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Electric and magnetic fields associated with a vertical fault. Coode, Alan Melvill 1963

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ELECTRIC AND MAGNETIC FIELDS ASSOCIATED WITH A VERTICAL FAULT by ALAN MELVILL COODE B . A . S c , U n i v e r s i t y 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 I n s t i t u t e of EARTH SCIENCES  We accept t h i s t h e s i s as conforming t o the required  standards  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963  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 of the requirements  f o r an advanced degree at the U n i v e r s i t y of  B r i t i s h Columbia, I agree  that 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 reference and study.  I f u r t h e r agr.ee that per-  mission f o r extensive copying 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 granted by the Head of my Department or by  h i s representatives,)  I t i s understood  that copying, or p u b l i -  c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department of'•• T K V S i t s  ' , ClMSTtToTg  The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date  aF  gv^TH  5ctetoces)  i  ABSTRACT  I n t e r e s t 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 r e c e n t l y r e v i v e d by the papers of  I. d ' E r c e v i l l e and G. Kunetz (1962)  (1962).  and D. Rankin  In the d e r i v a t i o n of h i s e q u a t i o n s Rankin used  d'Erceville s 1  t h e o r y which c o n t a i n s some f a l l a c i o u s assumptions. have been p o i n t e d out by J.T. Weaver (1962)  These  and a l s o i n t h i s  thesis. T h i s t h e s i s f o l l o w s the l i n e s of mathematical a t t a c k 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 a p p l y i n g the t h e o r y o f i n t e g r a l transforms t o 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 s s a t i s f i e d by l a n d and sea conductors.  The problem o f both a v e r t i c a l  f a u l t and a l s o a s l o p i n g f a u l t ,  i . e . 0 <(u ( 9 0 ° where  PC i s  the angle o f d i p of the f a u l t are c o n s i d e r e d . The r e s u l t s i n the g e n e r a l case are I n c o n c l u s i v e , no s o l u t i o n has been found and no s o l u t i o n i s suggested. The case o f ^ = 9 0 ° has proved to be e q u a l l y i n d e t e r m i n a t e , but a s o l u t i o n has been suggested, which,  although i t has  not been proved r i g o u r o u s l y , does not appear t o v i o l a t e any p h y s i c a l p r i n c i p l e s and a l s o seems t o r e p r e s e n t the f i e l d equations on the s u r f a c e of the l a n d and the sea.  iv  ACKNOWLEDGEMENTS  I wish t o acknowledge, with g r a t i t u d e , the h e l p , guidance  and i n f i n i t e p a t i e n c e g i v e n t o me by Dr. T.  Watanabe i n the development of t h i s t h e s i s . l i k e t o thank P r o f e s s o r J.A. Jacobs,  I a l s o should  f o r making i t p o s s i b l e  f o r me t o work i n the I n s t i t u t e , D r ' s . A. N i s h i d a and S.P. S r i v a s t a v a , f o r t h e i r i n t e r e s t and time  spent i n  d i s c u s s i o n , and l a s t l y Dr. J.T. Weaver f o r r e v i e w i n g work and o f f e r i n g s e v e r a l h e l p f u l s u g g e s t i o n s .  this  The work  i n t h i s t h e s i s has been supported by the N a t i o n a l Research Council.  ii  TABLE OP CONTENTS  ' 1  I  INTRODUCTION  II  INDUCTION EQUATIONS  3  III  PHYSICAL INTERPRETATIONS  13  IV  BOUNDARY CONDITIONS  19  V  BOUNDARY CONDITIONS FOR  VI  SOLUTION OF THE EQUATIONS FOR  K. = 9 0 °  24 = 90°  28 28 30  H-polarization E-polarization VII  PRACTICAL IMPLICATIONS  4 l  VIII  CONCLUSIONS  52  APPENDIX A l  SMELL'S LAW APPLIED TO A NONCONDUCTORCONDUCTOR BOUNDARY  54  APPENDIX A2  FRESNEL EQUATIONS  56  APPENDIX A3  PROOF OF DIV E = 0  58  APPENDIX A4  VALUES OF ^  APPENDIX A 5  SOLUTION OF THE EQUATIONS  AND J  FOR G ( y )  59  I N THE GENERAL CASE 60  iii  LIST OP FIGURES  I  C o - o r d i n a t e systems at a l a n d - s e a - a i r c o n t a c t .  II  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.  III  Magnetic f i e l d field  IV  8  generated from an e l e c t r i c  i n a r i g h t angled conductor.  15  Magnetic f i e l d s generated from e l e c t r i c In  two r i g h t angled conductors; J, >  the  v e r t i c a l magnetic f i e l d , H^,  fields  SO t h a t  , at the d i s -  c o n t i n u i t y i s d i r e c t e d upwards. V  3  15  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 r a y s at a boundary.  VI  20  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 r e c o r d s , adapted from U. Schmucker.  42  IX  Map  43  X  Schematic  of Southern C a l i f o r n i a . c r o s s - s e c t i o n at the sea c o a s t f o r  an o v e r s i m p l i f i e d c o n d u c t i v i t y c l i f f . XI  45  V e l o c i t y depth curve (shear waves) under the Canadian S h i e l d  (Model CANSD) and under the  ocean (Model 8099), a f t e r J . Brune and J . Dorman. XII(a)  I d e a l i z e d p l o t of 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 .  (b)  45  Probable v a r i a t i o n of  47 with period.  47  1. INTRODUCTION  I n t e r e s t 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 r e c e n t l y s t i m u l a t e d by the papers of I . d ' E r c e v i l l e and G. Kunetz  (1962)  (1962). The Weaver (1963)  and D. Rankin  i s an e x t e n s i o n of a paper by J.T.  p r e s e n t work who  was  l o o k i n g f o r a t h e o r e t i c a l e x p l a n a t i o n of an anomalously h i g h v e r t i c a l magnetic component of geomagnetic found at a l a n d - s e a c o n t a c t .  In 1961  micropulsations  D.A.  Christoffel,  J.A. Jacobs and J.A. Shand p u b l i s h e d a paper comparing the r a t i o s of the v e r t i c a l to the h o r i z o n t a l components of the magnetic f i e l d s at V i c t o r i a , B.C.,  which can be c o n s i d e r e d  to be s i t u a t e d at a l a n d - s e a c o n t a c t at the edge of the Pacific,  and R a l s t o n , which i s i n the Southern p a r t o f the  p l a i n s of A l b e r t a and thus i s removed from a l a n d - s e a contact.  The r e s u l t s showed t h a t there was  a definite  anomaly i n the magnitude  of the geomagnetic  the m i c r o p u l s a t i o n range  (0.001 cps  sea c o n t a c t .  to  3.0  vertical  amplitudes i n cps) at a l a n d -  T h i s c o n t r a d i c t s 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 o f the problem. I n t h i s t h e s i s the i n i t i a l e l e c t r o m a g n e t i c f i e l d i s assumed to be v e r t i c a l l y i n c i d e n t .  Even though  this  approximation i s not exact, i t i s found (see Appendix  1)  t h a t i t i s e q u i v a l e n t t o the c o n d i t i o n t h a t the displacement c u r r e n t i s n e g l i g i b l e i n Maxwell's e q u a t i o n s , i . e . t h a t I^H  »  I coe/  .  As a r e s u l t , the angle t h a t the plane of  2.  i n c i d e n c e 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 t o the d i s c o n t i n u i t y . T h i s p e r m i t s s i m p l i f y i n g assumptions t o be made and problem becomes two  the  dimensional.  The d i f f i c u l t i e s  a r i s i n g from v a r i a t i o n s of con-  d u c t i v i t y w i t h depth have been circumvented  by assuming t h a t  the c o n d u c t i v i t y does not vary f o r depths l e s s than 200 and t h a t the e l e c t r o m a g n e t i c f i e l d has been attenuated at t h a t depth, depth of 200 km.  "completely"  i . e . f o r t h e o r e t i c a l purposes  can be regarded as the depth to  with a i r ) .  a  infinity.  N e i t h e r l a n d nor sea have been assumed to be p e r f e c t d u c t o r s (although both may  km.  con-  be c o n s i d e r e d so when compared  3.  