i n t e r p r e t a t i o n of the data of magnetic surveys has been chosen as the p a r t i c u l a r subject of t h i s investigation, which consists of two parts. The f i r s t consists of the development of the equations f o r the magnetic force over a number of p a r t i c u l a r types of geological body, and of methods of int e r p r e t i n g the observed f i e l d , so that a more complete s p e c i f i c a t i o n of the p o l a r i z a t i o n of bodies i s possible from the survey data than i s commonly obtained at the present time. The second i s an application of the methods of i n t e r p r e t -ation of magnetic survey data to an actual case to i l l u s t r a t e the types of geological information that should be looked f o r i n magnetic survey data, with a view to increasing the usefulness of magnetic surveys i n solving geological problems. Consider f i r s t the i n t e r p r e t a t i o n of magnetic surveys. In t h i s work we are concerned only with the "anomaly f i e l d " of the earth,JNettleton, 1940, p.168J extracted from values of the t o t a l f i e l d observed along a well - controlled network of l i n e s over a plane at or above the earth's surface. Prom t h i s i t i s generally possible to extract quantitative information about p a r t i c u l a r bodies of magnetized rock. This may be done by f i t t i n g equations giving the t h e o r e t i c a l effects of various i d e a l i z e d bodies, taken to represent geological conditions, to the observed e f f e c t s and attempting to discover that body giving the best representat-ion. 3 F i n a l l y i t may be possible to evaluate parameters giving the position, s i z e , and magnetic properties of the actual body i n the subsurface. Equations f o r the magnetic f i e l d set up by bodies of various shapes and c h a r a c t e r i s t i c s have received considerable attention. Expressions f o r the components of the f i e l d i n the v e r t i c a l and horizontal directions are quite simple when the p o l a r i z a t i o n also i s along one of these directions, and these have been developed f o r a wide range of bodies JNettleton,1942j. For p o l a r i z a t i o n i n c e r t a i n other directions, but with measure-ments v e r t i c a l or horizontal, the equations can s t i l l be developed without d i f f i c u l t y , but are considerably more comp-l i c a t e d Heiland,1946, pp.389-400J . P r a c t i c a l d i f f i c u l t i e s i n handling the equations increase s t i l l more, when the directions of p o l a r i z a t i o n and of measurement are both required to be i n some d i r e c t i o n other than the v e r t i c a l or h o r i z o n t a l as i s the ease f o r aeromagnetic data. This accounts i n part f o r the very small number of e x p l i c i t equations f o r the i n t e r p r e t a t -ion of aeromagnetic data that have appeared i n the l i t e r a t u r e to date, a fact also remarked upon by Smellie |l956,p.102lj. This has l e d to a good many alternative methods, such as reducing to v e r t i c a l and horizontal components, ^Hughes and Pondrom, 1947 ^J, or constructing model f i e l d s , either experimentally or t h e o r e t i c a l l y from bodies with various values of the parameters j V a c q u i e r , 1 Q 5 1 ; Zietz and Henderson,1956 ^ . These methods suffer the l i m i t a t i o n [ t h a t they cannot be e a s i l y extended to cases where the p a r t i c u l a r parameters incorporated i n the models do not hold. 4 For a more general treatment, a f l e x i b l e set of equations r e l a t i n g f i e l d to body i s required-This has become more urgent by the recognition, l a r g e l y as the r e s u l t of experimental work on the magnetic properties of rocks, that a large v a r i e t y of directions of p o l a r i z a t i o n are found i n formations. At the same time the equations of inte r p r e t a t i o n as given i n the i i t e r a t u r e either do not allow f o r t h i s at a l l , or are not framed i n such a way as to allow f o r variations of t h i s additional parameter. Most attempts to incorporate a r b i t r a r y directions of measurement and p o l a r i z a t i o n begin by using e x p l i c i t l y the azimuths and i n c l i n a t i o n s of the directions involved, as f o r example, i n the developments of Sutton and Mumme |^  1957j -As we s h a l l see l a t e r , the shape of the anomalies does not depend e x p l i c i t l y upon these quantities, but upon ce r t a i n parameters combining these d i r e c t i o n s . Thus equations involving the directions themselves are unnecessarily complex. Once the combined parameters are formulated on the other hand^ i t i s possible to obtain quite simple expressions f o r the f o r c e - f i e l d over bodies from the simplest to more complex ones. In fa c t , the extra generality makes i t possible to include the e f f e c t s of a larger number of parameters, with greater o v e r - a l l s i m p l i c i t y , than i s possible when these more general quantities involving d i r e c t i o n are not employed. It i s one of the purposes of t h i s research to formulate these parameters, and apply them to the problems of magnetic i n t e r p r e t a t i o n . It i s possible by t h i s method not only to include the d i r e c t i o n of p o l a r i z a t i o n among 5 the parameters to be found, but also to derive expressions f o r the aeromagnetic anomalies over bodies which have not been treated previously i n the l i t e r a t u r e . In the second part, i n the application of the methods of magnetic int e r p r e t a t i o n to an acutal aeromagnetic survey, cert a i n general propositions about the v a r i a t i o n of the values of the physical properties of rocks and formations are applied to a p a r t i c u l a r area (Texada Island). It i s shown how to extract the most s i g n i f i c a n t geological information contained i n magnetic measurements. The d i s t r i b u t i o n of magnetization deduced i n t h i s way i s correlated with e x i s t i n g knowledge on the migration of ir o n during the formation of ore deposits on the island, and t h i s i n turn i s related to the s t r u c t u r a l h i s t o r y of the area as revealed i n the pattern of l i n e a r s observed on a e r i & l photo-graphs, and i n the general physiography of the i s l a n d . In the f i r s t period of the development of methods of i n t e r -preting magnetic surveys see, f o r example, Smythe [1896 j , He Hand |l 946j , and Nettleton £l942j , an extensive body of l i t e r a t u r e arose f o r use i n i n t e r p r e t i n g surveys making measure-ments of the v e r t i c a l and horizontal components of the magnetic force at the surface of the earth. With these l i m i t a t i o n s on the d i r e c t i o n of measurement, i t was possible to derive a set of equations f o r the magnetic f i e l d and i t s derivatives over bodies of a wide v a r i e t y of form. .However, newer instruments f o r measuring the f i e l d of the earth are not, l i k e the older ones, confined to measurements of the v e r t i c a l and horizontal components, and f o r t h i s reason "bring a new problem to magnetic i n t e r p r e t a t i o n . T h e f l u x g a t e magnetometer, f o r example, although capable of measuring any desired component of the f i e l d , i s much more accurate f o r measurements i n the d i r e c t i o n of the t o t a l field^Jakosky,1949; p.237], [Logachev, 1955;p.179] . Instruments based on nuclear free precession w i l l record measurements only along t h i s d i r e c t i o n £ Waters, 1956^ j . The problem of adapting the equations of in t e r p r e t a t i o n to be of use when measurements are made with these instruments may be approached i n a number of ways. The component, measured along the t o t a l f i e l d , of the force over a magnetized body may be. obtained from the components along the v e r t i c a l and horizontal dir e c t i o n s , i f t h i s force i s small with respect to the f i e l d due to the earth as a whole, by the following equation ^Vacquier, 1951; p.46j ^ T = C 0 S 1 + ^ Z S i n 1 where i i s the i n c l i n a t i o n of the earth's f i e l d (see Figure 1.1). Figure 1.1 - Components of the anomaly f i e l d . For any bodies where expressions f o r and have already been obtained, the transformation of the e x i s t i n g set of equations into those f o r the r e s u l t s of measurements with the fluxgate or nuclear free precession magnetometer presents no d i f f i c u l t y i n theory. In practice, l i t t l e d i f f i c u l t y i s encountered i n the case of bodies such as single poles or l i n e s and sheets of poles, "where no d i r e c t i o n of p o l a r i z a t i o n need be 7 assumed... Por bodies which must be considered as aggregates of dipoles, however, where a d i r e c t i o n of p o l a r i z a t i o n must be speci f i e d , the application of (1.1) becomes d i f f i c u l t except i n the simplest cases where p o l a r i z a t i o n can be taken to be v e r t i c a l or along the earth's f i e l d . A number of usef u l methods, based on these assumptions, have appeared such as f o r obtaining depths of polarized bodies if Henderson and Zietz, 1948, 1958; Smellie, 1956 fj , or f o r obtaining depth, i n t e n s i t y of magnetization and size of the body IVacquier, 1951; The assumption of these sp e c i a l directions encounters a fundamental objection a r i s i n g from studies of rock magnetism. For the basis of the long - held b e l i e f that p o l a r i z a t i o n generally l i e s along the earth's f i e l d i s the contention that the majority of rocks exhibiting magnetic properties owe t h e i r p o l a r i z a t i o n l a r g e l y to induction i n the earth's f i e l d . Factors such as remanent magnetization or demagnetization effects i n various directions, which would tend to cause the t o t a l vector of the i n t e n s i t y of magnetization to deviate from the t o t a l f i e l d d i r e c t i o n , are commonly thought to'be However, evidence to the contrary has been accumulating f o r a considerable period of time, and such cases have appeared since the e a r l i e s t days of geophysics (see f o r example Heiland Cl93(Tp. Cases of remanent magnetization several times as great as the induced component Kruglyakova, 1956 and i n n e g l i g i b l e directions other than 8 Buddington, 1957J have been found i n laboratory t e s t i n g of specimens, and the p o s s i b i l i t y that s i m i l a r cases might constitute a'sizeable proportion of the rocks of the crust has been recognized. The r e s u l t s of f i e l d surveys, i n which directions of p o l a r i z a t i o n other than along the earth's f i e l d , also confirm the laboratory work and add strength to the b e l i e f that such cases are r e l a t i v e l y widespread [Heiland, 1929; Garland, 1951 ; Mikov, 1952; Lundbak, 1 9 5 6 ] . The work of Balsley and Buddington J J 9 5 7 J » furthermore, i n c o r r e l a t i n g d i r e c t i o n of magnetization with metamorphic structures, i s a recent demonstration of the fact that magnetic properties are clo s e l y connected with geological conditions and a sensitive i n d i c a t o r of the l a t t e r . This has been known f o r some time, and confirms e a r l i e r work such as that by Davis p 9 3 5 J and his co-workers, or by Kalashnikov J J952] . This gives additional importance to d i r e c t i o n of p o l a r i z a t i o n as a parameter to be solved f o r i n the interpre-t a t i o n of magnetic surveys, i n d i c a t i n g as i t does that the magnitude of t h i s quantity i s dependent on geological conditions. The comparison of values of d i r e c t i o n of p o l a r i z a t i o n across an area may be a valuable in d i c a t o r of geological change. Any s i m p l i f i e d assumptions, then, as to the d i r e c t i o n of p o l a r i z a t i o n may render equations f o r the f i e l d s over various bodies inadequate to meet the requirements of actual conditions, and necessitate generalization of the equations to include any desired d i r e c t i o n . Among the f i r s t attempts at such a generalization were those of Mikov [1953] and Vo'skoboynikov [1955] who gave a t h e o r e t i c a l treatment of the effect of a r b i t r a r y d i r e c t i o n of p o l a r i z a t i o n on the v e r t i c a l and horizontal components of f i e l d over a horizontal cylinder. The equations f o r the bodies presented i n t h i s thesis were already developed when a paper appeared by Sutton and Mumme [l957], confirming the p o s s i b i l i t y of including the d i r e c t i o n of p o l a r i z a t i o n i n the equations of aeromagnetic interpretation, where the measurements have been taken with the fluxgate magnetometer. They considered the case of a single dipole and a l i n e of dipoles and developed equations f o r the magnetic force over these bodies when the d i r e c t i o n of p o l a r i z a t i o n i s a r b i t r a r y . They also showed the effect on p r o f i l e s over these bodies of various directions of p o l a r i z a t i o n and angles of i n c l i n a t i o n of the d i r e c t i o n of measurement. A method of determining depth to the l i n e of dipoles, using half-maximum width of the anomaly, i s given. The present thesis involves a d i f f e r e n t aspect of the ef f e c t of the d i r e c t i o n of p o l a r i z a t i o n , v i z . the combination of directions into shape factors, i t s effect on the peak and i n f l e c t i o n points of the p r o f i l e s , and methods of obtaining equations f o r more complex bodies and f o r higher derivatives of the f i e l d . The r e s u l t s of these studies are presented here. The method used i n the i n t e r p r e t a t i o n of the observed f i e l d determines those features of the t h e o r e t i c a l equations t be emphasized. The comparison of one curve with another (the observed p r o f i l e with the theoretical'type-curve) i s b a s i c a l l y 10 a process of c u r v e - f i t t i n g , and the methods of c u r v e - f i t t i n g , well-known i n mathematical analysis are applied i n one form or another. The choice of these methods governs the scheme of interpr e t a t i o n adopted. The method of elementary decompositions leads to the " d i r e c t " methods of interpretationJBullard and Cooper, 1948; Peters, 1948]; consideration of the p o s i t i o n of speci a l points such as peaks, h a l f maxima, i n f l e c t i o n points, zeros, minima i s t y p i c a l of the most frequently employed "in d i r e c t method" (examples arej^Nettleton, 1942; Smellie, 1956]). Derived quantities involving areas under the curve etc. lead to schemes of int e r p r e t a t i o n such as proposed by Kogbetlianz [1945] to determine the centres of gravity of a t t r a c t i n g bodies. Least squares f i t t i n g of t h e o r e t i c a l p r o f i l e s to observed ones has been developed by H a l l [1958^ . Special points on type curves and p r o f i l e s , e s p e c i a l l y the maxima and the i n f l e c t i o n points flanking them, are considered i n t h i s t h e s i s . This s e l e c t i o n of points to be considered was made because i n the i n t e r p r e t a t i o n of observed magnetic f i e l d s over the earth, these are i n general the only s p e c i a l points that can be located with-'qertainty. Furthermore, 'to a greater or l e s s e r accuracy, these points can be located f o r most anoma-l i e s ; the p o s i t i o n of the peaks-can always be found on the i n t e n s i t y maps, while f o r most areas, second v e r t i c a l derivat-ive maps can be constructed j~ Vacquier, 195l] . This i s done as routine .analysis i n modern practice, and the zeros give the p o s i t i o n of the i n f l e c t i o n points f o r elongated anomalies. Consequently,.the i n f l e c t i o n p o i n t s may be located by numerical methods, on, almost any p r o f i l e . 