ELECTRIC AND MAGNETIC FIELDS ASSOCIATED WITH A VERTICAL FAULT by ALAN MELVILL COODE B.A.Sc, University of B r i t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the In s t i t u t e of EARTH SCIENCES We accept t h i s thesis as conforming to the required standards THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library sh a l l make i t freely available for reference and study. I further agr.ee that per mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives,) It i s understood that copying, or publi cation of this thesis for f i n a n c i a l gain shall not be allowed without my written permission. Department of'•• TKVSits ' , ClMSTtToTg a F gv^TH 5ctetoces) The University of B r i t i s h Columbia,. Vancouver 8, Canada. Date i ABSTRACT Interest In the v e r t i c a l f a u l t problem f o r e l e c t r o magnetic f i e l d s has been recently revived by the papers of I. d ' E r c e v i l l e and G. Kunetz (1962) and D. Rankin ( 1 9 6 2 ) . In the derivation of h i s equations Rankin used d ' E r c e v i l l e 1 s theory which contains some f a l l a c i o u s assumptions. These have been pointed out by J.T. Weaver (1962) and also i n t h i s t h e s i s . This thesis follows the l i n e s of mathematical attack f i r s t employed by d ' E r c e v i l l e and Kunetz, and l a t e r developed by Weaver, i n applying the theory of i n t e g r a l transforms to the p a r t i a l d i f f e r e n t i a l equations s a t i s f i e d by land and sea conductors. The problem of both a v e r t i c a l f a u l t and also a sloping f a u l t , i . e . 0 <(u ( 90° where PC i s the angle of dip of the f a u l t are considered. The r e s u l t s i n the general case are Inconclusive, no solution has been found and no solution i s suggested. The case of ^ = 90° has proved to be equally indeterminate, but a solution has been suggested, which, although i t has not been proved rigourously, does not appear to vi o l a t e any physical p r i n c i p l e s and also seems to represent the f i e l d equations on the surface of the land and the sea. iv ACKNOWLEDGEMENTS I wish to acknowledge, with gratitude, the help, guidance and i n f i n i t e patience given to me by Dr. T. Watanabe i n the development of t h i s t h e s i s . I also should l i k e to thank Professor J.A. Jacobs, f o r making i t possible f o r me to work i n the Ins t i t u t e , Dr's. A. Nishida and S.P. Srivastava, f o r t h e i r i n t e r e s t and time spent i n discussion, and l a s t l y Dr. J.T. Weaver f o r reviewing t h i s work and o f f e r i n g several h e l p f u l suggestions. The work i n t h i s t h e s i s has been supported by the National Research Council. i i TABLE OP CONTENTS I INTRODUCTION ' 1 I I INDUCTION EQUATIONS 3 I I I PHYSICAL INTERPRETATIONS 13 IV BOUNDARY CONDITIONS 19 V BOUNDARY CONDITIONS FOR K. = 9 0 ° 24 VI SOLUTION OF THE EQUATIONS FOR = 9 0 ° 28 H - p o l a r i z a t i o n 28 E - p o l a r i z a t i o n 30 V I I PRACTICAL IMPLICATIONS 4 l V I I I CONCLUSIONS 5 2 APPENDIX A l SMELL'S LAW APPLIED TO A NONCONDUCTOR- CONDUCTOR BOUNDARY 5 4 APPENDIX A2 FRESNEL EQUATIONS 5 6 APPENDIX A3 PROOF OF DIV E = 0 5 8 APPENDIX A4 VALUES OF ^ AND J FOR G(y) 5 9 APPENDIX A 5 SOLUTION OF THE EQUATIONS IN THE GENERAL CASE 60 i i i LIST OP FIGURES I Co-ordinate systems at a land-sea-air contact. 3 II Diagram showing the transformation from one co-ordinate system to another. 8 III Magnetic f i e l d generated from an e l e c t r i c f i e l d i n a right angled conductor. 15 IV Magnetic f i e l d s generated from e l e c t r i c f i e l d s In two right angled conductors; J, > SO that the v e r t i c a l magnetic f i e l d , H^ , , at the d i s continuity i s directed upwards. 15 V Incident, transmitted and r e f l e c t e d rays at a boundary. 20 VI General f i e l d vector i n nonrectangular co-ordinates. 21 VII V e r t i c a l magnetic anomaly, a f t e r J.T. Weaver. 40 VIII Magnetogram records, adapted from U. Schmucker. 42 IX Map of Southern C a l i f o r n i a . 43 X Schematic cross-section at the sea coast f o r an oversimplified conductivity c l i f f . 45 XI V e l o c i t y depth curve (shear waves) under the Canadian Shield (Model CANSD) and under the ocean (Model 8099), af t e r J . Brune and J . Dorman. 45 XII(a) Idealized p l o t of skin depth vs. period f o r two co n d u c t i v i t i e s . 47 (b) Probable v a r i a t i o n of with period. 47 1. INTRODUCTION Interest i n the v e r t i c a l f a u l t d i s c o n t i n u i t y problem has been recently stimulated by the papers of I. d ' E r c e v i l l e and G. Kunetz (1962) and D. Rankin (1962). The present work i s an extension of a paper by J.T. Weaver (1963) who was looking f o r a t h e o r e t i c a l explanation of an anomalously high v e r t i c a l magnetic component of geomagnetic micropulsations found at a land-sea contact. In 1961 D.A. C h r i s t o f f e l , J.A. Jacobs and J.A. Shand published a paper comparing the r a t i o s of the v e r t i c a l to the horizontal components of the magnetic f i e l d s at V i c t o r i a , B.C., which can be considered to be situated at a land-sea contact at the edge of the P a c i f i c , and Ralston, which i s i n the Southern part of the pl a i n s of Alberta and thus i s removed from a land-sea con t a c t . The r e s u l t s showed that there was a d e f i n i t e v e r t i c a l anomaly i n the magnitude of the geomagnetic amplitudes i n the micropulsation range (0.001 cps to 3.0 cps) at a land- sea contact. This contradicts an assumption used by d ' E r c e v i l l e i n h i s t h e o r e t i c a l treatment of the problem. In t h i s thesis the i n i t i a l electromagnetic f i e l d i s assumed to be v e r t i c a l l y incident. Even though t h i s approximation i s not exact, i t i s found (see Appendix 1) that i t i s equivalent to the condition that the displacement current i s n e g l i g i b l e i n Maxwell's equations, i . e . that I^ H » I coe/ . As a re s u l t , the angle that the plane of 2. incidence makes with the d i s c o n t i n u i t y i s a r b i t r a r y and hence i t has been assumed to be at r i g h t angles to the d i s c o n t i n u i t y . This permits simplifying assumptions to be made and the problem becomes two dimensional. The d i f f i c u l t i e s a r i s i n g from variat i o n s of con d u c t i v i t y with depth have been circumvented by assuming that the conductivity does not vary f o r depths l e s s than 200 km. and that the electromagnetic f i e l d has been "completely" attenuated at that depth, i . e . f o r t h e o r e t i c a l purposes a depth of 200 km. can be regarded as the depth to i n f i n i t y . Neither land nor sea have been assumed to be perfect con ductors (although both may be considered so when compared with a i r ) . 