INDUCTION EQUATIONS  F o r the g e n e r a l case the c o o r d i n a t e axes 0 (X, Y, z) have been chosen w i t h the X - a x i s along the d i s c o n t i n u i t y j u n c t i o n of l a n d , sea and a i r , the Y a x i s a l o n g the s e a - a i r c o n t a c t , and the z a x i s along the l a n d - s e a c o n t a c t . case o f a v e r t i c a l d i s c o n t i n u i t y  I n the  ( &< = 9 0 ° ) , the axes are  r e c t a n g u l a r 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 F i g u r e 1 ) .  FIGURE 1  Co-ordinate  systems at a l a n d - s e a - a i r c o n t a c t .  I t i s assumed t h a t the i n c i d e n t e l e c t r o m a g n e t i c f i e l d i s a wave o f the form  H  L  = Ho e  4. where: Ha  = the constant, f i n i t e amplitude  of the magnetic  field ka=  the wave number i n a i r  i.e.  l<i - ^W/^o^o'  where  the p e r m e a b i l i t y and p e r m i t t i v i t y  and &  0  are  in air  A  h- = e  the u n i t v e c t o r i n the d i r e c t i o n of p r o pagation  and  CO = the frequency Upon s t r i k i n g the l a n d - s e a conductor the magnetic  f i e l d i s " s p l i t up" i n t o two p a r t s , the r e f l e c t e d and t r a n s mitted f i e l d s .  Both these f i e l d s are assumed t o vary as  follows L  tut  Hence i n the atmosphere the f i e l d i s g i v e n by  TT= TT + hT. = e "[ H. e ^ + i t - ^ t f ] lofc  L  A s i m i l a r s e t of equations h o l d s f o r the e l e c t r i c field,  assuming  5.  I t i s p o s s i b l e t o s i m p l i f y Maxwell's equations i n a homogeneous, i s o t r o p i c  V % hT=  (a)  ^7»H  (a)  i  +  =0  medium.  (b)  V x t=  (b)  V-E  -|5  (1)  =  (2)  I f the f i e l d s are v a r y i n g i n a p u r e l y p e r i o d i c manner then f o r the two dimensional case, e q u a t i o n 2(b) i s found t o be a u t o m a t i c a l l y equal t o zero, i . e .  (3) The proof of t h i s i s g i v e n i n Appendix A3. W i t h i n the conductors i t i s n e c e s s a r y t o c o n s i d e r the  ratio  unity.  JLL  , which i s found t o be much g r e a t e r than  Table I g i v e s the value o f t h i s r a t i o f o r l a n d and  sea r e s p e c t i v e l y : Table I Medium  Conductivity; mho/m  80xl0-  sea land  Dielectric constant^ Parad/m  10~ -105  6  cps  10  3xl0~ -4xl010  Frequency  10  +io  3  1 0  3  10+*  6.  Hence i t i s p o s s i b l e  i n both  c a s e s ' t o assume t h a t  displace-  ment c u r r e n t s may  n e g l e c t e d i n the conductors,  i . e . that  i n both  be  l a n d and  sea.  However t h e r e i s no  i g n o r i n g the displacement Such a j u s t i f i c a t i o n the e l e c t r i c if  we  treat  static  and m a g n e t i c  equations to f i n d  ^7x  = o H~  (a)  has  justification  - -  .  (1962).  w i t h more f u l l y  equations f o r the case  when  substitute  t h e n t h e e q u a t i o n s t o be  electric Cx. = 9 0 ° ,  rj*=  assuming we  cannot  field.  use  This point Ph.D.  atmosphere  (1)  -  (4)  fields  as  ( 5 )  i t i s possible diffusion  i.e.  f ^ q - t o  s o l v e d i n each  - ^  Further  \ 7 ^ = -|!  b  and  of  o r magneto-  i n Dr.. A. N i s h i d a ' s  ( )  express the magnetic  we  field),  Hence i n t h e n e u t r a l  V * H " = } f  If  Thus  the e l e c t r i c  Therefore using equations to  a solution  f o r u s i n g Maxwell's  remaining f i e l d .  to f i n d  in  atmosphere.  been o b t a i n e d .  ( f o r a magnetostatic  has been d e a l t Thesis  the  field  after  as p u r e l y e l e c t r o s t a t i c  t h e n t h e r e i s no  y * H  c u r r e n t i n an u n c h a r g e d  i s only possible  the f i e l d s  Justification  conductor  (6)  are  {7(a))  7.  and  In  ^  ^ " H - u ^ H .  (7(D)  )  the atmosphere the equations t o be s o l v e d can be d e r i v e d  i n a s i m i l a r manner and are  (a)  \7'E" = ^eco^E'  and (b)  2  7  ^ H ^ A ^ H  (8)  Because of the two d i m e n s i o n a l nature of the model a l l f i e l d v e c t o r s are independent  of x and we can f i n d two s o l u t i o n s of  equations ( 7 ) , one f o r E " - p o l a r i z a t i o n and another f o r H are  -polarization.  I n both cases the p o l a r i z e d f i e l d v e c t o r s  d i r e c t e d along the x - a x i s .  Hence f o r H  -polarization  and f o r E T - p o l a r i z a t i o n  These two cases are completely independent  of each o t h e r and  the s o l u t i o n of the d i f f u s i o n equations w i l l be o b t a i n e d f o r each case s e p a r a t e l y . For  the case of  H - p o l a r i z a t i o n the e q u a t i o n t o  be s o l v e d f o r z>0 i s  •  —  _  h ru  ^ = ^o  6  (ii)  8. and f o r  ET - p o l a r i z a t i o n the equation i s  (12)  So equation H and  when 9 0 ° > >  -polarization  0 ° «  ^ =• y +  F o r the cases  t h e y may be o b t a i n e d  (12) u s i n g t h e e q u a t i o n s  coordinate  See  f a r no m e n t i o n h a s b e e n made o f t h e d i f f u s i o n o f £ - and  from equations  (11)  o f t r a n s f o r m a t i o n f o r t h e two  s y s t e m s 0 (X, Y, Z) and G (x, y, z ) v i z .  2  ( 1 3 ) or  V = ^ "* \ cc^tr&c  Figure I I .  FIGURE I I  Diagram showing t h e t r a n s f o r m a t i o n co-ordinate  system t o  another.  f r o m one  (14)  9.  Therefore  (15)  and  and hence  T h e r e f o r e equations  (11) and (12) may be w r i t t e n  (16)  where  F  i s equal t o e i t h e r  E=x o r  Hat.  The value of the t r a n s m i t t e d magnetic f i e l d z=0 can be expressed and time,  "t  only, say ^ | 6 p ,  as a f u n c t i o n of  , f o r H - p o l a r i z a t i o n and the value of the trans-  m i t t e d e l e c t r i c f i e l d can be expressed ^  for  E -polarization,  say  Ci<£  assuming s o l u t i o n s of equations  and  across  ir\u2&  as another f u n c t i o n of  and time,  "b  .  ( 8 ) t o be of the form  Hence  10.  we can express the component 2  i n a i r (omitting the term  as b e f o r e ) as  The c o n d i t i o n which must be s a t i s f i e d f o r a s o l u t i o n of (8) i s therefore  (17) i.e.  and  n  =  n =  J / A f r ^ -  m*-'  LjrY^-jxe^ '  for  1  f  o  r  w  j^TT" < I l 1  M  (18)  The g e n e r a l e x p r e s s i o n f o r the r e f l e c t e d wave can be expressed by  (19) J-00  where "h,  i s g i v e n by (18). Since  H  where the subscript 2 refers to air, at z=0 for H -polarization 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.  For  H  -polarization or  Applying  AC*™)  a F o u r i e r transform,  as a f u n c t i o n of  we  7j -°  on  ;  can express the  function  , viz.  —PO  Since  + 00  where  v  i s the D i r a c d e l t a f u n c t i o n ,  Aim) ^£7^ ~^^ " ° ^ -^ s  )e  v/ —  H  (21)  oo  Hence knowing the f u n c t i o n and  (VVV4  by  s o l v i n g equations  (12) i t i s p o s s i b l e to express the magnetic and  f i e l d i n the a i r . be obtained lines,  and  for  A s i m i l a r f u n c t i o n d e f i n e d by E-polarization.  i t can e a s i l y be  00  The  proof  (11) electric  St^)  can  follows similar  shown t h a t (22)  12.  where C(^}ls  the value of the t r a n s m i t t e d e l e c t r i c  which corresponds t o  GjC^)> on z=0.  field,  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 g i v e n by:  E  x  =o  EL - 1  IH* ^ c = i  and f o r EE" - p o l a r i z a t i o n ,  i h f ^ oc.r  (24)  the magnetic f i e l d i n the atmo-  sphere i s g i v e n by:  H -S x  H «i-i^ w  Equations  x  and  Ha* I l l ^ E *  (25)  ( 9 ) , ( 1 0 ) , (24) and (25) may be g e n e r a l -  i z e d f o r the case of 0 <  90° by a p p l y i n g equations  (15).  13.  PHYSICAL INTERPRETATIONS  When a n e l e c t r o m a g n e t i c wave m e e t s a c h a n g e i n c o n ductivity  ( e . g . a t an a i r - s e a  propagated  through  ductivity  t h e new c o n d u c t i n g medium w i t h  where ^JL  o f ^2=J£_  contact) the incident  and  T*  o f t h e medium.  