11 Thus'the means are at hand of using t h e o r e t i c a l r e s u l t s concerning derivatives, and i n f l e c t i o n points, of the magnetic i n t e n s i t y over disturbing "bodies, to obtain information about the earth's crust. This gives additional j u s t i f i c a t i o n f o r such developments: apart from t h e i r t h e o r e t i c a l importance, they may be e a s i l y applied to the study of the earth. Prom t h i s study, i t w i l l become evident that i n addition to the parameters that are normally found to-day by geophysical in t e r p r e t a t i o n v i z . the depth, size, and i n t e n s i t y of magnetization, of the disturbing body, another parameter, related to the fundamental magnetic properties of the body v i z . the d i r e c t i o n of p o l a r i z a t i o n , may also be found, knowing the p o s i t i o n of the peak and i n f l e c t i o n points. Since these points are commonly presented i n maps covering wide areas, as part of a routine modern analysis of magnetic data, i t i s import:ant to emphasize again the possible a p p l i c a t i o n of the present study to the mapping of the variations i n the values of physical properties of the earth's crust over wide areas. 2. THE MAGNETIC FIELD OVER SOME ELEMENTARY BODIES -a. The point dipole - see Figure 2.2 The factors that must be allowed f o r are dictated by the method of surveying, and by the p a r t i c u l a r conditions of magnetization to be expected. The advent of airborne surveys makes i t necessary to allow f o r the point of observation to be anywhere i n space, and considering what has been learned about the magnetization of rocks, dipoles with axes i n any d i r e c t i o n may "be expected. Heiland £l946, pp.391-3] summarizes the application of dipoles to magnetic interp r e t a t i o n , the equations "being f o r s t r a i g h t - l i n e , meridional traverses, with v e r t i c a l or horizontal components of the magnetic i n t e n s i t y over dipoles polarized i n the d i r e c t i o n of the earth's f i e l d . Further generalization, to include meridional traverses over dipoles with axes along the earth's f i e l d , and measurements made of the component of i n t e n s i t y along t h i s d i r e c t i o n came with the advent of the airborne magnetometer,[Henderson and Zietz, 1948; Smellie, 1956] . With increasing knowledge about the directions of p o l a r i z a t i o n i n rocks, came a further step of generalization, to allow f o r an a r b i t r a r y d i r e c t i o n of p o l a r i z a t i o n . Examples of such a development are the paper of Sutton and Mumme £l957j and the present i n v e s t i g a t i o n . The l a t t e r develops the equations to a s l i g h t l y more general form, i n which they are suitable f o r further generalization to represent the f i e l d s over more complex bodies. These equations w i l l now be developed. Let a magnetometer Q at the point (x, y, z) measure jf) , the component i n the d i r e c t i o n -£l, m, nj of the f i e l d due to a dipole P of moment jX , polarized i n the d i r e c t i o n ^L, M, and located at the point (a, b, c ) . Then ^  I s given by [Jeans, 1948, p. 372] 13 Figure 2.1 -Traverse over dipole (2.1) where ^r; i s d i f f e r e n t i a t i o n i n the d i r e c t i o n a l , m, n j , that i s i n the d i r e c t i o n of the component to be measured, and i s d i f f e r e n t i a t i o n i n the d i r e c t i o n | l , M, NJ , that i s i n the d i r e c t i o n of the north pole of the dipole, and r 2 = (x - a ) 2 + (y - b ) 2 + (z - c ) 2 I f f and g are any two functions of x, y, and z: df = Ii df_ + M df +N bt ds dx by dz and = 1 frg + m d_g_ + n dg_ ^ t £x ay b z These expressions can be used to expand (2.1) i n the ((2.2) (2.3) (2.4) form: f 5 = AkC^x-a) 2 +QC22(y-b)2 + 0 C 3 3 ( Z . C ) 2 +oC 1 2(x-a)(y-b) + oC 1 3(x-a) (z-c) +CC23(y-b) ( z - c) (2.5) where oC1 -, = 2L1 - Mm - Nn oC ^ = 3(MI + Lm) OC22 = 2Mm - Nn - 1L <X13 = 3(N1 + Ln) CC 3 3 = 2Nn - 1L - Mm O C ^ = 3 ( N m + Mn) 14 ( i ) Observation i n a horizontal plane above a point dipole Taking the z-axis downward, and the dipole at (0, 0, h) we have the case of observation i n the x-y plane, which i s a distt-ance h above the dipole Thus i n (2.5), z = a = b = 0, and c = h, (2.6) and hence r-^ = A f l 1 x 2 • t a :22y 2 4 C C33 h 2 + q C12 xy - c C 1 3 Xh - f l £ 3 y h l ( x 2 + y 2 + h 2) 5 / 2 [ ( 2 > 7 ) J Some s i m p l i f i c a t i o n r e s u l t s p a r t i c u l a r l y i n performing actual computation, from writing distances i n units of h and going over to what we w i l l c a l l the "reduced form" of the expression f o r ^ . Lettin g F= - , = — (2.8) h n we have • (2.9) Thus the d i r e c t i o n of p o l a r i z a t i o n and of measurement, as well as depth, are among the unknown parameters of a body which determine the'shape of the corresponding anomaly. The various charts given by Vacquier £l95lj i l l u s t r a t e t h i s with prismatic bodies f o r the spe c i a l case £ L , M , N | = | l , m, n^ . The peak of the anomaly, and the l i n e of minima are given by the condition: (2.10) G>3 = dfr _ Q Prom (2.9) we obtain: (2.11) and (2.12) When substituted into (2.10) these give a pair of simultaneous t h i r d order equations i n ^  and/Y[ which could, i f required, be solved numerically to f i n d the positions of the peak and minima corresponding to given directions of p o l a r i z a t i o n and of measurement. If the axes are taken so that the y-axis i s along the magnetic meridian, (2.7) above i s equivalent to equation (6) of Sutton and Mumme ^1957J . In t h i s form i t i s suitable f o r ca l c u l a t i n g the actual contours over a point dipole, or as i s done by the above-mentioned authors, f o r studying the p r o f i l e s i n s p e c i a l directions such as along or perpendicular to the magnet ic' meridian. However, i f the equations f o r the dipole are to be integrated to obtain the p r o f i l e s over more complex bodies, the more general equation (2.7) i s preferable. 16 ( i i ) Observations along a traverse.. If Q (Figure 2.1) i s confined to the x-axls,, then/7]= 0 and the equations are further s i m p l i f i e d . Substituting r(\-0 into £. 9 )an d(2.11 )we obtain: 3« ^ s C u ff-Cf+C*] C,=^SL0 C2= ^ 2 - ( 2.13) and (2.14) from (2.14) we obtain further: Uoo,, ii2j i-20C ( f 3 + 3 ( / 0 C 2-7jP ^ / 5 ' C f - ^ C a ^ ) ? (2.15) The shape of the p r o f i l e , then i s determined by three quantities: h, C^, and If the p o s i t i o n of the dipole i n the horizontal plane can be determined from other methods, f o r instance from a gravity survey, or from geological i n d i c a t i o n s , then the o r i g i n of the co-ordinate system i n (2.13) to (2.15) may be f i x e d . Using the positions of peaks and i n f l e c t i o n points along one, or a number of p r o f i l e s through the o r i g i n , values of L, M, N, and h, can be determined without d i f f i c u l t y . I f the o r i g i n cannot be located, by independent means, a contour map of the magnetic values i n the vicinity,may be used to .locate t h i s o r i g i n . An i t e r a t i v e Table I - Derivatives of f i e l d values along traverse over a single dipole to follow page 1 6 . Referring to F i g u r e ' 2 . 2 , and writing D n J = d n ? , D ^ = d n 3 , dx 1 1 dz 1 1 we have: r3oC11x3i ~ 4^C^hx2 + (5*, ,- aC, 1 )h 2x +oC ^ h 15 33 13* r—"V i r ^ T T ? ! 7 ^ J D 2 J yXl 2oC^ 1x4-20JC1 31IX3+3( 10X33-^ -, )h2x2+15oC, 3 l i 3 x - ( 5 o C 3 3 - 2 o C J 1)h 2 2 D 2!J =^ [ ( 2 o C 3 r 5 o C n ) / + 1 ^ 1 3 x 3z + 3 (10cC 1 1-/bC 3 3) X z. - 2CbC 1 3xz 3 +12oC i 3z 4 17 process i s possible, s t a r t i n g from an a r b i t r a r y o r i g i n , f o r which the corresponding values of L, M, N and h are computed. These values, when substituted into (2.11)'and (2.12) give a new, more correct p o s i t i o n of the o r i g i n . The process may be continued f o r as long as i s necessary. The solution would be laborious, however, since new p r o f i l e s would have to be drawn fo r each re-determination of the o r i g i n . A single dipole approximates a compact, uniformly magnet-ized body at depth, with dimensions roughly the same i n a l l d i r e c t i o n s . Such bodies are encountered e s p e c i a l l y i n mining exploration (see, f o r example, Yungul [1956]), and f o r t h i s reason the equations developed f o r the dipole may on occasion be required f o r the solution of p r a c t i c a l problems. They are intended here, however, as the s t a r t i n g point f o r developing equations f o r the p r o f i l e s over more complex bodies. (b) The h o r i z o n t a l l i n e of dipoles (see Figure 2.3) -Let the magnetic moment have a constant l i n e density Ur^ per unit length, and consider a straight, horizontal traverse perpendicular to the l i n e and passing over i t at a height h. I f we take axes with z v e r t i c a l l y downward, and the traverse along Ox, then any element, length dt, of the l i n e at b = t can be considered as a dipole with magnetic momentyU^dt, producing a f i e l d dj^ at the magnetometer, given hy (2.5) i n which y = z = a = o » h = t , c = h , and r = x + 2 2 Yi + t . we have: (2.18 If^T-j f i - s "the force due to a l i n e terminating at y = i ^7t=. r ^ (2.19) Substituting from (2.18) into (2.19) and carrying out the integration we f i n d : where C^ = -Cg = (1L - nN) and C5= -2(lN + Ln). (2.21) 19 If 7\|->°0» w e have the case of an i n f i n i t e l i n e of dipoles, and g . v e n b y ; ^ - 2MLGA [ E 2 +A,g - 1 } h 2 /*2 . „^ 2 (2 .22) where A ,= = 2 ( 1 N + nL)/(nN - 1 L ). This may he c a l l e d the " p o l a r i z a t i o n function" . (2 .23) (c) The i n f i n i t e l i n e of dipoles This body, with p r o f i l e given i n ( 2 . 2 2 ) , i s of considerable use i n interpretation, and has been the object of considerable attention. The effect of an a r b i t r a r y d i r e c t i o n of p o l a r i z -ation on the areas under the p r o f i l e s of the v e r t i c a l and horizontal i n t e n s i t y of anomalies over t h i s body has been studied by Mikov [1953] and by Voskoboynikov [1955] , and a method for c a l c u l a t i n g the unknown parameters of the body i s given by these authors. P r o f i l e s of the t o t a l f i e l d of the anomaly f o r a number of directions of p o l a r i z a t i o n have been drawn by Sutton and Mumme [1957]. Additional quantities are necessary f o r the more complete analysis of p r o f i l e s , and these w i l l be developed i n l a t e r sections , 3. Extension and generalization of methods. The equations f o r the magnetic f i e l d and derivatives over two elementary bodies - the single dipole and the l i n e of dipoles, have been generalized to include cases where both the directions of p o l a r i z a t i o n and measurement are a r b i t r a r y . These developments are of value i n dire c t applications to geological conditions which can be approximated by these elementary forms. Within reasonable l i m i t s of accuracy, such elementary forms do have a wide application i n p r a c t i c a l i n t e r p r e t a t i o n j^Nettleton, 1942, p.293] , but i f the methods are to be used to study the d i r e c t i o n of p o l a r i z a t i o n over a wide v a r i e t y of geological situations, or i f generalizations are to be made, a wider range of bodies must be treated. These include cylinders of a r b i t r a r y cross section, t h i c k sheets, sloping steps and bodies derivable from the l a t t e r (synclines, a n t i c l i n e s , and other forms as shown f o r example by Heiland, |j946, p.396 )J. At a further stage of generalization, various parameters have been found which may be evaluated even though the p a r t i c u l a r form of the body i s not known |Kogbetlianz, Various aspects of the d i r e c t i o n of p o l a r i z a t i o n f o r elementary forms have been studied i n the past and these have already been mentioned. It i s believed that more complex bodies have not been treated at a l l i n t h i s way, and methods of performing such a generalization are now examined. Equations f o r a number of the more complex bodies may be derived from those f o r the single dipole, l i n e of dipoles and the t h i n sheet, by using these bodies as elements, and integrating between the required l i m i t s . For others where the actual integration i s d i f f i c u l t , charts are used as an aid i n the summation Logachev. 