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Reflected wave propagation in a wedge Ishill, Hiroshi 1969

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REFLECTED WAVE PROPAGATION IN A WEDGE by " HIROSHI I SHII B . S c , Tohoku U n i v e r s i t y , 1963 M.S c , Tohoku U n i v e r s i t y , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming to the THE UNIVERSITY OF BRITISH COLUMBIA September, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date 2-UU Sept . \°.t> °l i i ABSTRACT The behavior of e l a s t i c body waves i n a dipping layer overlying an e l a s t i c medium has been t h e o r e t i c a l l y investigated by a multiple r e f l e c t i o n formulation. Although the d i f f r a c t e d wave i s not included i n t h i s formulation, i t s importance i s studied by in v e s t i g a t i o n of the amplitude d i s c o n t i n u i t i e s within "the wedge. For a plane SH wave incident at the base of the dipping layer perpendicular to s t r i k e , a series solution has been obtained. Numerical values of the amplitude, phase and phase v e l o c i t y are calculated on the surface. For waves propagating i n the up-dip d i r e c t i o n the amplitude versus frequency curves for a constant depth to the interface change slowly with increasing dip for dip angles less than 20°. However for waves propagating i n the down-dip d i r e c t i o n the character of the amplitude curves change r a p i d l y . In these cases, i t i s found that the d i f f r a c t e d wave plays an impor- tant r o l e . In addition to s a t i s f y i n g the boundary conditions at the surface and the lower boundary of. the wedge, the d i f - fracted wave must also s a t i s f y additional conditions along a dipping interface between the wedge boundaries due to the geometrical nature of the r e f l e c t e d wave so l u t i o n . It i s found that the phase v e l o c i t i e s vary r a p i d l y with both period of the wave and depth to the in t e r f a c e . For incident plane P and SV waves, the complexity of the problem due to the converted waves does not allow the sol u t i o n to be expressed i n series form. However, a com- putational scheme has been developed which allows the i i i c a l c u l a t i o n of the d i s t u r b a n c e due to the m u l t i p l y r e f l e c t e d waves. For both i n c i d e n t P and SV waves, numerical values of displacements and displacement r a t i o s are c a l c u l a t e d on the s u r f a c e . I t i s found t h a t the displacement r a t i o s f o r i n c i d e n t SV waves are much more s e n s i t i v e to d i p than are there f o r i n c i d e n t P waves. For i n c i d e n t P and SV waves p r o p a g a t i n g i n the down-dip d i r e c t i o n with a propa- g a t i o n d i r e c t i o n oL, (3 = 120°, the amplitude r a t i o versus frequency curves f o r constant depth to i n t e r f a c e do not have s i g n i f i c a n t peaks f o r d i p angles g r e a t e r than 15°. The maximum d i s c o n t i n u i t i e s caused by the outgoing wave are a l s o c a l c u l a t e d to determine the r o l e of the d i f f r a c t e d wave. As s u b s i d i a r y problems the energy r e l a t i o n s between waves at an i n t e r f a c e between e l a s t i c media are determined i n terms of p r o p a g a t i o n d i r e c t i o n i n a c y l i n d r i c a l system and the complex p r o p a g a t i o n d i r e c t i o n i s i n t e r p r e t e d u s i n g the R a y l e i g h wave. The f i n a l study i s to determine by a r e f l e c t e d wave f o r m u l a t i o n the displacements due to p e r i o d i c and i m p u l s i v e l i n e sources of SH waves i n the wedge o v e r l y i n g an e l a s t i c medium. A formal s o l u t i o n i s found by which the c o n t r i b u t i o n s due to head and r e f l e c t e d waves are determined by e v a l u a t i o n of the i n t e g r a l s by the method o f s t e e p e s t descent. Using ray paths, the c o n t r i b u t i o n s i v of the i n t e g r a l s have been i n t e r p r e t e d . The range of e x i s t e n c e of head waves has been examined and the d i s c o n - t i n u i t i e s a s s o c i a t e d w i t h d i f f r a c t e d waves s t u d i e d . In the case of a f r e e or r i g i d lower boundary of the wedge, the d i s p e r s i o n r e l a t i o n has been determined. V TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v i i i LIST OF TABLES x i ACKNOWLEDGEMENTS x i i CHAPTER 1 GENERAL INTRODUCTION ' i 1.1 P r e l i m i n a r y Remarks 1 1.2 Summary of Previous Stud i e s 2 1.3 Scope of T h i s T h e s i s 4 CHAPTER 2 MULTIPLE REFLECTION OF PLANE SH WAVES BY A DIPPING LAYER 7 2.1 I n t r o d u c t i o n 7 2.2 Wave Equation and Fundamental S o l u t i o n 8 2.3 R e f l e c t i o n and R e f r a c t i o n C o e f f i c i e n t s 9 2.4 M u l t i p l e R e f l e c t i o n S o l u t i o n f o r a Wedge 14 2.5 Numerical Computations and D i s c u s s i o n 20 2.5.1 Amplitude D i s c t o n i n u i t y at Q-<^>^~J\Z 20 2.5.2 Surface Amplitude C h a r a c t e r i s t i c s 21cL 2.5.3 Phase V e l o c i t y at the Free Surface 30 CHAPTER 3 MULTIPLE REFLECTION OF PLANE P AND SV WAVES BY A DIPPING LAYER 32 3.1 I n t r o d u c t i o n 32 3.2 Equations of Motion and Boundary C o n d i t i o n s 32 3.3 R e f l e c t i o n and R e f r a c t i o n C o e f f i c i e n t s 36 3.4 Computation of Displacement i n the Case of a Dipping Layer 46 v i 3.5 Displacement D i s c o n t i n u i t i e s 50 3.6 Surface Displacements and Displacements R a t i o s 53 3.6.1 I n c i d e n t P 53 3.6.2 I n c i d e n t SV 56 CHAPTER 4 HEAD AND REFLECTED WAVES FROM AN SH LINE SOURCE IN A DIPPING LAYER OVERLYING AN ELASTIC MEDIUM 61 4.1 I n t r o d u c t i o n 61 4.2 Equation of Motion and Boundary C o n d i t i o n s 62 4.3 Steady State Plane Wave S o l u t i o n - 65 4.4 Formal Steady State S o l u t i o n f o r a Lin e Source 72 4.5 E v a l u a t i o n of the F i r s t S e r i e s Term of the I n t e g r a l - 73 4.5.1 C o n t r i b u t i o n from the Saddle Po i n t ( R e f l e c t e d Waves) 76 4.5.2 C o n t r i b u t i o n from the Branch Point (Head Waves) 78 4.6 A p e r i o d i c S o l u t i o n 80 4.7 I n t e r p r e t a t i o n of the T r a v e l Time, 82 4.8 Range of E x i s t e n c e o f Head Waves 84 4.9 D i s c o n t i n u i t i e s 88 4.10 D i s p e r s i o n Equation f o r the Lower Boundary Free and R i g i d 94 4.11 The H o r i z o n t a l Layer S o l u t i o n 96 4.12 Computation o f Displacement Seismograms 99 v i i CHAPTER 5 SUMMARY, CONCLUSIONS AND FURTHER STUDIES 105 5.1 Summary and Conclusi o n s 105 5.2 Suggestions f o r F u r t h e r Studies 109 BIBLIOGRAPHY 111 APPENDIX I ENERGY RELATIONS 113 APPENDIX II EXPRESSION OF A FREE.RAYLEIGH WAVE USING COMPLEX ANGLES 116 APPENDIX I I I EVALUATION OF THE SECOND SERIES TERMS OF THE INTEGRAL 120 v i i i LIST OF FIGURES FIGURE PAGE 2-1 C y l i n d r i c a l c o o r d i n a t e system ( T 0 2/) used i n t h i s problem. ' 10 2-2 R e f l e c t i o n and r e f r a c t i o n at a boundary i n c l i n e d at an a r b i t r a r y angle 6tk - H . 2-3 M u l t i p l e r e f l e c t i o n and r e f r a c t i o n f o r a wedge-shaped medium with a wave i n c i d e n t w i t h p r o p a g a t i o n d i r e c t i o n oL • 15 2-4 Displacement d i s c o n t i n u i t y along the edge of outgoing r e f l e c t e d wave f o r u n i t ampli- tude i n c i d e n t waves with pr o p a g a t i o n d i r e c - t i o n cL . " 22 2-5a Amplitude s u r f a c e f o r the parameters d i p angle and ( r=£/3jLH / C b i T f o r an i n c i - dent wave with p r o p a g a t i o n d i r e c t i o n d = 60°. 24 2-5b Amplitude s u r f a c e f o r the parameters d i p angle and < T ~ 2 s f 3 7 Z H / C b t T f o r an i n c i - dent wave with p r o p a g a t i o n d i r e c t i o n Oi = 120° . 25 2-6a Amplitude s u r f a c e f o r the parameters propa- g a t i o n d i r e c t i o n cL and fr=j2/?JJGH/CbiT f o r a h o r i z o n t a l boundary. 27 2-6b Amplitude s u r f a c e f o r the parameters propa- g a t i o n d i r e c t i o n oL and <S*-Zt/JJZ H / C b i T f o r a d i p angle 10°. 28 2-7 Amplitude s u r f a c e f o r the parameters d i p angle and T — J Z , J L T / C M T f o r an i n c i - dent wave with p r o p a g a t i o n d i r e c t i o n oL = 60°. 2- 8 Phase v e l o c i t y (Cu/Cbi ) curves versus CT = ̂ ^3 70H/6biT f o r a dip angle of 10° and p r o p a g a t i o n d i r e c t i o n cL . The t h i n h o r i z o n t a l l i n e s are the phase v e l o c i t i e s f o r the h o r i z o n t a l l y l a y e r e d case. 31 3- 1 R e f l e c t i o n and r e f r a c t i o n o f waves at a boundary i n c l i n e d at an a r b i t r a r y angle 8dL w i t h the nomenclature f o r angles between rays and the h o r i z o n t a l and boundary s u r f a c e s i n d i c a t e d . 37 i x 3-2 R e f l e c t i o n of waves at a f r e e s u r f a c e with nomenclature f o r angles between rays and the f r e e s u r f a c e i n d i c a t e d . 44 3-3 Flow diagram showing the computational scheme used to c a l c u l a t e the amplitudes and p r o p a g a t i o n d i r e c t i o n s of the r e f l e c t e d waves i n the wedge and thus the displacement and displacement r a t i o at any p o i n t . 49 3-4 Maximum displacement d i s c o n t i n u i t y of the r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from the e x i t i n g SV waves f o r an i n c i d e n t P wave wit h propa- g a t i o n d i r e c t i o n s oi - 60° and oi = 120°. 52 3-5 Maximum displacement d i s c o n t i n u i t y of the r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from the e x i t i n g SV waves f o r an i n c i d e n t SV wave wit h propa- g a t i o n d i r e c t i o n s = 60° and ^ = 120°. 54 3-6 H o r i z o n t a l and v e r t i c a l displacements versus the parameter (T = Z/TjZH/CaiT f o r i n c i - dent P waves w i t l i p r o p a g a t i o n d i r e c t i o n s cL = 60° and — 120 f o r the range of d i p angles 3°^6<K^30°. 55 3-7 Displacement r a t i o s V/H versus the parameter §- = zJKlLH/CatT f ° r i n c i d e n t P waves with p r o p a g a t i o n d i r e c t i o n s oC = 60° and oC « 120° f o r the range of d i p angles B°— 6̂ .= 3 0 ° 57 3-8 H o r i z o n t a l and v e r t i c a l displacements versus the parameter c? ~Zf3 7 Z \ A / C ( x \ T f o r i n c i - dent SV waves with p r o p a g a t i o n d i r e c t i o n s (S = 60° and ($,= 120° f o r the range of d i p angles $°£9<i&300. 58 3- 9 Displacement r a t i o s H/V versus the parameter 6~" -2,J~3K,H/Ca.iT f o r i n c i d e n t SV waves with p r o p a g a t i o n d i r e c t i o n s (3 = 60° and .|S = 120 y f o r the range of d i p angles S%d^k30°. 60 4- 1 Geometry of the problem: the l i n e source ( S ) i s l o c a t e d at (d, 0 ) and the r e c e i v e r ( R. ) at ( T , 0 ) i n the wedge bounded by the f r e e s u r f a c e ( Q=—S\ ) and the boundary (0 = 0ji, ) between the two media. 64 X 4-2 The oii-plane (0Cr = X + l ^ ) on which Re(A-s)7>0 and the regions of p o s i t i v e and negative I m ( A s ) , separated by the curves bg and L̂ g , indicated. Nota- t i o n : 5 - saddle-point; B , B' - branch points; L - o r i g i n a l path of integration; Lis - path of steepest descent through the saddle point; LM , Viz. - paths of branch l i n e i n t e g r a l ; U8 - branch cut Re(As)=0; and bg - curve along which lm( / l s ) = 0 • ^5 4-3 Basic ray paths used in physical i n t e r - pretation of contributions from branch and saddle points. 83 4-4 Ray paths of the head and r e f l e c t e d waves expressed by the f i r s t series term of the i n t e g r a l s . 85 4-5 Maximum value of Qx for which the head waves shown in Figure 4-4 ex i s t versus the r a t i o of source to observation distances. The observation and source points at 5° from the free surface. 87 4-6 Maximum value of the wedge angle ( 6 i + Qz ) for which the head waves of the types shown in F i g . 4-4 exist for an observation point at 5° from the free surface and d/T=lO.O. 89 4-7 D i s c o n t i n u i t i e s in medium (1) due to in t e r - action of the wave with the vertex. The lined areas indicate the regions for which the geo- metric wave from thelast r e f l e c t i o n exists with the term from which i t arises indicated in brackets. 90 4-8 Relative amplitudes of the displacement d i s - c o n t i n u i t i e s due to a plane i n i t i a l wave close to the x-axis for propagation upward ( 771 = -) and downward ( 771 = +) . 92 4-9 Coordinate system for the horizontal layer case with the source ( S ) at (dl,0 ) and , the receiver ( R. ) at . 97 4-10 Three cases for which th e o r e t i c a l seismograms were calculated. The parameters used were: Hi = 9.59 km, H^= 3.00 km, 0 - 99.6 km, ck - 10.0 km, and the displacement parameter C = 0.05 sec. 100 x i 4-11 Ray paths which c o n t r i b u t e to.the theo- r e t i c a l seismograms. 101 4-12 Displacements of the component waves f o r the geometries g i v e n i n F i g u r e s 4-10a, 4-10b and 4-10c. 102 4-13 S y n t h e s i z e d seismograms r e s u l t i n g from the displacements of F i g u r e 4-12. 104 A - l C o o r d i n a t e system used to c a l c u l a t e the complex angle of a f r e e R a y l e i g h wave. 117 A-2 The c^-i-plane showing branch cuts and i n t e g r a l paths f o r e v a l u a t i o n of the second s e r i e s term of the i n t e g r a l s . N o t a t i o n : B , C - branch p o i n t s ; £ - saddle p o i n t ; L - o r i g i n a l path of i n t e g r a t i o n ; L s - path of s t e e p e s t descent through saddle p o i n t ; and L L (I— 1 , Z • - -) - paths of branch l i n e i n t e g r a l . 123 A-3 Ray paths of the head waves expressed by the second s e r i e s term of the i n t e g r a l s with the f o u r combinations of 77L ( + , — ) and £ ( \ , Z ) c o r r e s p o n d i n g to the f o u r second s e r i e s terms of the i n t e g r a l in (A-3.1). 