c a n be e x p r e s s e d  by  G  H  )  V * " E T S (.1 w y l ^ c r - u i * ^ e )  However  so t h a t  equations  pretation  (7).  these  The e f f e c t  £"  e q u a t i o n s c a n be of t h i s  approximated  on a n y p h y s i c a l  inter-  i s t o h e a v i l y damp t h e e l e c t r o m a g n e t i c wave i n t h e  c o n d u c t i v e medium and t o p e r m i t a finite  and e l e c t r i c  i n t h e f o l l o w i n g manner:  V^a H = ( I > ^ and  a velocity  a r e t h e p e r m e a b i l i t y and c o n -  U s i n g Maxwell's e q u a t i o n s the magnetic fields  wave i s  time  to build  the c u r r e n t d e n s i t y t o take  up, w h i c h 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 b y e q u a t i o n s ( 7 ) .  The fields from  description  of the e l e c t r i c  and  associated with the current d e n s i t i e s  the vector p o t e n t i a l  magnetic  c a n be  found  14.  Since  and if  V * H afjr +  l ^ E  the g r a d i e n t o f the s c a l a r p o t e n t i a l of the e l e c t r o s t a t i c  field  i s small compared w i t h  -^-^  .  Hence i f we take a  "wedge" o f sea, o r t o s i m p l i f y i t , take the case when the angle  tx? = 90° then i t can be seen t h a t f o r v e r t i c a l p r o -  p a g a t i o n a c u r r e n t system i s set up, d e c a y i n g e x p o n e n t i a l l y with depth  (z).  T h i s c u r r e n t system r a d i a t e s waves i n a l l  d i r e c t i o n s , both i n the y - d i r e c t i o n and i n the n e g a t i v e z-direction  (reflected ray).  The r a d i a t i o n i n the y - d i r e c t i o n  i s not p a r t of the r e f l e c t e d r a y but i s r e f e r r e d . t o 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  field i s  g i v e n i n F i g u r e I I I , assuming t h a t the e l e c t r i c f i e l d i n the conductor i s d i r e c t e d along the x - a x i s .  I t w i l l be  n o t i c e d t h a t i n t h i s case ( i . e . £T - p o l a r i z a t i o n ) , the magnetic For  field  at the v e r t i c a l boundary has become  vertical.  the case of H - p o l a r i z a t i o n the c o r r e s p o n d i n g 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 a r i s e s of what  happens when two conductors are p l a c e d s i d e by s i d e . the angle ° C i s s t i l l  If  assumed t o be 90° then the magnetic  f i e l d f o r each.conductor,  separated as i n F i g u r e IV, i s  generated i n the same manner as b e f o r e but now there i s a mutual i n d u c t i o n f i e l d near y=0, which tends t o oppose the existing electric f i e l d  i n both c o n d u c t i n g media.  Upon  j o i n i n g the two conductors on the plane y=0 then there i s an  15.  H  \ I  reflected  o o o o o o o o o o o  0  H  diffracted  |  |  j, = O J E , , out of the paper  o 0 0  o  FIGURE I I I  Magnetic field  H  field  generated from an e l e c t r i c  i n a right  angled conductor.  H  r3 %  H  r  Z  11  \  ®  ®  ®  ®  ri  <g> E  J  i-  into the paper  H  medium 3  FIGURE IV  9  Magnetic  H  Z3  medium  I  f i e l d s generated from e l e c t r i c  i n two r i g h t  angled conductors;  the v e r t i c a l magnetic  f i e l d , H^.  c o n t i n u i t y i s d i r e c t e d upwards.  > ^  fields  so that  , at the d i s -  16.  electric  field  d i s c o n t i n u i t y between E g  there i s a d i f f u s i o n of e l e c t r i c  field  and  i n the  E|  y-direction,  i n the same manner as the t r a n s m i t t e d e l e c t r i c d i f f u s e s , u n t i l the value of the e l e c t r i c common value on The  and hence  field  field  reaches  a  y=0.  d i f f e r e n c e i n c u r r e n t d e n s i t y g i v e s r i s e to an  anomalous v e r t i c a l  \r\^  component of the magnetic f i e l d ,  .  There are three cases to c o n s i d e r : 1)  ^-3 ^ \i  when  i s downwards at the d i s c o n -  tinuity junction. |  2)  B 4  [i  i n which case the two media are  identical  (and t h e r e f o r e there i s no c o n d u c t i v i t y d i s c o n t i n u i t y ) 3)  j. i " N \ i  when  i s upwards (see F i g u r e I V ) .  Rather than t r y to compute the d i f f u s i o n of the f i e l d from f i r s t p r i n c i p l e s , In much the same way  field  same terms.  t  - r..  f, -  field  can be c o n s i d e r e d to be  i s attenuated  "attenuated" by  Hence the value of the e l e c t r i c  media 1 and 3 may  t  as the e l e c t r i c  guess i s made.  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  with depth, f o r electric  a reasonable  electric  be g i v e n  t  fields in  by:  (I - e * » > +  (. - e-«).  the  ^  e >\ ?  ^« , 0  r  o  (26)  £ * ^ 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, i s the value of E, y= - oo related  at y= +• «> £ >  i n media 1 and 3.  &> = J u.uoa~ I  and  t o the s k i n depth The  polarization  i s the value of E  3 o  /  same i d e a may  E*i  0  at  3  I t i s thus  . be a p p l i e d t o the case of H -  and a s i 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  rT -  0  -e*»)  f ^ e M  +  -e^  +  ^  ^  The r a t h e r a r b i t r a r y may  seem u n j u s t i f i e d  ^< o, .  ;  v  s  0  vj >o, -j  >  c h o i c e of E  and  ( 2 8 )  s  H  (29)  o  o  n  z=  at t h i s stage but i t w i l l be found t h a t  a f u n c t i o n of t h i s type i s needed l a t e r i n t h i s t h e s i s , if  even  i t r e p r e s e n t s a crude approximation t o the t r u e case. Summarizing, f o r the case of  three f i e l d s e x i s t :  W^,  and  .  ^  -polarization  The f i e l d s Hx  appear to be continuous f u n c t i o n s w i t h continuous  and first  d e r i v a t i v e s each approaching constant and known v a l u e s at ±°o  0  but the value of  appears t o have a d i s t r i b u t i o n  with a maximum (cusp) value at y=0  and t e n d i n g to zero at  18.  When c o n s i d e r i n g the case of a wedge there i s a l s o a t r a n s m i t t e d term t o be added t o 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 f l o w s a c r o s s the l a n d - s e a c o n t a c t . As the angle 0<. approaches zero i t i s found t h a t there i s l e s s and l e s s d i f f u s i o n because of the d i f f e r e n c e i n e l e c t r i c field  i n t e n s i t i e s i n the two media and more p r o p a g a t i o n o f  c u r r e n t as a simple t r a n s m i t t e d and r e f l e c t e d r a y problem. Hence i n the l i m i t i n g case o f h o r i z o n t a l l a y e r s ( L . Cagniard (1953), A.T. P r i c e  (1962)) there i s o n l y a problem o f t r a n s -  m i s s i o n and r e f l e c t i o n w i t h no d i f f u s i o n due t o a d i f f e r e n c e in electric field  intensity.  19.  BOUNDARY CONDITIONS  A solution Presnel  equations  of Maxwell's equations  i s g i v e n by the  a t y= ± oo f o r t h e a i r - l a n d ,  contacts.  I f i t i s assumed t h a t t h e i n i t i a l  the  ^  form:  -±-  then the transmitted  -  )  air-sea wave i s o f £  II  and r e f l e c t e d waves w i l l  v  p.  be o f t h e  form:  and  ' u-r  -  J  z  F o r E" - p o l a r i z a t i o n  C C,  where sea, and H  t>  (q,  z  H, —  At  ^ ( f  L  + f  r  £), H «  n  d  P  C  2  Eo  and  a  field  -polarization  a  and <^j a r e known c o n s t a n t s  a i r - l a n d boundary electric  IQJL+X.  £ , = C, E o  M -polarization  For  _^  (see Appendix  c a n be c a l c u l a t e d  2).  f o r the a i r -  Hence t h e m a g n e t i c  at y=±«o  f o r both  and E T - p o l a r i z a t i o n .  any b o u n d a r y ,  ) = ^  E  b  and  K x C H t t Hv}*h)JTt.  (30)  20.  FIGURE V  Incident,  transmitted  and r e f l e c t e d  rays at  a boundary.  For  H  where  -polarization  Hfct^)  Let us c a l l  equations  i s the magnetic f i e l d  t h i s f u n c t i o n on z = 0  and £|'6^ i n g e n e r a l .  sea E  these  give:  inside the ^-jt^in  A similar  conductor.  