1955, pp.77-80 ; 1945; 1948; Henderson and Zietz, 1957J . These charts are based on elementary bodies, and any equations f o r these which take into account the d i r e c t i o n of p o l a r i z a t i o n may be used to modify the charts to account f o r t h i s parameter, a. Bodies obtainable by i n t e g r a t i o n . In an e a r l i e r section, the expression was derived f o r the magnetic f i e l d observed on a traverse perpend-i c u l a r to the s t r i k e , over an i n f i n i t e l i n e of dipoles (eq. ,2.22). Expressing distance along the traverse i n terms of x and h rather than x/h and h as i n (2.22) we may write: with and as defined i n (2.21), This equation i s the basis of further generalization to long, c y l i n d r i c a l bodies with various cross - section shapes. The magnetic force due to any body may be obtained by integration over elementary volumes, each taken as a dipole [jeans, 1948, p.375j . Thus f o r a c y l i n d r i c a l body, a l i n e of dipoles i s an admissible element of i n t e g r a t i o n . • A l l forces derived w i l l be f o r p r o f i l e s along traverses perpendicular to the s t r i k e of the body, ( i ) The t h i n , dipping s e m i - i n f i n i t e sheet of dipoles (see Figure 3.4) ( 3 . D 22 The magnetic force at Q i s determined by the i n t e n s i t y of magnetization, I of the sheet, "by the distances x and h, and "by the directions of p o l a r i z a t i o n and measurement, expressed as d i r e c t i o n cosines with respect to the axes 0 x z. The derivation of the expression f o r the force i n terms of these quantities may he ca r r i e d out most e a s i l y by f i r s t deriving i t i n terms of distances and directions 1 1 1 r e l a t i v e to the axes 0 x z , This procedure avoids having to perform the laborious integrations which otherwise would be necessary. The transformations are as follows, r e f e r r i n g to Figures 3.1 and 3.2. 23 Figure 3.2 -Transformation of L and N x = x cos d + h sin d h =-x sin d + h cos d L = L cos d - N sin d N = L sin d + N cos d and similarly 1 1 = 1 cos d - n sin d n"*= 1 sin d + n cos d (3.2) (3.3) (3.4) Consider an elementary horizontal cylinder with axis 1 1 passing through the point (u' , h ), (Figure 3.1 ) cross sectional area t du, and a magnetic moment per unit length of I t du. By (3.1) the contribution of the cylinder to the force at Q in the direction | l , m , n j is given by d ^ = 2 I t oj (V + x 1 ; 2 + C^h1 ( u 1 + x 1 ) - C{h 1 2 du1 [(u1 + x 1 ) 2 + h12J Thus the force at Q due to the sheet is given by r ^ = 2 I t = I t , ,1 u'=-oo ,1 2c]x1 + c !h 1  4 5 x 1 2 + h 1 2 1_1 where C | , and C^ are referred to axes V'x'z'. The integration is simply done by ( 18 ) of appendix I . Substituting from (3.2) into (3.6) (3.5) (3.6) r^ 1= Iti 2fi|(x cos d + h sin d) + C^(-x sin d + h cos d)? ^ x 2 + h2 — (3.7) 24 *Jl = It (2cJ cos d - s i n d) x + (2cJ s i n d + cj! cos d)h x^Th2 " ~"(3.8) It can be shown from (3.3) and (3.4) that ,1 '4 ~ ~4 — ^ a « 5 C] = C, cos 2d + i C c s i n 2d and = - 2C^ s i n 2d + cos 2d (3.9) Substituting these expressions into (3.8) c o l l e c t i n g terms i n and C^, and using the i d e n t i t i e s : s i n d = s i n (2d-d) = s i n 2 d cos d - cos 2d s i n d cos d = cos (2d-d) = cos 2d cos d + s i n 2d s i n d, we f i n a l l y obtain: '"J = It C ? x + Co h 7 8 x z + ^ ( 3 ' 1 0 ) where C~ = 2C, cos d + Cr- s i n d 7 4 5 Cg =-2G^ s i n d + cos d (3-11) Similar to the expression f o r the p r o f i l e over a point dipole (2.9) or over an i n f i n i t e l i n e of dipoles (2.22), the corresponding expression f o r the t h i n , dipping sheet, (3.10) may be written i n reduced form. 2 Dividing numerator and denominator by h , and fa c t o r i n g out Cg, we have: <J, = it o8 I A2I+1 (3.12) where ^ = x/h, and ^ 2 = C7^GQ = A , tan d + 2 A, - 2 tan d (3.13) 25 The reduced form i s important, f o r i t separates the parameters into two groups: the "scale parameters", which are ef f e c t i v e i n determining the size of the anomaly (here I,t, Cg and h); and the "shape parameters", which are e f f e c t i v e i n determining "type curves", representing shape of the anomaly. Here J\-2 ^ s ^ e parameter of a family of type curves obtained by p l o t t i n g the function f[ 2 f + 1 against . . ( i i ) The thick, dipping polarized sheet: (see Figure 3.3) Figure 3 . 3 -Traverse over thick, dipping polarized Take as an element a t h i n sheet.whose top i s at (u, h) an4. width du s i n d, then from (3.10) the force at Q i s given by d 1 ^ = I s i n d C ?(x-u) + Cg! y(x- Cgh ( x - u ) 2 + h 2 du (3.H) The force due to the whole sheet i s J= l-dj1 = I s i n d u=-b C 7(x-u) + Cgh ( x - u ) 2 +• h 2 du I s i n d = + I s i n d |~C 7log e£(x-u) 2 + h 2 j + Cg tan x-u h -, +b -b C 7 l o g e ( x 2 + h 2 ) + Cg tan"' x/h J -1 ~|2 where 2 and 1 represent the distances from (-b,h) and (b,h) to Q. ( i i i ) The sloping step (see Figure 3,4) 0 ^ i . X Figure 3.4 Co-ordinate systems over sloping step» In this case the force *jf« obj»|lit»ed at Q i s that due to a sloping step of mean depth h and with the origin at a distance x from Q. 3* maj "be expressed i n terms of these distances and the direction of polarization of the step and the direction of measurement„ If the distances and directions were measured with respect to axes 0 x' z , however^ j* would he given by the expression for a thi^k polarised sheet of dip S » Furthermore distances an# directions relative to 0 x z* may be transformed into those relative to 0 x ja, by equations (3.2)," (3.3) and (3.4) with d .*» p . The fore a at Q i s thus given bys ' 0 ~ I sin L Orj 10g e {X 12 + h 1 2) + Cg tan""1 x1/^1 (3.16) 1 1 whe&e C^ = 20 ^ cos .1 1 + 0^ sin i and *'dg - -2cJ sinS" + C ^ cosS since equation (3.15) f o r the dipping sheet may be used with 1 1 1 distances and di r e c t i o n s r e f e r r e d to 0 x z . 27 There are some advantages i n expressing the slope of the face of the step i n terms of the angle j3 = 180° - £ rather than S . The quantities x 1 • 1 - 1 • ' 1 • 1 1 1 h , C^9 6^ referred to 0 x . z must he transformed into the corresponding quantities x, h, C^ and C^ referred to O x z. This i s f a c i l i t a t e d by n o t i c i n g that 1 1 -C 7 and -Cg (expressed i n terms of ^  ) are of the same form as the c o e f f i c i e n t s j f x and h respectively i n equation (3.8). These- were transformed into C 7 and Cg respectively, as given i n equation (3.11) by the same transformation :(with|3.=d) as those we are now applying. Thus i n the present case C 7 becomes •} —-G^and Cg becomes - Cg on transforming to axes 0 x z$ where C 7 * 2C 4 cos {3 + C^ s i n p (3.17) and Cg = -2C^ s i n p + C^ cos [3 F i n a l l y using equation (3.2) (with d =|3), equation (3.16) becomes: ^ 2> . , = 1 s i n 6 -C 7 log ( x 2 + h 2) - Co tan" Vx cosg+h s i n P •') 0r r l—^ 6 ° ^ - x sin(3 + h cosfV Since tan' -1 f x cos p + h s i n P ] = tan" 1 ( x/h .^xtanft "\ \ -x sin(3 + h cos py \ 1 - x/'h tan|3y = tan" 1 x/h + (3 j we f i n a l l y have, writing d f o r @ f o r the sake $ f uniformity; r^P~ I s i n d C ? l o g p ( x 2 + h 2)+ Cg tan" 1 x/h.' O _ 1 T i (3.19) Figure 3.5-Traverse over sloping step. z 28 (iv) Special cases. None of the equations f o r the force f i e l d over dipping sheets, or the sloping step appear i n the l i t e r a t u r e f o r ar b i t r a r y directions of p o l a r i z a t i o n and measurement. However, various s p e c i a l cases do exist a l l of which can he obtained from the more general equations given above, one of which follows as an example. 1. The t h i n , dipping sheet Equations f o r Z and H, the v e r t i c a l and horizontal components of S1 » f ° r a v e r t i c a l l y - p o l a r i z e d sheet are given where 2b since i s the thickness of the sheet, and oC i s the dip (the complement of d i n equation (3.10)). I, x and h have the same meaning. Thus i n the present notations and H = 41b sinoC h coscC- x sin«C Z = 2It h s i n d x cos d and. h + x H = 2 It -h cos d x s i n d (3.20) Z i s the value of ^ when 1 = 0 , n = H . For v e r t i c a l p o l a r i z a t i o n , L = 0, N =1 , making = -1 and = 0. Then Gj = -2 cos d and Cg = 2 s i n d, so that from (3.10), Z = 2It h s i n d - x cos d ,• as obtained by Logachev 0 The expression for H may be s i m i l a r l y obtained. b. A system of equations of i n t e r p r e t a t i o n . 29 Reviewing the equations f o r the f i e l d over an i n f i n i t e l i n e of dipoles (3.1), a t h i n , dipping, polarized sheet (3.10), and a thick, polarized sheet (whether traversed as a dipping sheet (3.13) or as a sloping step (3.17)),we may note that each equation i s derived from the preceding one, apart from transformations of axes Which change only the c o e f f i c i e n t s , hy integration with respect to x. The form of the equation f o r the f i e l d over a thick sheet i s j f 1 ^ cP, where c i s a constant, and It follows from t h i s and the r e l a t i o n s between equations 3.1, 3.10 and 3.17 that the form of the equations f o r any of these bodies or any order of t h e i r derivatives with respect to x may be obtained from (3.21) by d i f f e r e n t i a t i o n , , Expressions f o r the higher derivatives of E are given i n Appendix I, and the f i r s t eight are given i n Table I I . Comparing these with the equations f o r the various bodies, we see c e r t a i n s i m i l a r i t i e s , which are summarized i n the following table. h (3.21) b. A system of equations of i n t e r p r e t a t i o n . 29 Reviewing the equations f o r the f i e l d over an i n f i n i t e l i n e of dipoles (3.1), a t h i n , dipping, polarized sheet (3.10), and a thick, polarized sheet (whether traversed as a dipping sheet (3.13) or as a sloping step (3.17)),we may note that each equation i s derived from the preceding one, apart"from transformations of axes Which change only the c o e f f i c i e n t s , by integration with respect to x. The form of the equation f o r the f i e l d over a t h i c k sheet i s cf"= cF, where c i s a constant, and It follows from t h i s and the r e l a t i o n s between equations 3.1, 3.10 and 3.17 that the form of the equations f o r any of these bodies or any order of t h e i r derivatives with respect to x may be obtained from (3.21) by d i f f e r e n t i a t i o n . . Expressions f o r the higher derivatives of F are given i n Appendix I, and the f i r s t eight are given i n Table I i . Comparing these with the equations f o r the various bodies, we see c e r t a i n s i m i l a r i t i e s , which are summarized i n the following table. h (3.21) TABLE II - Derivatives of F =°Ctan~ 1x/h + |3log e ( x 2 + h 2) - to follow page 29 . Yl Yl Referring to Appendix I, eq. 10 , writing D P =d P , and 2 2 2 dx11 r = x + h , we have: n = 1 DP = r 2 1 2 £ x +cChJ n = 2 D ^ V - 2 px 2 + 20Chx - 2p h 2 j n 3 D 3F = r 6 * 2 p x 3 + 3oC hx 2 - 6p h 2 x - oC h 3 j n = 4 D 4F = - 6 J" r 8 l 2|3x 4 + 4oChx3 - 12(3* h 2 x 2 - 4oC h 3 x + 2(3 h 4 j n 5 D 5P = 24 So px 5 + 5oChx4 - 20(3 h 2 x 3 - 1 0 < £ h 3 x 2 T + 10/3 h 4 x +<Xh5 J n = 6 D 6F = - 120 j ?px 6 + 6oChx5 - 30p h 2 x 4 - 20cCh 3x 3 + 3 0 p h 4 x 2 + 6o(h5x - 2|Sh 6 j n 7 D 7P = 720 2^(3 r n 1 r x 7 + 7<*hx6 - 42(3 h 2 x 5 - 35oC h 3 x 4 + 70P h 4 x 3 + 2loCh 5 x 2 - 14ph6x n 8 D 8P = - 5040 1 r 1 6 l 2(3 x 8 + 8dC hx 7 - 56 p h 2 x 6 - 5&< h 3 x 5 + 140(3 h 4 x 4 + 56oCh5x3 - 56f3h 6x 2 - cKh7x + 20h8 ] TABLE III 30 9 = kD nF Body cO n = 2 k l i n e of dipoles C5 °4 1 i t t h i n , dipping sheet °8 C7/2 0 I s i n d thick,dipping sheet C8 °7/2 0 I s i n d sloping step C8 °7/2 We may also summarize the quantities represented by the d i f f e r e n t orders of D nF f o r the bodies, as i n Table IV. Using the re l a t i o n s between Jover the various bodies treated i n t h i s section and the function E as given i n Table I I I , i t i s quite easy to u t i l i z e Table II to construct tables of t31. and i t s derivatives (up to the third) f o r the various bodies. These are done i n Tables V to IX, both e x p l i c i t l y i n terms of x and h and i n reduced form. The f i r s t and second derivatives with respect to z, obtained by ( 15 ), Appendix I, are also included. TABLE IV 31 n l i n e of dipoles t h i n sheet thick sheet or step, (with boundary condi-tions) 0 area under p r o f i l e value of f i e l d area under p r o f i l e value of f i e l d slope of p r o f i l e value of f i e l d slope of p r o f i l e second deriva-t i v e , (with respect to x or z) slope of p r o f i l e second derivative (with respect to x or z) t h i r d derivat-ive (with respect to x) second derivative (with respect t-o x or z) t h i r d derivative (with respect to x) fourth derivat-ive (with respect to x or z) t h i r d derivative (with respect to x) fourth derivative (with respect to x or z) f i f t h derivat-ive (with respect to x) Table V - Derivatives of f i e l d values along traverse over i n f i n i t e l i n e of dipoles - to follow page 31 Referring to Tables II and I I I , and Figure 2 . 3 , writing Dn5= dn3> , D£5 = dn3>, we have: n n dx dz J = 2K \ V 2 + ° 5 l u c - ° 4 h 2 • ^ t 2 ,2x2 (x + h ) D j = " 2 / t L / 2 C 4 x 3 + 3 C 5 h x 2 - 6 C 4 h 2 x - C 5 h 3 J } ( x 2 + h 2 ) ^ J 2 G\h 3 - 3CVxh2 - 6C ,x 2h + Cp-x3  4 5 4 5 ( x 2 + h 2 } 3 7 D 2 J = D 2 g 7 = 1 2 / ^ L J ° 4 x 4 + 2 C 5 h x i - 6 C 4h^x^ - 2 C 5 h 3 x + G^h 2, 2 n n v3. / 2 ^ , 2 x 4 (x + h ) J D 3 J = -24UL- 2 C^x 5 + 5 G 5hx 4 - 2 0 C 4 h 2 x 3 - 1 OC 5h 3x 2 +1 OC 4h 4x+C 5h 5 7 I ? T ? ? : * Table VI - Derivatives of f i e l d values along traverse over i n f i n i t e l i n e of dipoles - reduced form: Table VII - Derivatives of f i e l d values along traverse over t h i n , dipping sheet - to follow page 31 Referring to Tables II and I I I , and Figure 3.1, writing B n 5 = ^ . ^ J = - ^ , we have* dxx dz A J = It J C ?x + Cgh x + h Dj = - 2It J G 7x 2 + 2Cohx - C 7 h 2 '7 8X ( x 2 + h 2 ) 2 D z J = = 2It I Cgh 2 + 2C 7hx - C ( x ^ T I ^ 8: D 2 J = D 2 , J = 2 I t / C 7x 3 + 3Cghx 2 - 3C ?h 2x - Cgh 3 (x 2 + h 2 ) 3 - 6 l t <> C 7x 4+ 4Cghx 3- 6C 7h 2x 2- 4C Qh 3x + C ? h 4 Table VIII - Derivatives of f i e l d values along traverse over th i n , dipping sheet - reduced form - to follow page 31 . Referring to Table VII, writing D ^ l f ) = d n5(£L. D n , J (p=d2f(?X dx n dz 1 1 we have: h ? 