3. INDUCTION EQUATIONS For the general case the coordinate axes 0 (X, Y, z) have been chosen with the X-axis along the di s c o n t i n u i t y junction of land, sea and a i r , the Y axis along the sea-air contact, and the z axis along the land-sea contact. In the case of a v e r t i c a l d i s c o n t i n u i t y ( &< = 9 0 ° ) , the axes are rectangular and are-renamed 0 (x, y, z ) . Thus GX and Ox are i d e n t i c a l as are 0Y and Oy (see Figure 1 ) . FIGURE 1 Co-ordinate systems at a land-sea-air contact. It i s assumed that the incident electromagnetic f i e l d i s a wave of the form H L = H o e 4. where: H a = the constant, f i n i t e amplitude of the magnetic f i e l d ka= the wave number i n a i r i . e . l<i - ^W/^o^o' where and & 0 are the permeability and p e r m i t t i v i t y i n a i r A h-e= the unit vector i n the d i r e c t i o n of pro pagation and CO = the frequency Upon s t r i k i n g the land-sea conductor the magnetic f i e l d i s " s p l i t up" into two parts, the r e f l e c t e d and trans mitted f i e l d s . Both these f i e l d s are assumed to vary as follows L tut Hence i n the atmosphere the f i e l d i s given by TT= TTL + hT. = elofc"[ H. e ^ + i t - ^ t f ] A si m i l a r set of equations holds f o r the e l e c t r i c f i e l d , assuming 5 . I t i s possible to simplify Maxwell's equations i n a homogeneous, i s o t r o p i c medium. (a) V % hT= i + (b) V x t = - | 5 (1) (a) ^7»H = 0 (b) V-E = (2) I f the f i e l d s are varying i n a purely periodic manner then f o r the two dimensional case, equation 2(b) i s found to be automatically equal to zero, i . e . (3) The proof of t h i s i s given i n Appendix A3. the r a t i o Within the conductors i t i s necessary to consider J L L , which i s found to be much greater than unity. Table I gives the value of t h i s r a t i o f o r land and sea respectively: Table I Medium Conductivity; mho/m sea land 10~5-10-6 D i e l e c t r i c c o n s t a n t ^ Parad/m Frequency cps 80xl0- 1 0 3 3xl0~ 1 0-4xl0- 1 0 3 1 0 + i o 10+* 6 . Hence i t i s p o s s i b l e i n both cases' to assume t h a t d i s p l a c e ment c u r r e n t s may be n e g l e c t e d i n the conductors, i . e . t h a t i n both l a n d and sea. However there i s no J u s t i f i c a t i o n i n i g n o r i n g the displacement c u r r e n t i n an uncharged atmosphere. Such a j u s t i f i c a t i o n i s o n l y p o s s i b l e a f t e r a s o l u t i o n of the e l e c t r i c and magnetic f i e l d has been o b t a i n e d . F u r t h e r i f we t r e a t the f i e l d s as p u r e l y e l e c t r o s t a t i c or magneto- s t a t i c then t h e r e i s no j u s t i f i c a t i o n f o r u s i n g Maxwell's equations to f i n d the remaining f i e l d . Thus assuming y * H = o ( f o r a magnetostatic f i e l d ) , we cannot use 7^x H~ - - . to f i n d the e l e c t r i c f i e l d . T h i s p o i n t has been d e a l t with more f u l l y i n Dr.. A. N i s h i d a ' s Ph.D. T h e s i s ( 1 9 6 2 ) . Hence i n the n e u t r a l atmosphere (a) V * H " = } f ( b ) \ 7 ^ = - | ! ( 5 ) T h e r e f o r e u s i n g equations (1) - (4) i t i s p o s s i b l e to express the magnetic and e l e c t r i c f i e l d s as d i f f u s i o n equations f o r the case when Cx. = 9 0 ° , i . e . I f we s u b s t i t u t e rj*= f ^ q - t o (6) then the equations to be s o l v e d i n each conductor are - ^ { 7 ( a ) ) 7. and ^ ^ " H - u ^ H . ( 7 ( D )) In the atmosphere the equations to be solved can be derived i n a sim i l a r manner and are (a) \7'2E" = 7^eco^E' and (b) ^ H ^ A ^ H (8) Because of the two dimensional nature of the model a l l f i e l d vectors are independent of x and we can f i n d two solutions of equations (7), one f o r E"-polarization and another f o r H - p o l a r i z a t i o n . In both cases the polarized f i e l d vectors are directed along the x-axis. Hence f o r H - p o l a r i z a t i o n and f o r ET-polarization These two cases are completely independent of each other and the solution of the d i f f u s i o n equations w i l l be obtained f o r each case separately. For the case of H - p o l a r i z a t i o n the equation to be solved f o r z>0 i s • — _ h r u ^ = ^ o 6 ( i i) 8. and for ET -polarization the equation i s (12) So f a r no mention has been made of the d i f f u s i o n e q u a t i o n when 9 0 ° > > 0 ° « F o r the cases of £ - and H - p o l a r i z a t i o n they may be ob t a i n e d from e q u a t i o n s (11) and (12) u s i n g the equations of t r a n s f o r m a t i o n f o r the two co o r d i n a t e systems 0 (X, Y, Z) and G (x, y, z) v i z . ^ =• y + 2 See F i g u r e I I . ( 1 3 ) or V = ^ "* \ cc^ tr&c (14) FIGURE I I Diagram showing the t r a n s f o r m a t i o n from one c o - o r d i n a t e system t o another. 9. Therefore and (15) and hence Therefore equations (11) and (12) may be written (16) where F i s equal to ei t h e r E=x or Hat. The value of the transmitted magnetic f i e l d across z=0 can be expressed as a function of only, say ^ | 6 p , and time, "t , f o r H-polarization and the value of the trans mitted e l e c t r i c f i e l d can be expressed as another function of ^ f o r E - p o l a r i z a t i o n , say Ci<£ and time, "b . Hence assuming solutions of equations ( 8 ) to be of the form and ir\u2& 10. we can express the component i n a i r (omitting the term 2 as before) as The condition which must be s a t i s f i e d f o r a solution of (8) i s therefore (17) i.e. n = J / A f r ^ - m*-' f o r and n = LjrY^-jxe^1' f o r w j^ TT"1 < I M l (18) The general expression f o r the r e f l e c t e d wave can be expressed by J-00 (19) where "h, i s given by (18). Since H where the subscript 2 refers to air, at z=0 for H -polar ization the tangential component of the magnetic field is continuous across the land-sea conductor i.e. (20) where h. is the unit vector normal to the surface. 11. or ; o n 7j -° For H - p o l a r i z a t i o n Applying a Fourier transform, we can express the function AC*™) as a function of , v i z . —PO Since + 00 v where i s the Dirac delta function, Aim) s^£7 )^e~^^ " H° ^ (VVV4-^ (21) v/ — oo Hence knowing the function by solving equations (11) and (12) i t i s possible to express the magnetic and e l e c t r i c f i e l d i n the a i r . A similar function defined by St^) can be obtained f o r E - p o l a r i z a t i o n . The proof follows s i m i l a r l i n e s , and i t can e a s i l y be shown that 00 (22) 12. where C(^}ls the value of the transmitted e l e c t r i c f i e l d , which corresponds to GjC^)> on z=0. Hence B ^ e * ( 2 3 ) i s the e l e c t r i c f i e l d i n the atmosphere ( z < 0 ) . For H - p o l a r i z a t i o n , the e l e c t r i c f i e l d i n the atmosphere i s given by: Ex = o EL - 1 IH* ^ c = i ihf^ oc.r and f o r EE" - p o l a r i z a t i o n , the magnetic f i e l d i n the atmo sphere i s given by: H x - S Hw«i-i^ x and Ha* I l l ^ E * (25) Equations ( 9 ) , ( 1 0 ) , (24) and (25) may be general ized f o r the case of 0 < 90° by applying equations ( 1 5 ) . (24) 1 3 . PHYSICAL INTERPRETATIONS When an e l e c t r o m a g n e t i c wave meets a change i n con d u c t i v i t y (e.g. at an a i r - s e a c o n t a c t ) the i n c i d e n t wave i s propagated through the new co n d u c t i n g medium wit h a v e l o c i t y of ^2=J£_ where ^JL and T* are the p e r m e a b i l i t y and con d u c t i v i t y of the medium. Us i n g Maxwell's equations the magnetic and e l e c t r i c f i e l d s can be expressed i n the f o l l o w i n g manner: V ^ a H = ( I > ^ - G ) H and V * " E T S (.1 wyl^cr - u i * ^ e ) £ " However so t h a t these equations can be approximated by equations (7). The e f f e c t of t h i s on any p h y s i c a l i n t e r p r e t a t i o n i s to h e a v i l y damp the e l e c t r o m a g n e t i c wave i n the conductive medium and to permit the c u r r e n t d e n s i t y to take a f i n i t e time to b u i l d up, which r e s u l t s i n the d i f f u s i o n phenomenon d e s c r i b e d by equations (7). The d e s c r i p t i o n of the e l e c t r i c and magnetic f i e l d s a s s o c i a t e d w i t h the c u r r e n t d e n s i t i e s can be found from the v e c t o r p o t e n t i a l 14. Since and V * H afjr + l ^ E i f the gradient of the scalar p o t e n t i a l of the e l e c t r o s t a t i c f i e l d i s small compared with -^-^ . Hence i f