l a n d , (^iCt^jin  case e x i s t s f o r  -polarization,  Let  the transmitted and  electric  In sea C , a n d In the previous  field C(i^),  on z = 0 I n l a n d be c a l l e d i n general.  s e c t i o n an i n t e l l i g e n t  g u e s s was  made o f t h e v a l u e s o f G,t^ ,(=> t^, C ^ h a n d C C^) b u t no use 3  will  be made o f t h i s u n t i l  i t i s necessary.  5  21.  At t h e l a n d - s e a c o n t a c t f o r H - p o l a r i z a t i o n t h e t a n g e n t i a l component i.e.  of the e l e c t r i c  field  i s continuous,  Ei£ i s c o n t i n u o u s .  FIGURE V I  General f i e l d  vector i n nonrectangular  co-ordinates.  Er? «(Cw!7>tC0<) E y and  E , - J - O b * - LH-^.11H* _1 i  so t h a t  S  Therefore  J- 3  and  For  E" 2  =  -  Hg, _  CJHx^C &  b  i s continuous  Y  Hx  X  E - p o l a r i z a t i o n t h e t a n g e n t i a l component  field  H  and a s i m i l a r  of the magnetic  set of equations  results.  22.  boundary 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><\ Hence t h e  H(li)  conditions  H = H3~<^Z)on  V=0.  H(iii) Etangential continuous, i.e. is  J_lb = J . < L b i on v = o 3  H(iv) HnHsi iti' ^Jds, 3H, and"bH all tend to zero as H(v) H,.= Ho, V*+«o and H^- ^MV«-°^which are all known, constant and finite. 3  Z—*>-+»o~  ) 0  For E"-polarization the boundary conditions are: E(i) transmitted electric field CCT) the electric field in the air o n Z = 0 - o o < ^ < + o o . 55  The  (  E(ii) E = E.s^Z) on Y = £ 3  E(iii)  = ^ E'  on V - o  E(iv) E*|4 E.,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 V- -oe^ which are all known, constant and finite. 3  O J  23.  I t w i l l be noted t h a t i n the case and 2=^.  and the boundary c o n d i t i o n s  (X= 9 0 °  s t i l l hold  true.  Y-  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  a l r e a d y been enumerated.  T h e s e c o n d i t i o n s w i l l now be  adapted  t o t h e c a s e <X = 9 0 ° .  ditions  across the land-sea  that  t h e v a l u e s o f G\0g and  the form in  of the e l e c t r i c  have  I n a d d i t i o n the boundary  and a i r c o n t a c t w i l l be g i v e n so Ci^)  c a n be computed  and m a g n e t i c f i e l d s  t h e land^-sea c o n d u c t o r s  con-  and e q u a t i n g  knowing  I n t h e a i r and  them on t h e  b o u n d a r y z=0.  For H  -polarization: H(i)  The t r a n s m i t t e d m a g n e t i c 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 t h e b o u n d a r y p l a n e  H(ii) H(iii)  H, ^ H =  —  o0<u^<+oo.  •  The t a n g e n t i a l component o f t h e e l e c t r i c  field  continuous, I.e.  -L l b s * _L a ^  ^3  H(> to  H(v)  1  o n  3  is  H(iv)  ^—0,  H  3 ;  ^7  £ Hi  b%  on  3H|, 3 f c l  zero as z tends  ^s-o.  s  • Jij,  and  to i n f i n i t y .  H , = - <^)» H o ^  f o r y e q u a l t o -f oe  H3  f o r y equal  -  ^ 3 Ho £  ^t) and  t o - 00 , where  t h e v a l u e s o f C-j, and <£ a r e known, c o n s t a n t and 5  finite.  3  all  25.. H(vl)  The r e f l e c t e d wave i n a i r i s propagated i n the n e g a t i v e  upwards,  direction.  H ( v i i ) The t a n g e n t i a l components of the e l e c t r i c  field  are continuous a c r o s s the land-sea and a i r contact, i . e .  For  ET-polarization: E(i)  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 equals the t o t a l electric field  i n the a i r which has the value  Ct^> on the boundary plane -^=- O ^ — oo <<^< + oo. E (ii)  E  = E > ^ C ^  S  on  E ( i i i ) The t a n g e n t i a l components o f the magnetic f i e l d are equal on the land-sea i n t e r f a c e , i . e . $£•1  E(iv)  -  Eo E  u. s e> .  on  3_EJ  <L§3>  3J ^  ^  <LE/  and  ^  to zero as z tends t o i n f i n i t y . E(v)  E  (  =  C, E  0  3 - ^-3 - ° ^ c  the v a l u e s of C, finite.  3__E  a5l lt e n d  3  U 0  f o r y equal t o -t- <*> and f o r y equal t o - co , where and C 3 are known, constant and  26.  E(vi)  The  r e f l e c t e d wave i n t h e  upwards,  E(vii)  i n the  t a n g e n t i a l components o f t h e  are  continuous across  c o n t i n u i t y of  field  ditions  i s on  d ' E r c e v i l l e and  E  air  components o f  condition  the  magnetic  of c o n t i n u i t y of  the  the  question  the  the  c o r r e c t boundary  e a r l i e r work h a s  - p o l a r i z a t i o n one  must assume t h a t  at  the  i^,  w h i c h amounts t o  assuming t h a t  pointed  component o f The out  following  i n the  the  i s zero  of  magnetic f i e l d this  the  case  i s zero  on  is the the  a s s u m p t i o n has, a l r e a d y  "Introduction"  thesis.  D.  above a u t h o r s assumes t h a t ,  f o r the  case  on  the  vertical surface.  then the  component  Prom t h i s he  function  which i s i n c o r r e c t .  of  of  surface  to t h i s  t h i s were so  constant,  the  fallacy  - p o l a r i z a t i o n , the  field  Kunetz say  con-  been i n e r r o r . solve  d e p e n d e n t on  G.  of  to  surface.  If  and  that  vertical  H  vertical  the  t h a t most o f  I.  not  land-sea  field  for ET-polarization.  field  It  the  magnetic  i.e.  the  i s e n s u r e d by  electric  direction.  The  contact,  The  negative  a i r i s propagated  the  Hence R a n k i n ' s  Rankin, of  electric  concludes  w o u l d be  been  that  equal to assumption  a  27-  that the  the v e r t i c a l surface  of the e l e c t r i c  z=0 i s i n c o r r e c t .  same i n c o r r e c t H  component  i s z e r o on  J . T . Weaver h a s a l s o  assumption i n s t a t i n g  -polarization  field  that  made t h e  o n z=0, ^ = O f o r  and g((-f^ _ r> f o r E" - p o l a r i z a t i o n .  28.  SOLUTION OF THE EQUATIONS FOR <X = 90°  H  -polarization When c>< = 90° the c o - o r d i n a t e system i s r e c t a n g u l a r  and the d i f f e r e n t i a l fied.  Equation  (11),  has to be s a t i s f i e d z=0.  e q u a t i o n t o be s o l v e d i s thus  simpli-  viz.  with boundary c o n d i t i o n s a l o n g y=0 and  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  p o s s i b l e t o use a F o u r i e r s i n e t r a n s f o r m (see Appendix A5 f o r the g e n e r a l c a s e ) .  A F o u r i e r cosine transform could  a l s o be used but i t would be n e c e s s a r y t o e s t a b l i s h the boundary c o n d i t i o n s i n a s l i g h t l y more c o m p l i c a t e d manner. The F o u r i e r s i n e transform, w i t h i t s i n v e r s e , can be expressed  as f o l l o w s  and  A p p l i c a t i o n o f the F o u r i e r s i n e t r a n s f o r m t o the d i f f e r e n t i a l e q u a t i o n g i v e s the f o l l o w i n g r e s u l t s :  29.  and ^  a  ^  l  +  where  I f we now l e t H * t f y 6 ) =  (53)  f'* *}*'1  £?(^) i t can be seen t h a t there i s  o n l y one unknown f u n c t i o n whereas f o r the g e n e r a l case there are two (see Appendix A 5 ) . The expressed  s o l u t i o n o f t h i s d i f f e r e n t i a l e q u a t i o n can be  as f o l l o w s :  H =r 3  + V s ^ ^ K ^  fJCKi^  (54(b))  where  ^  = ffi -  *  Co-)  }  K =^ (  6a)  (55)  ^ | cl^Q( h  $ and  -J^)  =  *z  y, ^ ^ 1=0 t N>3 T>  (  +<T ^/Co) - ^ 1^1°) 3  5  6  (57)  )  30.  E  -polarization S i m i l a r s o l u t i o n s e x i s t f o r the case of £ - p o l a r -  ization.  No proof w i l l be g i v e n but the r e s u l t s w i l l  be  stated:  '-  3  ML  L^e'^  $I  [K*  a  K  a n d  =  )3  ^)  «**  SjJ  where V * ^  w i t h  + M^^  +  &  >  3>1  8  (  a  )  )  ( 5 9 )  -  ( 6 o )  ()  isM+yfa-VsL*)  and e l e c t r i c f i e l d s ,  6l  f o r the magnetic  e x p l i c i t expressions f o r ^ICQ-)>  and jf-jt-fc) w i l l be  ^^C^")  obtained.  approximation  equal to one constant f o r y ^ O  i s that ^  a  n  d  ^  a r e  and to another f o r y ^ 0.  T h i s i s too simple an assumption,  and l e a d s to f a l s e  r e s u l t s ( i . e . a s t e p - f u n c t i o n f o r the magnetic and  physical  electric  The next estimate- would be to apply the  f u n c t i o n noted i n the p h y s i c a l i n t e r p r e t a t i o n with the values  5  (58(b))  , "i<»  (  Having obtained the b a s i c formulae  f i e l d s f o r z=0).  (  ^  * ^,  The f i r s t  o  V  31.  E,« = C E . M  = G Ho 3  H,o  and  The  3o  v a l u e s o f t h e L|^ 's and <J • s c a n t h e n be c a l c u l a t e d  pletely  as f o l l o w s .  Heaviside's  com-  operator formula f o r  exponentials i s  (61)  Applying  this  to  ! then  and Ct^  and n o t i n g t h a t  = at  ^ / t ^ , ^ ^ ,  ^  ,(^)  (62)  and  <f 0^ become 3  (63(a)) (A p r o o f  of t h i s  i s g i v e n i n A p p e n d i x A4).  Similarly  (63(b))  (64(a))  (64(b))  £W The may  q u e s t i o n of the v a l i d i t y of H e a v i s i d e ' s method  be r a i s e d but the r e s u l t s can be v e r i f i e d by  equations  ( 6 3 ( a ) , ( b ) ) and  substituting  (64(a),(b)) i n t o equations  (11)  (12).  F o r the case  y=0  (65(a))  (65(b))  and  ^ ^ o ) = ^ t ^ ) ( ^ ^ - H o  j  3  o )  (66(a))  (66(b))  Substituting  (63(a)) into equation  l e t t i n g <y*<p the boundary c o n d i t i o n l - U ^  for  (54(a)) c*=90  o  and  is  satisfied, i.e.  also  T h i s shows t h a t the suggested behaved at the l i m i t s I^-*±PO.  s o l u t i o n i s , at l e a s t , w e l l  33.  The unknown f u n c t i o n  may be found by e q u a t i n g the  t a n g e n t i a l components o f the e l e c t r i c f i e l d a t ( 0 , 0 , 0 ) . i.e.  i  — '  3  = - L &Bz  on  ^U^pT^  < 7<*» 6  Similarly, (67(b))  Hence the f u n c t i o n s  ^,(^)  ?  >  a  n  d  a  r  e  a l l known and may be computed. V a l u e s 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 s u b s t i t u t i o n and when one l e t s g-*0 (from the p o s i t i v e s i d e ) the i n t e g r a l should be equal t o ^ 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 c o n s i s t e n t .  Unfor-  t u n a t e l y , even i n the case o f t h i s r e l a t i v e l y simple estimate, the e v a l u a t i o n o f the i n t e g r a l i s extremely complex (due t o the presence  of branch p o i n t s ) and can o n l y be e v a l u a t e d by  n u m e r i c a l means.  34.  To ensure  that  Qi^  f u n c t i o n s on the s u r f a c e z=0,  and Ci^)  are the c o r r e c t  i t i s n e c e s s a r y 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 u s i n g the boundary c o n d i t i o n s H ( i ) , ( v i ) and  ( v i i ) f o r the magnetic f i e l d  boundary c o n d i t i o n s E ( i ) ,  ( v i ) and  ( v i i ) f o r the  and  electric  field. Applying c o n d i t i o n 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 t h a t : on  The value of to  I-l,  z=0  (68)  i s g i v e n by e q u a t i o n (54(a)), as z tends  zero from the p o s i t i v e s i d e , and the value of H  2  is  g i v e n i n p a r t by equation (21-), as z tends to zero from negative side.  the  Therefore:  y  (69)  -e  The o r d e r of d i f f e r e n t i a t i o n and i n t e g r a t i o n i s i n t e r changeable on the r i g h t hand s i d e and t a k i n g the l i m i t and i n t e g r a t i o n . incidence:  so i s the order of  Hence, assuming normal  35.  J — so  where  t i i = JyU-e <*>*' , from e q u a t i o n (1?)*  i s g i v e n by the e s t i m a t e .  and the value of  Having o b t a i n e d the value of  the r i g h t hand s i d e of the equation, t o prove t h a t the estimated value of  i s c o r r e c t , the value of the  hand s i d e must be computed and proved  left  to be equal t o the  r i g h t hand s i d e .  = |f[S,(H,.(.-e^)]^^a^ which, as  ^  tends t o i n f i n i t y ,  g i v e s the boundary c o n d i t i o n  H(v) upon i n t e g r a t i o n but otherwise, branch p o i n t .  (70)  i s an i n t e g r a l with a  I f the i n t e g r a l I i s g i v e n by  36.  then  I  where  J  and  I  = I, + = J f j~  x  • *  f  ^ W  ^  *°P^""^  H  a  *H  ^ »  f  ^  ?  *" fa ^ ~ H . > r P > / f » «Tht* 3  T h i s i n t e g r a l can be s u b d i v i d e d i n t o three more i n t e g r a l s , i.e.  let  I  a  s  I *+ I k + I*.  where •  H  t 0  ^ ,  H ^ e^' ^  i ^  (  and  1  (  a  )  )  (71(c))  These I n t e g r a l s possess complex b r a n c h - p o i n t s making mathematical  7  e v a l u a t i o n extremely c o m p l i c a t e d .  their  It i s  suggested t h a t these e q u a t i o n s can be b e t t e r s o l v e d on a computer and the r e s u l t s p l o t t e d on a graph.  i f the r e s u l t s  p l o t t e d from the r i g h t hand s i d e of the e q u a t i o n f a l l on the curve d e r i v e d from the l e f t hand s i d e then i t may be  assumed  t h a t the estimate s a t i s f i e s a l l boundary c o n d i t i o n s 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 s i m i l a r to H - p o l a r i z a t i o n , are  ^  M X - *  m  A  o  )  €  M  =  ~  e  ~ ^ J ( 7 ? )  I ~ I + I u.  and  s  where  I4  i s g i v e n by  2  ) e " ^ " ] siyy^eL^  ,  ^ >• O  (73)  and .  x  Jo  L  (N, +  J  S ^  X. Sj" £^«S^£j^dlsj  Again  J  4  =  + I  e  (74)  +  where I. = A f ^ E o - ^ E a o )  T  ^ ^  *  ""U<«> - ^(v.-fo ^  4> (>$,+  «;  ^  ^  | L i  ^ig  (74(a))  1  (74(b))  38.  The n u m e r i c a l i n t e g r a t i o n o f each of these e q u a t i o n s maybe obtained, by s e p e r a t i n g the f u n c t i o n s i n t o r e a l and imaginary p a r t s and e v a l u a t i n g 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 o f E - p o l a r i z a t i o n , from e q u a t i o n ( 1 0 ) t o be  Therefore:  e^^zj^dfs-  x  (75)  ^>,o  }  A s i m i l a r e q u a t i o n e x i s t s f o r y$ 0 . I f the estimate C*- ^ i s c o r r e c t 0  H  a n d  It w i l l  then  V ^ P ' ^ - O ^  j  (76)  _  ^ >x o  H^^p.a^-^e^ be n o t i c e d t h a t the v e r t i c a l magnetic  dependent of the frequency at y = 0 .  field  i s in-  Is proportional to  the square r o o t o f the frequency and so i s ( ^Lo) —  Ef,  0  )  and hence the o n l y frequency dependent term i s exp (-^p,ift) . T h i s r e s u l t was obtained, assuming t h a t  3  39.  on  z=0,  where B  i s c o n s t a n t , by J.T. Weaver  (1963)  and a l s o  by T. Watanabe i n a d e r i v a t i o n of e q u a t i o n s f o r magnetot e l l u r i c modeling ( p r i v a t e communication).  J.T. Weaver has  c a l c u l a t e d the r a t i o of the v e r t i c a l t o h o r i z o n t a l  magnetic  f i e l d over the l a n d - s e a c o n t a c t and h i s graphs are shown i n Figure V I I .  40.  3-0  20  30T  15 Distance - y (km)  —  5 coast  —  Distance +y FIGURE V I I  10 from  (km) from coast.  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  D e l t a has a l r e a d y been mentioned. thought, i s due  i n the P r a s e r R i v e r T h i s anomaly, i t i s  to r e l a t i v e l y c l o s e - s u r f a c e c o n d u c t i v i t y  v a r i a t i o n s and the emphasis, i f p l a c e d anywhere, must l i e on p e r i o d s i n the range 0<T^:50 seconds.  Longer p e r i o d s  occur l e s s f r e q u e n t l y and are l e s s a c c u r a t e l y p l o t t e d . F u r t h e r the theory so f a r expounded of l a t e r a l  skin-depth  c o u l d not be i n t e r p r e t e d f o r depths of over 5 km.  i n the  sea at an absolute maximum and t h e r e f o r e l o n g e r p e r i o d s would be of no i n t e r e s t  here.  Geomagnetic bays, with a p e r i o d 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 .  S t a t i o n s were set up on a l i n e  p e r p e n d i c u l a r to the sea coast f o r a d i s t a n c e of j u s t under 300  km.  at San F r a n c i s c o and La J o l l a .  Readings were a l s o  taken on the San Clemente I s l a n d s f o r the La J o l l a t r a v e r s e (see F i g u r e I X ) .  