2 + 1 ) D ^ - 2 I t C 8 | A # + 2 f - A 2 ( f + D 2 D Z ' J (f) .. = - 2ItCg D21(?) - 5(f) = 2"o8Mar3 + 3 f - 3 ^ 2 f - 1D3? (f) = - 6itCg_ J A 2 ? 4 + 4 l 3 - s y l 2 T 2 - 4 T ^ 2 h-Table IX - Derivatives of f i e l d values along traverse over thick, dipping dike or sloping step. - to follow page 31 • Referring to Tables II and I I I , and Figures 3 .3 and 3 . 5 , writing D n J = d 3 1? 7 , we have: dx n ^ / = I s i n d J Crj log e(x 2+h 2) + Cg tan' 1 - 1 x 4 1 I)J = I s i n d ) -C7x + Cgh 2 7 2 x + h , 2 D 2 = - I s i n d } C 7 x 2 .+ 2 C gxh - C ? h 2 X _ _ _ _ o J _ l 7 ( x 2 + h 2 ) 2 , 2 Reduced form of the equations. 1 2 .,N . . . . . ' : ™ r : : ^ r > ' ^ = I s i n d C g ) / \ 2 ( l o g ^ ( p 2 - r 1 ) ( 4 ' l ' o g e h 2 ) + t a n ~ r f 2 D > J = + I s i n d Cg I /[2f+ 1 ' 1 ' * 1 ?2 +1) , 2 D 2 ^ •= - I s i n d Cg f /\J2 + 2 f , - ^ 2 ? 32 ( i ) Calculation of the derivatives of F ^ C t a n ' 1 - ^ +(3loge ( x 2 + h 2) Referring to Table IV, i t i s evident that since each derivative of F corresponds to some feature of the p r o f i l e over each of the four bodies treated, c a l c u l a t i o n of a r e l a t i v e l y small number of values of these derivatives w i l l produce quantities which can be used i n a very much larger number of situations i n i n t e r p r e t a t i o n . Quite often when the basic parameters fo r a body are given, i t i s required to calculate the value of/the f i e l d and some of i t s derivatives at given points. Such calculations are laborious e s p e c i a l l y when an extra parameter such as d i r e c t i o n of p o l a r i z a t i o n i s to be taken into account, and the economy r e s u l t i n g from t r e a t i n g these four bodies as p a r t i c u l a r cases of a single function i s of great p r a c t i c a l value. These derivatives are expressed i n polynomial form -in the preceding tab l e s . This form i s best f o r i l l u s t r a t i n g the r e l a t i o n s between the f i e l d s and t h e i r derivatives over the various bodies. In a l a t e r section the significance of these equations w i l l be discussed. For the c a l c u l a t i o n of values of D nF, however, the trigonometric form ( 7 ) as derived i n Appendix I, i s more convenient. This i s : D nF = ( - l ) n _ 1 (n-1) I s i n n (9 joCsin nQ + 2Q cos n p { h n 1 •'. where cot 0 = x/h (7) Figure 5.1 - Relation of 0 to x and h. z Substitution of the appropriate parameters and m u l t i p l i c a t i o n of D nF by the corresponding value of k i n Table III w i l l give the value of jj * over the body required. A method of ca l c u l a t i o n using graphs s i n nQ s i n nQ and s i n n 0 c o s n$will be used. For lower values of n, these functions appear i n expressions f o r the v e r t i c a l and horizontal components of the anomalies over various bodies, and such curves have been published previously, Three of the curves f o r n = 1 and n =•• 2 appear, i n Nettleton 1 s "Master Curves" j^1942, P .299J. However, the system as a whole does not appear i n the l i t e r a t u r e , consequently these functions have been calculated and appear i n Plate I , plotted against .f , p o s i t i o n on traverse i n depth u n i t s . For each value of n, one of the functions i s equal to zero for^= 0 and the other has an absolute value of 1 at t h i s point. Thus f o r n = 1, s i n ^ s i n n#- sin 2# = 1 whenj= 0; at the same time, s i n ^ c o s n$= sin<9 oosQ= 0. For n = 2, s i n n@ sin nQ= s i n 2 # s i n 20= 0 when£= 0; n 2 at the same time, s i n Qaos n£= s i n Qoos 2Q- -1. values on curves to be prefixed by the following signs n fefl 1 + — 2 + — 3 — + 4 — + 5 — 34 Any of these functions which i s zero f o r 0 w i l l he denoted hy f Q , and those whose absolute value i s 1 at J= 0 by f . Graphs of f f o r various values of n are shown on Plate I, and of on Plate I I . These are plotted against | 1° I, the absolute value of]?, and represent f Q and f^ when preceded by the appropriate sign i n the "table of signs". Por example, f f o r n = 3 i s the negative of the values on the n = 3' curve of Plate I f o r ^ 0 , As a numerical example consider the horizontal l i n e of dipoles f o r which (refe r r i n g - t o Table III and (2.22)) J = i ^ { C 4 f l + V f 0 l f o r * -2 (3.22) h In c a l c u l a t i n g the value of ^  f or the same body at ^ = -0.23, i t i s seen from the "table of signs" that f^ i s equal to the negative of the value on the n = 2 curve of Plate II f o r |^ | = 0.23, and f i s the negative of thevalue on the n = 2 curve of Plate I at | J | = 0.23. Thus: 5 " 2 ^ L ^ -0.856C4 -0.412c 5 j (3.23) Si m i l a r l y , any of the quantities l i s t e d i n Table III may be calculated. ( i i ) An e x p l i c i t form f o r the derivatives of the f i e l d  over a thick, polarized sheet. Referring to Table VII, the expression f o r the derivatives of the f i e l d over a th i c k dipping dike or a sloping step are expressed as: Dng>= I s i n df Cgsin 1 1^ s i n nQ + C 7 s i n nQ cos nQ-1 7 2 (3.24) This i s convenient f o r ca l c u l a t i n g values of the derivatives, as outlined i n the preceding section. However, since the form of the equations a l t e r s considerably when the l i m i t s are inserted, t h i s must be done before examing t h e i r s i g n i f i c a n c e . •.• y^?--. (a) the thick, dipping dike For the f i r s t derivative Vj= I s i n d C ?x + Cgh 2~ + ii (3.25) i n s e r t i n g the l i m i t s D^= I s i n d C7(x+b)+Cgh _(x+b) 2+ h 2 - C 7(x-b)+Cgh (x-b) 2 + h 2 J = - I 2b s i n d = - 2bCgI s i n d C ?x 2+ 2CgXh - C 7 ( h 2 + b 2) x 4+ 2 ( h 2 - b 2 ) x 2 + ( h 2 + b 2 ) 2 A 2 f + 2 F - / 1 2 ( 1 + P 2 ) : (3.26) (3.27) _ t 4 + 2< 1-P 2)f 2+(1 +|S 2) 2 where ^ = x/h, p = b/h a n d / l 2 = tan d + 2 fo= 2(lN+nL) y\1 - 2 tan d nN-lL • (3.27) Applying (16) of Appendix I, we obtain further that: D25= I 2b si n d f2C 7x';+6Cghx'+-4C7 (h^+b2 )x3+4Cgh(h'c-b'" )x -2C ?(h 2+b 2)(3h 2-b 2)x-2Cgh(h 2+b 2) 2 '— ( x 4 * 2(h 2-b 2)x 2+(h 2+b 2) 2 (3.28) 36 (b) the, sloping, step Applying the corresponding l i m i t s to (3.25) f o r t h i s body, D j = I s i n d C 7(x+b) + Cg(h+b cot d) - C 7(x-b)+Cg(h-b cot d) (x+b) 2 + (h+b cot d)"2" 2 b l s i n d (x-b) 2+(h-b cot d ) 2 (3 .29) 2 (Cg-C 7cot d) x - 2(CgCot d-C 7)xh+ (C 7cot d-Cg)h +b 2csc 2d (C 7cot d + Cg^ _ x4+2 (h 2+b 2«b 2cot 2d)x 2-8b 2cot d xh+2b 2(b 2cot 2d-h 2j (3 .30) = 2(Cg-C 7cot d)bl s i n d h' where C f <AJ + ( ? 2 / \ 3 - 1 ) J 4 + 2 ( 1 + G ) ^ 2 - 8 ^ 2 c o t d -2C(22 ( 3 . 3 D = 1 - p 2 c o t 2 d /13 = - (A-]Cot d + co t 2 d - 1) 4. METHODS OP SOLUTION A system of equations f o r the magnetic force over several bodies has been derived i n terms of a number of new parameters, and a method of ca l c u l a t i n g the values of the f i e l d and i t s derivatives over a spe c i f i e d body has been given. It i s now necessary to discuss methods of applying these to magnetic maps, when the f i e l d i s given but the body i s unknown, so that they may be used to extract geological information from magnetic data. 37 Consider the l a t t e r to he.given i n the form of a contour map of magnetic f i e l d values, corrected f o r a l l other, influences and f i e l d s , except those due to magnetized geological formations and structures.' From t h i s i t i s possible to determine i n many cases the pos i t i o n of various bodies: r e l a t i v e to the plane of observation, t h e i r size, shape, average i n t e n s i t y of magnetization and average d i r e c t i o n of p o l a r i z a t i o n . If the values of observed at points i n the horizontal plane, were represented on a v e r t i c a l a x i S j a n undulating surface would re s u l t representing the f i e l d values. Information about the bodies connected with the various anomalies on the map may be.obtained by f i t t i n g the t h e o r e t i c a l expressions f o r the f i e l d s due to p a r t i c u l a r bodies, either to the three -dimensional surface f o r ^ , as i s done f o r example by Vacquier [l951^J , or as i s more common, to p r o f i l e s obtained from the surface by taking v e r t i c a l sections,in various d i r e c t i o n s . This l a t t e r , " t h e method of p r o f i l e s w i l l be adopted here. The most d i s t i n c t i v e points on a p r o f i l e are the maximaand minima. These mark the l o c a t i o n of anomalies, and separate one from the other. The minima however are not usually r e l i a b l e points, being subject to the disturbing influences of neighboring bodies. The maxima, on the other hand, are among the most r e l i a b l e points on the p r o f i l e . Their position are given by the condition that: -42" °- - (4 . 1 ) dx 38 The maxima and minima are flanked by inflection points, and those on each side of the peak are often easy to determine They l i e on the zero line on second derivative maps 1951 Elkins , and such maps are commonly constructed, especially in connection with aeromagnetic surveying. Being near the central portion of the anomaly, these points, like the maxima, are relatively free - (especially in the aeromagnetic anomalies) - from the disturbing influence of neighboring bodies. Their location i s found by solving the equation: d 2 ? = 0. dx 2 (4.2) The peaks of the second vertical derivative are easily located on maps of this quantity-, and the condition.for these points i s : d3tf* =0. dxdz 2 (4.3) For i n f i n i t e l y long bodies of uniform magnetization, d J = d ty by Laplace's equation, (4.4) dx 2 dz so that equation (4.3) becomes: _d2j =0. dx 3 (4.5) Since expressions for these derivatives have been obtained -conditions such as (4.2), (4.3) and (4.5) result i n considerable simplification, treatment of these special points is a suitable beginning in applying the new parameters to magnetic interpretation. a. Special points for the bodies of Table III The expressions f o r ^ a r e obtained from one of the derivatives of - F =oCtan^ ^ + (3 l o g e (x 2+h 2), and the special points from D nF = 0 . (4.6) From the trigonometric form of D nF, dn = n t a n " 1 (~ 2//\)' w ^ e r e (4.7) Q n i s the value of corresponding to x n, the po s i t i o n of the s p e c i a l point i n question, and has the value C^/C^, or 2Cg/C7 depending on whether the body i s a horizontal l i n e of dipoles, or whether i t i s one of the three sheet-like bodies. In Plate I I I , a graph of the p r i n c i p a l values of tan ~ (~2/^\ ) i s plotted against TV » from which the roots n of D F = 0 may be obtained f o r any value of n. For example, when'A = 1, the p r i n c i p a l value of tan ~ ^ ( - 2 / ^ ) may be read from t h i s curve as - 1.107. Other values are - 1.^07-TT , -1.107 - 2'TT , - 1.107-3IT etc. I f n = 3, f o r example, 3 values of s a t i s f y i n g (4.7)lie i n the range 0 to TT* These are , ^ r = - 0.369, - 1 .416, and - 2.4.63, hence = x 3 A = cot(9 3 =-2.58, - 0.16 and 1.24. For the l i n e of dipoles these would represent the po s i t i o n of a minimum, the peak and the other minimum respectively, ( i ) I n f i n i t e l i n e of dipoles and t h i n dipping dike Referring to Table IV, f o r these two bodies, putting n = 1 (4.7) gives f o r the dipping dike - — = f , rA Q \ which i s the value o f = x/h at which the p r o f i l e i n t e r s e c t s base l e v e l (^ ). tan-(2A) PRINCIPAL VALUES OF n0 WHEN DERIVATIVES OF THE FUNCTION F EQUAL ZERO 40 For n = 2, -1 '2 = £ tan (-2/^ ) (4.9) For the t h i n dipping dike, t h i s i s the condition for obtaining ^ at the maxima or minima ( ^ or ^ m ) , while f o r the l i n e of dipoles, i t gives,the points where the p r o f i l e cuts base l e v e l ^ 0 » primes marking points fo r f < 0 ) . ( i i ) the thick dipping dike and the sloping step The conditions f o r s p e c i a l points on p r o f i l e s over these bodies are most e a s i l y obtained from the e x p l i c i t forms given i n equations (3.26), (3.30) and (3.31). Considering only conditions f o r maxima and minima, f o r the t h i c k dipping dike: A 2 T 2 + 2 ? - / l 2 ( 1 + p 2 ) = 0 (4.10) and f o r the sloping step: +Atf + ( P 2 A 3 - i ) = o ( 4 . 1 D Solving the' l a t t e r equation t " "Al ± J ^ L + d " ( 3 2 A 3 ) (4.12) the positive sign before the root giving and the negative sign ^ m . Thus: ; ?M-F =/\|/l^"4"F 4 P 2 (4, c o t d + c o t 2 d - 1 ) ( 4 - 1 3 ) and from t h i s , 7\1 = - 2(j2 cot d +}f~Ap* cot2d+(fM-fm)k!-4-4|?k!(cot2d-1) (4.14) 41. This l a s t formula may be used to c a l c u l a t e / l ^ and hence the d i r e c t i o n of p o l a r i z a t i o n of a step of known dimensions, and position, when the positions of the maximum and minimum are also known. Special cases: When d = 90°, cot d = 0 and (4.12) becomes ? = ~ A i ± + (1 - (3 2), from which (4.15) A i = ±A/(|M - ? m ) 2 - 4(1 - | 3 2 ) (4.16) When d = 45°, cot d = 1 and (4.12) becomes f = " A i , ± ^A^4+ (1 + A 1(3 2), from which A, ="2P 2± «/^ 4 + ( ? M - f m ) 2 - 4 (4.17) 5. CALCULATION OF PARAMETERS OF BODIES The equations developed i n the preceding sections may be used to calculate various unknown parameters of ce r t a i n bodies from t h e i r magnetic p r o f i l e s . Methods of doing t h i s w i l l be i l l u s t r a t e d i n the case of the i n f i n i t e l i n e of dipoles, the t h i n dipping sheet, and the sloping step. For each of these bodies, the p o l a r i z a t i o n function , the dip d of any surfaces bounding the body 3their depths and magnetic moment per unit length along the s t r i k e combine i n a way t y p i c a l of the p a r t i c u l a r body to determine the -sise and shape of the anomaly. 4-2 a. The polarization function This function depends upon the direction of polarization, the direction of measurement and the angle at which the magnetic meridian cuts the strike, of the body. It is most simply expressed in terms of direction cosines, and as defined in section (2), Ai = 2(1N +• Ln) ' nN - 1L (2.23) 1 and n are always known and A| is determine iii from the anomaly, leaving L and N as the' unknown quantities. The relation of these to, the direction of polarization is shown in figure ( 5 . 