we take a "wedge" of sea, or to simplify i t , take the case when the angle tx? = 90° then i t can be seen that f o r v e r t i c a l pro pagation a current system i s set up, decaying exponentially with depth (z). This current system radiates waves i n a l l directi o n s , both i n the y - d i r e c t i o n and i n the negative z-direction ( r e f l e c t e d ray). The rad i a t i o n i n the y - d i r e c t i o n i s not part of the r e f l e c t e d ray but i s referred.to as a d i f f r a c t i o n term. The d i r e c t i o n of the magnetic f i e l d i s given i n Figure I I I , assuming that the e l e c t r i c f i e l d i n the conductor i s directed along the x-axis. I t w i l l be noticed that i n t h i s case ( i . e . £T - p o l a r i z a t i o n ) , the magnetic f i e l d at the v e r t i c a l boundary has become v e r t i c a l . For the case of H - p o l a r i z a t i o n the corresponding e l e c t r i c f i e l d becomes v e r t i c a l . The problem now arises of what happens when two conductors are placed side by side. If the angle °C i s s t i l l assumed to be 90° then the magnetic f i e l d f o r each.conductor, separated as i n Figure IV, i s generated i n the same manner as before but now there i s a mutual induction f i e l d near y=0, which tends to oppose the ex i s t i n g e l e c t r i c f i e l d i n both conducting media. Upon joi n i n g the two conductors on the plane y=0 then there i s an 15. \ I H diffracted | | H reflected j , = O J E , , out of the paper o o o o o o o o o o o 0 o 0 0 o FIGURE III Magnetic f i e l d generated from an e l e c t r i c f i e l d i n a right angled conductor. H r3 H ri medium 3 9 % H Z r \ 1 1 i - J H H Z 3 ® ® ® ® <g> medium I E into the paper FIGURE IV Magnetic f i e l d s generated from e l e c t r i c f i e l d s i n two right angled conductors; > ^ so that the v e r t i c a l magnetic f i e l d , H^. , at the d i s continuity i s directed upwards. 16. e l e c t r i c f i e l d d i s c o n t i n u i t y between Eg and E| and hence there i s a d i f f u s i o n of e l e c t r i c f i e l d i n the y-d i r e c t i o n , i n the same manner as the transmitted e l e c t r i c f i e l d d i f f uses, u n t i l the value of the e l e c t r i c f i e l d reaches a common value on y=0. The difference i n current density gives r i s e to an anomalous v e r t i c a l component of the magnetic f i e l d , \r\^ . There are three cases to consider: 1) ^-3 ^ \i when i s downwards at the discon t i n u i t y junction. 2) | 4 B [ i i n which case the two media are i d e n t i c a l (and therefore there i s no conductivity discontinuity) 3) j. i " N \ i when i s upwards (see Figure IV). Rather than t r y to compute the d i f f u s i o n of the e l e c t r i c f i e l d from f i r s t p r i n c i p l e s , a reasonable guess i s made. In much the same way as the e l e c t r i c f i e l d i s attenuated with depth, f o r E - p o l a r i z a t i o n , the d i f f u s i o n of the e l e c t r i c f i e l d can be considered to be "attenuated" by the same terms. Hence the value of the e l e c t r i c f i e l d s i n media 1 and 3 may be given by: tt - r.. ( I - e * » > + ^ e?>\ ^ « 0 , r o (26) f, - (. - e-«). t £ * ^ o « o , 3 . 0 (2 7> where 1^1^) i s the value of the e l e c t r i c f i e l d on y=0, E*i0 i s the value of E, at y= +• «> > £ 3 o i s the value of E 3 at y= - oo and &> = J u.uoa~ I i n media 1 and 3. I t i s thus related to the skin depth / . The same idea may be applied to the case of H - p o l a r i z a t i o n and a si m i l a r d i f f u s i o n of H*. found to e x i s t i n the form K - f t 0 -e *») + f ^ e M ; s^< o, v . 0 ( 2 8 ) rT - -e^ + ^ ^ > vj >o, -j s o ( 2 9 ) The rather a r b i t r a r y choice of E and H o n z = 0 may seem u n j u s t i f i e d at t h i s stage but i t w i l l be found that a function of t h i s type i s needed l a t e r i n t h i s thesis, even i f i t represents a crude approximation to the true case. Summarizing, f o r the case of ^ - p o l a r i z a t i o n three f i e l d s e x i s t : W^, and . The f i e l d s Hx and appear to be continuous functions with continuous f i r s t d erivatives each approaching constant and known values at ±°o but the value of appears to have a d i s t r i b u t i o n with a maximum (cusp) value at y=0 and tending to zero at 18. When considering the case of a wedge there i s also a transmitted term to be added to the d i f f u s i o n of an e l e c t r i c f i e l d which flows across the land-sea contact. As the angle 0<. approaches zero i t i s found that there i s less and l e s s d i f f u s i o n because of the difference i n e l e c t r i c f i e l d i n t e n s i t i e s i n the two media and more propagation of current as a simple transmitted and r e f l e c t e d ray problem. Hence i n the l i m i t i n g case of horizontal layers (L. Cagniard (1953), A.T. Price (1962)) there i s only a problem of trans mission and r e f l e c t i o n with no d i f f u s i o n due to a difference i n e l e c t r i c f i e l d i n t e n s i t y . 19. BOUNDARY CONDITIONS A s o l u t i o n of Maxwell's equations i s g i v e n by the P r e s n e l equations at y= ± oo f o r the a i r - l a n d , a i r - s e a c o n t a c t s . I f i t i s assumed t h a t the i n i t i a l wave i s of the form: ^ -±- - ) I I £ v p. then the t r a n s m i t t e d and r e f l e c t e d waves w i l l be of the form: and ' _^ u - r - z J IQJL+X. F o r E" - p o l a r i z a t i o n £ , = C , E o a n d P C 2 E o F o r M - p o l a r i z a t i o n H , — £), H « and where Ct>Cz, (q, and <^ja are known c o n s t a n t s f o r the a i r - sea, a i r - l a n d boundary (see Appendix 2 ) . Hence the magnetic and e l e c t r i c f i e l d can be c a l c u l a t e d at y=±«o f o r both H - p o l a r i z a t i o n and E T - p o l a r i z a t i o n . At any boundary, ^ ( f L + f r ) = ^ E b and K x C H t t Hv}*h)JTt. (30) 20. FIGURE V I n c i d e n t , t r a n s m i t t e d and r e f l e c t e d rays at a boundary. F o r H - p o l a r i z a t i o n these equations g i v e : where Hfct^) i s the magnetic f i e l d i n s i d e the conductor. L e t us c a l l t h i s f u n c t i o n on z= 0 ^ - j t ^ i n l a n d , (^iCt^jin sea and £|'6^ i n g e n e r a l . A s i m i l a r case e x i s t s f o r E - p o l a r i z a t i o n , L e t the t r a n s m i t t e d e l e c t r i c f i e l d on z= 0 In l a n d be c a l l e d and In sea C , a n d C(i^), i n g e n e r a l . In the p r e v i o u s s e c t i o n an i n t e l l i g e n t guess was made of the v a l u e s of G,t^ ,(=>3t^, C^h and C5C^) but no use w i l l be made of t h i s u n t i l i t i s necessary. 21. At the land-sea c o n t a c t f o r H - p o l a r i z a t i o n the t a n g e n t i a l component of the e l e c t r i c f i e l d i s continuous, i . e . Ei£ i s continuous. FIGURE VI General f i e l d v e c t o r i n n o n r e c t a n g u l a r c o - o r d i n a t e s . Er? «(Cw!7>tC0<) E y and so t h a t E , - J - O b * - LH-^.11H* S_1 i H Y T h e r e f o r e E" 2 = - CJHx^C & b H x and J - 3 H g , _ X F o r E - p o l a r i z a t i o n the t a n g e n t i a l component of the magnetic f i e l d i s continuous and a s i m i l a r set of equations r e s u l t s . 22. Hence the boundary c o n d i t i o n s required to solve Maxwell's equations are, for (-(-polarization: H(i) The transmitted magnetic field, ^»^=the magnetic field in the air on the boundary plane 2?= o,-a><\ H(li) H = H3~<^Z)on V=0. H(iii) Etangential i s continuous, i.e. J_lb3= J . < L b i on v = o H(iv) HnHsi iti' ^ Jds, 3H, and "b H 3 all tend to zero as Z — * > - + » o ~ H(v) H,.