A study of the v e r t i c a l component of  magnetic f i e l d s f o r w e l l behaved bays, i . e . the  electric  f i e l d v e c t o r p a r a l l e l s the coast, showed t h a t at the coast there was  the  sea  a pronounced d i s t u r b a n c e but the f u r t h e r  i n l a n d the r e c o r d s were taken the s m a l l e r the magnitude of the d i s t u r b a n c e u n t i l at a d i s t a n c e of 100 d i s t u r b a n c e disappeared I f one  km.  or l e s s the  (see magnetogram r e c o r d s F i g u r e V I I I ) .  assumes t h a t the theory as p r e v i o u s l y d e s c r i b e d i s  42.  Aug. 3 0 . I 9 6 0  H  Magnetogram r e c o r d s , adapted from U. Schmucker.  FIGURE IX  Map  of Southern C a l i f o r n i a .  44.  correct  one i s l e d t o t h e c o n c l u s i o n  diffusion the  current  arising  conductivity.  from l a t e r a l  I f this  magnitude e s t i m a t i o n s  that  i sa l a t e r a l  inhomogeneities i n  i s the case then crude order o f  may be c a r r i e d o u t t o f i n d  imate dimensions o f t h i s c o n d u c t i v i t y  -  ^  uD = 3600 c p s , a n d  i staken as the l a t e r a l  the  d i f f u s i o n , the conductivity,  the  relation  t h e approx-  anomaly.  I f ju. = 1 . 2 5 7 x 1 0 6 henry/m., ^ = 100 km. where  there  dimension o f  , c a n be d e t e r m i n e d  from  oi cr" is  f o u n d t o be o f t h e o r d e r S.P.  Srivastava's  o f IO"-  Ph.D.  1  thesis  mho/m. (1962)  contained  a  study o f t h e subsurface c o n d u c t i v i t y o f t h e planes o f Alberta using by  thep r i n c i p l e s of magnetotellurics  L. Cagniard  (1953).  Srivastava  concluded  as expounded  that  for  survey the d i s t r i b u t i o n o f the c o n d u c t i v i t y with depth be  summarized  10  5 t o 90 km., t o 150 km.,  T.  Rikitake  - 1  mho/m.  10"^  mho/m.  1 0 " mho/m. 1  (1951) c o n s i d e r e d  corresponding to Srivastava's he 100  placed km.  could  as f o l l o w s :  0 t o 5 km..,  90  his  h i s depth o f contact  a two l a y e r m o d e l ,  second and t h i r d  layers, but  c l o s e r t o 400 km. t h a n t o  45.  LAND  SEA  100  x l-  Q. UJ  Q  !__ LOW VELOCITY 4 cr=IO" mhos LAYER  200  CONDUCTIVITY 1  CLIFF  <r =10"' mhos  L  300 4001-  FIGURE X  Schematic c r o s s - s e c t i o n oversimplified  a t the sea coast f o r an  conductivity  cliff.  SHEAR VELOCITY (km/sec) 3-0 4-0 5-0  I0O  ..j  8 0 )9 tSE A)  E  CA K  (LA ND)  X  cZ  UJ  30O  o  400  500 FIGURE XI  ;  I  V e l o c i t y depth curve (shear waves) under the Canadian S h i e l d  (Model CANSD) and under the  ocean (Model 8099), a f t e r J . Brune  and J . Dorman.  46.  The  initial  of 1 0  value  d e p t h o f 0 t o 5 km., i s n o t influence  theoverall  _ 1  mho/m., r a n g i n g  from a  t h i c k enough t o a p p r e c i a b l y  effective  c o n d u c t i v i t y f o r the  90  km.  at  a d e p t h o f 9 0 km. o r more i s much l a r g e r t h a n t h e  Therefore  ductivity  to a f i r s t  .  =  t  cr—  ^gok-v.  this  Pacific there is (1)  stratification Coast  and v a l u e s  found t o e x i s t . The h e i g h t ducting  _ O  J OO  o f t h e c o n d u c t i v i t i e s and depths Srivastava's a striking  of the surface  layer (IO  - 1  o f the E a r t h  I.e.  the value  d e p t h w o u l d be o f t h e o r d e r  be (2)  (The  t o lower the  The v e r t i c a l imation,  under the sea.  alternative to raising conductivity  from a " c l i f f "  field  would  of the skin-  o f 75 km. r a t h e r the  than  depth  would  contrast.)  anomaly w o u l d a r i s e ,  where t h e v a l u e  the  above t h e c o n -  mho/m.) w o u l d be o f t h e o r d e r o f  considerably,  km.  similarity  Two p o i n t s must be e m p h a s i z e d :  km. a n d h e n c e t h e anomalous v e r t i c a l  flatten  100  con-  o f c o n d u c t i v i t y i s extended t o the  a r e compared w i t h  100  the conductivity  above, i . e . ^  If  approximation,  first  i n the  i n the  first  c o n d u c t i v i t y a t depth;  o f c r a t 200 km. w o u l d be I O  l a n d and o f t h e o r d e r  approx-  -  1  mho/m.  o f 1 0 " ^ mho/m. u n d e r  47.  . Z , DEPTH  IN  km  1000  500-  60 3 (3600 sec)  100 (IQOOOsec)  T  FIGURE X I I ( a )  Idealized for  two  plot  of skin  depth  VERTICAL MAGNETIC ANOMALY  vs. period  conductivities.  CONTINENT  SEA AH  VT  ~1  215 km  315 km  °2  PERIOD FIGURE  Xll(b)  Probable  variation  of H  2  with  period.  IN sec  2  48.  One  i s l e d t o the c o n c l u s i o n  c o n d u c t i v i t y w i t h d e p t h m i g h t be It  would  seem l o g i c a l  information seismic  t o t r y and c h e c k  L. Sykes e t a l .  this  (1962)  depths of have  60  and  215  i s a low v e l o c i t y km.  J . Brune  there  the Canadian S h i e l d .  i s a low v e l o c i t y  km,.  derived  have p r o p o s e d t h e s e a , and  a have  l a y e r between the  (1963)  and J . Dorman  Their  results  and  show  that and  l a y e r must go f r o m deep t o  s h a l l o w e r depths as i t p a s s e s f r o m under  the C o n t i n e n t s to  under  shallow to  t h e Ocean,  depths as might  instead be  of passing from  expected from a simple study of  d e p t h s and c o n d u c t i v i t y . either  t h e shape  or l i n e s  other or that the p i c t u r e  o f t h e low v e l o c i t y  the l a y e r s  I t would  clusion  i s correct  seem l o g i c a l  concluded that  are independent of  t o assume t h a t  but the p o s s i b i l i t y Assuming  and t h e i s o c o n d s p a r a l l e l  the v e r t i c a l magnetic  skin  each  and t h e i s o c o n d s a r e r e l a t e d  completely r u l e d out.  layer  deeper  l a y e r and t h e " i s o c o n d s " ,  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  fied.  be  Hence i t must be  of equal c o n d u c t i v i t y ,  and  i s oversimplithe l a t t e r  con-  o f t h e former, c a n n o t  that  t h e low  velocity  each o t h e r the morphology  anomaly f o r v a r i o u s p e r i o d s may  found from the.diagrammatic  X.  from  l a y e r b e t w e e n t h e d e p t h s o f 115  Hence t h e low v e l o c i t y  of  with other  suggested a v e l o c i t y depth curve f o r the mantle  c r u s t under  315  there  i n Figure  result  and c r u s t  v e l o c i t y depth curve f o r the mantle under suggested that  the v a r i a t i o n  as i l l u s t r a t e d  about the E a r t h ' s mantle  sources.  that  graphs of depth v s . p e r i o d  be and  of  49.  vertical will  field  vs. period  be n o t i c e d  that  (see Figures  there  are f i v e  w h i c h c o r r e s p o n d t o ( 1 ) no v e r t i c a l (3) f i e l d  field,  decline, five (2)  field,  b a y s and ( 5 ) s t o r m s . XII(a),  ( 3 ) and ( 4 ) g e o m a g n e t i c  i s t h e s k i n d e p t h and hence i s a The two c u r v e s ^, a n d ^ a r e  i d e a o f t h e shape o f t h e v e r t i c a l m a g n e t i c f o r a given  T h i s m i g h t be a f i r s t velocity  depth curve  Assuming t h a t land the  i s  conductivity  function.  as i t passes under the sea c o a s t .  the depth t o the bottom o f the l a y e r under the  Figure  skin  depth  However,  although  t o make good e s t i m a t e s o f t h e c o n d u c t -  ivity  a t 1 0 0 a n d 2 0 0 km. u n d e r t h e l a n d  basic  concept  should  ( 2 1 5 km.),  i f t h i s picture of the conductivity  i s assumed t o be c o r r e c t .  may n o t be a b l e  z  Xll(b).  i n t e r p r e t a t i o n of the l a t e r a l  becomes somewhat c l o u d e d  obtained  to obtain  approximation t o the junction of the  second graph i s obtained,  distribution  constant  anomaly a s i t  step  I), ( 3 1 5 km.) a n d u n d e r t h e s e a i s I b  The  one  d e p t h and  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  varies with period  These  micropulsations,  p l o t s f o r two d i f f e r e n t p l a n e s o f i n f i n i t e  an  ( 2 ) growth o f t h e  The d e p t h ^ p l o t t e d i n t h e f i r s t  function of conductivity.  conductivity.  on t h e graphs,  and ( 5 ) no f i e l d .  