1 ) Figure 5.1 - Direction cosines of polarization vector. If i is the inclination and a the azimuth of the polarization, p p .. • * L = cos i sin a_ P P M = cos i .cos a P P N = sin i P - --• For infinitely long, uniform bodies, M - 0. This is the case for a l l the bodies treated in this section. 43. L an N are then not independent,for 2 2 L + N = 1 The c a l c u l a t i o n of L an N from A .j i s f a c i l i t a t e d by defining a quantity K = A-| N - 21 A-,1 + 2 N Substitution into ( 2 . 2 3 ) gives _J _K i/r + K * ' L = V 1 + K 2 A graph of N and L against K (Plate IV) i s given f o r use i n c a l c u l a t i n g these quantities, b. Calculation of the parameters ( i ) The i n f i n i t e l i n e of dipoles ( r e f e r to figure 2.2 and equation ( 2 . 2 2 ) ) The d i r e c t i o n of p o l a r i z a t i o n , which can be represented °yAi» ~bhe depth h and the magnetic moment per unit length , are the unknown parameters. The following three quantities,observable on the p r o f i l e , are s u f f i c i e n t to determine the unknowns: x M - x r = w\ x M. - ,x I = w, J M - r j j (or\^ ). These are shown i n figure 5.2. Figure 5.2 -Quantities observed on p r o f i l e . *!> XM *I X (5 .2 ) ( 5 . 3 ) ( 5 . 4 ) 44 Referring to equations ( 4 . 1 0 ) and (4*11) i s given by-one of the solutions of tan 3(9 = - 2. » where cot & = x ( 5 . 5 ) Ai 11 and x-j- and X j are two of the solutions of tan 4 Q = - 2 ( 5 . 6 ) Consequently, w and w' are functions of A* alone, IT h as i s w/w. A l l these quantities may be calculated from the equations above, and are shown i n Plate V plotted against/\ 1 . It should be noted that w and w^  are so related that the value of one f o r -A-\ i s equal.to that of the other f o r • Thus w/w' f o r -A.^ i s equal to the inverse of the value of w/w' at/1^ . w/w1 can be calculated from the p r o f i l e , and the corresponding value of/\^ obtained from the graph. Knowing A •]» the graphs supply values of w/h or w'/b-j since w and w' are known from the p r o f i l e , h can thus be calculated. From equation ( 3 . 2 2 ) f o r an i n f i n i t e l i n e of dipoles i t follows that ( 5 . 7 ) Sinceyt^ i s the only unknown, i t can be calculated from the above equation. DIRECTION COSINES OF POLARIZATION VECTOR FOR BODIES WHERE M«0 -21 tan if where f? is the dip of the magnet-ization vector below the x-axis O ' 2. «~ f 5 * m p < 45 ( i i ) The th i n , dipping, polarized sheet (r e f e r to figure 2.4. and equation (3.10) ). The d i r e c t i o n of p o l a r i z a t i o n , represented hy^^, the depth h, dip d and the magnetic moment per unit surface area of the sheet, I t , are the unknown parameters. Exactly the same quantities as f o r the l i n e of dipoles may he measured on the p r o f i l e (Figure 5.2). For t h i s body, r e f e r r i n g to equations (4.9) and (4.10) x^ i s one of the solutions of tan 2p= - 1 , cot $ = x/h (5.8) A2 and x-j. and Xjl are two of the solutions of tan 30 = - _1_ . (5.9) These may be used to calculate values of w , and w/w , h h which are functions o f / l 2 alone. These quantities are a l l shown on Plate VI plotted against[i^* These can be used i n the same manner as the corresponding graphs f o r the l i n e of dipoles, to calculate h and A 2 f o r the dipping sheet, given a p r o f i l e over that body. Referring to equation (3.12) i t follows that f o r t h i s hody, | ^ M ~ ? I = I t ° 8 ( ( f l V - ^ > I + / y f o > M - ( f o > l J } (5.10) This may be used to calculate I t . Since / 1 2 = /[ 1 tan d + 2 A 1 - 2 tan d (3.13) PLATE V to follow p. 4 5 SHIFT OF PEAK AND INFLECTION POINTS SEPARATION OF PEAK AND INFLECTION POINTS PLATE VI to follow p.45 I "01 average of distances.inrlection -rpoints to peak, per unit height of 0-2 Dipping sheet A 3 V\a for sheet, A**A| for M lAl 4 6 8 10 46 " b o t h t h e p o l a r i z a t i o n f u n c t i o n a n d t h e d i p f o r t h i s b o d y c o m b i n e j o i n t l y t o d e t e r m i n e t h e s h a p e o f t h e p r o f i l e . T h u s u n l e s s d i s k n o w n f r o m s o m e o t h e r , s o u r c e o f i n f o r m a t i o n /\ . j a n d h e n c e t h e d i r e c t i o n o f p o l a r i z a t i o n c a n n o t b e o b t a i n e d . ( i i i ) T e s t s f o r t y p e o f b o d y " ^ P r o f i l e s o v e r a n i n f i n i t e l i n e o f d i p o l e s a n d a d i p p i n g s h e e t a r e v e r y s i m i l a r ( c o m p a r e f i g u r e s . n P l a t e f l l . a n d t e s t s t o d i s t i n g u i s h o n e f r o m t h e o t h e r a r e n e c e s s a r y . T h e r e a r e a n u m b e r o f w a y s i n w h i c h t h i s m a y b e d o n e . 1. H e i g h t o f i n f l e c t i o n p o i n t s a n d p e a k R e f e r r i n g t o e q u a t i o n (3 .22) i t f o l l o w s t h a t JM-% " <F1>lt - ' V i *An r<fo>l|-.<fo>l] = ~ <*1>M--An<VM ( 5 . 1 1 ) w i t h n = 1 i n t h e e a s e o f a t h i n , d i p p i n g s h e e t , a n d n - 2 f o r a l i n e o f d i p o l e s . T h i s r a t i o d e p e n d s t h e n , a l o n e , a n d m a y b e c a l c u l a t e d f o r v a r i o u s v a l u e s o f t h e s e p a r a m e t e r s , a n d p l o t t e d a s i n P l a t e s V a n d V I . T h e s e g r a p h s m a y b e u s e d t o d e t e r m i n e h o w c l o s e l y a p r o f i l e c o r r e s p o n d s t o a l i n e o f d i p o l e s o r a t h i n d i p p i n g s h e e t w h e n o n e o r t h e o t h e r i s s u s p e c t e d . C o n v e r s e l y , i f t h e t y p e o f b o d y i s k n o w n o r a s s u m e d , t h e c o r r e s p o n d i n g v a l u e o f ' w h e n / l " i s k n o w n , m a y b e u s e d t o f i n d t h e b a s e l e v e l o f p r o f i l e s o n w h i c h o n l y JM ~ "Jm i s k n o w n . 47 2. Other s p e c i a l points and slopes As further tests to determine whether the type of "body assumed corresponds to the p r o f i l e , other s p e c i a l points and slopes may he calculated from the parameters, and compared with the observed p r o f i l e . ( i ) Zero points and width of anomaly at base_level Putting ^fequal to zero i n equation 2.22 gives T1 - H7?~TT .'•*A1. t -I (5.12) so that 5 0 -s_ = "'+ 4 , i n the case of the l i n e of dipoles. For the dipping dike, the. p r o f i l e cuts the base l e v e l at one point, ?o = " - J - (4.8) ( i i ) Slopes at any point on the p r o f i l e For the l i n e of dipoles, (n=2), r e f e r r i n g to section 3, ^ - - % C 4 { f o +4_ F 1 ] " k 3 I 2 > (5.13) and f o r the t h i n lipping sheet, (n = 1) / " J , 1 = - £ ^ 8 _ p o *A2 t, | C5.14) These equations may be used to obtain the slope at any point, using the calculated parameters. Comparison with 48 observed slopes i s a test of the accuracy of the interpretation. 6. EXAMPLES OP CALCULATION OP THE PARAMETERS IN THE CASE OP AN INFINITE LINE OF DIPOLES. • Plate VII shows a theoretically - calculated profile over an inf i n i t e line of dipoles. It i s instructive to calculate the parameters of the body from such a 'perfect" profile. Assume that the profile i s over a body in high magnetic latitudes-, where to a good approximation, 1 = 0 and n = 1 . a. Calculation of depth and polarization function -r Assume that the inflection points have been located as shown. Then: w =0.68 miles, w' = 0.60 miles, Knowing both w, and w', two estimates may be made of h. Both of these give h = 1.50 miles. Also from Plate V = -0.33 miles, Xj = + 0.35 miles and x.j. =-0.93 miles. Referring to section (3) and Plates I and II : ^M - ? I = 3352T- and J m - ft/ = f 9 0 ^ . Adopting the procedure of section 5, ! w/w' = 1.13, and hence - 1.50. From this value o f / L , w = 0.45 and w" = 0.40. 1 Hb~ ~E • ' thus - 0.866 - 0.75 (0.396) U . ^ C ^ ( 1 - 1 6 3 ) } Anomaly field PLATE VII a t 2 A°4 ^ - 0.856 + 0.75 (0.412) j - 2/^ 0 4 (0.542) h h 2 and at | / » ' J = J ( ^ 4 j -0.324 - 0.75(0.649)| = - g t ^ . (0.810) ^ ^ M - ^ I = - 2A\ | (0.621)] = - 335/ = - M ^(0.353)^= - 1 9 0 ^ and in both cases 3^ L^ 4 * ~ 539^T • v • b. Calculation of the polarization was found to be 1.50, and i t was assumed that 1 = 0, n = 1. Hence for equation (5.3), Z = Aj „ 0.15. From plate IV, L = 0.60 N = 0.80 C 4 = 1L - nN =-0.80. The intensity of magnetization can now be computed from the v^lue of 2 ^ L C 4 f g j _ v i n g , jlL = 5.39 x (1 ;5) 2x (1.609) 2 x 10 7 = 1.98 x 10 1.60 c. g.s. unit s. c. Determination of base level 8 Prom Plate V ? M - % = 0.55, % , = 0.31 *JM 3M for/\^ = 1.5 . Substituting the observed values of ~ a n d " ^ i ' * t i i e 1 3 8 , 8 , 3 l e v e l is.found to be 610*^ below the peak. 50 From equation (5.12), ijQ may be calculated to be - 0f50, giving x Q = -0.75 miles as observed. To summarize: from the distances between peak and inflection points on the observed profile, values of a number of parameters have been obtained as follows -j/\l> polarization function: 1.50, h, depth to line : 1.50 miles, {L,0,N] direction cosines of polarization vector [ 0.60, 0,0.80 jjy^L, magnet-Q i c moment per unit length of l i n e : 1.98 x 10 c.g.s.. units. These are the parameters that were used to calculate the original Profile. d. Calculation of slopes at the inflection points It i s a useful check to compute these slopes from the calculated parameters and compare with the observed values. The inflection points occut at "fj = + 0.23 and =* - 0.62. Referring to equation 5.13 and Platesl and I I } T)J = 719 { - 0.75 (.725) - 0.580-j ^mile = -8l0*/mile at X j , and I(51 = 719 { - 0.75 (-0.054)+ 0.6l6j^mile = + 470 ^/miie at X j / . 7. . - EXAMPLE,IK THE CASE-OF A THIN., BIPPIHg, POLARIZED SHEET • - i • A a. Calculation of h and/Yg Plate VII.. shows a theoretically-calculated profile over .; a thin, dipping sheet w ithA 2 = -0.85 and h = 0.50 miles. These constants were chosen to produce a curve of similar size and shape to that of Figure 6.1, il l u s t r a t i n g the similarity that can exist between profiles over these two bodies. Assuming that the inflection points have been located as shown, w; = 0.32 miles, w'= 0.30 miles As outlined in section 5, the solution proceeds as follows: w/w1 = 1.07 ; consequently^2 = - 0.Q5 , and w:/h- •= 0.65, w^ h = 0.60. Knowing both w and w', two V.." ^'' - , estimates may be made of h. Both of these give h = 0.50 miles. Also Table VI gives ^ = -0.35,fj + 0.30, f j / =-0.95, leading to =-0.18 miles,. X j = 0.15 miles, and x' = - 0.48^$liles. Referring to section 3 and Plates I and II, at (7.1) = ItC + 0.85(0.312)+ 0.890 V = 1.155 ItC, 8 at = ItCg j -0.85 (0.274) + 0.918 ( = 0.686 ItC, h ~ l J E '.8 > J = ItC 8 + 0.85 (0.499) •+ 0.527 | = 0.950 ItC 8 h" Thus -Jz = 0. 469 ItCg 5M -3v= ° - 2 0 5 I T C 8 h and i n both cases ItCg = 512$'. h b. Calculation of the dip and p o l a r i z a t i o n Referring to equations 3.12 and 3.13jthe shape of the p r o f i l e depends only on h a n d / ^ . Furthermore,/\ 2 i s a function of both the p o l a r i z a t i o n function and the dip. Thus neither of these l a s t two quantities can be determined from the magnetic p r o f i l e unless the other i s known from some independent source of information. As an example, suppose the dip of the sheet was suspected to be 65° S. The p o l a r i z a t i o n function i n terms ofA^  and d i s given hy: A-, = 2 ( A 2 tan d + 1) ? ( 7 # 2 ) A 2 ~ t a n d thus A 1 = 2 ( - 0.85(2.14) + 1 ) = 0.55. I f the - 0.85 - 2.14 magnetic l a t i t u d e and s t r i k e of the body are such that 1 = 0.31 and n = 0.95, then k =-0.05, corresponding approximately to v e r t i c a l p o l a r i z a t i o n , with N = 1 and L = 0. In t h i s oase C^ = - 0.95, C^ = -0.62, and Cg = 1.46. Thus I t , the magnetic moment per unit area of face of the sheet i s : 53 It = 512 x 0.50 x 1.61 = 282 c.g.s units-1.46 C» Determination of the base level Prom Plate VI, JM = 0.41 and Jyf-^I = 0»18 f o r / \ 2 = - 0.85. Substituting the observed values of and^ f f i » ; f c i i e t a s e level i s found to be 585^ below the peak. Prom equation (4.8) T =—1 = 1.18, as observed on the profile. As To summarize Prom the distances measured on the profile a number of parameters of the body have been found as follows: A 2 = - 0.85 h, depth to upper edge = 0,50 miles. Assuming a dip of 65° Sy and knowing 1 and n, the direction of polarization is given by L = 0, N = 1, corresponding to a polarization function of 0.55. The quantity It i s equal to 282 c.g.s units. These are the parameters from which the original profile was calculated, d. Calculation of slopes at the inflection points. Referring to section 3, 54 = - ItCg f ^ + f j for n = 2 . (7.3) 1? F or | = T x = + 0.30, J1 = _ 1024 - 0.85(-0.766)+ 0,504 ] ^/mile = - 1185 ^/mile Por T = Tj/ =-0.95, 5' - - 1024 { - 0.85 (-0.026) - 0.525^ ^/mile = 515 X/mile. These slopes are observed on the profile. 8. APPLICATIONS TO THE INTERPRETATION OF AEROMAGNETIC MAPS. a. Anomalies approximating profiles over in f i n i t e lines of  dipoles or thjn ? dipping sheets. s Plate VIII shows the interpreted magnetic trends in a port -ion of an aeromagnetic map of the La-Plonge area of Saskatche-wan, Canada £ Geological Survey of Canada, 1952^ 1 . Two trends are shown, marking the peaks of anomalies of a type which frequently occurs over one or other of the bodies treated i n this section. profile along a traverse ABC, chosen so that i t crosses the bodies perpendicular to the strike i s also shown i n Plate VIII The two peaks are numbered 1 and 2 respectively, and the i n f l e c t -ion points...flanking these, ,asrdetermined by numerical analysis, are marked. I90d 1800 1700 Anomaly field In gammas PLATE VIII - tofollow p.54 Bi - j l i iMfl f rilpnU»»\ Ik) depth in miles Interpreted profile, |*l mile ^ line of dipoles 0 top of sheet * line of dipoles » B.L..thin. dipping shaat •top of sheet •line of dipoles AEROMAGNETIC CONTOURS AND PROFILE LA- PL0N6E, SASKATCHEWAN I07»35' {traced from Geological Survey mop 561 6) 55 (1) Analysis of anomaly 1 as being due to a l i n e of dipoles If t h i s i s due to a l i n e of dipoles, then since w = w = 0.38 miles , 7 \ 1 = 0, w/h,= w1/h = 0.42 (Plate VI). Consequently h = 0.91 miles. >orA T - 0, % - % .•?,-?!>.. 