130 LIST OF TABLES TABLE 1 N o t a t i o n used i n F i g u r e 3-3. PAGE 50 x i i ACKNOWLEDGEMENTS I wish to express my s i n c e r e thanks to Dr. R. M. E l l i s f o r h i s guidance and encouragement and f o r many hours o f d i s c u s s i o n d u r i n g the course of the e n t i r e i n v e s t i g a t i o n . Thanks are due to Drs. R. D. R u s s e l l and D. E. Smylie f o r re a d i n g the manuscript and Dr. G. K. C. Cla r k e f o r h i s comments on Chapter 4. I a p p r e c i a t e the constant i n t e r e s t and encouragement of Dr. R u s s e l l d u r i n g my s t u d i e s at the U n i v e r s i t y o f B r i t i s h Columbia. H e l p f u l d i s c u s s i o n s are acknowledged with my c o l l e a g u e Mr. 0. G. Jensen, who a l s o p r o v i d e d me with h i s program f o r plane waves i n c i d e n t on a h o r i z o n t a l l y l a y e r e d system. I would l i k e to express my a p p r e c i a t i o n to P r o f e s s o r Akio T a k a g i , C h i e f , A k i t a Observatory, Tohoku U n i v e r s i t y f o r g r a n t i n g e d u c a t i o n a l leave and to P r o f e s s o r Z i r o Suzuki who suggested s t u d i e s at the U n i v e r s i t y of B r i t i s h Columbia. T h i s manuscript was typed by Miss J u d i Kalmakoff. T h i s study was supported by the N a t i o n a l Research C o u n c i l (Grant A-2617) to Dr. R. M. E l l i s and the Defence Research Board of Canada (Grant 9511-76) to Drs. R. M. E l l i s and R. D. R u s s e l l . A U n i v e r s i t y of B r i t i s h Columbia Graduate F e l l o w s h i p d u r i n g the second year of t h i s study i s g r a t e f u l l y acknowledged. CHAPTER 1 GENERAL INTRODUCTION 1.1 P r e l i m i n a r y Remarks E l a s t i c waves p l a y very important r o l e s i n the d e t e r m i n a t i o n o f ' t h e c r u s t a l s t r u c t u r e and the i n t e r n a l c o n s t i t u t i o n o f the e a r t h . In the past decade new ana- l y s i s techniques coupled with advances i n i n s t r u m e n t a t i o n have l e a d to r a p i d expansion of our knowledge concerning the s e i s m i c p r o p e r t i e s of the e a r t h . However, the ana- l y s e s are r e s t r i c t e d by the l i m i t e d number of models which are a v a i l a b l e - mainly f o r h o r i z o n t a l l y l a y e r e d s t r u c t u r e s . In earthquake seismology, s u r f a c e waves have been p a r t i - c u l a r l y u s e f u l f o r i n t e r p r e t a t i o n as they y i e l d an average s t r u c t u r e over the p r o p a g a t i o n path and hence the h o r i - z o n t a l l y l a y e r e d f o r m u l a t i o n has proved to be adequate i n most cases. However, body wave a p p l i c a t i o n s (e.g., Phinney (1964), E l l i s and Basham (1968), Ibrahim (1969)) have only been moderately s u c c e s s f u l as body wave amplitudes depend on a l o c a l i z e d r e g i o n beneath the o b s e r v a t i o n p o i n t which may be g e o l o g i c a l l y complex as i n d i c a t e d by the r e f l e c t i o n s t u d i e s of Clowes et a l (1968) . Hence i t i s necessary and important to i n v e s t i g a t e the behavior of waves i n a d i p p i n g l a y e r to o b t a i n an understanding of the more complex models. 2 1.2 Summary of Previous S t u d i e s The i n t e r p r e t a t i o n of h o r i z o n t a l l y l a y e r e d s t r u c - t u r e s has been dominated by the t h e o r e t i c a l s t u d i e s of H a s k e l l (1953, 1960, 1962). He c o n s i d e r e d an input wave at the base of a h o r i z o n t a l l y l a y e r e d system and by apply- ing the boundary c o n d i t i o n s o b t a i n s propagator m a t r i c e s which c a r r y the displacements and s t r e s s e s from one boundary to the next e v e n t u a l l y o b t a i n i n g a r e l a t i o n between the i n p u t wave at the lower boundary and the s u r f a c e motion. For i n c i d e n t P and SV waves, the frequency domain input f u n c t i o n may be e l i m i n a t e d by t a k i n g the r a t i o of the v e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t s . For i n c i d e n t P, the e x p e r i m e n t a l V/H r a t i o versus frequency, and f o r i n c i - dent SV, the experimental H/V r a t i o , can then be compared with t h e o r e t i c a l H a s k e l l r a t i o s to determine c r u s t a l s t r u c - t u r e . H a s k e l l ' s f o r m u l a t i o n i s a l s o a p p l i c a b l e to s u r f a c e wave s t u d i e s . S e v e r a l s t u d i e s f o r n o n - p a r a l l e l boundaries have been done, mainly r e l a t i n g to s u r f a c e waves. Hudson (1963), Nagumo (1961) and Sato (1963) d e a l t with SH waves i n a wedge-shaped medium. Hudson s t u d i e d SH waves from a l i n e source i n a wedge-shaped medium with a r i g i d lower s u r f a c e . He o b t a i n e d a s o l u t i o n composed of m u l t i p l y r e f l e c t e d 3 and d i f f r a c t e d waves. Using t h i s s o l u t i o n he i n v e s t i - gated the e f f e c t of d i f f r a c t i o n at the apex of the wedge by means of an approximate form of the d i f f r a c t e d p u l s e and found that the d i f f r a c t e d wave amplitude decreases as ( w n e r e T and T, are the d i s t a n c e s of the source and o b s e r v a t i o n p o i n t , from the v e r t e x ) . Nagumo co n s i d e r e d two dimensional e l a s t i c wave p r o p a g a t i o n i n a l i q u i d l a y e r o v e r l y i n g a r i g i d bottom. He found that mode s o l u t i o n s e x i s t . From the s o l u t i o n he i n v e s t i g a t e d d i s p e r s i o n r e l a t i o n s o f the wave. Sato s t u d i e d the d i f - f r a c t i o n problem of SIT waves at an obtuse-angled corner due to i n c i d e n t plane SH p u l s e p a r a l l e l to one of the f r e e boundaries and c a l c u l a t e d d i f f r a c t e d wave forms which he found d i m i n i s h e d r a p i d l y away from the v e r t e x . Lapwood (1961) , Kane and Spence (1963) , Hudson and Knopoff (1964) , McGarr and Alsop (1967) and others have s t u d i e d R a y l e i g h wave t r a n s m i s s i o n i n a wedge-shaped medium. Lapwood i n v e s t i g a t e d wave forms from a l i n e p u l s e source on one of the f r e e boundaries of a r i g h t angle, u s i n g i n t e g r a l t r a n s f o r m a t i o n and approximation procedures. Kane and Spence (1963) and Hudson and Knopoff (1964) c o n s i d e r e d R a y l e i g h wave t r a n s m i s s i o n on e l a s t i c wedges with f r e e boundaries. The f i r s t authors employed an i t e r a - t i o n procedure and the l a t t e r employed a Green's f u n c t i o n technique i n order to c a l c u l a t e t r a n s m i s s i o n c o e f f i c i e n t s 4 of the Rayleigh wave. Using an approximate v a r i a t i o n a l method, McGarr and Alsop (1967) computed the r e f l e c t i o n and transmission c o e f f i c i e n t s for Rayleigh waves normally i n c i - dent on v e r t i c a l d i s c o n t i n u i t i e s . Conversely, there have only been a few studies (Fuchs (1966)) and Kane (1966)) on the e f f e c t of non-parallel boundaries for body waves; nevertheless body waves constitute an i n i t i a l section of a seismogram which i s very often used i n analyses. Fuchs synthesized seismograms due to a primary P s i g n a l propa- gating along the median plane i n a s o l i d wedge with free boundaries, by taking a summation of r e f l e c t e d waves. He determined the dispersion of the body waves and p a r t i c l e motion. Kane employed a tree diagram which i s obtained by r e f l e c t i n g the wedge rather than the rays and a vector which c a r r i e s nine pieces of data. Thus, he calculated t h e o r e t i c a l seismograms due to an input plane P pulse for the teleseismic response of an array of stations located on a uniformly dipping crust. In t h i s way he demonstrated the s i g n a l d i s t o r t i o n e f f e c t s of the geometry. However the amplitude c h a r a c t e r i s t i c s which are used for i n t e r p r e t a t i o n of c r u s t a l structure were not investigated nor was the d i f f r a c t e d wave. 1.2 Scope of This Thesis The objective of t h i s thesis i s to extend the theory of body wave propagation i n a dipping structure using a re- f l e c t e d wave formulation. Although the forms of the d i f - fracted waves are not investigated, determination of the amplitude d i s c o n t i n u i t i e s due to the r e f l e c t e d wave within the wedge indicates i t s importance. 5 F i r s t , i n Chapter 2, a plane S H wave i n c i d e n t at the base of a d i p p i n g l a y e r i s c o n s i d e r e d as i n t h i s case no converted waves are p r e s e n t . A s o l u t i o n by m u l t i p l e r e f l e c t i o n i s ob t a i n e d and the amplitude charac- t e r i s t i c s and phase v e l o c i t y c a l c u l a t e d on the s u r f a c e i n terms of depth to the i n t e r f a c e , p e r i o d of the wave, and d i p angle. The d i s c o n t i n u i t i e s which r e s u l t from the l a s t r e f l e c t i o n and which are r e l a t e d to d i f f r a c t e d waves are determined. T h i s development i n Chapter. 2 serves as a guide f o r s o l v i n g the more d i f f i c u l t problems of Chapters 3 and 4. In Chapter 3, the co r r e s p o n d i n g problem i s s t u d i e d f o r i n c i d e n t P and S V waves. The complexity does not a l l o w a s e r i e s s o l u t i o n to be obtained; however, a com- p u t a t i o n a l scheme i s developed which allows the c a l c u l a - t i o n o f the displacement and phase v e l o c i t i e s . As sub- s i d i a r y problems , the energy r e l a t i o n s between waves at a boundary are gi v e n i n terms of the pr o p a g a t i o n d i r e c t i o n and the complex p r o p a g a t i o n d i r e c t i o n i n t e r p r e t e d . In Chapter 4, p r o p a g a t i o n of S H waves from a p e r i o d i c and i m p u l s i v e l i n e source i n a d i p p i n g l a y e r o v e r l y i n g an e l a s t i c medium i s i n v e s t i g a t e d u s i n g a r e f l e c t e d wave f o r m u l a t i o n . The c o n t r i b u t i o n s due to head and r e f l e c t e d waves are determined by e v a l u a t i n g the i n t e g r a l s by the method of s t e e p e s t descent and a 6 comparison made with a h o r i z o n t a l l y layered case through the case of numerical examples. The range of existence of head waves i s determined and the d i s c o n t i n u i t i e s associated with d i f f r a c t e d waves studied. The study i s summarized and suggestions made for further investigations i n the f i n a l chapter. The theory for multiply r e f l e c t e d waves as developed i n t h i s thesis could serve as a useful s t a r t i n g point for the study of d i f f r a c t i o n . Techniques such as the geometrical theory of d i f f r a c t i o n as developed by K e l l e r (1962) appear to be applicable; however, they may not be p r a c t i c a l due to the complexity introduced. In th i s theory for small wavelengths, K e l l e r uses d i f f r a c t i o n laws s i m i l a r to laws of r e f l e c t i o n and r e f r a c t i o n which are derived from Fermat's p r i n c i p l e . Away from the d i f f r a c t i n g surfaces, he i s able to use d i f - fracted rays just l i k e ordinary rays. By the use of the r e f l e c t e d wave so l u t i o n and such a d i f f r a c t e d wave procedure, i t may be possible to obtain a more s a t i s f a c t o r y d e s c r i p t i o n of e l a s t i c waves i n a wedge. 7 CHAPTER 2 MULTIPLE REFLECTION OF PLANE SH WAVES BY A DIPPING LAYER 2.1 I n t r o d u c t i o n The c a l c u l a t i o n of the amplitude c h a r a c t e r i s t i c s of waves propagating i n h o r i z o n t a l l y l a y e r e d media has been g r e a t l y s i m p l i f i e d by the matrix f o r m u l a t i o n of H a s k e l l (1953, 1960, 1962). The a p p l i c a t i o n of t h i s f o r m u l a t i o n has proved to be a powerful method f o r d e t e r - mining the c r u s t and upper mantle s t r u c t u r e u s i n g s u r f a c e waves. However, body wave a p p l i c a t i o n s have only been moderately s u c c e s s f u l . Even though f o r s u r f a c e waves the r e g i o n a l s t r u c t u r e may conform c l o s e l y enough to the l a y e r e d theory to a l l o w a s u c c e s s f u l i n t e r p r e t a t i o n , the body wave amplitudes may not be u s e f u l f o r i n t e r p r e t a t i o n as they depend onl y on a l o c a l i z e d area beneath the s t a t i o n which may be g e o l o g i c a l l y complex. I t i s , t h e r e f o r e , important to study the e f f e c t of d i p p i n g boundaries on the charac- t e r i s t i c s observed at the s u r f a c e . Fernandez and Careaga (1968) have suggested that a model of t h i s type may be r e q u i r e d to e x p l a i n body wave o b s e r v a t i o n s at La Paz. As an i n i t i a l study of body waves i n t e r a c t i n g with a wedge o v e r l y i n g an e l a s t i c medium, a l l waves i n t e r n a l l y r e f l e c t e d between the f r e e s u r f a c e and the d i p p i n g l a y e r due to a plane SH wave i n c i d e n t on the wedge p e r p e n d i c u l a r to the d i r e c t i o n o f s t r i k e w i l l be c o n s i d e r e d . The o b j e c t i v e 8 i s to c a l c u l a t e the amplitude c h a r a c t e r i s t i c s i n terms of d i s t a n c e from the v e r t e x , depth from the s u r f a c e , and the p e r i o d of the wave. On the b a s i s of the r e s u l t s of pre- v i o u s workers, i t i s expected that the m u l t i p l y r e f l e c t e d waves w i l l p l a y the most important r o l e i n a seismogram at o b s e r v a t i o n p o i n t s d i s t a n t from the v e r t e x and w i l l be e x p l i c i t l y i n v e s t i g a t e d i n t h i s study. For the d i f f r a c t e d wave, the boundary c o n d i t i o n s are expressed and c a l c u l a - t i o n s made to i n d i c a t e i t s importance i n p a r t i c u l a r s i t u a - t i o n s . T h i s simple case i n which there i s no c o u p l i n g between wave types serves as a guide f o r s o l v i n g the more d i f f i c u l t problems of i n c i d e n t P and SV waves as w e l l as being o f i n t e r e s t i n i t s own r i g h t . F u r t h e r , as s u r f a c e wave, r e f r a c t e d wave and r e f l e c t e d wave components are obtained by e v a l u a t i n g the c o n t r i b u t i o n of p o l e s , branch p o i n t s and saddle p o i n t s r e s p e c t i v e l y i n terms of m u l t i p l e r e f l e c t i o n , the s o l u t i o n of the pr e s e n t problem i s an impor- t a n t step l e a d i n g to the s o l u t i o n of these more complex problems. 