would c o r r e s p o n d t o ( 1 )  long period micropulsations,  graph, F i g u r e  regions  It  m a g n i t u d e r e a c h e s a maximum and s t a r t s t o  ( 4 ) decay o f the f i e l d  regions  X l l ( a ) and ( b ) ) .  n o t be d i s c a r d e d .  from the attenuation  and t h e s e a ,  the  I n f o r m a t i o n c a n be  of the v e r t i c a l  component, b u t  50.  t h e e x a c t t h e o r y must be It  i s pertinent,  and J . O l i v e r mantle  reworked  at t h i s p o i n t ,  (i960),  of the P a c i f i c  who  for this  suggested a s t r u c t u r e  B a s i n f r o m s u r f a c e wave t h e low  Ewing  f o r the  that  r e g i o n of the upper mantle  e x t e n d s u p w a r d s t o much  that  and t h a t  shear v e l o c i t i e s  c o v e r i n g t h e whole  a r e somewhat  lower  down t o a b o u t  i s f o r r e a d i n g s o f the magnetic spectrum of p e r i o d s ,  24 h o u r s , f o r d i s t a n c e s o f 0 t o 1000 surface  from 1 second t o  km.  from the coast  evidence of F i g u r e XII(b) can  s e e n f r o m t h e magnetogram r e c o r d s o f U. V e r t i c a l magnetic  anomalies a r i s e  of the l a n d - s e a c o n t a c t .  c a n n o t be  field  on  of the sea.  Corroborative  from  micropulsations  Unfortunately There  component o f g e o m a g n e t i c  t h e p r e s e n c e o f some s o r t  be  Schmucker.  seen f r o m Schmucker's r e c o r d s .  anomaly i n t h e v e r t i c a l dicating  about  km.  g r e a t e s t need  because  to  (Moho)  km/sec a t d e p t h s o f  than i n the s u b - c o n t i n e n t a l mantle  400  beneath  b a s i n are i n t e r p r e t e d  shear v e l o c i t y below the M  d e c r e a s e s t o a b o u t 4.3 60 km  velocity  R a y l e i g h wave d i s p e r s i o n d a t a f o r  t h e p a t h s on t h e P a c i f i c indicate  than  upper  dispersion:  . . . R a y l e i g h waves i n d i c a t e  the c o n t i n e n t s .  the  case.  t o q u o t e J . Dorman, M.  shallower depth beneath the oceans  The  special  of step  these i s an bays, i n -  ( i t i s more  51.  probably a s l o p i n g c o n t a c t ) i n the c o n d u c t i v i t y , but no v e r t i c a l component anomaly i s found i n storms,  indicating  that the c o n d u c t i v i t y I s l a t e r a l l y homogeneous at g r e a t e r depths.  52.  CONCLUSIONS  The v a l u e s o f Q  and  t  show t h a t Q, — 2 and imation the r e s u l t Weaver ( 1 9 6 3 ) ,  and h e n c e t h a t Q, = <$ =» A 3  i s valid.  the f u n c t i o n  ( o b t a i n e d from Appendix I I )  (q C y ) =  =  to a f i r s t  , a constant,  An improvement ^1  ^  ^3  ^  >  as used by J.T.  w o u l d be t o w r i t e  0  <0  b u t t h i s w o u l d be p h y s i c a l l y u n r e a l i s t i c . estimate of G  Hence t h e  w h i c h seems t o be p h y s i c a l l y  been suggested t o g i v e  approx-  sound, h a s  the s o l u t i o n a d d i t i o n a l  accuracy  even  though i t h a s n o t been checked m a t h e m a t i c a l l y . It  C *  i s found  that  -4=-7 i\  C  and  5  —-L=,  0 <  V^ri  err  and h e n c e  ; C  Thus t h e t e r m C  3  z  =* ^ 1 I O  i s dominant.  This fact  i s important i n  t h e c o m p u t a t i o n o f Ct^) . Graphs o f t h e v e r t i c a l magnetic f i e l d the case o f  E " - p o l a r i z a t i o n have b e e n c a l c u l a t e d b y J . T .  Weaver f o r t h e a p p r o x i m a t i o n s c i t e d  above and t h e s e  been i n c l u d e d h e r e w i t h h i s v e r y k i n d VIII). and  arising i n  consent  The g r a p h s have b e e n c a l c u l a t e d f o r  have  (see F i g u r e = 0.01 c p s .  co = l c p s . and a r e b a s e d on Weaver's e q u a t i o n a s s u m i n g  53.  constant v a l u e s a c r o s s the d i s c o n t i n u i t y as opposed t o a v a r y i n g value of  (T(i^).  The theory of l a t e r a l s k i n depths has been t e n t a t i v e l y a p p l i e d t o two cases, f i r s t l y t o the v e r t i c a l magn e t i c anomaly a r i s i n g from the F r a s e r D e l t a experiment and secondly t o the magnetogram r e c o r d s c o l l e c t e d by U. Schmucker. Although  there seems t o be some doubt as t o whether the  theory may be f r e e l y a p p l i e d i n the l a t t e r case, one can conclude  t h a t there i s some s o r t o f l a t e r a l c o n d u c t i v i t y i n -  homogeneity.  I t s exact shape and form i s not c l e a r , but  two models have been proposed, the f i r s t b e i n g a simple 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 i n f o r m a t i o n obtained by s e i s m o l o g i s t s on the low v e l o c i t y  layer.  con-  54. APPENDIX A l  S n e l l ' s Law a p p l i e d  t o a Nonconductor-conductor Boundary  Couple.*  Couple-*  According to Snell's l<,  s i 9, n  (cx.+'u^O s i o  or  =  vMtwe.  Law k  a, =  t  "sin  e  0  (<*i  SinG-o  Al(i)  However, n o t i n g t h a t , f o r a i r c<  and,  =  2  coJjZ^?  f o r a conductor  Al(ii)  <*, = ^, = ^J!^2^  we c a n e x p r e s s © , a s a f u n c t i o n  o f <*,,  Hence  ^  -sin The  G, -  condition,  Al(iii) p i , and 9  4  n  I J / ^ T pfc—  T-i-rr-  W  Al(iv)  f o r which t h e d i s p l a c e m e n t c u r r e n t  i s negligible  i s M»|i-w4 and t h i s i s s a t i s f i e d  on t h e l a n d - a i r  boundary.  i n A l ( i v ) s i n 6 | i s g i v e n by  Using t h i s condition  L  and  sea-air  l  J  ^  Al(v)  assuming  Since  t h e e x p r e s s i o n f o r sin "©, i s t h e p r o d u c t o f two 2  55.  factors,  i t s vanishing  can e q u a l l y w e l l be the r e s u l t of sit-i^S,,— O  •l.e.Q^O.  Hence the c o n d i t i o n t h a t ^L»|-has the same e f f e c t  as assuming  t h a t the wave i s propagated almost v e r t i c a l l y and  t h a t o n l y h o r i z o n t a l components of the e l e c t r i c  and magnetic  fields exist.  P h y s i c a l l y i t i s p o s s i b l e t o see t h i s as f o l l o w s : As soon as the wave passes i n t o the c o n d u c t i n g medium, the f r e e charges, l y i n g at random throughout the volume, migrate to the boundaries and s e t up an e l e c t r i c f i e l d , to the s u r f a c e s , E  N  .  component of the f i e l d  This f i e l d  opposes  which i s normal  any v e r t i c a l  i n the conductor, i . e . E  N  =E , .  = C>  hegah>/£ cKaf-geS  A t7  H  1  1  h H — I  I- -f  K  1—|  h  V  "  56.  APPENDIX  Fresnel  Equations  If E The  L  A2  = E  o  we assume an i n i t i a l  e  ,  wave t o be o f t h e f o r m  HL-J^  n  a  * E l  A  2  (i)  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 o f t h e f o r m  E -  E,  t  H t -  e  & - v U E  A 2  t  (ii)  and E  The and  ^  E  -  ^  o n l y unknown f u n c t i o n s a r e C", and C  z  Cq,  and 6 j f o r a  A2(iii) for E-polarization  H - p o l a r i z a t i o n where C^,CX) G,  T T , = I^JT. ,  and at  >  e  H  a n d <S a r e d e f i n e d t  = <q H.  a  a  the boundary  ^ ( E .  +  r ^ ^ ^  a n <  a  Pi^+TT^.H",  E'-polarization  For therefore  E"  1c-polarization  0  ET  ?  E ,  n.» E = O 0  A2(iv)  57.  c o Go E - ^os©,. E »  and  a  from e q u a t i o n A 2 ( i v ) .  we  Or> ©^E,  2  Noting that  find  ^ - .^r.  ^  k  '  6  ,  ,  • •  E  f t  and  and hence the value of C, and  may be o b t a i n e d .  n o t i c e d t h a t k, i s complexjbut  It will  be  l<j.is r e a l .  ^polarization  For  |-( =.Q  H-polarization  H« * ^*  and i n t h i s case  Eo  Noting that:  0  • fiT*.  =  " ^ \ ~  =  ^"  »  E", =  -J±^>< £ ,  *Tt  k,  and it  £-*-*TT  Ei »  0  i s found t h a t  cos e TTL - o o s e i * 0  ir  from e q u a t i o n A 2 ( i v ) , U  and  L_|  Z./.^  =  A*^'  and hence  k,* V  ^  0  6  0  cos^o -  and hence the value of  o n e , M. s o l v i n g f o r H,  **<t  and  ,  kjtf-  kl *\^G  e  Hjwe TT  r  A  M +H^K  0  and £j^can be o b t a i n e d .  T^j~  find,  58.  