= Ho, V*+«o and H^ - ^M0)V«-°^ which are all known, constant and finite. F o r E"-polarization the boundary conditions are: E(i) The transmitted electric field CCT)55the electric field in the air o n Z = 0 ( - o o < ^ < + o o . E(ii) E3= E.s^Z) on Y = £ E(iii) = ^ E' on V - o E(iv) E*|4 E3.,dEi ^Si and LEa all tend to zero as Z —* M ( <* o ) . E(v) E i - ^ E o , V'-*»-foo and E 3 = £ 3 E O J V- -oe^ which are all known, constant and finite. 23. It w i l l be noted that i n the case (X= 9 0 ° Y- and 2=^. and the boundary conditions s t i l l hold true. 24. BOUNDARY CONDITIONS PGR 0<= 90° The boundary c o n d i t i o n s f o r the g e n e r a l case have a l r e a d y been enumerated. These c o n d i t i o n s w i l l now be adapted t o the case <X =90° . In a d d i t i o n the boundary con d i t i o n s a c r o s s the land-sea and a i r c o n t a c t w i l l be g i v e n so that the v a l u e s of G\0g and Ci^) can be computed knowing the form of the e l e c t r i c and magnetic f i e l d s I n the a i r and i n the land^-sea conductors and eq u a t i n g them on the boundary z=0. F o r H - p o l a r i z a t i o n : H ( i ) The t r a n s m i t t e d magnetic f i e l d e q u a l s the t o t a l magnetic f i e l d i n the a i r which has the value on the boundary plane ^ — 0 , — o0<u^<+oo. H ( i i ) H, ^ H3= o n 1 • H ( i i i ) The t a n g e n t i a l component of the e l e c t r i c f i e l d i s continuous, I.e. - L l b s * _L £H ^ 3 a ^ ^7 b% i on ^s-o. H ( i v ) H ( > H 3 ; 3H|, 3 f c l s • J i j , and ^ t ) 3 a l l tend to zero as z tends t o i n f i n i t y . H(v) H , = - <^ )» H o ^ f o r y equal t o -f oe and H 3 - ^3 Ho £ f o r y equal to - 00 , where the v a l u e s of C-j, and <£5 are known, constant and f i n i t e . 25.. H(vl) The r e f l e c t e d wave i n a i r i s propagated upwards, i n the negative d i r e c t i o n . H ( v i i ) The tangential components of the e l e c t r i c f i e l d are continuous across the land-sea and a i r contact, i . e . For E T-polarization: E ( i ) The transmitted e l e c t r i c f i e l d equals the t o t a l e l e c t r i c f i e l d i n the a i r which has the value Ct^> on the boundary plane -^ =- O ^ — oo <<^< + oo. E ( i i ) E S = E > ^ C ^ on E ( i i i ) The tangential components of the magnetic f i e l d are equal on the land-sea interface, i . e . $£•1 - 3 _ E J on u. s e> . E ( i v ) Eo E3J <L§3> <LE/ and 3__E5 all tend ^ ^ ^ 3 U to zero as z tends to i n f i n i t y . 0 E(v) E( = C, E0 f o r y equal to -t- <*> and 3 - ^ -3 c- ° ^ f o r y equal to - co , where the values of C, and C 3 are known, constant and f i n i t e . 26. E ( v i ) The r e f l e c t e d wave i n the a i r i s propagated upwards, i n the n e g a t i v e d i r e c t i o n . E ( v i i ) The t a n g e n t i a l components of the magnetic f i e l d are continuous a c r o s s the l a n d - s e a and a i r c o n t a c t , i . e . The c o n t i n u i t y of the v e r t i c a l components of the magnetic f i e l d i s ensured by the c o n d i t i o n of c o n t i n u i t y of the e l e c t r i c f i e l d f o r E T - p o l a r i z a t i o n . I t i s on the q u e s t i o n of the c o r r e c t boundary con d i t i o n s t h a t most of the e a r l i e r work has been i n e r r o r . I. d ' E r c e v i l l e and G. Kunetz say t h a t to s o l v e the case of E - p o l a r i z a t i o n one must assume t h a t at the s u r f a c e i s not dependent on i ^ , which amounts to assuming t h a t the v e r t i c a l component of the magnetic f i e l d i s zero on the s u r f a c e . The f a l l a c y of t h i s assumption has, a l r e a d y been p o i n t e d out i n the " I n t r o d u c t i o n " to t h i s t h e s i s . D. Rankin, f o l l o w i n g the above authors assumes t h a t , f o r the case of H - p o l a r i z a t i o n , the v e r t i c a l component of the e l e c t r i c f i e l d i s zero on the s u r f a c e . Prom t h i s he concludes t h a t I f t h i s were so then the f u n c t i o n would be equal to a constant, which i s i n c o r r e c t . Hence Rankin's assumption 27- t h a t the v e r t i c a l component of the e l e c t r i c f i e l d i s zero on the s u r f a c e z=0 i s i n c o r r e c t . J.T. Weaver has a l s o made the same i n c o r r e c t assumption i n s t a t i n g t h a t on z=0, ^ = O f o r H - p o l a r i z a t i o n and g((-f^ _ r> f o r E" - p o l a r i z a t i o n . 2 8 . SOLUTION OF THE EQUATIONS FOR <X = 90° H - p o l a r i z a t i o n When c>< = 90° the co-ordinate system i s rectangular and the d i f f e r e n t i a l equation to be solved i s thus simpli f i e d . Equation ( 1 1 ) , v i z . has to be s a t i s f i e d with boundary conditions along y=0 and z=0. Because of the s i m p l i f i e d d i f f e r e n t i a l equation i t i s possible to use a Fourier sine transform (see Appendix A5 f o r the general case). A Fourier cosine transform could also be used but i t would be necessary to e s t a b l i s h the boundary conditions i n a s l i g h t l y more complicated manner. The Fourier sine transform, with i t s inverse, can be expressed as follows Application of the Fourier sine transform to the d i f f e r e n t i a l equation gives the following r e s u l t s : and 29. and ^ a ^ l + where f'*1*}*'- ( 5 3 ) I f we now l e t H* t fy6) = £?(^) i t can be seen that there i s only one unknown function whereas f o r the general case there are two (see Appendix A 5 ) . The solution of t h i s d i f f e r e n t i a l equation can be expressed as follows: H 3=r fJCKi^ + V s ^ ^ K ^ ( 5 4(b)) where ^ = ffi - * Co-) } K ( = ^ 6a) ( 5 5 ) ^ | cl^ hQ( $ * z ( 5 6 ) and -J^) = y, ^ ^ 1=0 t N>3 T> +<T3^/Co) - ^ 1^ 1°) ( 5 7 ) 30. E - p o l a r i z a t i o n Similar solutions ex i s t f o r the case of £ -polar i z a t i o n . No proof w i l l be given but the r e s u l t s w i l l be stated: ML L^ e'^ + M ^ ^ > V o ( 5 8 ( a ) ) 3 '- $ I [K* «** + & (3>1 , "i<» (58(b)) where V * a ^ SjJ ^ ( 5 9 ) w i t h K)3 = - ( 6 o ) a n d ^) * ^, isM+yfa-VsL*) (6l) Having obtained the basic formulae f o r the magnetic and e l e c t r i c f i e l d s , e x p l i c i t expressions f o r ^ I C Q - ) > ^ ^ C ^ " ) and jf-jt-fc) w i l l be obtained. The f i r s t approximation i s that ^ a n d ^ a r e equal to one constant f o r y^O and to another f o r y ^ 0. This i s too simple an assumption, and leads to f a l s e physical r e s u l t s ( i . e . a step-function f o r the magnetic and e l e c t r i c f i e l d s f o r z=0). The next estimate- would be to apply the function noted i n the physical i n t e r p r e t a t i o n with the values 31. and E,« = C E . M 3 o = G 3 Ho H,o The v a l u e s of the L|^ 's and <J • s can then be c a l c u l a t e d com p l e t e l y as f o l l o w s . H e a v i s i d e ' s o p e r a t o r formula f o r e x p o n e n t i a l s i s (61) A p p l y i n g t h i s t o and Ct^ and n o t i n g t h a t ! = at then ^ / t ^ , ^ ^ , ^ ,(^) and <f30^ become (62) (A proof of t h i s i s g i v e n i n Appendix A4). (63(a)) S i m i l a r l y (63(b)) (64(a)) £W (64(b)) The question of the v a l i d i t y of Heaviside's method may be raised but the r e s u l t s can be v e r i f i e d by substituting equations ( 6 3(a),(b)) and (64(a),(b)) into equations ( 1 1 ) ( 1 2 ) . For the case y=0 and ^ ^ o ) = ^ t ^ ) ( ^ ^ - H o j 3 o ) ( 6 5(a)) ( 6 5(b)) ( 6 6(a)) ( 6 6(b)) Substituting ( 6 3(a)) into equation ( 5 4(a)) and l e t t i n g <y*<p the boundary condition l -U^ f o r c* = 9 0 o i s s a t i s f i e d , i . e . also This shows that the suggested solution i s , at l e a s t , well behaved at the l i m i t s I^-*±PO. 33. The unknown function may be found by equating the tangential components of the e l e c t r i c f i e l d at ( 0 , 0 , 0 ) . i . e . i — ' = -L &Bz on 3 ^U^pT^ < 67<*» Simi l a r l y , (67(b)) Hence the functions ^ , ( ^ ) ? > a n d a r e a l l known and may be computed. Values of the e l e c t r i c and magnetic f i e l d may now be found by d i r e c t substitution and when one l e t s g-*0 (from the p o s i t i v e side) the i n t e g r a l should be equal to ^ C ^ ) f o r H - p o l a r i z a t i o n and Cc<£) f o r E - p o l a r i z a t i o n i f the a n a l y t i c a l procedures employed are s e l f consistent. Unfor tunately, even i n the case of t h i s r e l a t i v e l y simple estimate, the evaluation of the i n t e g r a l i s extremely complex (due to the presence of branch points) and can only be evaluated by numerical means. 