2.2 Wave Equation and Fundamental S o l u t i o n In t h i s problem wi t h a d i p p i n g boundary i t i s found convenient to choose a c y l i n d r i c a l c o o r d i n a t e system ( T , 0 H, ) r e l a t e d to a c a r t e s i a n system ( % } -"fr ^ 2L ) as 9 shown i n F i g u r e 2 - 1 . For a plane SH wave propagating i n the x-y p l a n e , the motion i s independent of Z and the displacement has o n l y a z-component. Assuming a time v a r i a t i o n of the form > the equation of motion ~ct~9±F  ( 2 , 1 ) becomes i n c y l i n d r i c a l c o o r d i n a t e s fr* + ^ f r + ^ f e r + K ) u = " 0 ( 2 . 2 ) where £ b = o ; / C b We choose as the fundamental s o l u t i o n of t h i s e quation A • (3 which i s a plane wave of ampli- tude A p r o p a g a t i n g i n the cL s d i r e c t i o n . The only non-zero component of s t r e s s i s 2.3 R e f l e c t i o n and R e f r a c t i o n C o e f f i c i e n t s We now c o n s i d e r two e l a s t i c media d i v i d e d by 0= 0^ w i t h waves from medium ( 2 ) i n c i d e n t on the i n t e r f a c e ( F i g u r e 2 - 2 ) . The s o l u t i o n s i n media ( 1 ) and ( 2 ) can be w r i t t e n as   12 . L - f e b , r c o s ( 0 - | s ; itbzrcos(e-oL). L%irco$(e-r) .IXz~ AL - o -4- Afji'S The boundary c o n d i t i o n s at the i n t e r f a c e r e q u i r e c o n t i n u i t y i n displacement and s t r e s s . At O — Bd. wo can th e r e - f o r e w r i t e (2.5) ffao)|~( Pae) z The c o n d i t i o n of e q u a l i t y o f phase at G — OeL leads to £ b l c o s ( e d - p ) = ^ c o s ( e ^ - ^ ) = ^ c o S ( e a - T ) ( 2 . 6 ) Using (2.3) and s u b s t i t u t i n g (2.4) i n t o (2.5), we o b t a i n A r j i . = S S i n(6di- op - A S (n(9^- g> AL A S i n ( 0 d - {$)- S s i n ( O d - T ) Arf-- , 5 s i n ( G d - ^ ) - g s i n ( e d i - r ) A i A s i n ( ' 0 A - p ) - S S / n ( 0 d - T ) where & = C^Z/CM and % — //(z//M-\ Using (2.6) and the geometric r e l a t i o n s h i p s between the angles cL, $ t T a n d t h e angles ; [,p ; l T ( F i g u r e 2-2), we have (2.7) 13 7 0 (2.8) (2.9) is = ed + -^- - s i n ' ^ c o s C e a - o O ) and Sin (ed-p) = .-7 i - ( i / ^ ) c o s a ( ^ - o c ) with e d .<oc< 6a -v- .̂ c F i n a l l y , s u b s t i t u t i n g (2.9) i n t o (2.7) we o b t a i n A n s I n ( 0 d - oC) + CI / S) cos a ( " ^ " o Q " AL S i n ( 0 d - o / o - ( | / g ) y A - - c o s - ( 0 d - 5 y Arf • _ • a s in (eA -oO A c S i n ( 0 d - o C ) - ( 1 / S ) ^A A - c o s ^ O c i - . o c ) Thus, we have been able to denote the r e f l e c t i o n and r e f r a c t i o n c o e f f i c i e n t s i n terms of the i n i t i a l propa- g a t i o n d i r e c t i o n , the d i p angle, and the e l a s t i c c o n s t a n t s . In the case where the waves are i n c i d e n t on the boundary from medium (1), the same process y i e l d s the f o l - lowing e q u a t i o n s : (2.10) AL A $ m ( 0 d - o O - r § / 7 ^ A * C 0 S * ( 0d.-op (2.11) A f f = ' 2&S\n(6<k~oc) A i ~ A S i n ( 0 d - o l ) i - \SJ \- A ^ c o s ^ - o O 14 and ' (2.12) P = - i r ^ + S i r r ^ A C o s c e ^ - o c ) ) w i t h 0 ^ H - T t < cL < . 0 * + £ 7 G I f A A C O S a ( 0 d - o C ) > I then J | - A a C O S * ( e A ~ o O ~ must be r e p l a c e d by - I J A^COS^CO^ — Oi) ~ T f o r t h e s o l u t i o n to remain f i n i t e at i n f i n i t y . For waves i n c i d e n t on the f r e e s u r f a c e , we have (2.13) T = Z7L - cL w i t h 0 < OL < J b 2.4 M u l t i p l e R e f l e c t i o n S o l u t i o n f o r a Wedge A ^u"^ b^TCOS ( e - oi) Consider a wave A ^ ' O i n c i d e n t on the boundary Q= 0^ from medium (2) (Fi g u r e 2-3) and assume a r e s u l t i n g r e f l e c t e d wave ^'' and r e f r a c t e d wave of the forms V , » '/ A o l t b ^ T C O S ( 0 - T , ) n « A l - A - k - e i * b ' r c o 8 ( e - ^ I M * 2-3. M u l t i p l e r e f l e c t i o n and r e f r a c t i o n f o r a wedge--. E i g . 2 Siped m e d i u m w i t k a Wave i n c i d e n t with propaga- t i o n d i r e c t i o n cC 16 Using equations (2.8) and (2.10), the boundary c o n d i t i o n s are s a t i s f i e d f o r A " _ sin(e^-oO-r ( i / S ) v / A a " ~ c o s 2 - ( 6 d - o C ) A , - s s n ( e d - ^ ) ~ ( ! / s ) / A " - c o s - ( a a " o C ) Tj = ZdcK-i-Z'JZ-oL (2.14) and /\ = 2$\n(6rj-cL) S i n ( e d - o C ) - ( i / § ) / ^ c o s ^ c e ^ - o c ) Pi = Od -T - Sin'(•S'COS(edL-oC)) (2.15) To s a t i s f y the boundary c o n d i t i o n at 9 = 0 , we assume a r e f l e c t e d wave i> b,rcos(e- r , o By eq u a t i o n (2.13), the boundary c o n d i t i o n i s s a t i s f i e d p r o v i d e d t h a t To s a t i s f y the boundary c o n d i t i o n s at 0 = 0 ^ , we must assume the r e f l e c t e d and r e f r a c t e d waves l - f e b | T C O S ( 0 - T i ) A*/A',- A L - 6 ^ / / = A l - A (• AL' S L^ B^ T G O S^ 0~ ^ 17 which s a t i s f y the boundary c o n d i t i o n s f o r •* A s i n ( 0 d - V H 8/1 - A acos*(aL- r / ) (2.16) and (2.17) u s i n g equations (2.11) and (2.12). These steps are then repeated. However, i t i s not an i n f i n i t e process f o r i t terminates whenever- ^ < V*, < X + 6JL o r IL < T ^ < JO + 0dL i n these cases the wave propagates down the wedge without f u r t h e r c o l l i s i o n with the boundaries. The l a s t term o f the s e r i e s i n medium 1 i s of the form A , ( 7 c A . ) e ^ r c ° S ( e - ^ where L - Tl m a x 18 T h i s g i v e s r i s e to a d i s c o n t i n u i t y i n the displacement, at 0 = <£> — JZ . T h e r e f o r e , the s o l u t i o n f o r the d i f - '71 f r a c t e d waves must be of a form that w i l l g ive c o n t i n u i t y of displacement and s t r e s s at 0 = (h — JG as w e l l as s a t i s f y - i n g the boundary c o n d i t i o n s at Q — 0 and 0 = 0̂  . As we s h a l l see i n the next s e c t i o n , the d i s c o n t i n u i t y at ®~ $ri~ ~"S ""n m o s t c a s e s S m a H i n d i c a t i n g t h a t the m u l t i p l e r e f l e c t i o n s o l u t i o n u s u a l l y dominates the seismo- gram. F u r t h e r , Sato (1963) has shown that the amplitude of the d i f f r a c t e d wave decreases r a p i d l y a\\Tay from the r e g i o n o f the ray theory d i s c o n t i n u i t y and hence w i l l be s m a l l at s u r f a c e p o i n t s d i s t a n t from the v e r t e x . We then w r i t e M ^ A ^ f T t A j e ^ 0 5 ^ ^ K=A,f : ( JcA w )e^ T C 0 S ( 9 _ T - ) and (2.18) N = A , i ( ^ A j e ^ i r c o s ( e ' T - } (2.19) 19 where A ~ , : _ (2.20) (2.21) \ s in (e * - o l ) - ( c o s ^ e I = ^ ft- e A + f - - s i n ^ i - c o s c e A - o c ) ) L TTt max |_j — TTt max ] (2.22) (2.23) (2.24) The s o l u t i o n can then be w r i t t e n i n one of two forms depend- ing on whether the l a s t r e f l e c t i o n i s from 0=0 or Q _= 0̂  . In the f i r s t case we have U i = -N,+ Ni ' f o r o i e 4 < - } c A / , (2.25) ' ^ i = N i + M , . ' ^or r L ~ ^ ^ e ^ e A with 20 and i n the second case we have W I T H TO < TJ, < The amplitude A, the phase (H) , and the phase v e l o c i t y C v ( i n the d i r e c t i o n 0 = constant) may be w r i t t e n as: ( 2 . 2 6 ) A - y H r C e C U O * * I m ( U ( f ( 2 . 2 7 ) ( 2 . 2 8 ) v- / R e ( u , ) - r x ; — ~ l m ( ^ i ) — x / ( 2 - 2 9 ) 2.5 Numerical Computation and D i s c u s s i o n For the numerical computations, the values chosen f o r the parameters were JJLX = ( t 8 $Z and Ct>z/Cb\-\ which correspond to the c r u s t - upper mantle model used by H a s k e l l (1960) . 2.5.1 Amplitude D i s c o n t i n u i t y at As d i s c u s s e d i n the p r e v i o u s s e c t i o n , the l a s t r e f l e c t i o n , which does not c o l l i d e w i t h a boundary, gi v e s r i s e to a displacement d i s c o n t i n u i t y and corre s p o n d i n g to 21 "tKis~a ^''-function in the stress at @ ~ 3^ • The magnitude of the displacement disc o n t i n u i t y versus dip angle i s shown i n Figure 2-4 for various angles of incidence as applicable to teleseismic waves. In the case where the magnitude i s small the r e f l e c t e d wave solution adequately describes the physical problem. However, i f the discon- t i n u i t y i s large, then a d i f f r a c t e d wave with a large ampli- tude in the region of d — '^—TC i s required to provide continuity i n displacement and st r e s s . We see that for the incident wave propagating in the up-dip d i r e c t i o n (oL<\£]00 ), the di s c o n t i n u i t y i s small for dip angles less than 15°. However, for incident waves propagating in the down-dip d i r e c t i o n (OC>^0 ) the dis- placement d i s c o n t i n u i t y i s large for some ranges of small dip angles. In these cases the d i f f r a c t e d wave i s impor- tant because the i n t e r n a l l y r e f l e c t e d wave propagates out of the wedge af t e r a small number of r e f l e c t i o n s . How- ever, for surface points distant from the vertex, i t i s expected that the r e f l e c t e d wave amplitude w i l l give a good approximation i n most regions to the true amplitude as the dis c o n t i n u i t y surface becomes distant from the free surface. It should also be pointed out that i n addition to the di s c o n t i n u i t y within the wedge, d i s c o n t i n u i t i e s are 21a generated by the vertex on r e f l e c t i o n of the incoming wave and each r e f r a c t i o n into medium (2). In the r e f l e c t e d wave theory these appear as displacement and stress d i s - c o n t i n u i t i e s radiating from the vertex. The e f f e c t of these w i l l not normally be large on the surface of the wedge except close to the vertex as the amplitude decreases rather r a p i d l y with distance and the wave w i l l be p a r t i a l l y r e f l e c t e d at the lower boundary of the wedge. However i n any study of d i f f r a c t e d waves t h e i r r e l a t i v e impor- tance should be investigated. 2.5.2 Surface Amplitude Ch a r a c t e r i s t i c s One e f f e c t of in t e r e s t i s the e f f e c t of a va r i a - t i o n of dip angle on the amplitude c h a r a c t e r i s t i c s at the surface for a constant depth to the boundary and fixed 22 O o CO Q L L J Q DIP ANGLE (DEGREES) Fig» 2-4 0 Displacement discontinuity along the edge of out-going reflected wave for unit amplitude incident waves with propagation direction cLo 23 p r o p a g a t i o n d i r e c t i o n . In F i g u r e s 2-Sa and 2-5b the ampli- tude s u r f a c e s are p l o t t e d f o r the parameters d i p angle and JO H/C b t T ( H i s depth to the boundary) f o r oL = 60° and 120°. For f i x e d H and Cbl these are then amplitude s u r f a c e s f o r v a r y i n g d i p angle and p e r i o d . For an i n c i d e n t wave with p r o p a g a t i o n d i r e c t i o n o(, = 60°, we see that the amplitude c h a r a c t e r i s t i c s change s l o w l y with i n c r e a s i n g d i p angle except i n the range near 25°. However, f o r an i n c i d e n t wave with cL = 120° the amplitude c h a r a c t e r i s t i c s change r a t h e r r a p i d l y with f r e - quency over the range of d i p angles c o n s i d e r e d . The curves 2-5a and 2-5b are t y p i c a l of the c h a r a c t e r i s t i c s f o r i n c i - dent waves pro p a g a t i n g i n the up-dip and down-dip d i r e c t i o n r e s p e c t i v e l y . One of the reasons f o r t h i s i s i n d i c a t e d i n F i g u r e 2-4. For a p r o p a g a t i o n d i r e c t i o n cL = 60°, the amplitude d i s c o n t i n u i t y i s small at 0 = ^ — 7 0 f o r Q^<^Z\° but f o r a p r o p a g a t i o n d i r e c t i o n oL = 120°, the d i s c o n t i n u i t y i s s i g n i f i c a n t over most o f the range of d i p angles c o n s i d e r e d . T h i s i n d i c a t e s t h at i n these r e g i o n s of r a p i d changes of the amplitude c h a r a c t e r i s t i c s , the m u l t i p l e r e f l e c t i o n s o l u t i o n as presented here does not f u l l y d e s c r i b e the p h y s i c a l s i t u a t i o n but t h a t the d i f - f r a c t e d wave i s l i k e l y to p l a y an important r o l e . From a p h y s i c a l v i e w p o i n t , t h i s l a r g e r d i f f r a c t e d wave f o r waves pr o p a g a t i n g i n the down-dip d i r e c t i o n a r i s e s as the wave 24 PROPAGATEON DIRECTION » SO° 2 JlwH Cb, T 2-5a a Amplitude surface f o r the parameters dip angle and =JZ/3"7GH/CbiT f ° r a n i n c i d e n t wave with propagation d i r e c t i o n oC=60°<> 25 o ro BamndiAiv P i g . 