0.40. Since J* IFtil the i n f l e c t i o n points are 82^T below the peak.,^^ =. 205^T. 1 Wilting 2^LC^ = k i n (5.7), we have whenA 1 =0: " M "3*1 = k f ( f 1 V " (f 1 ^ i J • From the curve f o r n = 2, Plate II, {JM = k (.1 - 0.60). since the peak i s at f = 0 and = w/h = 0.42. Thus k = B2& = 205^T, equal t o ^ i n t h i s case. 0.40 As a test of the accuracy of the parameters obtained, the values ^ at various points on the traverse may be computed. Substituting the value of k obtained above into (5.7) forA-j=0, *Ji = 205 f - j ( f ) f o r n = 2, where f= x/h = x/0.91 • (8.1) Values of ^ » calculated from (8.1) are plotted on the p r o f i l e i n Plate VIII. A 1 = 0 means thafcL = 0, N = 1, and that p o l a r i z a t i o n i s v e r t i c a l . (2) Analysis of anomaly 2 as due to a l i n e of dipoles Prom a numerical analysis of the p r o f i l e , there are three points which might possibly be i n f l e c t i o n points.. That to the south, numbered 1 i s d e f i n i t e , while 2 and 3 mark the possible l i m i t s of the one to the north, obscured perhaps by a minor trend p a r a l l e l i n g the main one. Assuming that there a c t u a l l y exists a single i n f l e c t i o n point on the north, obscured by a minor 56, trend, i t i s possible to obtain f i r s t an approximate solution, and then to follow this with an adjustment of the parameters obtained to give more exact values. Inflection point 1 i s situated so that the ratios between w and w1, and^jj "5i a n d^M ~ ^ ' i / ' m a ' f c c n those expected i f the profile were due to a line of dipoles or a thin, dipping sheet, i f the other inflection were at point 4 as marked. Assuming.that the body i s a. line of dipoles, w = 0.60 miles, w1 =0.52 miles, w/w1 = 1.15 and from Plate VI,/\ 1 = ,1.75, w/h =. 0.46, w1/h = 0.40, Referring to Plate V, fM =-0,25, ^ j'= - 0.65, ^fj = 0.22. Thus h =. 1.30 miles and X j j =. 0.30. miles. These are f i r s t approximations and give: 3 = k{ 0.88 f Q ( p + f - , C p j for n = 2 (8 .3) where T = x ° T30 Is a reliable point and.for this value, =0.22, tiz = - 0.53 k. JQ -Jj = k (- 1 + 0.53) = -0.47k, where j»0 i s .the value of the f i e l d at x = 0. rfQ - ^ j - 7QgT|| thus k = - 149^; the base level i s 1492'below'J^, at 1690#1 To adjust the parameters, choose two widely-separated points on the profile and procede as follows. Consider points x and - x, and the corresponding f i e l d v a l u e s a n d g ; - . ^ + = k[4iJo + + f i + } writing * X ' « f o l l o w s t h a t A l = f1 + " X f l " (8.4) ~2* T T f. * - X f. - .. (8.5) .57 Two successive applications of (8.5) gives the corrected value Values of y calculated from (8.6) are shown on the p r o f i l e i n Plate VIII, as well as the composite curve f o r both bodies. 3. Analysis of the anomalies as due to t h i n , dipping sheets A preliminary t r i a l shows that each anomaly can also be f i t t e d by the t h e o r e t i c a l curve over a t h i n , dipping sheet. As i n the case of assuming lanes.; of dipoles, these predict about equal base l e v e l s f o r both anomalies, about lOO^f below those obtained when assuming l i n e s of dipoles. It i s reasonable then, to assume that the anomalies are due to either two l i n e s of dipoles or two dipping sheets, rather than to one of .each. In the analysis assuming l i n e s of dipoles i t was possible to treat each anomaly separately, since the f i e l d due to such bodies drops off s u f f i c i e n t l y r a p i d l y with distance, that neighboring bodies do not materially d i s t o r t the anomalies. Por sheets, however, as can be, seen from Plates I and I I , the f i e l d drops off much more slowly with distance, and f o r anomalies as close together as those treated i n the present section, correction must be made for the effect of the neighboring body. Thus J= - 144 (0.55 f Q (f) + f 1 ( f.» f o r n = 2 (8.6) As a start of the solution, let us find the thin, dipping sheet which i s the best-fit to anomaly- 2, ignoring the effect -t of the other sheet. Por w = 0.60 miles, w =0.54- miles, we obtain /\ 2 = - 0.55. For the dipping sheet, the equation for . adjustment of the parameters, correspinding to (8.5) i s .: — f 1 + ~ X f 1 for n = 1. f Q + - X f 0 ~ (8.7). Making adjustment, allowing for the effect of the sheet giving best-fit to anomaly 1, and then adjusting the latter so as to allow for the sheet connected with anomaly 2, and so on may be continued as an iteration^ which converges to the following values.of the parameters for the sheets: for anomaly 2, A 2 - - 0.45, h = 0.98 miles, and = - 0.23 miles; for anomaly 1 / \ 2 = °» n = °»53 miles. The profile oyer the bodies, and the predicted base levels are shown on Plate VIII. 4. Comparison of the interpretations In the area under consideration, the inclination, of the earth 1s. magnetic f i e l d i s 79°N, and the declination is ;N21°E. ThisN is the direction of the component, of the f i e l d measured by the airborne magnetometer. Thus i n Figure 2.2, i = 79° for both anomalies; for anomaly 1, a = 10° and for anomaly 2, a = 0°. a. Anomaly 1 as due to a line of dipoles -A-, =0, h = 0.91 miles. The peak i s ver t i c a l l y above the li n e . 1 = cos, 79° sin 10° = 0.033, n = sin 79° - 0.982. Thus K = °'07 = 0.036, and referring to Plate IV, L k 0, N = 1, and 1.96 polarization i s roughly v e r t i c a l . Base level i s calculated to be at 1705^ . b. Anomaly 2 as due to a line of dipoles -= 1.10, h = 1.36 miles. The peak is shifted 0.25 miles to the southeast of the li n e . 1 = cos 79 sin 0° = 0, n = sin 79° = 0.982. Thus K - 1.10 (0.982) = 0 .55. From Plate IV, 1.964 _ L = 0.48, N = 0.87, and <p = 61.1 . Base level i s calculated to be at 1720^ T . c. Anomaly 1 as due to a thin, dipping sheet = 0» h = 0.53 miles, and the peak i s directly above:, the top of the sheet. Base level i s calculated to be aj 1 6 0 1 $ . / \ 2 = °» referring to (3.13), means that -'2//^ = tan d. Measurement along the earth's f i e l d (as with the airborne magnetometer) means that 1&0 and n& 1, making/^., ai 2L.- Thus. - tan d = tan ^  ,, (where is as defined i n Plate IV), and the polarization is directed downwards along the dip of the,sheet. d. Anomaly 2 as due to a thin,,dipping sheet = ~* 0»4-5, h = 0.98 miles, and the peak i s shifted 0.23^miles to the southeast of;the top of the body. Base level is calculated to be at 1607$% For vertical polarization, as an example, L = 0, N = 1. Since for this body, 1 = 0 this would mean that /V -j - 0. By rearranging (3.13) we; see that d = tan - 1 f 7 \ 1 A 2 - 2 2A 2 + A " (8.8) and we would have d = tan-Y_ T j I2__\ = 6 5. 8o.' #. V-2x0.45/ 60 Thus i f the body were polarized entirely by induction in the earth's f i e l d , i t would be dipping 65 .8°SE to give/lg ="0*45. e. Summary and most l ike ly interpretation Near the peaks of the prof i les , both bodies give an equally^ close f i t . The predicted profi les di f fer at the outer parts of the anomaly, however, and in the values of the base levels that would result from each assumption. This affords a possib-i l i t y of distinguishing between these,two proposed interpretations. The profi les corresponding to the dipping sheets combine to f i t the observed profi le very closely, and the predicted base level of about 160oK i s close to that observed nearby in areas re lat ive ly free.of anomalies. It appears that this interpretation i s the more l i k e l y . 9. THE GEOLOGICAL SIGNIFICANCE OF MAGNETIC DATA -AN INTERPRETATION OF AN AIRBORNE SURVEY OVER... TEXADA ISLAND, B . C . The mathematical models of the preceding sections are based on the assumption that the f ie lds observed oyer magnetized formations reflect geological conditions. Areas where good magnetic surveys exist, and where significant comparisons can be made with geological data afford a test of this hypothesis, and a guide in applying quantitative methods of interpretation. In February, 1957., the B r i t i s h Columbia Department of Mines relaaeed Map A.M.57-3, an aeromagnetic survey of a portion of Texada Island, B .C . As conditions are favourable in this area 61 f o r the observation of the geology, and since the o r i g i n a l magnetometer traces and f l i g h t data were available at the Department of Mines,, i t was decided to use t h i s survey as a test both of the degree to which geological information can be r e f l e c t e d i n magnetic data and of the a p p l i c a b i l i t y of some of the methods of in t e r p r e t a t i o n developed i n previous sections. The contours of equal magnetic i n t e n s i t y shown on the map may be taken to represent the attractions of magnetized rocks at and below the surface of the ground. In d e t a i l , these form closures - l o c a l i z e d "anomalies" or area of high or low f i e l d , , r e f l e c t i n g l o c a l concentration or other variations, i n the magnetization of the, rocks. On a broader scale, they divide into zones, each having a c h a r a c t e r i s t i c average l e v e l of f i e l d . The entire area i s divided into magnetic zones on t h i s basis, and the d i v i s i o n s between zones are considered to represent l a t e r a l d i s c o n t i n u i t i e s i n magnetic properties. Some of these postulated d i s c o n t i n u i t i e s are tested using magnetic p r o f i l e s across them, and analyzed by the methods developed i n the previous sections. F i n a l l y , comparison i s made with the known geology of the island, and with the r e s u l t s of a s t r u c t u r a l study of the area carried out with the a i d of a e r i a l photographs. When such zoning of magnetic properties i s observed to occur, i t i s l i k e l y to have considerable geological s i g n i f i c a n c e . Quite apart from the general geological environment,.. many factors such as the i n t r i n s i c magnetic moments of the various ions composing the minerals, the c r y s t a l structure, the past states of i temperature, pressure and stress a l l influence, the amount, kind, and s t a b i l i t y of the magnetization. It would thus be surp r i s i n g i f the properties peculiar to each region were not r e f l e c t e d also i n i t s magnetic co n s t i t u t i o n . Experience with magnetic surveys shows that t h i s i s true i n very many cases. Texada Island i s a l o c a l i t y where the geological structure i s well expressed i n topographic 1 features; d i f f e r i n g rock types occupy areas of d i f f e r i n g elevation; f a u l t s and other r i fractures are c l e a r l y v i s i b l e , there are many exposed areas where the .trends of the formations may be plotted from the a i r . . Under such conditions i t has become widespread practice to obtain much of the geological information from a study of a e r i a l photographs. Such studies have been found to b e s p a r t i c u l a r l y valuable i n correlating, aeromagnetic trends with.geological formations, and correlations 5f t h i s type were found i n the present study. Some.of the main magnetic zones are bounded by major l i n e a r features determined from the a e r i a l photographs. Smaller l i n e a r features are often followed by minor trends i n the magnetic values. Thus an airphoto study can possibly supply a geological explanation of some of the magnetic trends, a. Analysis of the magnetic trends Most of t h i s analysis was carried out on data obtained from the aeromagnetic map, which"-presents magnetic contours on a scale of mile = 1 inch, ;and with a contour i n t e r v a l of 25^f. The map was made from continuous magnetic p r o f i l e s with a l i n e spacing of \. mile and with 500 feet ground clearance, recorded with a fluxgate magnetometer towed behind a f f i x e d wing a i r c r a f t . 63 Anomalies of p a r t i c u l a r interest were studied d i r e c t l y on the o r i g i n a l magnetometer and radio altimeter records, made available by the kind co-operation of Dr„. H. Sargent, Cheif Mineralogist, B r i t i s h Columbia Department of Mines. It i s possible to divide magnetic trends and anomalies into four main groups as follows: large, d i s t i n c t anomalies with height exeeding 500$; smaller anomalies i n the range 200-500 0 ; smaller anomalies s t i l l , w i t h height below 2000 ; and f i n a l l y , straight, continuous lows, often apparently i s o l a t e d , but usually associated with the la r g e r anomalies. Trends were divided according to t h i s c r i t e r i o n , those over 500$^ height being c a l l e d " f i r s t order anomalies", those with height i n the range 200-500^ "second order anomalies", and those below 200^- " t h i r d order anomalies", ( i ) The f i r s t order anomalies These f a l l into long, even, narrow trends, interrupted only by occasional cross displacements. This may be taken as an i n d i c a t i o n of a system o r i g i n a l l y uniform, but l a t e r complicated by f a u l t i n g . The f i r s t - o r d e r trends l i e i n two be l t s , one near the northeast coast of the island, and averaging a f i e l d at the peaks with a -standard deviation ' of 450^ among the highs along i t s length.; This is,shown as Zone I on maps 3 and 4, The second belt of f i r s t - o r d e r anomalies l i e s near the southwest coast of the i s l a n d and averages a 3500 f i e l d at the peaks, with a standard deviation of 400^1 This i s shown as Zone I I I . These are separated by a b e l t , marked as Zone II on the maps, which, i s free of major, anomalies. 