34. To ensure that Qi^ and Ci^) are the correct functions on the surface z=0, i t i s necessary to equate the magnetic and e l e c t r i c f i e l d s i n the conducting media to the magnetic and e l e c t r i c f i e l d s i n the a i r using the boundary conditions H ( i ) , (vi) and ( v i i ) f o r the magnetic f i e l d and boundary conditions E ( i ) , (vi) and ( v i i ) f o r the e l e c t r i c f i e l d . Applying condition H(v l i ) f o r H - p o l a r i z a t i o n i t i s found that: on z=0 (68) The value of I-l, i s given by equation (54(a)), as z tends to zero from the pos i t i v e side, and the value of H 2 i s given i n part by equation (21-), as z tends to zero from the negative side. Therefore: y (69) - e The order of d i f f e r e n t i a t i o n and integration i s i n t e r changeable on the right hand side and so i s the order of taking the l i m i t and integration. Hence, assuming normal incidence: 35. J — so where t i i = JyU-e <*>*' , from equation (1?)* and the value of i s given by the estimate. Having obtained the value of the right hand side of the equation, to prove that the estimated value of i s correct, the value of the l e f t hand side must be computed and proved to be equal to the right hand side. = |f[S,(H,.(.-e^)]^^a^ ( 7 0 ) which, as ^ tends to i n f i n i t y , gives the boundary condition H(v) upon integration but otherwise, i s an i n t e g r a l with a branch point. If the i n t e g r a l I i s given by 3 6 . then I = I , + ^ where J x = J f j~ W ^ *H ^ ? and I • * f * ° P ^ " " ^ H a ^ » f *" 3fa ^ ~ H.>rP> / f » «Tht* This i n t e g r a l can be subdivided into three more int e g r a l s , i . e . l e t I a s I * + I k + I* . where • H t 0 ^ , H ^ e^' ^ i ^ ( 7 1 ( a ) ) and (71(c)) These Integrals possess complex branch-points making t h e i r mathematical evaluation extremely complicated. I t i s suggested that these equations can be better solved on a computer and the r e s u l t s plotted on a graph. i f the r e s u l t s plotted from the right hand side of the equation f a l l on the curve derived from the l e f t hand side then i t may be assumed that the estimate s a t i s f i e s a l l boundary conditions and d i f f e r e n t i a l equations. 37. The equations f o r E" -p o l a r i z a t i o n , which are si m i l a r to H -p o l a r i z a t i o n , are ^ m M X - * A o ) € M = ~ e ~ ^ J ( 7 ? ) and I ~ I s + I u. where I 4 i s given by 2 )e"^"] siyy^eL^ , ^ >• O (73) and . x Jo L (N, + S ^ J X. Sj" £^«S^£j^dlsj (74) Again J 4 = + I e + where I . = A f ^ E o - ^ E a o ) * ^ ^ (74(a)) T ^ ^ ""U<«> - ^(v.-fo ^ | L i ^ i g (74(b)) 4> (>$,+ «; 1 38. The numerical integration of each of these equations may be obtained, by seperating the functions into r e a l and imaginary parts and evaluating each i n t e g r a l on a computer. The v e r t i c a l magnetic f i e l d can be found, f o r the case of E - p o l a r i z a t i o n , from equation (10) to be Therefore: x e ^ ^ z j ^ d f s - } ^>,o (75) A s i m i l a r equation e x i s t s f o r y$ 0 . I f the estimate C*-0^ i s correct then H V ^ P ' ^ - O ^ j _ (76) ^ >x o a n d H ^ ^ p . a ^ - ^ e ^ It w i l l be noticed that the v e r t i c a l magnetic f i e l d i s i n dependent of the frequency at y=0. Is proportional to the square root of the frequency and so i s ( ^Lo) — Ef, 0 ) 3 and hence the only frequency dependent term i s exp (-^p,ift) . This r e s u l t was obtained, assuming that 3 9 . on z = 0 , where B i s constant, by J.T. Weaver ( 1 9 6 3 ) and also by T. Watanabe i n a derivation of equations f o r magneto t e l l u r i c modeling (private communication). J.T. Weaver has calculated the r a t i o of the v e r t i c a l to horizontal magnetic f i e l d over the land-sea contact and h i s graphs are shown i n Figure VII. 40. 3-0 20 15 10 5 Distance -y (km) from coast 30T — — Distance +y (km) from coast. FIGURE VII V e r t i c a l magnetic anomaly, a f t e r J.T. Weaver. 41. PRACTICAL IMPLICATIONS The v e r t i c a l anomaly found i n the Praser River Delta has already been mentioned. This anomaly, i t i s thought, i s due to r e l a t i v e l y close-surface conductivity variations and the emphasis, i f placed anywhere, must l i e on periods i n the range 0<T^:50 seconds. Longer periods occur l e s s frequently and are l e s s accurately plotted. Further the theory so f a r expounded of l a t e r a l skin-depth could not be interpreted f o r depths of over 5 km. i n the sea at an absolute maximum and therefore longer periods would be of no i n t e r e s t here. Geomagnetic bays, with a period of about 1 hour (36OO seconds) have been recorded by U. Schmucker on magnetograms i n C a l i f o r n i a . Stations were set up on a l i n e perpendicular to the sea coast f o r a distance of just under 300 km. at San Francisco and La J o l l a . Readings were also taken on the San Clemente Islands f o r the La J o l l a traverse (see Figure IX). A study of the v e r t i c a l component of the magnetic f i e l d s f o r well behaved bays, i . e . the e l e c t r i c f i e l d vector p a r a l l e l s the coast, showed that at the sea coast there was a pronounced disturbance but the further inland the records were taken the smaller the magnitude of the disturbance u n t i l at a distance of 100 km. or l e s s the disturbance disappeared (see magnetogram records Figure V I I I ) . If one assumes that the theory as previously described i s 42. Aug. 3 0 . I 9 6 0 H Magnetogram records, adapted from U. Schmucker. FIGURE IX Map of Southern C a l i f o r n i a . 44. c o r r e c t one i s l e d t o the c o n c l u s i o n t h a t there i s a l a t e r a l d i f f u s i o n c u r r e n t a r i s i n g from l a t e r a l inhomogeneities i n the c o n d u c t i v i t y . I f t h i s i s the case then crude order of magnitude e s t i m a t i o n s may be c a r r i e d out to f i n d the approx imate dimensions of t h i s c o n d u c t i v i t y anomaly. I f ju. = 1.257 x 1 0 - 6 henry/m., uD = 3600 cps, and ^ = 100 km. where ^ i s taken as the l a t e r a l dimension of the d i f f u s i o n , the c o n d u c t i v i t y , , can be determined from the r e l a t i o n oi cr" i s found to be of the order of IO"-1 mho/m. S.P. S r i v a s t a v a ' s Ph.D. t h e s i s (1962) c o n t a i n e d a study of the subsurface c o n d u c t i v i t y of the pl a n e s of A l b e r t a u s i n g the p r i n c i p l e s of m a g n e t o t e l l u r i c s as expounded by L. Cagniard ( 1 9 5 3 ) . S r i v a s t a v a concluded t h a t f o r h i s survey the d i s t r i b u t i o n of the c o n d u c t i v i t y with depth c o u l d be summarized as f o l l o w s : 0 to 5 km.., 1 0 - 1 mho/m. 5 to 90 km., 10"^ mho/m. 90 to 150 km., 1 0 " 1 mho/m. T. R i k i t a k e (1951) c o n s i d e r e d a two l a y e r model, cor r e s p o n d i n g to S r i v a s t a v a ' s second and t h i r d l a y e r s , but he p l a c e d h i s depth of co n t a c t c l o s e r t o 400 km. than t o 100 km. 4 5 . 