2-5b. Amplitude surface f o r the parameters dip angle and 6^=£/3 70H/CbiT f o r an i n c i d e n t wave with propagation d i r e c t i o n oi=[ZQ° • 26 on l y c o l l i d e s with the boundary a very few times before i t propagates out of the wedge and hence t h i s wave, which leads to the d i f f r a c t e d wave, i s s t i l l o f s i g n i f i c a n t ampli- tude. Hence we see t h a t except f o r l a r g e dip angles and waves wit h l a r g e i n c i d e n t angles p r o p a g a t i n g i n the down- d i p d i r e c t i o n the r e f l e c t e d wave s o l u t i o n adequately des- c r i b e s the p h y s i c a l problem. A l s o o f i n t e r e s t are the amplitude s u r f a c e s f o r the parameters cT~ and p r o p a g a t i o n d i r e c t i o n f o r a constant d i p angle. These are shown i n F i g u r e s 2-6a and 2-6b f o r d i p angles o f 0° and 10°. These two graphs are s i m i l a r i n the range of p r o p a g a t i o n d i r e c t i o n 45° to 90° i n that f o r i n - c r e a s i n g p r o p a g a t i o n d i r e c t i o n t h i s amplitude versus curves o s c i l l a t e more r a p i d l y . I t should be noted t h a t the curve f o r a d i p angle of 10° and pr o p a g a t i o n d i r e c t i o n oL = 45° has one a d d i t i o n a l o s c i l l a t i o n between = 0 and cT" = 49 as compared to the curve with no d i p . For p r o p a g a t i o n d i r e c t i o n s g r e a t e r than 110° the amplitude curves change r a p i d l y w i t h i n c r e a s i n g angle. Again i t should be noted from F i g u r e 2-4 t h a t we expect the d i f f r a c t e d wave to p l a y a s i g n i f i c a n t r o l e f o r t h i s range of propaga- t i o n d i r e c t i o n s . F i n a l l y , the amplitude s u r f a c e s f o r the parameters -r; = ^JSL£- and d i p angle ( F i g u r e 2-7) are c o n s i d e r e d . The DIP ANGLE = 0 ° P i g 0 2-6&0 Amplitude surface for the parameters propagation direction oL and 6"=Zj3 7tH/C\,\T f o r a horizontal "boundary. DIP ANGLE • 10° 4 - i UJ Q • 2 0. < 135 / / / / 45 60 0 7 14 21 28 35 42 49 2 7 3 T T H C b , T ... _ P i g . 2-6b. Amplitude s u r f a c e f o r the parameters p r o p a g a t i o n d i r e c t i o n cL and (T = £/3~X.H/GbiT f o r a d i p angle 0^=10°. PROPAGATION DIRECTION - 6 0 ° 14 21 28 35 42 49 56 63 70 77 2 IT r Pig* 2-7. Amplitude surface f o r the parameters dip angle and T=2JDr/CbiT an inc i d e n t wave with propagation d i r e c t i o n ©1=60° * 30 major f e a t u r e of these curves i s the i n c r e a s e i n the r a t e of o s c i l l a t i o n f o r the amplitude versus . T curves as the dip angle i n c r e a s e s . Hence f o r a constant d i s t a n c e from the ve r t e x the s p e c t r a l c h a r a c t e r o f a seismogram i s expected to change r a p i d l y with, a changing dip angle. 2.5.3 Phase V e l o c i t y at the Free Surface In F i g u r e 2-8, the phase v e l o c i t y curves are p l o t t e d f o r a d i p angle of 10° f o r v a r i o u s p r o p a g a t i o n d i r e c t i o n s . These d i f f e r markedly from the phase v e l o c i t y f o r the h o r i z o n t a l l y l a y e r e d case ( t h i n l i n e s ) as d i s p e r s i o n i s pr e s e n t which depends oh both the p e r i o d of the wave and depth to boundary. The amplitude of the phase v e l o c i t y o s c i l l a t i o n s i n c r e a s e s w i t h i n c r e a s i n g p r o p a g a t i o n angle u n t i l an angle i s reached which corresponds to v e r t i c a l i n c i d e n c e f o r the h o r i z o n t a l l y l a y e r e d case. Beyond t h i s angle the o s c i l l a t i o n s then decrease. C l e a r l y , measure- ments of phase v e l o c i t y on a wedge-shaped medium w i l l d e v i a t e markedly from the h o r i z o n t a l l y l a y e r e d case due to v a r i a t i o n s with both p e r i o d of the wave and depth to the i n t e r f a c e . 30a An e x p l a n a t i o n of the l a r g e v a r i a t i o n s i n the phase v e l o c i t y which occur f o r i n c i d e n t waves i n the up- d i p d i r e c t i o n i s as f o l l o w s . In t h i s case waves propa- gate toward the v e r t e x and then have t h e i r d i r e c t i o n r e v e r s e d and propagate out of the wedge. For s m a l l i n c i - dent angles ( oC = 75°) , the amplitudes upon r e v e r s a l of d i r e c t i o n w i l l s t i l l be l a r g e r e s u l t i n g i n a s i g n i f i c a n t c o n t r i b u t i o n to the phase v e l o c i t y w i t h l i t t l e change i n the t r a n s f e r f u n c t i o n compared to the h o r i z o n t a l l y l a y e r e d s i t u a t i o n . 31 A1I0013A 3SVHd Pigo 2-8o Phase v e l o c i t y GV/C-bi curves versus 6"=£>/3 70H/CbiT f o r a dip angle of 10° and propagation d i r e c t i o n cC o The t h i n h o r i z o n t a l l i n e s are the phase v e l o c i t i e s f o r the h o r i z o n t a l l y layered case* 32 CHAPTER 3 MULTIPLE REFLECTION OF PLANE P AND SV WAVES BY A DIPPING LAYER 3.1 I n t r o d u c t i o n , In the p r e v i o u s chapter the problem of a plane SH waves i n c i d e n t at the base of a d i p p i n g l a y e r was con- s i d e r e d . A s o l u t i o n by m u l t i p l e r e f l e c t i o n f o r waves i n c i d e n t at any angle p e r p e n d i c u l a r to s t r i k e has been o b t a i n e d f o r the amplitude, phase and phase v e l o c i t y . In t h i s chapter the case of i n c i d e n t P and SV waves i s c o n s i d e r e d . Experimental i n v e s t i g a t o r s who use d i s p l a c e - ment c h a r a c t e r i s t i c s to i n t e r p r e t c r u s t a l s t r u c t u r e (e.g., Phinney, 1964) have been l i m i t e d i n t h e i r c a l c u l a t i o n s to h o r i z o n t a l l y l a y e r e d s t r u c t u r e s . Consequently, the present a n a l y s i s w i l l expand the number of models a v a i l a b l e f o r i n t e r p r e t a t i o n purposes. 3•2 Equations of Motion and Boundary C o n d i t i o n s In t h i s problem, i t i s again convenient to choose a c y l i n d r i c a l c o - o r d i n a t e system ( T Q ) % ) r e l a t e d to a c a r t e s i a n system ( % , ̂ , % ) as shown i n F i g u r e 2-1. For plane P and SV waves propagating i n the x-y plane, the motion i s independent of & and the displacement has only T and 0 components. The equations of motion i n c y l i n d r i - c a l c o o r d i n a t e s a r e : 33 ^ at * ~ U + W a r T ae - . C 3" 1 } where: 2CO a = -™(T.lAe) 7p " 9 0 ^ (3.4) and i s d e n s i t y ; and y/. , Lame's c o n s t a n t s ; r\Xr and , displacements i n the T and 9 d i r e c t i o n s . The s t r e s s components are expressed by = » ( M i _ Jig. , J _ 9 U ^ Using equations (3.3) and (3.4) i n the equations of motion, we o b t a i n r 9 * ® • .-a i r / . V 3 * 0 i ' ^ I 1 • a - ^ \ „ 7 1 at* a r 1 T r f ae* ' 1 J 34 and i n the s t r e s s r e l a t i o n s , we o b t a i n 9 T (3.9) (3.10) where" J - ^ L and Ci , the P and S wave v e l o c i t i e s r e s p e c t i v e l y . Assuming a time v a r i a t i o n ~ c *() of the form ^ , equations (3.1), (3.2), (3.7) and (3.8) become 1 a® u T where ^ .2. b ^& CO a. Q CT. Gb and V (3.11) (3.12) (3.13) (3.14) 9 + I 2 a r 2 1 r a r ' r^sd* S u b s t i t u t i n g (3.11) and (3.12) i n t o (3.9) and (3.10), we have (3.16) 35 We choose as the fundamental s o l u t i o n of equations (3.13) and (3.14) , a n d F ' 6 plane waves pr o p a g a t i n g i n the cyC and |3 d i r e c t i o n s r e s p e c t i v e l y . S u b s t i t u t i n g the fundamental s o l u t i o n s i n t o (3.11), (3.12), (3.15) and (3.16) we have the f o l l o w i n g e x p r e s s i o n s f o r displacements and s t r e s s e s : 2i • , i'kbTcosfe-6) Z L Fs i r i ( e-0)e P (3.i7) e- s'\n(_e-oL)Ql . Kb (3.18) - ^ c : F c o s ( e - p ) s m ( e - p ) e i * b r c o s ( 0 ' p ) fe = -z f c%{ E cos(e-OL)S i n ( e e ^ T C 0 S ( 0 - + ( l ~ z c o s ^ e - p ) ) F e l l b r c o S ( e ^ H (3.20) 36 3.3 R e f l e c t i o n and R e f r a c t i o n C o e f f i c i e n t s In t h i s s e c t i o n the r e f l e c t i o n and r e f r a c t i o n c o e f f i c i e n t s i n terms of the i n i t i a l p r o p a g a t i o n d i r e c t i o n and the e l a s t i c constants w i l l be c o n s i d e r e d . F i r s t con- s i d e r two e l a s t i c media separated by 0= 0̂  with waves from medium (2) i n c i d e n t on the i n t e r f a c e ( F i g u r e 3-1). The s o l u t i o n s of the equations of motion, (3.13) and (3.14), i n media (1) and (2) can be w r i t t e n as lto^TCOS(e~oLrJi) (3.21) o3a,= D i n e ^ T C O S ( 9 ^ + [ ) r t e where 60 The boundary c o n d i t i o n s at are (3.22) re, = r e 37 CD CO b cT CD ti CD Reflection and refraction of waves at a boundary inclined at an arbitrary angle 0* with the nomen- clature for angles between rays and the horizontal and boundary surfaces.indicated. 38 For a s o l u t i o n o f the form (3.21), the d i s p l a c e - ments and s t r e s s e s i n medium (1) and medium (2) are from e x p r e s s i o n s (3.17) to (3.21) n - [ - L a r n c f f l ^ . L % a \ r c o s ( e - d r £ ^ - o ^ C i n C Q S C e - o O e ^ 0 0 3 0 6 - ^ - ^ p i n S i n ( e - P ) e ^ T C 0 S ( e - ^ - t D r A s i n(e- prJt)&1 W c o s C e - r<-CM -t-l^— C r x s in(e~c^rx)6 Khz. 39 - ^P .Cb l co sCe -p^ s i nCe -MB^e^ 1 ™^ "^ (3.24) + { I - * c o s * C e - M 6 ^ r C ° S ( e _ c o s C e - ^ s i n O - o o C i n e ^ 0 0 5 ^ " ^ + { i - ^ e - p i D I „ e l * w r C 0 S ( e - p ) + ( i ^ - M } ^ t , ' t t 0 5 ( ' ' . ' r t ) A p p l i c a t i o n o f the boundary c o n d i t i o n s (3.22) leads imme- d i a t e l y to the e q u a l i t y of phase which f o r i n c i d e n t P or SV waves y i e l d s r e s p e c t i v e l y i a . c o s ( e * - o i ) j = ^ f l 2 C 0 S ( e A _ ^ ) = ^cosceu-Pm) .cs-25) ft b 4 cos (e*- p) J = t a , c o s ( 0 A - i b l c os (©A - p r 5) which i s S n e l l ' s Law expressed i n a c o s i n e form. The boundary c o n d i t i o n s (3.22) then y i e l d the f o l - lowing equations 40 CD. crT S ^ CO. V- ccx. I CD • f— i I 8 5 ' 0 0 o _ o , i I 8 ^ o C C u I CO O O -6 C D 8 X CD I CD cO 8 . E Q_| CO c X CO. CD C O CO o o < cn. I CD o OQ. CQ. CO o O CD c\> ,5 CA5 ^ l CD CO cA> y I CD CO O o t 'co 9 5 co o o y o ll cS 3 o o O C N ( Q4 CAi II •Q_i CD Sr CD 41 To s o l v e equations (3.26) i n the case o f e i t h e r an i n c i d e n t P or SV wave, the angles oiRSL » $TJL ' oCr-J- a n d §r£ "lust D e determined i n terms o f the i n c i d e n t and boundary a n g l e s . For a P wave i n c i d e n t from medium ( 2 ) , the f o l l o w i n g geometric r e l a t i o n s h i p s are e v i - dent from F i g u r e 3-1. I 06= 0^ + "^7 70 - boL oCT:c = 0^ + -Jr JO — LoCr̂ . f o r 0^<^oC^ 0^4" ' l J s ^ n S (3.25) we then o b t a i n ztr^ e ^ ^ j r - 7 l t S \ n [ { c O S ( 3 A - c t ) } ^ Z e d + 27l~oC $rSL= % + X x " t 5 f n ~ 1 { t c ^ / C a 2 ) C O S ( 0 ^ - o 6 ) } _ ( (3.28) ^ r * * 5 x 7 0 ™ s i n l(u{/Ca2)CoS(0d-cC)} where the p r i n c i p a l value o f the i n v e r s e s i n e i s taken i n each case. I t i s e a s i l y v e r i f i e d t h a t (3.28) a l s o holds f o r A.-^.-25 <c (X,<^ • For i n c i d e n t SY waves, we f i n d (3.29) 42 (3.29) Hence, by s e t t i n g D i n — 0 i n (3.26) and s u b s t i t u t i o n o f (3.28), the r e f l e c t i o n and r e f r a c t i o n c o e f f i c i e n t s f o r an i n c i d e n t P wave can be determined. S i m i l a r l y , by s e t t i n g Q-ft = 0 and the use of (3.29) allows us to determine the r e f l e c t i o n and r e f r a c t i o n c o e f f i c i e n t s f o r an i n c i d e n t SV wave. For P waves i n c i d e n t on the boundary from medium (1) we determine the f o l l o w i n g e x p r e s s i o n s f o r the a n g l e s . OLTJL= dd -f \ j c - s i n '{COS (0A- oi)}^ze^-i-zjc-oi P r X ^ ^ + i r ^ - S i n ' l { ( G b , / C a , ) C O S ( 0 ^ ~ o 6 ) } ^ J (3.30) cLr$ = 6 A + - j r ^ * Sin ^{{Uz/c^COSidcK-oL)} Pr± = 6* + j r ^ + 5 m H { ( C b^/Ca | ) C O S ( 0 < A - o ( . ) } and f o r S waves ^ r x ^ ^ ^ i : ^ - - S I n , {(G a (/c b l )C0S(e^-(S)} o (3 ^ = 6* H- JG + sy n ' { ( G WC b , ) c o s f ^ - - jS;} Equations (3.26) may then be used to determine the r e f l e c - t i o n and r e f r a c t i o n c o e f f i c i e n t s f o r waves i n c i d e n t from medium (1) by the use of (3.30) and (3.31) and the f o l l o w - ing s u b s t i t u t i o n s : and Cb2~^ Obi ' s n o u l d ^ e noted t h a t Ar^- a n d 3r$- are i n t h i s case amplitudes i n medium (2) and C-rJL a n ^ DTJ)_ amplitudes i n medium (1) . F i n a l l y , we c o n s i d e r waves i n c i d e n t on the f r e e s u r f a c e 0 — 0 (Figure 3-2). The s o l u t i o n s of (3.13) and (3.14) i n medium (1) can be w r i t t e n as © , = A i n e - f - A r x 6 - i t b \ r c o s ( e - § ) ' • • r , t b i r c o s ( e - p r j L ) (3.32) F o l l o w i n g the same procedure as before we f i n d the equa- t i o n s between the amplitude c o e f f i c i e n t s to be ~~COSoLrjzS\noLrJL ^ cos (3TJL si n iV •\-ZC0S*pTJL • - 4 - v b * C o s p s f n g For P wave i n c i d e n c e the f o l l o w i n g r e l a t i o n s are seen to h o l d £ C O S I co§oLS\r\cL (3.33) and the f r e e s u r f a c e i n d i c a t e d . 