APPENDIX A 3  Proof of d i v ~E=0  (1)  In the case of E - p o l a r i z a t i o n the e l e c t r i c  only one non-zero  component,  which i s zero because E  x  i s  has  f i e l d has  two  and so,  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 non-vanishing components E / and Hat  field  which are connected w i t h  as f o l l o w s  (These equations are d e r i v e d from Maxwell's E q u a t i o n s ) But  V ~ F =  4-  59.  APPENDIX A4  V a l u e s o f i|> and Q  t h e n r e p l a c i n g H|  If  jr^'tj <^ | * t h e  functions for  by  no  Now  convergent therefore  similarly  and  t h e summation  can  ) then  found.  £ioJ  ^^u>^,^u  so t h a t P  i s convergent  I f i t i s divergent (i.e.|^j-|>|  s o l u t i o n c a n be  e"  C^)  the value of  series  simply expressed.  GiC^  2  +  ° y ^ i ^7  &  1  = j ^ ^y^, - * 3  Therefore f o r a l l ^ ^ o  and '  the  series  i  60. APPENDIX A 5  S o l u t i o n of the Equations  In General: H  -polarization  A s o l u t i o n o f t h e g e n e r a l c a s e , when o<<^<90  0  o n l y be t e n t a t i v e l y whereas the case f o r detail  suggested  K=90° h a s  and l e f t  In general  The  found  i n the general  t h i s case  i n a d d i t i o n t o the second  respect  t o t h e V and t h e 2  term.  This leads to d i f f i c u l t i e s  partial  differential  i s used  t h e n two unknown f u n c t i o n s r e s u l t ,  be  about t h e  case.  g e n e r a l equations d i f f e r s from  &C=90°, b e c a u s e  terms,  been worked o u t i n r e a s o n a b l e  and c o n c l u s i o n s c a n be drawn f r o m  type o f f i e l d  will  the case of  differentials  with  co-ordinates i t has a cross-product  equation.  i n the s o l u t i o n of the  I f the Laplace  transform  which both need t o  calculated.  If  is  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,  applying i t to the general equation it  i s found  (16),  then  forH-oolarization,  that  17  61,  as-  ^cy  n  2  A5(ii)  T h i s e q u a t i o n i s an o r d i n a r y d i f f e r e n t i a l and,  ,  H* -  of this  k*  where  e q u a t i o n w o u l d be  + k,  e  = Sj"  and  Setting ^ j s a ^ l s  a  +V/t^  C, = ^  ^»J L l y * - - ^  '  1  A5(v)  , A s s u m i n g t h a t VC^  i s f i n i t e at  1  , t h e b o u n d a r y c o n d i t i o n H ( v ) c a n be a p p l i e d t o show  0  k =0 z  .  The c o n d i t i o n  R*£>| i s  <  ^  tends  t o a minimum v a l u e  (zero) as  2j~  tends  t o a maximum v a l u e  (infinity)  When ^  A5(iv)  we g e t ,  Hence, when 1 ^ , 6 ) s J 0  - H ^ ' H * = OCV)  _ Z^oo^fix  A solution  that  in V  placing  we f i n d  T= +  equation  has a value  outside this  limit,  S" < i  JlJfi^tf^ *  tends as  so  that  t o z e r o , and tends  i.e..^*^  t o 90°.  || sin^ z  then  2ccs<*. t h e p r o b l e m becomes much more c o m p l i c a t e d . c h o i c e s d e p e n d i n g u p o n t h e f u n c t i o n k\C^)  T h e r e w i l l be two .  Either  Kt « k  2  =*0  62.  or n e i t h e r K, nor  k* are zero,  the c o n d i t i o n as  i n f i n i t y b e i n g s a t i s f i e d by the to produce a f i n i t e number. t h a t both }Cj A theory  and  ka  infinities  An i n t e l l e g e n t guess might i t can o n l y be  w i l l be developed assuming t h a t <  tends to  s u b t r a c t i o n of two  are zero but  zero i n s i d e the domain ^  V  J-  •  and  be  a guess.  at l e a s t i s a t e n t a t i v e approach  V C Y ) w i l l always be g i v e n  to the problem w i l l be g i v e n .  which Is the p a r t i c u l a r i n t e g r a l of the  by:  differential  equation, assuming t h a t OC^can be expressed as a F o u r i e r s e r i e s and  t h a t there  The  are no  values  singularities.  of k|  and  , can be computed  by  a p p l y i n g boundary c o n d i t i o n s H ( i i ) -  and  i^-v,  applying  ^  |<3  to)  ^  =  -v^o>  A  5  (  v  l  )  H(iii)  ^V.co) (£,-*,) -wi- Vi^XgitV^-riV^^gaV; ^ 1  A 5 ( v l l )  I t i s apparent t h a t 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 f u n c t i o n of U(s^)  .  ^CV^)  i n t u r n , I s a f u n c t i o n of the magnetic f i e l d on the 2i»0 OCV)  .  But  the value onZ=-0  cannot be p l a c e d  u n t i l the p a r t i a l d e r i v a t i v e dM*  Hence, although  appear inUC$.  The  , two  In the  and has  seperate u n c a l c u l a t e d  function  surface function  has been computed.  i s o n l y a f u n c t i o n of V  c a l c u l a t e d along 2 = 0  ,  been  functions  can be d e f i n e d by  the  63.  following conditions:  <scv>- ^5Ho  and  ,  Liv-n - L U m  and l a s t l y  The  y -  3  function  -  - oo  Ll**,  JL  Uv*  $ t ^  ^jj"^^'^  l s  n o - f c  ^ ~  % t<>,P)  n e a r l y so easy to  d e f i n e ; i t appears t h a t i t can o n l y be c a l c u l a t e d a f t e r v a l u e s of H,  and  r-^have been computed, which i n d i c a t e s t h a t  a second a l t e r n a t i v e s o l u t i o n of the p a r t i a l equation  the  i s necessary  differential  before a complete s o l u t i o n of  this  case, u s i n g t h i s method, can be found.  E*-polarization  The  which can be equation f o r  relevant equation  is  " s o l v e d " i n the same way  as the p a r t i a l  H - p o l a r i z a t i o n u s i n g the L a p l a c e  differential  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 differentials  to o b t a i n the  equation  to o r d i n a r y  64.  where NJ*» a f - p V n f c  a  n  <  i  A5(ix) The  solution of this  differential  equation  i s t h e same  as b e f o r e , i . e .  A5(x) ^ , = Sj i^rpc-L  Once a g a i n in  and  =• " S ^  —  i fSJ i s  a p p l y i n g c o n d i t i o n E ( v ) makes  t h e c o r r e c t domain.  The e v a l u a t i o n o f K  and i C j i s  4  o b t a i n e d t y a p p l y i n g t h e b o u n d a r y c o n d i t i o n s E ( i i ) and then  E(iii).  k|-^p  and ^ * ! ^ 7 % ) - S (o) 5  Application of E ( i i l )  ^  g i v e s the' v a l u e  o f |(^) a s  ^ + 6",  when " S , * ^  .  Once more t h e s o l u t i o n o f t h e i n t e g r a l , equation value  i s dependent on t h e - v a l u e  (or function) represented  o f £lV^ and wJ=*|  b y C(Y) may be  differential .  The  expressed  by a s e r i e s o f l i m i t i n g c o n d i t i o n s i n t h e same way a s GCd ,  65.  i.e.  CCV)=C,Eo  ,  V-  +  »  and  However the problem of d e f i n i n g the value of ^£^2M still  remains e s s e n t i a l l y u n s o l v a b l e u n l e s s a subsiduary  solution  i s added.  *  66.  BIBLIOGRAPHY  Brune,  J . , and J . Dorman, B u l l . S e i s . Soc. Am.,  Cagniard, L., Geophysics, 18,  605,  53,  167,  1963.  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 F r a s e r D e l t a Experiment of i960, P a c i f i c Naval L a b o r a t o r y Report 6l-5, August d ' E r c e v i l l e , I . , and G. Kunetz, Geophysics, 2J_, 651,  1961.  1962.  Dorman, J . , M.Ewing and J . O l i v e r , A s s o c i a t i o n de Seismologie et de Physique de l ' I n t e r i e u r de l a T e r r e , Resume 166, H e l s i n k i , i960. N i s h i d a , A., Ph. D. T h e s i s , U n i v e r s i t y of: B.C., Rankin, D., Geophysics, 27,: 666,  1962.  1962.  R i k i t a k e , T., B u l l . E a r t h . Res. I n s t . , XXVIII, p a r t I I , page  263,  1951.  S r i v a s t a v a , S. P.,  Ph. D. T h e s i s , U n i v e r s i t y 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 F a u l t Problem i n M a g n e t o t e l l u r i c theory, P a c i f i c Naval L a b o r a t o r y T e c h n i c a l 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 , E l e c t r o d y n a m i c s of Continuous Media, V o l . 8, Pergamon P r e s s , i960. P r i c e , A. T.,  Quart. J . Mech. and App. Math., 3, 385,  P r i c e , A. T.,  J . Geophys. Res.,  67,  1907,  1950..  1962.  S t r a t t o n , J . A., E l e c t r o m a g n e t i c Theory, McGraw-Hill, Inc., 1941.  

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