100 200 x l - Q. UJ Q 300 4001- FIGURE X SEA LOW VELOCITY LAYER LAND ! _ _ cr=IO"4 mhos 1 CONDUCTIVITY CLIFF <r =10"' mhos L Schematic cross-section at the sea coast f o r an oversimplified conductivity c l i f f . SHEAR VELOCITY (km/sec) 3-0 4-0 5-0 I0O E X cZ 30O UJ o 400 500 80 )9 ..j CA tSE A) K (LA ND) ; I FIGURE XI Ve l o c i t y depth curve (shear waves) under the Canadian Shield (Model CANSD) and under the ocean (Model 8099), a f t e r J . Brune and J. Dorman. 46. The i n i t i a l value of 1 0 _ 1 mho/m., rang i n g from a depth of 0 to 5 km., i s not t h i c k enough to a p p r e c i a b l y i n f l u e n c e the o v e r a l l e f f e c t i v e c o n d u c t i v i t y f o r the f i r s t 90 km. The r e f o r e to a f i r s t approximation, the c o n d u c t i v i t y at a depth of 90 km. or more i s much l a r g e r than the con d u c t i v i t y above, i . e . ^ = . _ J cr— t O OO ^gok-v. I f t h i s s t r a t i f i c a t i o n of c o n d u c t i v i t y i s extended t o the P a c i f i c Coast and va l u e s o f the c o n d u c t i v i t i e s and depths there are compared wi t h S r i v a s t a v a ' s a s t r i k i n g s i m i l a r i t y i s found to e x i s t . Two p o i n t s must be emphasized: (1) The h e i g h t of the su r f a c e o f the E a r t h above the con d u c t i n g l a y e r ( I O - 1 mho/m.) would be of the order of 100 km. and hence the anomalous v e r t i c a l f i e l d would f l a t t e n c o n s i d e r a b l y , I.e. the value of the s k i n - depth would be of the or d e r of 75 km. r a t h e r than 100 km. (The a l t e r n a t i v e t o r a i s i n g the depth would be to lower the c o n d u c t i v i t y c o n t r a s t . ) (2) The v e r t i c a l anomaly would a r i s e , i n the f i r s t approx imation, from a " c l i f f " i n the c o n d u c t i v i t y at depth; where the value of cr at 200 km. would be I O - 1 mho/m. under the l a n d and of the order of 1 0 " ^ mho/m. under the sea. 47. . Z , DEPTH IN km 1000 500- 60 T 3 (3600 sec) 100 VT IN sec2 (IQOOOsec) FIGURE X I I ( a ) I d e a l i z e d p l o t o f s k i n depth v s. p e r i o d f o r two c o n d u c t i v i t i e s . SEA CONTINENT A H VERTICAL MAGNETIC ANOMALY 215 km °2 ~1 315 km FIGURE X l l ( b ) PERIOD Probable v a r i a t i o n of H 2 w i t h p e r i o d . 48. One i s l e d to the c o n c l u s i o n t h a t the v a r i a t i o n of c o n d u c t i v i t y with depth might be as i l l u s t r a t e d i n F i g u r e X. I t would seem l o g i c a l to t r y and check t h i s r e s u l t w i t h o t h e r i n f o r m a t i o n about the E a r t h ' s mantle and c r u s t d e r i v e d from seismic sources. L. Sykes e t a l . (1962) have proposed a v e l o c i t y depth curve f o r the mantle under the sea, and have suggested t h a t there i s a low v e l o c i t y l a y e r between the depths of 60 and 215 km. J . Brune and J . Dorman (1963) have suggested a v e l o c i t y depth curve f o r the mantle and c r u s t under the Canadian S h i e l d . T h e i r r e s u l t s show t h a t there i s a low v e l o c i t y l a y e r between the depths of 115 and 315 km,. Hence the low v e l o c i t y l a y e r must go from deep to shallower depths as i t passes from under the C o n t i n e n t s to under the Ocean, i n s t e a d of p a s s i n g from shallow to deeper depths as might be expected from a simple study of s k i n depths and c o n d u c t i v i t y . Hence i t must be concluded t h a t e i t h e r the shape of the low v e l o c i t y l a y e r and the "isoconds", or l i n e s of equal c o n d u c t i v i t y , are independent of each ot h e r or t h a t the l a y e r s and the isoconds are r e l a t e d and the p i c t u r e of the c o n d u c t i v i t y d i s t r i b u t i o n i s o v e r s i m p l i f i e d . I t would seem l o g i c a l t o assume t h a t the l a t t e r con c l u s i o n i s c o r r e c t but the p o s s i b i l i t y of the former, cannot be completely r u l e d out. Assuming t h a t the low v e l o c i t y l a y e r and the isoconds p a r a l l e l each o t h e r the morphology of the v e r t i c a l magnetic anomaly f o r v a r i o u s p e r i o d s may be found from the.diagrammatic graphs of depth vs. p e r i o d and 4 9 . v e r t i c a l f i e l d vs. p e r i o d (see F i g u r e s X l l ( a ) and ( b ) ) . I t w i l l be n o t i c e d t h a t there are f i v e r e g i o n s on the graphs, which correspond to ( 1 ) no v e r t i c a l f i e l d , ( 2 ) growth of the f i e l d , ( 3 ) f i e l d magnitude reaches a maximum and s t a r t s to d e c l i n e , ( 4 ) decay of the f i e l d and ( 5 ) no f i e l d . These f i v e r e g i o n s would correspond to ( 1 ) m i c r o p u l s a t i o n s , ( 2 ) l o n g p e r i o d m i c r o p u l s a t i o n s , ( 3 ) and ( 4 ) geomagnetic bays and ( 5 ) storms. The depth ^ p l o t t e d i n the f i r s t graph, F i g u r e X I I ( a ) , i s the s k i n depth and hence i s a f u n c t i o n of c o n d u c t i v i t y . The two curves ^, and ^ are p l o t s f o r two d i f f e r e n t p l a n e s of i n f i n i t e depth and constant c o n d u c t i v i t y . By i n t e r p o l a t i o n , i t i s p o s s i b l e to o b t a i n an i d e a of the shape of the v e r t i c a l magnetic anomaly as i t v a r i e s w i t h p e r i o d f o r a g i v e n c o n d u c t i v i t y step f u n c t i o n . T h i s might be a f i r s t approximation t o the j u n c t i o n of the v e l o c i t y depth curve as i t passes under the sea c o a s t . Assuming t h a t the depth to the bottom of the l a y e r under the l a n d i s I), ( 3 1 5 km.) and under the sea i s I b z ( 2 1 5 km.), the second graph i s obtained, F i g u r e X l l ( b ) . The i n t e r p r e t a t i o n of the l a t e r a l s k i n depth becomes somewhat clouded i f t h i s p i c t u r e of the c o n d u c t i v i t y d i s t r i b u t i o n i s assumed to be c o r r e c t . However, although one may not be able t o make good e s t i m a t e s of the conduct i v i t y at 1 0 0 and 2 0 0 km. under the l a n d and the sea, the b a s i c concept should not be d i s c a r d e d . I n f o r m a t i o n can be obtained from the a t t e n u a t i o n of the v e r t i c a l component, but 50. the exact theory must be reworked f o r t h i s s p e c i a l case. I t i s p e r t i n e n t , at t h i s p o i n t , t o quote J . Dorman, M. Ewing and J . O l i v e r ( i 9 6 0 ) , who suggested a s t r u c t u r e f o r the upper mantle of the P a c i f i c B a s i n from s u r f a c e wave d i s p e r s i o n : ...Rayleigh waves i n d i c a t e t h a t the low v e l o c i t y r e g i o n of the upper mantle extends upwards to much shallower depth beneath the oceans than beneath the c o n t i n e n t s . R a y l e i g h wave d i s p e r s i o n data f o r the paths on the P a c i f i c b a s i n are i n t e r p r e t e d t o i n d i c a t e t h a t shear v e l o c i t y below the M (Moho) decreases to about 4.3 km/sec at depths of about 60 km and t h a t shear v e l o c i t i e s are somewhat lower than i n the s u b - c o n t i n e n t a l mantle down to about 400 km. The g r e a t e s t need i s f o r r e a d i n g s of the magnetic f i e l d c o v e r i n g the whole spectrum of p e r i o d s , from 1 second to 24 hours, f o r d i s t a n c e s of 0 to 1000 km. from the coast on the s u r f a c e of the sea. C o r r o b o r a t i v e evidence of F i g u r e XII(b) can be seen from the magnetogram r e c o r d s of U. Schmucker. V e r t i c a l magnetic anomalies a r i s e from m i c r o p u l s a t i o n s because of the land-sea c o n t a c t . U n f o r t u n a t e l y these cannot be seen from Schmucker's r e c o r d s . There i s an anomaly i n the v e r t i c a l component of geomagnetic bays, i n d i c a t i n g the presence of some s o r t of step ( i t i s more 51. probably a sloping contact) i n the conductivity, but no v e r t i c a l component anomaly i s found i n storms, i n d i c a t i n g that the conductivity Is l a t e r a l l y homogeneous at greater depths. 