4 5 COScLrsL— COSdL £0S ^ r J L — (Cb i /Ca\) COS oi S i n o d = ^ i - c o s v T od r J L= -§-J"c + 5\r\]{cosoi}-=Z7io-oL and f o r an i n c i d e n t SV wave C 0 S o ^ = ( C A i / C b » ) C 0 S ( 3 C O S p r j i = COS £ S i n p - / " P c o s 2 ^ " cCrx = x X . + 51 n" 1 { (GA>/Cb i ) c O S £ > Using (3.34) and (3.35) i n (3.33), we o b t a i n f o r i n c i d e n t P waves ^ •__ 4^bfcos^sino(,/i-i/^cos :cxl ~ (\-z v£ cos* oiT ^ V . (3.36) p ;<lcoSct S i not ( I - zVbi COS*oQ ^ r ^ c o s 2 - ^ s inoty i -• v £ c o s ^ -t (1 ~ z v b T c o s V J * i n 46 and f o r i n c i d e n t S waves A -zv^cosps\r\$(\-2coszp) rjL cosa p s i n (3 / - c o s ^ + ( i - z cosAp " ^ 1 n (3.37) P ^ O — ~ L K . In e x p r e s s i o n s (3.36) and (3.37), i t should be noted t h a t when the argument A i n a square r o o t i s n e g a t i v e , then \f~/\ must be r e p l a c e d by -~L\J~/\ f o r the s o l u - t i o n to remain f i n i t e at i n f i n i t y . As a check on the ampli- tude c o e f f i c i e n t s i n t h i s form, the energy f l u x equations have been d e r i v e d i n Appendix I. <> 3.4 Computation of Displacement i n the Case of a Dipping Layer To deteirnine the amplitude at any p o i n t i n a wedge we must sum the complex amplitudes o f a l l waves which a r r i v e . From (3.23) we see t h a t t h i s r e q u i r e s the c a l c u l a t i o n of the amplitude c o e f f i c i e n t and p r o p a g a t i o n d i r e c t i o n f o r each wave. In the process o f computation, complex angles have been employed i n order t h a t the cases f o r t o t a l r e f l e c t i o n and i n c i d e n t angles g r e a t e r than the c r i t i c a l angle are auto- m a t i c a l l y i n v o l v e d i n r e s u l t s . In Appendix I I , i n v e s t i g a - t i o n o f a Ray l e i g h wave w r i t t e n i n terms of complex angles 4 7 shows t h a t the r e a l p a r t o f the angle i n d i c a t e s the propa- g a t i o n d i r e c t i o n and the imaginary p a r t g i v e s the decrease of amplitude. As a P and S wave a r i s e from the i n i t i a l r e f r a c - t i o n and from each r e f l e c t i o n from the f r e e s u r f a c e and the boundary between media, the rays i n c r e a s e i n number 71+ I as Z where fl i s the order of r e f l e c t i o n . The r e f l e c t i o n process i s terminated whenever the p r o p a g a t i o n d i r e c t i o n i s between 70 and 70+'9^ f o r i n t h a t case the wave propagates out of the wedge. For computation pur- poses a f u r t h e r a r t i f i c i a l t e r m i n a t i o n was i n t r o d u c e d by n e g l e c t i n g a l l waves whose amplitude was l e s s than (o—{CT^ (the displacement amplitudes are normalized by the d i s p l a c e - ment amplitudes which the i n c i d e n t wave would have on the f r e e s u r f a c e i n the absence of the boundary). For computational purposes, i t i s important to note t h a t the amplitudes and p r o p a g a t i o n d i r e c t i o n f o r a wavefront i s the same at a l l p o i n t s . Hence, i f the ampli- tudes ( K ) and p r o p a g a t i o n d i r e c t i o n s ( TTl ) are determined f o r a l l waves r e v e r b e r a t i n g i n the wedge, then the t o t a l amplitude of motion at any p o i n t may e a s i l y be c a l c u l a t e d . F u r t h e r , as the Kf and rffl of r e f l e c t e d waves depend d i r e c t l y on M and TTL of the input wave, i t i s important at the i n i t i a l r e f r a c t i o n and each r e f l e c t i o n to s t o r e Kl and f o r waves which may generate f u r t h e r waves. The r a t h e r 48 complex computation scheme used, can best be understood by examination of the flow c h a r t (Figure 3-3). M and TVb f o r the r e f r a c t e d waves are f i r s t c a l c u l a t e d and s t o r e d i n v e c t o r s i n order that they l a t e r can be used to c a l - c u l a t e the amplitude and phase at any p o i n t i n the wedge. As the P wave i s to be f o l l o w e d through the wedge, the amplitude and p r o p a g a t i o n d i r e c t i o n of the S wave ( H$ and OTls ) a r e a l s o t e m p o r a r i l y s t o r e d i n a v e c t o r which w i l l l a t e r be used to i n v e s t i g a t e waves due to S \</ave c o n v e r s i o n , f̂ l and T)1 are then c a l c u l a t e d and s t o r e d f o r the ray which propagates through the system as P (at the same time s t o r i n g f^g. and TTLs i n the s u b s i d i a r y v e c t o r ) u n t i l P e i t h e r propagates out of the system or the amplitude i s l e s s than £ . The waves generated by the S wave of t h i s order and h i g h e r order waves they may generate are then examined (with a t t e n t i o n again f i r s t f o cussed on the P), then those generated by the next lowest order u n t i l f i n a l l y the r e f r a c t e d S and i t s r e s u l t i n g waves are examined. Upon completion of t h i s c a l c u l a t i o n the v e r t i c a l and h o r i z o n t a l displacement and the v e r t i c a l - h o r i z o n t a l displacement r a t i o may then be c a l c u l a t e d f o r d i f f e r e n t In these computation the values chosen f o r the parameters were C b i / C a i = 0. £ 7 8 < t , Caz/C&\~ I .^£>7 Cbz./Ga\= 0.73 II and | . I'7 £ • which c o r - respond to cr u s t - u p p e r mantle model employed by H a s k e l l (1962). No Y e s llnout parameters LN=i Determine M and m f o r : r e f r a c t e d P . and S and store i n STP , STS. Store M s i n RECS(N). Determine whether r e f l e c t i o n from f r e e surface or boundary. Determine M and m f o r r e f l e c t - ed P and 8 f o r i n c i d e n t P and store i n STP,STS. Store Ms,ms i n RECS(N). Mo Determine wether r e f l e c t i o n from f r e e surface or boundary. Determine M and m f o r r e f l e c - ed P and S f o r i n c i d e n t S (using Ms and m5 from RECS(N)) and store i n STP and STS. Store Ms,ms i n RECS(N) . Yes i) < b VO No Yes N = N- No Yes M = 0 C a l u c u l a t i o n of t o t a l d i s p l a c e - ment and v e r t i c a l - h o r i z o n t a l displacement r a t i o F i g . 3-3. Flow diagram showing the computational scheme used to c a l u c u l a t e the amplitudes and propa- g a t i o n d i r e c t i o n s of- the r e f l e c t e d waves i n the wedge and thus the displacement and d i s - placement r a t i o at any p o i n t . Notation i s given i n Table 1. I E P r i n t [End] 50 Table 1.' N o t a t i o n used i n F i g u r e 3-3. STP(K,2) - complex storage matrix f o r p amplitudes and propagation d i r e c t i o n s STS(K,2) - complex storage matrix f o r S amplitudes and propagation d i r e c t i o n s RECS(L,2) - complex matrix to temporarily, r e t a i n S amplitudes and propagation d i r e c t i o n s of S rays which may generate f u r t h e r s i g n i f i c a n t amplitudes N-1 - no. of r e f l e c t i o n s a wave has undergone M - amplitude m - complex propagation d i r e c t i o n S u b s c r i p t s p and s i n d i c a t e P and S wave types 3.5 Displacement D i s c o n t i n u i t i e s As d i s c u s s e d i n the p r e v i o u s chapter, the l a s t r e f l e c t i o n which does not c o l l i d e with boundaries gi v e s r i s e to a d i f f r a c t e d wave which i n the r e f l e c t e d wave s o l u t i o n appears as a displacement d i s c o n t i n u i t y . When the displacement d i s c o n t i n u i t y i s s m a l l , o n l y a smal l d i f f r a c t e d wave i s r e q u i r e d to p r o v i d e c o n t i n u i t y i n d i s - placement and s t r e s s and hence the r e f l e c t e d wave s o l u t i o n adequately d e s c r i b e s the p h y s i c a l problem. Large d i s c o n - t i n u i t i e s w i l l r e q u i r e l a r g e d i f f r a c t e d waves; however, at l a r g e d i s t a n c e s from the v e r t e x the s o l u t i o n i s s t i l l expected to be adequate as d i f f r a c t e d waves decrease r a p i d l y w i t h d i s t a n c e . 51 For i n c i d e n t P waves the magnitude of the maximum d i s c o n t i n u i t y of both the r a d i a l component from the l a s t P wave and the t a n g e n t i a l component from the e x i t i n g S wave i s shown i n Fig u r e 3-4 f o r i n c i d e n t waves with propa- g a t i o n d i r e c t i o n s cL - 60° and 120°. S e v e r a l p o i n t s should be noted. The d i s c o n t i n u i t y i n the case of down- dip p r o p a g a t i o n i s much l a r g e r than f o r the up-dip d i r e c - t i o n . T h i s i s expected s i n c e fewer r e v e r b e r a t i o n s occur b e f o r e the wave propagates out of the wedge. The d i s c o n - t i n u i t y from P waves i s r e l a t i v e l y l a r g e i n comparison to th a t f o r S waves and r a p i d changes i n the amplitude r e s u l t as the maximum d i s c o n t i n u i t y i s a s s o c i a t e d with d i f f e r e n t e x i t i n g waves f o r d i f f e r e n t d i p ang l e s . The p a r t i c u l a r l y r a p i d decreases observed r e s u l t when an SV wave g e n e r a t i n g an e x i t i n g r e f l e c t e d P wave reaches the c r i t i c a l a ngle. The maximum P wave d i s c o n t i n u i t y at the next c a l c u l a t e d p o i n t ( c a l c u l a t i o n i n t e r v a l = 0.25°) i s then due to another wave which may be of much lower amplitude. 52 30nindlAIV 3AI1V~I__ Pig., 5-4o Maximum displacement discontinuity of the radi a l component from the exiting P waves and tangential component from the exiting S.Y waves for an incident P wave with propagation directions <x>60° and cC=12.0° » 53 a n incIdent SV wave (Figure 3-5), the, d i s - placement d i s c o n t i n u i t i e s are n e g l i g i b l e for dip angles less than 21° for an incident wave with (9= Q0° i n d i - cating that t h i s solution very c l o s e l y approximates the complete s o l u t i o n . Again the disco n t i n u i t y i s larger for the incident wave propagating i n the down-dip d i r e c t i o n . However i n the case for j3 = 120° the large d i s c o n t i n u i t y for the outgoing P wave i s for large dip angles rather than the smaller dip angles ( O^^JZO" ) as found for the incident P wave case. The physical reason for t h i s i s not c l e a r . As discussed i n Chapter 2, d i s c o n t i n u i t i e s exist i n medium (2) due to the r e f l e c t i o n of the incident wave and r e f r a c t i o n of waves back into the lower medium. Except close to the vertex, the amplitudes of the r e s u l t i n g d i f - fracted waves are expected to be small. 3.6 Surface Displacements and Displacement Ratios 3.6.1 Incident P Horizontal and v e r t i c a l displacements are plotted versus the parameters (T=^/3~JGH/CaiT * C H i s depth to the interface) and i l l u s t r a t e d i n Figure 3-6. For an 53a i n i t i a l p r o p a g a t i o n d i r e c t i o n of 60° the v e r t i c a l component changes very s l o w l y w i t h i n c r e a s i n g d i p angle. The h o r i - z o n t a l displacement changes r a t h e r more r a p i d l y ; however the major changes i n c h a r a c t e r occur f o r 0̂  JZ>0° where from F i g u r e 3-4, we see that the r o l e of the d i f f r a c t e d ray becomes important. For QC = 120°, a more r a p i d change i n c h a r a c t e r o f both the v e r t i c a l and h o r i z o n t a l s u r f a c e s 54 3Cin±ndlAIV 3AI1V13H Pig,* 3-5c Maximum displacement d i s c o n t i n u i t y of the r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from the e x i t i n g ST waves, f o r an i n c i d e n t S¥" wave with propagation d i r e c t i o n s @=60°and p=120° 0 55 in u o \~ O <_ U> ce O 1-o IU ce a z o to o cc CL o < CC < X o I-z U J s UJ o < _l CL </> a < o I- Ui > o <0 o H o LU CC Q Z o < a. o cc CL to O co cc Ui I-o < CC <. X (J U J £ UJ o < _ J 0. CO o z o N CC o X o z o t-o 00 cc UJ I-o < ce < X o a U J 5 Z g B ^ Y- CL < co O a § 3 CL H Q: > oo o . & O 5 CM £ < cc < X o 1-z U J s UJ o < _ l CL oo Q z o I-o UJ cc a z g CO s o cc CL z o N ce o X «> * M O 3anindwv Pig. 3 - 6 . Horizontal and v e r t i c a l displacements versus the parameter <y=;i/3JtH/Ca,T for incident P waves with propagation directions oC=60° and oC~120 for the range of dip angles 5 ° ^ © d ^ 3 0 ° • i s e v i d e n t with i n c r e a s i n g d i p angle. T h i s i s p a r t i c u l a r l y - t r ue i n the h o r i z o n t a l where f o r 0dL/M7° t n e s u r f a c e i s r a t h e r f e a t u r e l e s s . However i t should be noted that the d i s c o n t i n u i t y curves i n t h i s case i n d i c a t e d t hat the d i f f r a c t e d wave may be important over most of the range of d i p a n g l e s . The displacement r a t i o s V/H which are of i n t e r e s t i n p r a c t i c a l a n a l y s i s are shown i n F i g u r e 3-7. For the i n i t i a l p r o p a g a t i o n d i r e c t i o n oL - 60° , the r a t i o s are very s i m i l a r f o r d i p angles l e s s than 10°. For d i p angles g r e a t e r than 10°, the peaks move to l a r g e r values of (5s and i n c r e a s e markedly i n amplitude. For oL = 120°, the r a t i o s change much more r a p i d l y even at small dip angles w i t h the peaks moving to i n c r e a s i n g 6̂  a g a i n . However i n t h i s case the amplitude decreases u n t i l f o r 9<K~=£Q° the V/H r a t i o i s almost constant f o r v a r i a b l e 6̂ 3.6.2 I n c i d e n t SV From F i g u r e 3-8, i t i s seen that the displacement surfaces- e x h i b i t s i g n i f i c a n t c h a r a c t e r . One p a r t i c u l a r f e a t u r e i s that f o r dip angles g r e a t e r than 18°, the p e r i o d of the v a r i a t i o n of h o r i z o n t a l displacement becomes shor t and the c o r r e s p o n d i n g amplitude small f o r |3 = 60°. I t should be noted t h a t t h i s change occurs b e f o r e d i f f r a c t e d waves become INCIDENT P WAV Lb PROPAGATION DIRECTION • 60 ° PROPAGATION DIRECTION =120° - C O I T c a i T Pig. 3-7« Displacement ratios ?/Bversus the parameter 6"=A^3JcH/CaiT for incident P waves with propagation directions oC=60° and o(=1200 for the range of dip angles 5% 0 ^ 30°. 58 59 s i g n i f i c a n t . For (B = 120° the h o r i z o n t a l displacement o s c i l l a t i o n s become very small f o r d i p angles g r e a t e r than 10° w h i l e the v e r t i c a l displacement o s c i l l a t i o n s more s l o w l y decrease i n amplitude with i n c r e a s i n g d i p angle and at the same time the p e r i o d of the v a r i a t i o n lengthens. The above f e a t u r e s are most evident i n the d i s - placement r a t i o curves H/V (Figure 3-9). For even small d i p angles marked d i f f e r e n c e s from the h o r i z o n t a l l y l a y e r e d curves are e v i d e n t . At the l a r g e r d i p angles f o r propaga- t i o n d i r e c t i o n |3 = 60° we see the r a p i d o s c i l l a t i o n s due to the h o r i z o n t a l component and f o r J Q — 120 0 the r a t i o becomes f e a t u r e l e s s . INCIDENT SV WAVES P R O P A G A T I O N DIRECTION = 6 0 ° P R O P A G A T I O N DIRECTION = 1 2 0 ° 60 DIP =30° 25° 20" 10' ' a i T 5 5 0 DIP =30° 25° 20« 10' 'at T Pig. . . 