64 The boundaries between the zpnes are p a r a l l e l e d by remarkably continuous and regular b e l t s of lows averaging a 3 0 0 0 ^ f i e l d , with standard deviation df 400^. An examination of the northern belt indicated that two systems of sharp, straight boundaries running east-west and north-south respectively cut through the b e l t s and further subdivided them. These sub-di v i s i o n s are shows as l a , I b , I c , Id, and l e . ( i i ) Second and t h i r d order anomalies These add d e t a i l to the zones and from the basis of f i n a l d i v i s i o n s into regions of d i f f e r i n g magnetization, as shown on Map 4. (b) Analysis of s t r u c t u r a l trends V e r t i c a l a e r i a l photograph coverage on a scale of -§• m;Lle = 1 inch was obtained, and the whole area studied -under a stereoscope. Many fractures, straight scarps and other features suggestive of a f r a c t u r i n g pattern were v i s i b l e , and these were plotted, using the standard c r i t e r e a and techniques of analysis of f r a c t u r i n g patterns from airphoto l i n e a r s , such, as those employed previously by the author (Hall,1950) (see also Wilson, 1948), and used i n recent years as preliminary data i n planning Geological Survey mapping. This pattern i s shown on Map 2. Comparison with Map 1 shows a,great s i m i l a r i t y between t h i s pattern and the d i v i s i o n s made on. the basis.of the magnetic map, and suggests cert a i n ideas regarding the s t r u c t u r a l h i s t o r y of the region, which are outlined i n the following section. (c) Comparison of magnetic and s t r u c t u r a l data 65 The most s t r i k i n g feature of the two sets of data i s the d i r e c t i o n of trends. The major boundaries between the d i f f e r e n t magnetic regions are p a r a l l e l e d by s t r u c t u r a l trends, and the directions of minor magnetic trends within the regions are frequently p a r a l l e l e d by the smaller.linears. There i s no doubt that the d i v i s i o n s appearing on the magnetic map r e f l e c t geological boundaries, and that magnetic contrastSjrepresenting segregation of magnetization i n varying degrees from one geological d i v i s i o n to the next, are r e f l e c t e d i n the magnetic f i e l d values, This c o r r e l a t i o n with rock type i s remarked on~ i n a publication describing ground checks of some of the anomalies, that,,were made by geologists from the B r i t i s h Columbia Department  of Mines jj958j[ , This gives support to the idea that s i g n i f i -cant geological information may be obtained by c a l c u l a t i n g the magnetization of various bodies and zones from the anomalies over them. The major trends suggest the following s t r u c t u r a l h i s t o r y . The three major belts shown i n Map 2 were o r i g i n a l l y continuous. Subject to thrusting from thr southwest, they were displaced along a set of transcurrent f a u l t s i n a north-south and east-west d i r e c -t i o n , s u f f e r i n g a displacement of, west side north e s p e c i a l l y along a l i n e between Vanada and Welcome Bay and again on a l i n e between Davie and Mouat Bays. This would mean that he strong magnetic trend running on the south side of the i s l a n d i s displaced f i r s t at Welcome Bay and then again at Mouat Bay, u n t i l i t runs into the s t r a i t s of Georgia i n the v i c i n i t y of Davie" Bay. 66 S i m i l a r l y the magnetic trend from Limekiln hay to P r i e s t Lake i s dispalced so as to run on the south side of Comet mountain, to the south side of Mount Davies and ultimately into Sabine channel'north of Jedediah i s l a n d . The above analysis was based on theories of f r a c t u r i n g and displacement a-s put forward by Wilson [j 948J , which appears to explain the observed.pattern s a t i s f a c t o r i l y . An a l t e r n a t i v e theory, put forward by Moody and H i l l s [l956Jwas also t r i e d as a possible expalanation but without success. The p o s s i b i l i t y of such a s t r u c t u r a l d i s l o c a t i o n i s confirmed by the topography of the island, as show i n Map 5. Variations i n topography r e f l e c t the d i v i s i o n s made on the basis of the magnetic and s t r u c t u r a l trends. Por example, the southern boundary of.Zone I from Comet mountain south i s followed c l o s e l y by the 1000-foot contour; the highland on the north west t i p of the i s l a n d f a l l s i n magnetic d i v i s i o n I l i a ; and Zone II i s followed by a s t r i p of low land. (d) Quantitative calculations f o r magnetic p r o f i l e s The foregoing q u a l i t a t i v e comparisons leave no dpuht that . the magnetic contours contain s i g n i f i c a n t geological information. Certain i n t e r e s t i n g points are raised, which can be tested by quantitative analysis of the magnetic values. This w i l l help s e t t l e some points raised i n the interpretation, and w i l l serve to indicate the value of the methods of c a l c u l a t i o n developed i n the e a r l i e r sections, when applied to actual cases. ( i ) Magnetic high, over Comet mountain 67 The two magnetic highs over t h i s mountain are the highest on the sheet, and magnetic Zone lb (Map 3) comprising them i s the most magnetic of a l l the divisions.. The boundary between lb and Ic constitutes a major magnetic d i v i s i o n , apparently connected with anomaly east of Pocohontas Bay. This r i s e s to 4924^ and has the appearance of an anomaly o r i g i n a t i n g from a pair of sloping steps, as shown i n Plate IX. This figure shows also the topographic p r o f i l e along f l i g h t l i n e 21, which was flown with a constant ground clearance of 500 feet as checked with the radio altimeter record. Consequently the airborne traverse i s as shown, and i n r e l a t i o n to the topographic p r o f i l e , i s the equivalent of a horizontal traverse over a pair of sloping steps, the south-east one co-inciding with the major boundary mentioned above. This boundary i s represented on the a e r i a l photographs by a prominent l i n e a r . The step to the north-west i s s u f f i c i e n t l y f a r removed not to a f f e c t values over the one at the boundary, which we are interested i n , as was found by t r i a l c a l c u l a t i o n . Consequently, the l a t t e r step may be treated independently. In previous sections the following equation was obtained f o r the force f i e l d over a step: 3* = I s i n d £ c 7 log e(x 2+h 2)+C g tan" 1 x (3.19) 2 and from equation (3.31) the condition f o r maxima and minima over the p r o f i l e may be obtained, v i z : 1600-1200-800 Elevation in feet PLATE IX to follow p. 67 2980 5960' 8940 TOPOGRAPHIC PROFILE OVER I-b AND r-C l»,920f lASOCl d l s t ance along flight line 21 BOUNDARY BETWEEN Anomaly field in gammas 600-400-200 0--200 • -400--600 1 -2000 calculated profile values from observed profile. projection of direction of measurement -1000 lobo 1= 5x IO"3c.g.s.u A , = 0.68 ^ slight increase in I direction of polarization 2000 3000 CROSS SECTION OF POSTULATED BODY, CALCULATED PROFILE, AND OBSERVED VALUES (EAST OF COMET MOUNTAIN, TEXAOA ISLAND, BC.) (section taken along flight line 21, northwest'of fiducial 409) X (feet) i h= 1000 where 68 f 2 +A1? + <p2A3'- D > © ( 9 . D A 3 = - (A<i cot d + cot 2 d - 1), p = b/h, and ^  = x/h. •A step with the observed thickness of the topographic feature w i l l not produce an anomaly with the observed positions' of maximum and minimum for any value bf^/\.^. This may be seen from Plate IX : no value of A1 w i l l result i n both^jj and Xm satisfying this equation. If the boundary were the surface trace of a magnetic boundary going to depth, the anomaly would be the result, approximately, of the magnetic attraction of a thicker step. i A few t r i a l s for the thickness show that for b = 550 » values of andT m result which both satisfy (9.1) forA-j = 0 . 6 8 , the latter value corresponding to L = 0,19> N = O.98, a direction of polarization approximately that of induction in the earth's f i e l d . On this assumption, the f i e l d values at various positions of the body can be calculated. In equation (3.29), l e t : X = x - b cot d , Y = x + b cot d h-b h + b then, inserting the limits, ^J?= I sin d i C ? l o g e X2+ 1 + C ? l o g c h - b I T Y2+ 1 F T ~ T + C g (tan" 1 X - tan~ 1Y) J = I sin d< C ? l o g e X2+1 + C ? l o g Q h-b + Cgtan"1 I - Y L 2) C T Y 2 +1 1 •+ XYJ Since d = 45°, and the dip, declination and strike of body are 69 such as to give 1 = 0.13 and n = 0.95, =-0.89, = - 0.50 for polarization along the earth's f i e l d , and C 7 = - 1.83, and Substitution into equations (9.2) gives values of j»at various points along the profile, and these are shown in Plate IX together with the observed aeromagnetic profile taken from the original magnetometer records. The closeness of f i t shows that the hypothesis of a marked contrast in magnetic properties at boundaries like that tested can be supported by exact calculations. The method of calculating anomalies for arbitrary directions of. polarization and measurement proves to be quite applicable in such tests. ( i l ) Magnetic high over Pochontas Mountain This high, rising to 4000^ l i e s on the boundary between Zones I and II. The drop-off from the high to the adjascent low (3000$) appears to be connected with;the boundary. Topography i s steep here, however, and there i s the possibility that the anomaly i s due to topography without any particular magnetic contrast ^ B . 0. Department of Mines, 1958, p. 22J . Equation (9.1) may be used to test this hypothesis. The topographic profile shown on Plate X indicates that" "the feature may be treated as a step. The positions of measurement of the maximum and minimum are shown. betweem maximum and minimum, and i s found.to be 4.97 x 10""-' d.g.s.u'. 70 In a similar analysis to that in the previous section i t can he shown that equation (9.1) cannot he satisfied for any value ,and that therefore the anomaly is not due to a uniformly magnetized topographic feature. The only alternative is a true lateral contrast of magnetic properties between the two zones, ( i i i ) Postulated fault line west of Ponchontas Mountain The boundary between 1-5 and 9 and J-6 and 8, Map 4 shows prominently on the magnetic contours and on the aerial photographs, and i s one of the main set of posutlated transcurrent fractures in section c. This was, also postulated as a fault by geologists of the Brit i s h Columbia Department of Mines during ground checks on the aeromagnetic map JB.C Department of Mines, 1958, P»22^ . Plate X shows an example of a profile across this zone, traced from the original magnetic records. There are a number of similar anomalies in the v i c i n i t y of the zone, a l l with shape suggesting their connection with a sloping step. When f i t t e d by steps with i = 90°, these indicate t a contrast going to a depth of about 500 ., with a difference in the intensity of magnetization of 5 x 10*"^c.g.s.units or more, , i f polarization i s taken to be parallel to the earth's f i e l d . These results are consistent with the idea that a fault i s connected with the zone. ( e) General summary An interpretation of certain features of the aeromagnetic map of Texada Island has been made. This shows that the magnetic anomalies reflect geological conditions. Some of the equations of intepretation developed in earlier sections are applied to AEROMAGNETIC PROFILE ACROSS POSTULATED FAULT ZONE EAST OF POCOHONTAS MOUNTAIN, TEXADA ISLAND, B.C. (Proj'ecred in a direction perpendicular to the strike of the anomalies) — taken from original magnetometer record, line 14 Fiducial 2770 SOUTHWEST SIDE OF POCOHONTAS MOUNTAIN to pographic prof i le (N. 60° E. direction of induced magnetization 1500' magnetic contrast as interpreted from the magnetic anomalies O > = H o m •p x 3 .71 test hypotheses a r i s i n g from the comparison of the magnetic r e s u l t s with the geology. This i l l u s t r a t e s how quantitative methods of analysis of magnetic f i e l d s when s u f f i c i e n t l y f l e x i b l e , oan be used to provide geological information, and check geological theories. 10. SUMMARY AND CONCLUSIONS Expressions f o r the magnetic force and i t s derivatives over a number of uniformly magnetized bodies have been derived f o r the case when both the directions of p o l a r i z a t i o n and measurement are a r b i t r a r y . This provides a greater generality i n the equations f o r inte r p r e t i n g magnetic anomalies than i n the forms previously published i n the l i t e r a t u r e . In the present treatment the equations may be applied equally well to v e r t i c a l or h orizontal directions of measurement or to the t o t a l f i e l d d i r e c t i o n as measured by nuclear free precession magnetometer or by the type of fluxgate magnetometer used i n present day aeromagnetic surveying. In addition, any a r b i t r a r y d i r e c t i o n of p o l a r i z a t i o n aay be incorporated into the c a l c u l a t i o n of the f i e l d , or as an unknown parameter. Por each of the bodies treated: the single dipole, the i n f i n i t e l i n e of dipoles, the t h i n dipping sheet, the thi c k , dipping sheet, and the sloping step, expressions f o r the f i e l d and i t s derivatives have been expressed i n a reduced form, which spearates those parameters of the body which determine the shape of.the anomaly, from those_which determine i t s s i z e . Since in c l u s i o n of the d i r e c t i o n of p o l a r i z a t i o n gives a more complete 72 set of parameters than those given i n previous treatments, i t i s possible to make a more thorough assessment of the geological information obtainable from an analysis of the shape and size of magnetic anomalies. It i s concluded that from magnetic anomalies, the depth, pole strength, and d i r e c t i o n of p o l a r i z a t i o n of a single dipole or a horizontal l i n e of dipoles can be determined. For dipping sheets or a sloping step, depth, pole strength per unit surface area, and a p o l a r i z a t i o n function combining the d i r e c t i o n of p o l a r i z a t i o n and the dip of the i n c l i n e d faces, can be determined. The p o l a r i z a t i o n function, and the pole strength (which i s determined j o i n t l y by the size and the i n t e n s i t y of magnetization of the body) can y i e l d values of t h e i r component quantities only when values of one or other of them are obtained from some other source. Methods are presented f o r the solution of these parameters using data obtained from magnetic anomalies. Examples are given, and applications to magnetic surveys i n the v i c i n i t y of La l a Plonge, Saskatchewan, and Texada Island, B r i t i s h Columbia, i l l u s t r a t e t h e i r use i n the int e r p r e t a t i o n of aeromagnetic surveys. The equations f o r f i e l d s and t h e i r derivatives over i n f i n i t e l i n e s of dipoles, dipping sheets and sloping steps are expressed as derivatives of a single function, and a general expression f o r derivatives of any order i s given. Thus a single set of equations can be used to compute values of the f i e l d s and t h e i r derivatives over these bodies. Graphs are presented as an aid to t h e i r computation. 