52. CONCLUSIONS The val u e s of Qt and (obta i n e d from Appendix I I ) show t h a t Q, — 2 and and hence t h a t to a f i r s t approx i m a t i o n the r e s u l t Q, = <$3 =» A , a constant, as used by J.T. Weaver (1963), i s v a l i d . An improvement would be to w r i t e the f u n c t i o n (q C y ) = ^1 ^ > 0 = ^3 ^ < 0 but t h i s would be p h y s i c a l l y u n r e a l i s t i c . Hence the estimate of G which seems to be p h y s i c a l l y sound, has been suggested to gi v e the s o l u t i o n a d d i t i o n a l accuracy even though i t has not been checked mathe m a t i c a l l y . I t i s found t h a t C * - 4 = - 7 and C5 0 < —-L=, i\ err V^ri and hence ; Cz =* ^ 1 I O Thus the term C 3 i s dominant. T h i s f a c t i s important i n the computation of Ct^) . Graphs of the v e r t i c a l magnetic f i e l d a r i s i n g i n the case of E " - p o l a r i z a t i o n have been c a l c u l a t e d by J.T. Weaver f o r the approximations c i t e d above and these have been i n c l u d e d here with h i s very k i n d consent (see F i g u r e V I I I ) . The graphs have been c a l c u l a t e d f o r = 0 . 0 1 cps. and co = l cps. and are based on Weaver's e q u a t i o n assuming 53. constant values across the disco n t i n u i t y as opposed to a varying value of (T(i^). The theory of l a t e r a l skin depths has been tenta t i v e l y applied to two cases, f i r s t l y to the v e r t i c a l mag netic anomaly a r i s i n g from the Fraser Delta experiment and secondly to the magnetogram records c o l l e c t e d by U. Schmucker. Although there seems to be some doubt as to whether the theory may be f r e e l y applied i n the l a t t e r case, one can conclude that there i s some sort of l a t e r a l conductivity i n homogeneity. I t s exact shape and form i s not clear, but two models have been proposed, the f i r s t being a simple con d u c t i v i t y " c l i f f " and the second r e l y i n g on information obtained by seismologists on the low v e l o c i t y layer. 5 4 . APPENDIX A l S n e l l ' s Law a p p l i e d to a Nonconductor-conductor Boundary C o u p l e . * C o u p l e - * vMtwe. A c c o r d i n g to S n e l l ' s Law l<, s i n 9, = k t "sin e 0 or (cx.+'u^O s i o a , = (<*i S i n G - o However, n o t i n g t h a t , f o r a i r c< 2 = coJjZ^? and, f o r a conductor <*, = ^, = ^J!^2^ we can express©, as a f u n c t i o n of <*,, pi, and 9 4 Hence ^ n - s i n G, - I W J / ^ T pfc— T - i - r r - A l ( i ) A l ( i i ) A l ( i i i ) A l ( i v ) The c o n d i t i o n , f o r which the displacement c u r r e n t i s n e g l i g i b l e i s M»|i-w4 and t h i s i s s a t i s f i e d on the l a n d - a i r and s e a - a i r boundary. Using t h i s c o n d i t i o n i n A l ( i v ) s i n l 6 | i s g i v e n by L J ^ A l ( v ) assuming Since the e x p r e s s i o n f o r sin 2"©, i s the product of two 5 5 . factors, i t s vanishing can equally well be the resu l t of sit - i^S,,— O •l.e.Q^O. Hence the condition that ^ L»|-has the same e f f e c t as assuming that the wave i s propagated almost v e r t i c a l l y and that only horizontal components of the e l e c t r i c and magnetic f i e l d s e x i s t . As soon as the wave passes into the conducting medium, the free charges, l y i n g at random throughout the volume, migrate to the boundaries and set up an e l e c t r i c f i e l d , which i s normal to the surfaces, E N . This f i e l d opposes any v e r t i c a l component of the f i e l d i n the conductor, i . e . E N = E , . P h y s i c a l l y i t i s possible to see t h i s as follows: = C> h e g a h > / £ c K a f - g e S A t7 K H 1 1 h H — I I- -f 1—| h V " 56. APPENDIX A2 F r e s n e l E q u a t i o n s I f we assume an i n i t i a l wave to be of the form E - L = E o e , HL-J^ n a * E l A 2 ( i ) The t r a n s m i t t e d and r e f l e c t e d waves w i l l be of the form E t - E , e H t - & - v U E t A 2 ( i i ) and E ^ E - e > ^ A 2 ( i i i ) The o n l y unknown f u n c t i o n s are C", and Cz f o r E - p o l a r i z a t i o n and Cq, and 6 j a f o r H - p o l a r i z a t i o n where C^,CX) G, and <S t are d e f i n e d and T T , = I^JT. , H a = < q a H . at the boundary ^ ( E . + r ^ ^ ^ a n < a P i ^ + T T ^ . H " , A 2 ( i v ) E ' - p o l a r i z a t i o n F o r 1 c - p o l a r i z a t i o n n.» E 0 = O t h e r e f o r e E"0 ET ? E , 57. and co Go E a - o^s©,. E 2» Or> ©^E, from e q u a t i o n A 2 ( i v ) . Noting that we f i n d ^ - .^r. k ^ 6 ' , , • • E f t and hence the value of C, and may be obtained. I t w i l l be noticed that k, i s complexjbut l<j.is r e a l . and ^ p o l a r i z a t i o n For H-polarization |-( 0 =.Q and i n t h i s case H« * ^* " ^ \ ~ • fiT*. = Noting that: E o = ^" » E", = -J±^>< £, *T t k, and E i » £-*-*TT0 i t i s found that cos e 0TTL - oose i ri* one , M. **<t M e+H^K from equation A 2 ( i v ) , and hence solving f o r H , and Hjwe f i n d , U Z./.^ k,* ^ 0 6 0 r TT and L_| = A*^'V c o s ^ o - A , kjtf- kl *\^G0 T^j~ and hence the value of and £j^can be obtained. 58. APPENDIX A 3 Proof of div ~E=0 (1) In the case of E-polarization the e l e c t r i c f i e l d has only one non-zero component, and so, which i s zero because E x i s independent of ac . (2) In the case of H - p o l a r i z a t i o n the e l e c t r i c f i e l d has two non-vanishing components E / and which are connected with H a t as follows (These equations are derived from Maxwell's Equations) But V ~ F = 4-59. APPENDIX A4 Values of i|> and Q f u n c t i o n s f o r GiC^ then r e p l a c i n g H| by the value of C^) I f jr^'tj <^ | * the s e r i e s i s convergent and the summation can simply expressed. I f i t i s d i v e r g e n t ( i . e . | ^ j - | > | ) then no s o l u t i o n can be found. Now ^ ^ u > ^ , ^ u £ioJ 2 + ° y ^ i ^7 & e" 1 so t h a t P = j ^ ^y^,3- * T h e r e f o r e f o r a l l ^ ^ o the s e r i e s i convergent and t h e r e f o r e ' s i m i l a r l y 60. APPENDIX A5 S o l u t i o n of the Equ a t i o n s In Ge n e r a l : H - p o l a r i z a t i o n A s o l u t i o n of the g e n e r a l case, when o<<^<900 w i l l o n l y be t e n t a t i v e l y suggested and l e f t In g e n e r a l terms, whereas the case f o r K=90° has been worked out i n reasonable d e t a i l and c o n c l u s i o n s can be drawn from t h i s case about the type of f i e l d found i n the g e n e r a l case. &C=90°, because i n a d d i t i o n t o the second d i f f e r e n t i a l s w i t h r e s p e c t to the V and the 2 c o - o r d i n a t e s i t has a c r o s s - p r o d u c t term. T h i s l e a d s t o d i f f i c u l t i e s i n the s o l u t i o n of the p a r t i a l d i f f e r e n t i a l e q u a t i o n . I f the L a p l a c e t r a n s f o r m i s used then two unknown f u n c t i o n s r e s u l t , which both need to be c a l c u l a t e d . i s d e f i n e d as the L a p l a c e t r a n s f o r m w i t h i t s Inverse, then