3-9. Displacement r a t i o s H /V versus the parameter fr--£/3JoH/C*fT f o r i n c i d e n t ST waves with propagation d i r e c t i o n s (3 =60° o and ̂ =120° f o r the range of dip angles 5%QA^30° » 61 CHAPTER 4 HEAD AND REFLECTED WAVES FROM AN SH LINE SOURCE IN A DIPPING LAYER OVERLYING AN ELASTIC MEDIUM 4.1 I n t r o d u c t i o n A number of workers (e.g., Lapwood (1961), Hudson (1963), and Hudson and Knopoff (1964)) have i n v e s t i g a t e d the p r o p a g a t i o n of s u r f a c e and d i f f r a c t e d waves i n wedge- shaped media. Hudson (1963) p o i n t e d out t h a t the s o l u t i o n i n the case of a r i g i d lower boundary c o u l d be d i v i d e d i n t o two p a r t s - the m u l t i p l y r e f l e c t e d and d i f f r a c t e d wave s o l u t i o n s . However, the e a r l y p a r t of the seismogram con- s i s t i n g o f head and m u l t i p l y r e f l e c t e d waves which are o f t e n used i n i n t e r p r e t a t i o n has not been w e l l s t u d i e d f o r a d i p p i n g l a y e r o v e r l y i n g an e l a s t i c medium. In the present chapter the author w i l l e s t a b l i s h one method of s o l u t i o n and w i l l t h e o r e t i c a l l y i n v e s t i g a t e the problem f o r an SH l i n e source i n an e l a s t i c wedge o v e r l y i n g an e l a s t i c medium. In Chapter 2, a s o l u t i o n f o r the problem of m u l t i p l e r e f l e c t i o n of plane SH waves by a d i p p i n g l a y e r has been found. By i n t e g r a t i o n of a s o l u t i o n of t h i s type, the d i s - turbance due to a l i n e source i s sought which does not i n - clude the d i f f r a c t e d wave term. However, f o r a t r a n s i e n t i n p u t , an o b s e r v a t i o n p o i n t d i s t a n t from the v e r t e x r e c e i v e s the r e f l e c t e d and r e f r a c t e d waves e a r l i e r than the d i f f r a c t e d 62 waves which r e s u l t from c o l l i s i o n s with the v e r t e x . There- f o r e the present s o l u t i o n should apply to the composition of the i n i t i a l s e c t i o n of the seismogram. The formal s o l u t i o n i s e v a l u a t e d by the s t e e p e s t descent technique as recommended by Honda and Nakamura (1954) f o r e v a l u a t i o n of branch l i n e i n t e g r a l s and as a p p l i e d by Emura (1960) and o t h e r s . In t h i s way the wave forms and the ranges of e x i s t e n c e o f the head waves are determined f o r v a r i o u s d i p angles f o r comparison w i t h the case of a h o r i z o n t a l l a y e r . For a l l computations the f o l l o w i n g e l a s t i c parameters cor- responding to those of H a s k e l l (1960) are used i n the S wave v e l o c i t y i n the upper l a y e r Cbl = 3.64 km/sec, the v e l o c i t y r a t i o ^-Cbz/c^— 1.2-7 > a n c * the r i g i d i t y r a t i o t i o n s of p o l e s , the s u r f a c e wave problem reduces to f i n d - ing p o l e s of the f i n i t e s e r i e s e x p r e s s i o n of our s o l u t i o n . F u r t h e r , the d i s c o n t i n u i t i e s i n displacement a s s o c i a t e d w i t h the d i f f r a c t e d waves have been found and hence t h i s problem i s separated from the d e t e r m i n a t i o n of the s o l u - t i o n due to other waves. The s o l u t i o n of the prese n t prob- lem i s t h e r e f o r e an important step f o r the c o n s i d e r a t i o n of s u r f a c e and d i f f r a c t e d waves. 4.2 Equation of Motion and Boundary C o n d i t i o n s S =/<*//<i= 1.88 As s u r f a c e waves are ob t a i n e d from the c o n t r i b u - t e p r o p a g a t i o n of SH waves through a system con- 6 3 s i s t i n g of an e l a s t i c medium of r i g i d i t y jX\ > d e n s i t y f , and a d i p angle 6, •+• d& > o v e r l y i n g an e l a s t i c medium of r i g i d i t y JJLZ and d e n s i t y j \ (Figure 4-1) w i l l be c o n s i d e r e d . .The f r e e s u r f a c e i s Q — —Q\ and the boundary between the e l a s t i c media i s Q — . A car- t e s i a n system ( % } % ) i s r e l a t e d to the c y l i n d r i c a l c o o r d i n a t e s ( T '0 , E ) by the standard r e l a t i o n s h i p s %~TCO$Q » ^ = T S l n 0 ' a n d 2 , = 2 . The motion i s generated by a l i n e source ( S ) °f SH waves lo c a t e d , at ( ck. 7 0 ) i n the c y l i n d r i c a l c o o r d i n a t e system. For the above problem, the motion i s independent of E i , and the displacement has only a z-component. Then, assuming a time v a r i a t i o n of the form Q, , the e q u a t i o n of motion V U , = 7 ^ - r - T l / = l , £ (4.1) O b i <?"D becomes i n c y l i n d r i c a l c o o r d i n a t e s where C^i= sjj^lffl i s t n e v e l o c i t y of the S waves and Free s u r f a c e S ( d 5 o ) (2) Pigo 4-lo Geometry of the problems.the lin e source ( s ) i s located at (d,0) the receiver (R) at (r,0) i n the wedge bounded by the free surface (0=~6,) and the boundary (0=Sb) between the two media0 65 The only non-zero component o f s t r e s s i s The boundary c o n d i t i o n s then become (Pae),= 0 o.t 0 = ~ e l (4.4) and (kt d = Qz (4.5) To s o l v e the l i n e source problem, a plane wave s o l u t i o n s a t i s f y i n g the boundary c o n d i t i o n s w i l l f i r s t be o b t a i n e d . The l i n e source s o l u t i o n can then be obtained by i n t e g r a t i o n of t h i s s o l u t i o n with r e s p e c t to the c y l i n d r i c a l a n gle. 4.3 Steady State Plane Wave S o l u t i o n The i n i t i a l displacement due to a plane wave i s expressed i n the form - ^ b , { ( d - ^ ) C o S o i . i , + |^- |5Jno6Lj- = A.: • e 6 6 where c(.i i s the angle between the x - a x i s and. wave normal of the SH waves and may take on complex values i n the eva- l u a t i o n of the e f f e c t due to a l i n e source. For 07 0 IXQ r e p r e s e n t s waves downgoing from the x - a x i s , while f o r 0 "x. 0 7 riiQ r e p r e s e n t s waves upgoing from the x - a x i s . In those cases where the waves i n t e r a c t with the boundaries, the former c o l l i d e s with the d i p p i n g boundary f i r s t and the l a t t e r w i t h the f r e e s u r f a c e f i r s t . In Chapter 2, the r e f l e c t i o n and r e f r a c t i o n o f SH waves i n a d i p p i n g l a y e r has been i n v e s t i g a t e d i n d e t a i l . The s o l u t i o n by m u l t i p l e r e f l e c t i o n i s o b t a i n e d i n the same manner. For any o b s e r v a t i o n p o i n t i n the wedge, four e x p r e s s i o n s are r e q u i r e d to express the motion depending on whether the i n i t i a l d i r e c t i o n o f the wave i s p o s i t i v e or n e g a t i v e and whether the f i n a l r e f l e c t i o n i s from the boundary (1) or the f r e e s u r f a c e ( 2 ) . Using the same pro- cedure as the d e r i v a t i o n o f (2.18) and (2.19), these expres- s i o n s are ^ ( N ) = A ^ l S N e (4.7) 4(N)=A,C(70A^)e S2(H')=A;E:(£A)e t t 67 where the maximum numbers of r e f l e c t i o n s from the boundary |\j and ^ v ] / are seen by examination of the phase i n (4.7) to (4.10) to be determined by K, + e * Z z f N H M + ̂ N G ^ Jt-Q\ ( 4 . i i ) or (4.12) (4.13) and [\j/ by or 7 0 + ^ ^(M'+Oe.+^N'e*^ (4.14) The e x p r e s s i o n s f o r the d i r e c t wave and the wave once r e f l e c t e d from the f r e e s u r f a c e are from (4.6) C _ A J>~^b\^CO$(cii+\Q\)~itb\(kCO$cll 7)0-r\'v,& (4.15) and A ) 0 — r\i& (4.16) 68 The r e f l e c t i o n c o e f f i c i e n t s f o r a wave which s t a r t e d as a downgoing wave with r e s p e c t to the x - a x i s are s i m i l a r to e x p r e s s i o n (2.20); namely tf ASin(&+-*(1HX (4.17) and f o r a wave which s t a r t e d as an upgoing wave with r e s p e c t to the x - a x i s i s * As In{ e ^ l l e ^ ^ (4.18) z\ and § are d e f i n e d by A=Cb^/Cbl a n d &~Mz/Ml The s o l u t i o n on and c l o s e to the s u r f a c e due to the i n i t i a l d i s t u r b a n c e and s a t i s f y i n g the boundary c o n d i t i o n s (4.4) and (4.5) i s , (using the same method as f o r the d e r i v a t i o n of 2.25 and 2.26) 14 = So + S o + S * ( N -1) + 5 ^ (N - 0 + -ST(N- 0 + S l ( M - o (4.19) f o r c o n d i t i o n s (4.11) and (4.13) < = 5 o + ^ + S!(N-O + S1(N-O+S~(N ')+SI(N /; (4.20) 69 f o r c o n d i t i o n s (4.11) and (4.14) u ; = 5 0+ s> s>)+ SI(N)+s;v-o+ si (4.21) f o r c o n d i t i o n s (4.12) and (4.13) < = S 0 + S~ + S*(N) + Sl (N)+ S~(N')+ Sl(lM') (4.22) f o r c o n d i t i o n s (4.12) and (4.14). I t should be p o i n t e d out t h a t these formal s o l u t i o n s are not a p h y s i c a l s o l u - t i o n f o r an i n c i d e n t plane wave ©.£ a p a r t i c u l a r r e a l angle. oCl but are the plane wave forms s a t i s f y i n g the boundary c o n d i t i o n s from which the l i n e source s o l u t i o n w i l l be ob t a i n e d . The l a s t terms of the s e r i e s e x p r e s s i o n s (4.7)- (4.10) give r i s e to d i s c o n t i n u i t i e s i n displacement and s t r e s s which serve as boundary c o n d i t i o n s f o r the d i f - f r a c t e d waves. These w i l l be i n v e s t i g a t e d i n d e t a i l i n a l a t e r s e c t i o n . N e g l e c t i n g these l a s t terms of the s e r i e s , u;=50+So+s>-i)+jSi(N-o + sr(N'-o+sioi'-o i s v a l i d everywhere i n medium 1. For e v a l u a t i o n of the displacement due to a l i n e source, i t i s convenient to express formulae (4.7)-(4.10) and (4.15) and (4.16) as 70 5 > ) = A ± ( 7 t A ; ) e ^ , R ; | C O S K _ e ; i ) 71=1 H=i k / ^=1 \fc=i 7 where p + = (4.26) b> - A l 6 ( 4 . 2 7 ) b ^ A - G ( 4 . 2 8 ) (4.29) 71 n2 . • (4.30) t & n 6 u i = • (4.31) t a n 8 ^ . 2 = (4.32) (4.33) t a n e 0 = r s i n | e | / { d L - rco$\e\] t a n 6 j = T S i n f ^ i - e ) / { A - r c o s ( ^ a , + e ? } . 72 4,4 Formal Steady State S o l u t i o n f o r a Li n e Source In order to g e n e r a l i z e the r e s u l t s to the case of a l i n e source, the operator TC + I oo ~kb\j c^cxl ̂  (4.35) -loo i s a p p l i e d to the plane wave s o l u t i o n . In p a r t i c u l a r , the displacement IXo due to the i n i t i a l d i s t u r b a n c e can be w r i t t e n u s i n g ( 4 . 6 ) as 7t+loo -Loo ~t0O Equation (4.36) can e a s i l y be d e r i v e d from the r e s u l t s of Nakamura (1960). When {̂ ,0 i-s l a r g e , (4.36) can be a p p r o x i - mated by the asymptotic formula, U o = / \ / ^ l y _ e ~ ^ b | P ^ - t f 1 (4.37) l' (to which are the outgoing waves from the l i n e source. 73 Using (4.19) to (4.22) and (4.23) to (4.28), the s o l u t i o n i n the wedge corres p o n d i n g to a l i n e source (4.36) can be obtained as 4.5. E v a l u a t i o n of the F i r s t S e r i e s Term of the I n t e g r a l In t h i s s e c t i o n the i n t e g r a t i o n of the terms which are produced by waves which are r e f l e c t e d once by the boun- dary between the media are e v a l u a t e d . (As a guide to the e v a l u a t i o n o f the h i g h e r order s e r i e s terms, the c o n t r i b u - t i o n s due to waves twice r e f l e c t e d from the boundary bet- wreen the media are c a l c u l a t e d i n the Appendix I I I . ) From (4.23)-(4.26) and (4.38), we see that they have the f o l - lowing forms: f (4.38) (4.39) - loo where (4.40) 74 and J c . = 1 Z a n ( i 7 Y l ~ H" ., — • From the euqations ( 4 . 1 7 ) and ( 4 . 1 8 ) , 0 ^ 0 ^ and $ = 201+0.* and R,̂ and are given by the equations ( 4 . 2 9 ) to ( 4 . 3 2 ) . For t h i s i n t e g r a t i o n , the o r i g i n a l path L i s taken i n the plane f o r which R.6(A-s)/ >0 . The in t e g r a n d of ( 4 . 3 9 ) c o n t a i n s the two-valued f u n c t i o n \$~\J\— A^COS^^^-rol*) and i t s branch p o i n t s are giv e n by the r e l a t i o n C0S(^ 1 T U-+-c/_r,) " and are t h e r e f o r e l o c a t e d on the r e a l a x i s of the o i l -plane at the p o i n t s g ( 6 B = 6 0 ~ 0,̂  ) , B'(©B' = ^ ~ 6 0 ~ ^ ' ° ' ) 1 " ( F i g u r e 4 -2 ) where COS 6 o = ~ TL , t n e r e f r a c t i v e index. To f a c i l i t a t e e v a l u a t i o n , i t i s assumed that the medium i s very s l i g h t l y a b s o r p t i v e by s e t t i n g -yi— T l o ~ i < 0 where £, i s a very small p o s i t i v e q u a n t i t y . (This assumption does not a f f e c t the f i n a l r e s u l t s which correspond to £ — ^ Q but i s only a technique to f a c i l i t a t e e v a l u a t i o n of the i n t e g r a l s . ) The branch p o i n t B i s then d i s p l a c e d by [ 3 / ^ 1 — p a r a l l e l to the p o s i t i v e imaginary a x i s on the o t ^ - p l a n e . We choose the branch cut giv e n by R*Q[/\-S)—0 which i s d e f i n e d by: cos ( x + ^ ) s (n (x+$ m ) cos h u- s i n h y = n 0 1 c o s t * + 0*) co s h " ^ - s i nA(#+<f>™) s ! n rf ̂ > n 0 C 4 • 4 1 } y 76 In F i g u r e 4-2, L B i s 0 , ^ B i s ln\(Xs)=0 and the signs of Im(As)in the O^t-plane are i n d i c a t e d by p l u s and minus. When f o r l a r g e d i s t a n c e s from the o r i g i n , £ b^b{ R-IACOS(O(.I- eJJ) vanishes along the path L , the path, of i n t e g r a t i o n can t h e r e f o r e be s h i f t e d on the Riemann s u r f a c e s . The r e g i o n where I m (COS (OCL — 8|j2.)J<^ 0 O T P, vanishes at a l a r g e d i s t a n c e from the o r i g i n i s shown by h a t c h i n g . The o r i g i n a l path L , along which the r e l a t i o n s I m(Sl)ldl)<0 a n c i I m ( s ) ^ 0 h o l d , can be r e p l a c e d by L$ and ( L i , L>z. ) >' where passes through the saddle p o i n t and ( L i , L.2.) goes around the branch p o i n t B • The d o t t e d l i n e s denote t h a t they are on the second Riemann sheet where RLs^A-s")^ 0 . Each of them i s drawn along the path of s t e e p e s t descent, cosOfc -Ocosh^ . l <4'42) and c o 5 ( x - 0 ^ ) c o s h ^ = c o s ( 0 B - e^) (4.43) r e s p e c t i v e l y . 