73 Points at which f i e l d s or t h e i r derivatives are zero are given s p e c i a l consideration;, equations and a graph are given from which any of these points can he e a s i l y computed. These values give f o r changing directions of measurement and p o l a r i z -ation the s h i f t of base l e v e l s , maxima and minima and i n f l e c t i o n points. The base l e v e l i s shown to be a useful i n d i c a t o r of the type of body causing the anomaly. F i n a l l y as an application of magnetic interpretation, to geological problems an i n t e r p r e t a t i o n of an aeromagnetic survey of Texada Island, B r i t i s h Columbia, i s given. Here i t i s shown that cer t a i n zones may be outlined, each with a c h a r a c t e r i s t i c l e v e l of magnetization. The boundaries between these zones are found to co-incide with s t r u c t u r a l trends observed on a e r i a l photographs of the area. That the boundaries represent true l a t e r a l contrasts of magnetic properties, and not topographic e f f e c t s i s proved i n a number of cases by the application of the quantitative methods developed i n e a r l i e r section. It i s f i n a l l y concluded that evidence of f a u l t i n g can be deduced from the aeromagnetic map and the a e r i a l photographs and a s t r u c t u r a l h i s t o r y of the r e g i o n i s suggested on the basis of these observed patterns of f a u l t i n g . 74 APPENDIX I Derivatives and integrals of special- functions Techniques of analyzing geophysical force-field data require the numerical computation of the derivatives often to quite high order, of the observed distribution of f i e l d values, and the matching of these to the theoretically -calculated derivatives of various t r i a l distributions of matter. [Ev.len, 1936; Henderson and Zietz, 1949; Elkins,1951 general methods for obtaining derivatives of higher orders, as the labour Involved in successive differentiations i s usually prohibitive. There i s l i t t l e reference in the geophysical literature to methods of obtaining higher derivatives of the functions commonly required i n interpretation. Related functions have been dealt with elsewhere, and references are given by Schwatt [1924; p3^ and by Hobson [^ 1931 ? p 124~|, but none of these are directly applicable to the•functions appearing in equations for magnetic interpretation. . Tables I to IX show that functions of tjie type The progress of theoretical work in geophysical interpretation depends to some extent upon the av a i l a b i l i t y of JL» _ L (b + m m r r ,+b x ), log.(x +h ) and tan x/h frequently occur. Methods of obtaining higher derivatives of these w i l l now be developed. 1. Higher derivatives of 1 : m r —. . . . . . . 2 2 2 Consider the function y = J _ , where r =x +h , and m and •:. r m 2 h are constants. For y = f (u) where u = x , we may write [Gibson, 19315 p.78^] : £jr = ^ n(n - 1)...(n - 2 r + 1 ) ( 2 x ) n ~ 2 r (u) dx11 Z__a r! (1) r**o where q = -|n i f n i s even, = -g-(n-1) i f n i s odd. Since f (u) = 1 , i t can be U2^2)^2 ; shown that f ^ n ~ r ) ( u) = ( - 1 ) n " r m(m+2).. .(m+2(n-r-1)) 1 2 n " r ( x W ^ f i f we l e t 0 = cot" 1x/h (2) and substitute from (2) i n t o ^ O ) we f i n a l l y obtain: d i = s i n m + n ef > c o s n " 2 r ] (3) dx11 h E _n ,_m+n ( / ^ whereoCr= n(n-1)(n-2).;.(n-2r+1)m(m+2).. .(m+2(n-r-1)) ( - 1 ) r 2 r r! . In the spe c i a l case m = 2, (3) may be shown to reduce to: [ Gibson, 1931; p. 85] 1 \= (-D n n! s i n n + 1 A s i n ( n + 1 ) 0 (4) writing d ny = D ny. The related function x may be shown to dx11 77? have derivatives given by [~Gibson, 1931, P*8.5~j : Vn( x V l d L C - n l s i n n + 1 c 9 c o s ( n + 1 ) 0 (5) These w i l l be used i n the following section. 76 —1 2 2 2. Higher derivatives of F =OCtan~ x/h +(3log (x +h ) DnF = D n " Y OC h + 2px ) \x 2+h 2 x 2+h 2 / (6) .= (-1) n" 1(n-1)! s i n n 6 ^ C s i n n 0 + 2 (3 cos n $ j (7) substituting into (6) from (4) and (5) Thus DnF = 0 i f = 1 tan ~ 1 ( - 2/ft ) (8). n where /\ = 3 The above expression for DnF may be written in terms of x and h by expanding sin nQ and cos nd in terms of sinQ and cos@ and substituting h and x ( ( 7 ^ T 1 / 2 respectively for the last quantities. Thus oCsin nQ + 2 j8 cos n(Q ~<6 (?)x n" 1h - (?)x*" 3h 3 + .... ? leading to: ^ • ( ~ 1) n" 1(n - 1)l f 20 x n U ( n ) b x n " * 1 - 2 ^ ( ? ) h 2 x n " 2 ( x W ) n [ -oC(5)h3xn-3 •+ 2(3( n)h 4x n" 4 +oC(£) h 5x n~ 5 - ..... (10) These derivatives for n = 1 to 8 are given in Table III. 3. Derivatives of ^ = b Q + .,...bnxn Writing ^ as ^ = f (x) J (11) 77 When f(x) i s . a polynomial, b Q+ ... + h n x n , t h i s leads to the expression: where P., i s defined by the array: 0 1 4- 5 0 b . h V 0 0 0 0 1 - km 0 2 b ^ 0 0 0 2 0 0 3b3hz 0 0 3 0 0<T>) b z o o 4 •0. 0 0 (z-m)b3 O 5bshz . r . • f ' 2 2 2 where h = r - x . As an example of the method, suppose i t i s required to obtain the expression f o r D ^ i n Table I from the expression f o r ^ . Referring to the Table, we, have from the integrand. b 0 = o C 3 3 h 2 b 1 =-oC 1 3h b2 = o C 1 1 m = 5 applying (15) we obtain: 78 2 \ a Q = b.jh = -cC^lr* a f = -b0m+2b2h2 - -(5°C 3 3 - 2<£^ 1 )h 2 a 2 = (1-m)b^ = + 40C13h a 3 = (2-m)b2=- 300^ and DJ = -it f 3oC11 x 3- 4fX, 1 3hx 2+ 5(<* 33-2^C11) h2x+oC 1 3 h 3 / r l ;. (x 2 +.. y 2 +h 2) ^  l as given. 4. Derivatives off^r = b^ +...,.+ b„xn o n (CQ+..*.....+ c^x ) The results of the preceding section may he generalized as follows: d_3»= d f brt+ ......+ h x n —^ Y o n dx dx / „ •. . \_ n mx L (. c_ + ......+ c x ; ^ a_ +..... +a„ ._• • o n+m-r1 m* ' J I ( e f t + . . . . + b / j ^ o m that F.^ is defined by the array: j= O I Z 3 4 It can be shown K J - v c , b 0 6,c0 0 0 0 O O 1 wcxb0 (i-v)c,b, zbzCo 0 0 o O a -3-VCjbo (i-2*W>, ll-vKbx Z\HC0 O 0 O s '4vcA0-^k C2-2v)Cibl C3-vKb3 UtC* 0 o The above i s of use i n obtaining derivatives i n the case of the sloping step. 5. Integration If the f u n c t i o n 7 ^ = a +. + a x 1 1 i s known to be J o n m r the d e r i v i a t i v e of a function of s i m i l a r form, i . e . i f •Jdx = b Q + ...,...+ b n - 1x' n-1 (17) then the c o e f f i c i e n t s a^ . may be obtained from the b^ by means of (15). A set of equations may be obtained expressing a^ . i n terms of the h^-j and h^. + 1. These equations may be solved to obtain expression f o r the b^ i n terms of the c o e f f i c i e n t s a^ . , as follows: b ^ = a n-1 n n-m-1 b ^ = a * n-2 n-1 n-m 2, V3 - I ^ ^ V l - an-2. (n-m-1) .' " \-4 = (n-2)h 2b n, 2 - a ^ 3 (n-m-2) (18) Example: From.Table II we see that: ir - 2px2- 2oChx+2gh2 dx = 20x -wCh . r 4 """T2" (19) thus i n (17) a Q = 2ph 2, • a 1 = - 20Ch, a2=-2p, m = 4, n = 2 Hence from (18) b 1 = =2^ = 2p; b Q = j&h =tfh -1 2 as i n (19)« 8Q REFERENCES BALSLEY, J.R., ana A.P. BUDDINGTON, Remanent magnetism of the Russell belt of gneisses, northwest Adirondack Mountains, New York, Advances in Physics, v.6, pp.317-322 1957 BRANT, A.A., Some limiting factors and problems of mining geophysics, Geophysics, v.13, pp.556-581, 1948. . BRITISH COLUMBIA DEPARTMENT OF MINES, Airborne magnetometer surveys, 1956-57,Victoria, B.C., 34 pp., 1958. BRODING, R.A. et.al., Magnetic, well logging, Geophysics, v.17 , pp.1-26, 1952. BULLARD, E.C. and R.I.B. COOPER, The determination of masses necessary to produce a given gravitational f i e l d , Roy. Soc. London. Proc. A, v.194, pp.332-347, 1948. DAVIS, C.W. Geological Significance of magnetic properties of minerals, Economic Geology, v.30, pp 655-662, 1935. ELKINS, T.A., The second derivative method of gravity interpretation, Geophysics, v.16, pp.29-50, 1951. EVJEN, H.M., The place of the vertical :gradient in gravitational interpretations, Geophysics, v.1, pp.127-136, 1936. GARLAND, G.D., Combined analysis of gravity and magnetic anomalies, Geophysics, v.16, pp.51-62,.1951. 81 GIBSON, G.A., Advanced Calculus, MacMillan, London, 503 pp.,1931 HALL, D.H., The linears of Northern Saskatchewan, M.A. thesis, University of Toronto, 56pp., 1950./' HALL, D.H., Least squares in magnetic and gravity interpretation, Trans. American Geophysical Union, v. 39, pp. 35-39* 1958. HAWES, J., A magnetic study of the Spavinaw granite area, Oklahoma, Geophysics, v.17, pp. 27-55, 1952. HE I LAND, C.A., CW. HENDERSON, and J.A. MINKOVSKY, Geophysical investigations at Carihou, Colorado, U..S. Bureau of Mines, Technical Paper No. 439, pp. 1-45, 1929. HEILAND, C.A.,' Possible causes of abnormal polarization of magnetic formations, Zeits fur Geophysik, v 6, pp.228-232 (1930). HEILAND, C.A., Geophysical exploration, Prentice-Hall, New York, 1013 pp., 1946. HENDERSON, R.G. and I. ZIETZ, Analysis of total magnetic-intensity/ anomalies produced by point and line sources, Geophysics, v.13, pp. 428-436, 1948. HENDERSON, R.G. and I ZIETZ, Computation of second vertical derivatives of geomagnetic fields, Geophysics, v. 14 #p.508-51.6,1949 HENDERSON, R.G. and I. ZIETZ, Graphical calculations of total intensity anomalies of three dimensional bodies, Geophysics, v. 22, pp, 887-904, 1957. 82 HENDERSON, R.G. and.I. ZIETZ, Magnetic-Doublet Theory in the analysis of total - intensity anomalies. Geological Survey Bulletin 1052-D, pp.159-186, United States Government Printing Office, Washington, 1958. HOBSON, E.W., The theory of spherical and ellipsoidal harmonics, Cambridge, 496 pp., 1931* HUGHES, D.S. and W .Li PONDROM, Computation of vertical magnetic anomalies from total magnetic measurements, Trans. American Geophysical Union, v, 28, pp.193-197, 1947 . JAKOSKY, J.J., Exploration Geophysics, Trijay, Los Angeles, 1161 pp., 1949. JEANS,, J.H., The mathematical theory of electricity, and magnetism, Cambridge University Press, 645 pp.,1948. KALASHNIKOV, A.G. and S.P. KAPITSA, Magnetic Susceptibility of rocks under elastic stresses, Akad. Nauk. S.S.S.R. Doklady torn 86 no 3, pp. 521-523, 1952(in Russian) Or (abstract only in English) Geological Survey Bulletin, Geophysical Abstracts No 151 - 14025, Department of the Interior, Washington, D.C. KOGBETLIANZ, E. G., Quantitive interpretation of maps of magnetic and gravitational anomalies by mathematical methods, Quarterly of Applied Mathematics, v.3, pp.55-75, 1945'» KOGBETLIANZ, E.G., Quantitative interpretation of magnetic and gravitational anomalies, Geophysics, v.9, p»463 1948. 83 KRUGLYAKOVA, G.I., The effect of accessory minerals on the production of reverse polarity in rocks, Akad. Nauk., SSSR. Izv. Ser. geofiz, No. 2, pp. 214-217, (1956) (in Russian). LOGACHEV, A,A., Textbook of magnetic exploration, Moscow, Gosgeoltekhizdat, 302 pp.,1955, LUNDBAK, A., Combined analysis of gravimetric and magnetic anomalies and some paleomagnetic results, Geophys. Prosp. v 4, pp. 226-235, 1956. MIKOV, D.S., The determination of the magnitude and direotion of the intensity of magnetization of disturbing bodies from the results of magnetic and gravity surveys, Izv. Akad. Nauk SSSR, Ser. gaofiz, no 5, pp 55-6, 1952. MIKOV, D.S., Determination of the direction of polarization of disturbing bodies from the data of the magnetic survey, Akad. Nauk. S.S.S.R, Izv. Ser. geofiz, no.5, pp.418-423, 1953 (in Russian). MILLER, E.T. and M, EWING, Geomagnetic measurements in the Gulf of Mexico and in the vicinity"of Karyn Peak, Geophysics, v 21, pp. 406 -432, 1956. MOODY, J.D. and M.J. HILL, Wrench-fault tectonics, Bu l l . Geological Soc. America, v.67, pp. 1207-1246, 1956. NETTLETON, L.L., Geophysical prospecting for o i l , McGraw-Hill, New York, 444pp., 1940. L 84 NETTLETON, L.L., Gravity and Magnetic calculations, Geophysics, v.7, pp. 293-303, 1942. PATERSON, N.R. and C.W. FAESSLER, Determination of body parameters of a magnetic inclined dike, 27th Annual Meeting, Society of Exploration Geophysicists, 1957? t i t l e only, Geophysics, v.23, p.405, 1958. PETERS, L.J., Magnetic theory and i t s practical applications, Geophysics, v. 14, PP. 290-320, 1948. 3LICHTER, L.B., Certain aspects of magnetic surveying, „Trans. American Inst. Mining Engineers, v. 81, Geophysical Prospecting, pp.238-253 1929 SCHWATT, I.J., An introduction to the operations with series, University of Pennsylvania Press* Philadelphia, 287 pp., 1924* SMELLIE, D.W., Elementary approximations in aeromagnetic interpretations, Geophysics, v»21, no. 4, pp. 1021-1040, 1956. SMYTHE, H.L., Magnetic observations in geological mapping, Trans. American Inst. Mining Engineers, v.26, pp.640-709, 1896. STRAKHOV, V.N., The determination of certain basic parameters of magnetized bodies from magnetic observations, Akad. Nauk. SSSR, Izv. Ser. geofiz, no. 2, pp. 144-<146, 1956. (in Russian). SUTTON, D.J. and W.G. MUMME, The effect of remanent magnetization on aeromagnetic interpretation, Australian Journal of Physics, vol. 10, no.4. pp. 547-557, 1957. VACQUIER, V., N.C. STEENLAND, R.G. HENDERSON, and I. ZIETZ, Interpretation of aeromagnetic maps, Geol. Soc. Amer. Memoir No. 47, 151 pp.., 1951. VOSKOBOYNIKOV, G.M., On the direction of magnetization of disturbing bodies in magnetic surveys, Akad. Nauk. S.S.S.R. Izv. Ser. geofiz. no.5, pp. 483-485, 1955 (in Russian). WATERS, G.S. and £. PHILLIPS, A new method of measuring the Earth 1s magnetic f i e l d , Geophys. Prosp. V 4, pp. 1-9, 19560 WILSON, J.T., Some aspects of geophysics in Canada with special reference to structural research in the Canadian Shield, Trans. American Geophysical Union, v. 29, pp. 691-726, 1948. II «i YUNGUL, S., Prospecting for Chromite with the gravimeter and magnetometer over rugged topography in east Turkey, Geophysics, v.21, pp. 433-454, 1956. ZIETZ, I and R.G. HENDERSON, A preliminary report on model studies of magnetic anomalies of three-dimensional bodies, Geophysics v. 21, pp. 794-814, 1956, 

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