a p p l y i n g i t t o the g e n e r a l e q u a t i o n (16), f o r H - o o l a r i z a t i o n , i t i s found t h a t The g e n e r a l equations d i f f e r s from the case of I f 17 61, as- 2 n ^cy A 5 ( i i ) T h i s equation i s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n i n V , and, p l a c i n g we f i n d _ Z ^ o o ^ f i x - H ^ ' H * = OCV) A s o l u t i o n of t h i s e q u a t i o n would be H* - k* e + k, a +V/t^ A 5(iv) where = Sj" and C, = ^ ^»J L l y * - - ^ 1 ' A5(v) S e t t i n g ^ j s a ^ l s we get, Hence, when 1^ , 6) s J , Assuming t h a t VC1^ i s f i n i t e at T= + 0 0 , the boundary c o n d i t i o n H(v) can be a p p l i e d t o show t h a t kz = 0 . The c o n d i t i o n <R*£>| i s S"i< JlJfi^ tf^ * s o t h a t ^ tends to a minimum value (zero) as tends to zero, and 2j~ tends to a maximum value ( i n f i n i t y ) as tends to 90° . When ^ has a value o u t s i d e t h i s l i m i t , i . e . . ^ * ^ | | z sin^ then 2ccs<*. the problem becomes much more com p l i c a t e d . There w i l l be two c h o i c e s depending upon the f u n c t i o n k\C^) . E i t h e r Kt « k2 =*0 6 2 . or neither K, nor k* are zero, the condition as V tends to i n f i n i t y being s a t i s f i e d by the subtraction of two i n f i n i t i e s to produce a f i n i t e number. An i n t e l l e g e n t guess might be that both }Cj and ka are zero but i t can only be a guess. A theory w i l l be developed assuming that at least i s zero inside the domain ^ < J- • and a tentative approach to the problem w i l l be given. V C Y ) w i l l always be given by: which Is the p a r t i c u l a r i n t e g r a l of the d i f f e r e n t i a l equation, assuming that OC^can be expressed as a Fourier series and that there are no s i n g u l a r i t i e s . The values of k| and , can be computed by applying boundary conditions H ( i i ) - i^-v, to) |<3 = ^ -v^o> A 5 ( v l ) and applying H ( i i i ) ^ V^.co) (£,-*,) -wi- Vi^XgitV^-riV^^gaV;1^ A 5 ( v l l ) It i s apparent that the i n t e g r a l d i f f e r e n t i a l equation has been expressed as a function of U(s^ ) . C^V^) , i n turn, Is a function of the magnetic f i e l d on the surface 2 i » 0 . But the value onZ=-0 cannot be placed In the function OCV) u n t i l the p a r t i a l derivative dM* has been computed. Hence, although i s only a function of V and has been calculated along 2 = 0 , two seperate uncalculated functions appear inUC$. The function can be defined by the 63. following conditions: <scv>- ^ 5 H o , y - - oo and Liv-n -L - L l** , JL ^ and l a s t l y U m 3 - Uv* $ t ^ ~ % t<>,P) The function ^jj"^^'^ l s n o - f c nearly so easy to define; i t appears that i t can only be calculated a f t e r the values of H, and r-^have been computed, which indicates that a second alternative solution of the p a r t i a l d i f f e r e n t i a l equation i s necessary before a complete solution of t h i s case, using t h i s method, can be found. E *-polarization The relevant equation i s which can be "solved" i n the same way as the p a r t i a l d i f f e r e n t i a l equation f o r H-polarization using the Laplace transform and inversion. The p a r t i a l d i f f e r e n t i a l s can be converted to ordinary d i f f e r e n t i a l s to obtain the equation 64. where NJ*» af-pVnfc a n < i The s o l u t i o n of t h i s d i f f e r e n t i a l e q u a t i o n i s the same as b e f o r e , i . e . A5(ix) A5(x) ^ , = Sj i^rpc-L and =• " S ^ — Once again a p p l y i n g c o n d i t i o n E(v) makes i f S J i s i n the c o r r e c t domain. The e v a l u a t i o n o f K4 and iCj i s ob t a i n e d t y a p p l y i n g the boundary c o n d i t i o n s E ( i i ) and then E ( i i i ) . k | - ^ p and ^ * ! ^ 7 % ) - S 5 (o) A p p l i c a t i o n of E ( i i l ) g i v e s the' value of |(^) as ^ ^ + 6", when " S , * ^ . Once more the s o l u t i o n of the i n t e g r a l , d i f f e r e n t i a l e q u a t i o n i s dependent on the-value of £lV^ and wJ=*| . The value (or f u n c t i o n ) r e p r e s e n t e d by C(Y) may be expressed by a s e r i e s of l i m i t i n g c o n d i t i o n s i n the same way as GCd , 65. i . e . CCV)=C,Eo , V - + » and However the problem of defining the value of ^£^2M s t i l l remains e s s e n t i a l l y unsolvable unless a subsiduary * solution i s added. 66. BIBLIOGRAPHY Brune, J., and J. Dorman, B u l l . Seis. Soc. Am., 53, 167, 1963. Cagniard, L., Geophysics, 18, 605, 1953 C h r i s t o f f e l , D. A., J. A. Jacobs, and E. J . J o l l e y , J . K. Kinnear and J. A. Shand, The Fraser Delta Experiment of i960, P a c i f i c Naval Laboratory Report 6l-5, August 1961. d ' E r c e v i l l e , I., and G. Kunetz, Geophysics, 2J_, 651, 1962. Dorman, J., M.Ewing and J. Oliver, Association de Seismologie et de Physique de l ' I n t e r i e u r de l a Terre, Resume 166, Hel s i n k i , i960. Nishida, A., Ph. D. Thesis, University of: B.C., 1962. Rankin, D., Geophysics, 27,: 666, 1962. Rikitake, T., B u l l . Earth. Res. Inst., XXVIII, part II, page 263, 1951. Srivastava, S. P., Ph. D. Thesis, University of B. C., 1962. Sykes, L., M. Landisman and Y. Sato, J. Geophys. Res., 67, 5257, 1962. Weaver, J. T., A Note on the V e r t i c a l Fault Problem i n Magnetotelluric theory, P a c i f i c Naval Laboratory Technical Memorandum 62-1. Weaver, J. T., Canadian J. of Phys., 41, No. 3> 1963. General References: Landau, L. D., and E. M. L i f s h i t z , Electrodynamics of Continuous Media, V o l . 8, Pergamon Press, i960. Price, A. T., Quart. J. Mech. and App. Math., 3, 385, 1950.. Price, A. T., J. Geophys. Res., 67, 1907, 1962. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, Inc., 1941.
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Electric and magnetic fields associated with a vertical fault. Coode, Alan Melvill 1963-10-31
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Title | Electric and magnetic fields associated with a vertical fault. |
Creator |
Coode, Alan Melvill |
Publisher | University of British Columbia |
Date Issued | 1963 |
Description | Interest In the vertical fault problem for electromagnetic fields has been recently revived by the papers of I. d'Erceville and G. Kunetz (1962) and D. Rankin (1962). In the derivation of his equations Rankin used d'Erceville1s theory which contains some fallacious assumptions. These have been pointed out by J.T. Weaver (1962) and also in this thesis. This thesis follows the lines of mathematical attack first employed by d'Erceville and Kunetz, and later developed by Weaver, in applying the theory of integral transforms to the partial differential equations satisfied by land and sea conductors. The problem of both a vertical fault and also a sloping fault, i.e. 0 < α < 90° where α is the angle of dip of the fault are considered. The results in the general case are Inconclusive, no solution has been found and no solution is suggested. The case of α = 90° has proved to be equally indeterminate, but a solution has been suggested, which, although it has not been proved rigourously, does not appear to violate any physical principles and also seems to represent the field equations on the surface of the land and the sea. |
Subject |
Faults (Geology) Earth sciences -- Mathematical models Magnetic fields -- Mathematical models Electromagnetic fields -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053073 |
URI | http://hdl.handle.net/2429/38455 |
Degree |
Master of Science - MSc |
Program |
Geological Sciences |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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