4.5.1 C o n t r i b u t i o n from the Saddle Point ( R e f l e c t e d Waves) From (4.42), L,s makes an angle ^jp- with the x- a x i s . In the neighbourhood of the saddle p o i n t the 77 f o l l o w i n g approximations are v a l i d : <*l-e£= ftf* (4.44) and C0StdLi-da)=\-l?7z (4.45) The contour i n t e g r a l along Ls f o r (4.39) i s then (4.46) Expanding the term A, i ^ l ) near the saddle p o i n t and u s i n g Watson's lemma ( J e f f r e y s , 1956) we o b t a i n ^ S - A i / A, ( 0 ( J J 6 (4.47) where, i f 0 ( J L > 6 B S i n ( 0 , w + Q - S ^ / l / A ^ C o S 2 ( 0 , w + 9£) and, if Q,̂  < 6g (4.49) 78 where 4.5.2 C o n t r i b u t i o n from the Branch Point (Head Waves) Next the c o n t r i b u t i o n from the i n t e g r a l along L| i s c o n s i d e r e d . L| and L,̂  are taken along the path of s t e e p e s t descent around B a n ^ tend to ĵp- + 0 (^ 4~ L<2° . S e t t i n g S in (^VoU) ' w e h a v e f r o m ( 4 - 3 9 ) , u L l L a = t b , A i x / 0 m , Tn. (4.5 0) I t should be noted that along near B that and I m ( A - S ) > 0 . ( 4 ' 5 1 ) From (4.43) along Lî , we can w r i t e c o s C ^ r ~ 0 ^ ) - c o s ( e B - 0 a ) - i T : x>o C 4 . 5 2 ) t h e r e f o r e C U E , = L ^ / S i n ( o C c - e ^ ) (4.53) 79 along {JZ near B . P u t t i n g ot I = 0 B + W-+ i , V (4.54) f o r very small X{ and V" , we have approximately ( u + l u ) s i n ( e 8 - e u ) = L ' " c ( 4 # 5 5 ) On the other hand, along L^. we have Sm ( cxU+ (f^)- S \n ( 0 o + VL+ IV") Zl - I/A* - j r i U + L U ) (4.56) A and from (4.55) Hence, i n the l i m i t when along , we have u s i n g (4.51) M •V5in(0B-61^) ' (4.58) As CO 80 we have | -IK R £ COS (66-6,7)+ i- ( i - i / A i r { 5 i n ( e B - e - ) p ' e L ^ (4.60) 4.6 A p e r i o d i c S o l u t i o n For computation of s y n t h e t i c seismograms, i t i s convenient to choose a displacement o f the form 0 ( t ) = - ^ A > 0 , O O Performing the o p e r a t i o n 0 - 0 0 on (4.37), (4.47) and (4.60) we o b t a i n the f o l l o w i n g solu- t i o n s (1) D i r e c t Waves y - - / y ^ . A \ 81 where t D = d o / C U c = - A ' i (2) Waves r e f l e c t e d once from the i n t e r f a c e X J O A A T ( C ) 7TV Cos {|- tan l l ^ i ^ L - t - 7 0 c A. f o r 9 | i > $B Uc=-A t A -r-^jpj* x C O S ) 4 - t a n ' - ^ ^ - i - ^ - t i t \ £ L i & ~ y^[SL/C'b\ . (3) Head Waves (4.62) (4.63) U = A- ZJZJC-S / C b i l"2 / A ( I ~ I / A ^ ( R - ) ^ {sin(e6-e-)}3/^ x A | m ^ { i T p ^ y r cosl-Ltan <^&L + f jcj (4.64) where f L „ R * CO S(6 8 - 6^) u « i « — - • . Cbi " H - | A. 82 4.7 I n t e r p r e t a t i o n of the T r a v e l Time In a study of wave p r o p a g a t i o n from a p o i n t source i n a h o r i z o n t a l l a y e r , Honda and Nakamura (1954) e v a l u a t e d the c o n t r i b u t i o n s from branch p o i n t s and a saddle p o i n t and a s s o c i a t e d the time f a c t o r s of these c o n t r i b u t i o n s with head waves and r e f l e c t e d waves. S i m i l a r l y , i n t h i s s e c t i o n we w i l l determine the r e f l e c t e d and head wave t r a v e l times and f i n d t h a t they are g i v e n by ^p\jL a n d H ^ l i l ' the time f a c t o r s of the saddle p o i n t and branch p o i n t con- t r i b u t i o n s . Consider f i r s t the path SAR. i - n F i g u r e 4-3. Then ' S A + AR. S'R. S A f l C b , Cbi __ JJTcosdz-rcos(ea^ejj^-f {AsV/^Ws^iel^e)}" c b| + A l s o , the t r a v e l time along the path S8CR, i s given by / S B C R C b t ' Cb, eisin e A (4.65) . r - , + ^ k c o s e * - r c o s ( & - e ; 6 ^ s i n e ^ r s i n C e ^ e ) ] + r s i n ( 0 ^ - e ) t a n ( e B +e * ) J c b i S m ( 0 B + 0 2 ) 4-3o Basic ray paths used i n physical interpretation of contributions from branch and saddle pointso 84 G ( c i c o s 0 ^ - r c o s ( e ^ e ) ) cos ( e B + e*) +(dsiK \0 A +r5in(e a-e))5in^B-f & 0 } (4.66) -bl s i n c e cos(e B +6;0== CM/CM. , ra Hence we have v e r i f i e d the i n t e r p r e t a t i o n of $y,\SL a n d tn tj,. 0 as the r e f l e c t e d and head wave r e s p e c t i v e l y f o r /)TL= 4~ a ^ d X — I . In a s i m i l a r manner, D"fc, 0 and H " ^ J J I c a n ^ e i n t e r P r e t e d f o r d i f f e r e n t values of 0TL and J l as shown i n Fig u r e 4-4. O b v i o u s l y , f o r the o b s e r v a t i o n p o i n t d i s t a n t from the v e r t e x , these waves a r r i v e e a r l i e r than the d i f f r a c t e d waves which are pro- duced by c o l l i s i o n s o f waves wit h the v e r t e x . Hence t h i s s o l u t i o n should adequately d e s c r i b e the e a r l y s e c t i o n of the seismogram. 4.8 Range of E x i s t e n c e o f Head Waves The range of e x i s t e n c e o f head waves can be deter- mined by c o n s i d e r i n g the process by which the i n t e g r a l (4.39) i s e v a l u a t e d . In F i g u r e 4-2, when ©B/'^IJZ. » w e c a n f ° r m a c l o s e d contour which connects with the o r i g i n a l path and which i n c l u d e s c o n t r i b u t i o n s of the saddle p o i n t and the branch p o i n t on a p p l i c a t i o n o f Cauchy's theorem. On the 17V other hand when 0g<^ Q\j> > a c l o s e d contour cannot be ( a ) (b) ( c ) (d) T?i£ 4-4• Ray paths of the head and reflected waves expressed by the f i r s t series term of the integrals. CO 86 made without e x c l u d i n g the i n t e g r a l around the branch p o i n t . T h e r e f o r e i n the f i r s t case we have c o n t r i b u t i o n s from both the branch p o i n t (head waves) and saddle p o i n t ( r e f l e c t e d waves) while i n the second case there i s only a saddle p o i n t c o n t r i b u t i o n ( r e f l e c t e d waves). I t i s c l e a r t h a t the c r i t i c a l c o n d i t i o n to decide the e x i s t e n c e of head waves i s D B - 0|JL • O b t a i n i n g UB A N C L Djj2 by the use of (4.17), (4.IS) and (4.29), (4.30), (4.31), (4.32) r e s p e c t i v e l y and us i n g the c r i t i c a l c o n d i t i o n we o b t a i n the f o l l o w i n g t r a n s c e n d e n t a l e q u a t i o n s : t a n 1 - r s i n J ^ g j + g g g H ^ _ r o s ~ U „ ft d-rcosUdi+zo^-T-e) ~ A ^ C4.es) (k-rcostze^^^J"c ~K-dz-ze{ ( 4 . 6 9 ) _^ Q T4.70) In F i g u r e 4-5, the range of e x i s t e n c e of head waves whose paths are i l l u s t r a t e d i n Fig u r e 4-4 i s shown f o r 0 ( = ^ ° and the o b s e r v a t i o n p o i n t on the l i n e 8~Q. The a b s c i s s a i s the r a t i o o f source to o b s e r v a t i o n p o i n t d i s t a n c e s from the apex while the o r d i n a t e i s the maximum fo r which head waves e x i s t . As expected from the small P i g . 4-5. Maximum value of 6Z f o r which-the head waves shown i n P i g . 4-4. e x i s t v e r s u s the r a t i o of source t o o b s e r v a t i o n d i s t a n c e s . The o b s e r v a t i o n and source p o i n t s are 5° from the f r e e s u r f a c e . 88 path d i f f e r e n c e , the range of e x i s t e n c e of (a) and (b) are c l o s e as are (c) and (d). The range of e x i s t e n c e a l s o decreases with i n c r e a s i n g number of r e f l e c t i o n s and d e c r e a s i n g r a t i o cL/T For an o b s e r v a t i o n p o i n t at 5° from the f r e e s u r f a c e and a s o u r c e - v e r t e x to o b s e r v a t i o n - v e r t e x d i s t a n c e r a t i o o f 10.0, F i g u r e 4-6 shows the range of e x i s t e n c e of head waves with changing S\ , which corresponds to a change of depth of the l i n e source. I t i s noted that with i n - c r e a s i n g 0 | , the d i p angle f o r which the head waves (a) and (b) e x i s t l i n e a r l y i n c r e a s e s while f o r head waves'of type (c) and (d) i t l i n e a r l y d e creases. 4.9 Discont i n u i t i e s D i s c o n t i n u i t i e s i n displacement and s t r e s s i n medium (1) a r i s e because the f i r s t c o l l i s i o n of the wave with a boundary changes from the f r e e s u r f a c e to the boundary between the media as oi\, passes through zero. T h i s c o l l i - s i o n w i t h the vert e x r e s u l t s i n a d i f f r a c t e d wave which i s not c o n s i d e r e d i n t h i s s o l u t i o n . When both i n i t i a l l y up- goihg and down-going waves and the i n t e r f a c e of the l a s t r e f l e c t i o n are c o n s i d e r e d , we have four cases of the com- b i n a t i o n o f the d i s c o n t i n u i t y as shown i n F i g u r e 4-7. The c r o s s - h a t c h e d areas i n d i c a t e the regions f o r which the 0 5 SO 15 9. (DEGREES) F I g o 4-6. Maximum value of the wedge angle (6|+6A) for which the head waves of the types shown i n Pig* 4-4» exist for an observation point at 5° from the free surface and d/r= 1 0 o 0 » (a) ( b ) o (c) (d) P i g , 4_7 . D i s c o n t i n u i t i e s i n medium (l)d u e to i n t e r a c t i o n of the wave wit h the v e r t e x . The l i n e d areas i n d i c a t e the r e g i o n s f o r which the geometric wave from the l a s t r e f l e c e x i s t s with the term from which i t a r i s e s i n d i c a t e d i n b r a n k e t s . 91 geometric wave from the l a s t r e f l e c t i o n e x i s t s . From equa- t i o n s (4 .11) - (4 .14) the equations of these l i n e s of d i s - c o n t i n u i t y are • + ( O S , ® S i o) s; (ft) si e = 2 ( N - i ) 6 i - i - x N e ^ - 7 c ( 4 . 7 i ) .6= 7 C - z N 0 i - 2 N e ^ ( 4 . 7 2 ) e = j^-ziu'+oe^-zti'e^ ( 4 . 7 4 , The d i s c o n t i n u i t i e s i n d i c a t e a d i s c r e p a n c y of my s o l u t i o n from the complete s o l u t i o n of the p h y s i c a l prob- lem. In order to o b t a i n a q u a n t i t a t i v e e s t i m a t i o n of the displacement d i s c o n t i n u i t i e s , plane waves i n c i d e n t toward the v e r t e x and propagating at very small angles upward (m = -) and downward (m=+) have been examined. The r e s u l t - ing d i s c o n t i n u i t i e s are shown i n F i g u r e 4-8. For t h i s geometry the r e f l e c t e d wave s o l u t i o n i s a good approximation to the complete s o l u t i o n f o r 02,<C!3°. However f o r 0^,^2,1° the d i f f r a c t e d wave p l a y s an impor- t a n t r o l e . However, t h i s f o r m u l a t i o n should adequately d e s c r i b e the e a r l y p a r t of the seismogram as the d i f f r a c t e d waves from the v e r t e x w i l l a r r i v e l a t e r than the i n i t i a l phases. I t i s seen t h a t c o i n c i d e n c e of the d i s c o n t i n u i t i e s i n F i g u r e s 4-7b and 4 - 7 c leads to at l e a s t p a r t i a l c a n c e l - l a t i o n of the d i s c o n t i n u i t i e s . Two s p e c i a l cases are of F i g * 4 - 8 „ Relative amplitudes of the displacement disconti- nuities due to a plane i n i t i a l wave close t o t h e x-axis for propagation upward (m=-) and downward (m=+)o 93 int e r e s t as t o t a l c a n c e l l a t i o n r e s u l t s . (1) Lower boundary free or r i g i d When the lower boundary of the wedge is either free or r i g i d then A-fo and A-fe, are +1 or -1 respectively, The condition 6> + 0 * = N + N ' leads to (4.71) = (4.74) or (4.72) = (4.73) so that the two li n e s of the d i s c o n t i n u i t i e s are coincident and no d i s c o n t i n u i t i e s e x i s t . Hence the solution is complete and no d i f f r a c t e d waves e x i s t . (2) Surface Source If the l i n e source i s placed i n the surface ( 6,= 0 ) then from (4.17) and (4.18) A + t = A-fe. For the p a r t i c u l a r s i t u a t i o n N + N' • we have (4.71) = (4.74) or (4.72) = (4.73) and the two d i s c o n t i n u i t i e s coincide, hence i n th i s case no discon- t i n u i t i e s exist in medium (1). In t h i s discussion the d i s c o n t i n u i t i e s in medium (2) are again expected to be less important than those i n medium (1). 94 4.10 D i s p e r s i o n Equation f o r the Lower Boundary Free and R i g i d In t h i s s e c t i o n the d i s p e r s i o n e quation i s d e r i v e d f o r a d i p p i n g s t r u c t u r e i n the simple case where medium (2) i s e i t h e r a i r or r i g i d . When medium (2) i s a i r or r i g i d , Â > and -A-^ become +1 or -1 r e s p e c t i v e l y . For 2Tn.(9\-Td&)<^ | w e c a n w r i t e cosC^i+^m(0,+e4> = cosoCL-^7n(e,+eA)sinoCu In t h i s case S*(N)=AL-e • K ± 0 6 (4.75) <VKIWA P ' " ^ 6 (4.76) 95 S i m i l a r l y the e x p r e s s i o n s f o r S,(N) a n c ! Ŝ CH) c a n °e ob t a i n e d . Operating w i t h -loo p o l e s appear from the r e l a t i o n I \y — \J which y i e l d s Sin(-feb,rce,+ejsmoCL)=o c o s ( f e i T ( 0 , + 6^)s incx :L) = o For r e a l oLi , we can then w r i t e (niz (4 where C^i i s the phase v e l o c i t y as COSoil ~ Cb\/C>n ( 0<Coci<CJC ) . T h i s e x p r e s s i o n i s the same as that o b t a i n e d by Nagumo (1961) f o r a s l o p i n g r i g i d bottom. F u r t h e r , i f we put T ( G i-t©^)~H ( H i s the depth i n the case of a h o r i z o n t a l l a y e r ) , the d i s p e r s i o n r e l a t i o n (4.77) c o i n c i d e s with that o f the h o r i z o n t a l l y l a y e r e d case. Nagumo (1961) has c a l l e d Cyi and %L = - ^ - s — the formal phase and group v e l o c i t y to d i f f e r e n t i a t e from the observed v e l o c i t i e s . 96 4.11 The H o r i z o n t a l L a y e r S o l u t i o n I t i s i n t e r e s t i n g to d e r i v e the s o l u t i o n f o r a h o r i z o n t a l l a y e r u s i n g my method as the t r a n s i t i o n o f the s o l u t i o n t o the h o r i z o n t a l l a y e r case may suggest a method f o r o b t a i n i n g the s u r f a c e wave s o l u t i o n s f o r the d i p p i n g l a y e r . For the d i f f r a c t e d wave pro b l e m , i t i s u s e f u l t o st u d y t h i s t r a n s i t i o n as the q u a n t i t a t i v e and q u a l i t a t i v e b e h a v i o u r o f the d i s c o n t i n u i t i e s as they approach zero f o r ze r o d i p a n g l e may i n d i c a t e the n a t u r e o f the d i f f r a c t e d s o l u t i o n . The same ( X-, ̂  ) c o o r d i n a t e system i s used w i t h the x - a x i s now b e i n g h o r i z o n t a l ( F i g u r e 4-9). The so u r c e i s p l a c e d at ( cL , 0 ) i n the l a y e r o f t h i c k n e s s H-H1 + H2. Employing the same p r o c e d u r e as f o r the d i p p i n g l a y e r , we o b t a i n the d i s p l a c e m e n t f o r the time v a r i a t i o n s 70 + loo e -f-e , < ~ 7 A uf ~i1lb|{(4-;<)cos Wat . -L^bi{(^^cosoCu+(^(nH,+ n H z ) + ^ ) s i n o C L } " T C • T O - it b ! {(<L-PQ COSoLl + (Z( (71.+ 0 H , + UH_)+U )5 ' Hoci} I + e. J J -i&blRoCOS(cLi~60) -HLb\RoCOS {d-i-6~) 6 + 6 bl I n i -loo (4.78) F r e e s u r f a c e S (d 9o) x < X- x R ( x 8 y ) ( ! ) 7 T T T (2) y Pigo 4-9» Coordinate system for the horizontal layer case - with the source (S) at (d ,0) and the receiver (R) at ( x B y ) 9 98 (4.78) \ O "I- g j- &0C where A , = A S i n o C o - W i - ^ c o s V . t R X L I = / W ^ M ^ H r HO - l j (4.79) tan d~^=(z H,+^) / (ck- PC) t(\ndnr{z(w~HO~^/(ck-x) tan dtz^ (znH + VrVfa-pc) (4.80) As e q u a t i o n (4.78) i s o£ the same form as (4.38), formulae (4.62) or (4.63) and (4.64) can be a p p l i e d f o r the r e f l e c t e d and head waves r e s p e c t i v e l y . T h e r e f o r e the v a r i a t i o n s of D7Tl <m. the waveforms depend only on the values rtfljj. , 6 ^ and QQ . Equation (4.78) can a l s o be d e r i v e d from (4.38), the formal d i p p i n g l a y e r s o l u t i o n , i f we take the l i m i t as rs in6 i=Hi <xr\A rs\n9^H (4.81) 99 In the case of a h o r i z o n t a l l a y e r , s u r f a c e waves appear from c o n t r i b u t i o n s of p o l e s . As our s o l u t i o n t r a n s - forms to the h o r i z o n t a l l a y e r s o l u t i o n , we c o u l d i n v e s t i g a t e s u r f a c e waves i n the case of a d i p p i n g l a y e r i f the f i n i t e s e r i e s e x p r e s s i o n of our s o l u t i o n can be changed i n t o a compact form which corresponds to a normal mode e x p r e s s i o n . 4.12 Computation of Displacement Seismograms For the d i r e c t wave, head waves, and waves once r e f l e c t e d from the boundary, displacements have been c a l - c u l a t e d f o r the three cases shown i n F i g u r e 4-10. E l a s t i c c o n s t a n t s are again those employed by H a s k e l l (1960) . Ray paths of waves which C o n t r i b u t e to the seismogram are shown i n F i g u r e 4-11. The component waves are shown on the time-displacement p l o t o f Fi g u r e 4-12 and the a r r i v a l times c o r r e s p o n d i n g to the ray paths i n d i c a t e d by l e t t e r e d arrows. A d e t a i l e d f e a t u r e i s the small amplitude o f the head waves compared to the d i r e c t and r e f l e c t e d waves. T h i s i s expected from i n s p e c t i o n o f equations (65) and (66) which show t h a t the head waves decrease as [ / ( and the r e f l e c t e d waves as I / ( " R ^ A ) ' Although the t r a v e l times changed s i g n i f i c a n t l y , the wave forms of the r e f r a c t e d and r e f l e c t e d waves do not undergo l a r g e changes f o r the three cases i l l u s t r a t e d . 100 F i g . 4-10. Three cases f o r which t h e o r e t i c a l seismograms were c a l u c u l a t e d . The parameters used were: H,= 9.59 km, -HA= 3.00 km, D= 99-6 km, d= 10.0 km, and the displacement parameter c= 0.05 sec. 101 102 Pig„ 4-12. Displacements of the component waves f o r the geometries given i n Pi g s . 4-10a„. 4-10b 9 and 4-10c o 103 F i g u r e 4-13 r e p r e s e n t s the seismograms s y n t h e s i z e d from the components of F i g u r e 4-12. The seismograms look very d i f f e r e n t . . However, the d i f f e r e n t a r r i v a l s are a l l r e c o g n i z a b l e except f o r the head wave (b) i s embedded i n the wave forms of the d i r e c t wave (c) and the r e f l e c t e d wave (d) i n the case of the h o r i z o n t a l l a y e r . A very n o t i c e a b l e f e a t u r e i s the l a t e a r r i v a l o f the r e f l e c t e d wave (e) i n the case of the h o r i z o n t a l l a y e r . More m u l t i p l y r e f l e c t e d waves w i l l appear as l a t e r phases. As the d i s - tance between the o b s e r v a t i o n p o i n t and the ve r t e x i s 10.0 km and the v e l o c i t i e s of medium (1) and medium (2) are 3.64 km/sec and 4.62 km/sec r e s p e c t i v e l y , the d i f f r a c t e d waves h a r d l y c o n t r i b u t e to the s e c t i o n of the seismogram shown here as the d i f f r a c t e d wave a r r i v i n g 4 to 5 sec a f t e r the f i r s t a r r i v a l i s due to a head wave of small amplitude i n t e r a c t i n g w i t h the v e r t e x . 1 0 4 (a) 23 * 25 a Travel time (sec) 29- (b) d 23 t a 25 29 F i g * 4-13» Synthesized seismograms r e s u l t i n g from the d i s - placements of F l g o 4-12o 105 CHAPTER 5 SUMMARY, CONCLUSIONS AND FURTHER STUDIES 5.1 Summary and Con c l u s i o n s In t h i s paper, the behavior of e l a s t i c waves i n a d i p p i n g l a y e r o v e r l y i n g an e l a s t i c medium has been i n v e s t i g a t e d i n terms of body waves i n order to expand the models a v a i l a b l e f o r the i n t e r p r e t a t i o n of c r u s t a l s t r u c t u r e . In Chapter 2, the r e f l e c t e d wave s o l u t i o n f o r a plane SH i n c i d e n t at the base of a d i p p i n g l a y e r and per- p e n d i c u l a r to s t r i k e has been developed and numerical examples p r e s e n t e d . For waves propagating i n the up~dip d i r e c t i o n w i t h angle of i n c i d e n c e i n the range of that f o r t e l e s e i s m i c S waves (45°<^ oi <C 75°), i t i s found t h a t the r e f l e c t e d wave s o l u t i o n c l o s e l y approximates the com- p l e t e s o l u t i o n f o r smal l d i p angles as the boundary con- d i t i o n s are approximately s a t i s f i e d . However, f o r waves pro p a g a t i n g i n the down-dip d i r e c t i o n , the displacement d i s c o n t i n u i t y along the edge of the f i n a l wave which does not c o l l i d e with the i n t e r f a c e s becomes l a r g e . In t h i s case the wave has r e v e r b e r a t e d o n l y a very few times w i t h i n the wedge and hence i s s t i l l of s i g n i f i c a n t amplitude. The s i z e o f t h i s d i s c o n t i n u i t y l i a s been determined and hence serves as a guide as to whether the ray s o l u t i o n 106 i s a p p l i c a b l e . For a t r a n s i e n t input to the wedge, the r e f l e c t e d waves w i l l a r r i v e e a r l i e r than the d i f f r a c t e d waves and hence even f o r l a r g e d i s c o n t i n u i t i e s , t h i s type of s o l u t i o n should apply to the composition of the i n i t i a l s e c t i o n of a seismogram. The d i f f r a c t e d wave, must pr o v i d e c o n t i n u i t y i n displacement and s t r e s s along the edge of the f i n a l wave as w e l l as those imposed at. the s u r f a c e and the boundary between the media. In Chapter 3, the behaviour of P and SV waves i n - c i d e n t at the base of a d i p p i n g l a y e r and p e r p e n d i c u l a r to s t r i k e has been i n v e s t i g a t e d by means of a r e f l e c t e d wave s o l u t i o n developed u s i n g a c y l i n d r i c a l c o o r d i n a t e system. Due to the complexity of t h i s problem, a s e r i e s s o l u t i o n i s not presented as was done f o r the SH problem; however, a computational scheme i s gi v e n by which the amplitudes and p r o p a g a t i o n d i r e c t i o n s of a l l the c o n t r i - b u t i n g waves are determined. In t h i s way the displacement at any p o i n t i n the wedge due to r e f l e c t e d waves may be found. Numerical examples of displacements and d i s p l a c e - ment r a t i o s at the s u r f a c e are presented f o r i n c i d e n t waves pr o p a g a t i n g i n both up-dip (oC, ^ = 60°) and down-dip (oL, ^ = 120°) d i r e c t i o n s . I t i s found that the displacement r a t i o s versus frequency curves f o r constant depth to i n t e r f a c e become f l a t f o r i n c i d e n t P and SV waves propagating i n the down-dip 107 d i r e c t i o n f o r d i p angles g r e a t e r than 15°. T h i s i s very d i f f e r e n t from the case of up-dip d i r e c t i o n . For the P wave pro p a g a t i n g i n the up-dip d i r e c t i o n ( = 60°), the peaks are l a r g e f o r l a r g e d i p angles and f o r d i p angles g r e a t e r than 10° the peaks s h i f t to lower frequency and become narrower with d e c r e a s i n g d i p . A f e a t u r e of p a r t i - c u l a r note i s that the H/V displacement r a t i o curves f o r i n c i d e n t SV are much more s e n s i t i v e to small changes of d i p at s m a l l d i p angles than are the V/H displacement r a t i o curves f o r i n c i d e n t P waves. I t appears t h e r e f o r e t h a t a study of SV waves would be more l i k e l y to y i e l d informa- t i o n c o n c e r n i n g d i p p i n g i n t e r f a c e s than would P waves. For waves- propagating i n the down-dip d i r e c t i o n , i t i s found that the displacement d i s c o n t i n u i t y may be l a r g e even f o r small dip angles i n d i c a t i n g that the d i f f r a c t e d wave i s of s i g n i f i c a n t amplitude. However, s i n c e the r e f l e c t e d waves w i l l a r r i v e e a r l i e r than the d i f f r a c t e d waves f o r a t r a n s i e n t input to the wedge, the r e f l e c t e d wave s o l u t i o n s h ould again apply to the composition of the i n i t i a l sec- t i o n o f the seismogram. The complex p r o p a g a t i o n d i r e c t i o n used i n t h i s chapter has been i n t e r p r e t e d i n Appendix II u s i n g the example of a f r e e R a y l e i g h wave to show that the r e a l p a r t of the angle i n d i c a t e s the p r o p a g a t i o n d i r e c t i o n and the imaginary p a r t g i v e s the decrease of amplitude. In Chapter 4, the p r o p a g a t i o n of, SH waves from a 108 l i n e source i n a clipping l a y e r o v e r l y i n g an e l a s t i c medium has been i n v e s t i g a t e d u s i n g m u l t i p l e r e f l e c t i o n f o r m u l a t i o n . A formal s o l u t i o n which does not i n c l u d e d i f f r a c t e d waves has been o b t a i n e d . The f i r s t two s e r i e s terms of the i n - t e g r a l have been e v a l u a t e d u s i n g the method of s t e e p e s t descent to o b t a i n displacements f o r both a harmonic and an a p e r i o d i c time v a r i a t i o n and c o n t r i b u t i o n s have been i n t e r - p r e t e d u s i n g ray paths i n terms of head and r e f l e c t e d waves. I f i n the i n t e g r a l the branch p o i n t s are s m a l l e r than the saddle p o i n t s , head waves do not appear. Hence the range of e x i s t e n c e of the v a r i o u s types of head waves may be determined. Using the same technique, the s o l u t i o n i n the case of a h o r i z o n t a l l a y e r has a l s o been found and compari- son made with the d i p p i n g l a y e r through numerical examples. The wave forms of the a r r i v a l s do not. d i f f e r g r e a t l y ; however, the c h a r a c t e r of the s y n t h e t i c seismogram markedly changes due to changes i n a r r i v a l times. D i s c o n t i n u i t i e s i n displacement which are a s s o c i a t e d with the d i f f r a c t e d wave have been s t u d i e d . For s p e c i a l cases i t i s found t h a t the r e f l e c t e d wave s o l u t i o n i s the complete s o l u t i o n . In the other cases, t h i s s o l u t i o n can be a p p l i e d to the i n i t i a l s e c t i o n of the seismogram. 109 5,2 Suggestions f o r F u r t h e r S t u d i e s As a r e s u l t of t h i s study, the f o l l o w i n g l i n e s of i n v e s t i g a t i o n are suggested.: (1) The c a l c u l a t i o n of a s y n t h e t i c seismogram at a s t a t i o n i n a wedge with an e l a s t i c base f o r an i n c i d e n t plane wave p u l s e . (2) The c a l c u l a t i o n of the amplitude c h a r a c t e r i s - t i c s of a m u l t i p l e r e f l e c t i o n i n the case of both d i p p i n g and h o r i z o n t a l l a y e r s by a combination of the technique developed i n t h i s t h e s i s and H a s k e l l ' s method ( H a s k e l l , 1953). (3) The problem of P and SV l i n e sources i n a wedge o v e r l y i n g an e l a s t i c medium i n terms of head and r e f l e c t e d waves n e g l e c t i n g the d i f f r a c t e d waves. (4) The exact s o l u t i o n i n terms of m u l t i p l y r e - f l e c t e d waves and m u l t i - r e f l e c t e d head waves i n the case of a l i n e source i n a d i p p i n g l a y e r with an e l a s t i c base f o r t r a n s i e n t time v a r i a t i o n s u s i n g the method of Cagniard (1962). (5) An i n v e s t i g a t i o n o f s u r f a c e wave p r o p a g a t i o n i n the presence of a d i p p i n g l a y e r o v e r l y i n g an e l a s t i c medium. In the case o f a h o r i z o n t a l l a y e r , s u r f a c e waves appear from a c o n t r i b u t i o n o f p o l e s . When the d i p angle approaches zero, the s o l u t i o n found i n Chapter 4 reduces 110 to the case of a h o r i z o n t a l l a y e r . Hence s u r f a c e waves i n the presence of a d i p p i n g l a y e r c o u l d be i n v e s t i g a t e d i f the f i n i t e s e r i e s s o l u t i o n can be w r i t t e n i n a compact form which, corresponds to a normal mode e x p r e s s i o n . ( 6 ) An.attack on the problem of d i f f r a c t e d waves u s i n g the m u l t i p l e r e f l e c t i o n wave s o l u t i o n and the d i s - c o n t i n u i t i e s found i n t h i s s o l u t i o n which are r e l a t e d to the d i f f r a c t e d , waves. I l l BIBLIOGRAPHY Cagnia r d , L., 1962. R e f l e c t i o n and r e f r a c t i o n o f progres- s i v e s e i s m i c waves, McGraw-Hill, New York. Clowes, R. M., Kanasewich, E. R., and Cumming, G. L,, 1968. Deep c r u s t a l s e i s m i c r e f l e c t i o n s at n e a r - v e r t i c a l i n c i d e n c e , Geophysics, 3_3, 441-451 . E l l i s , R. M. and Basham, P. W. , 1968. C r u s t a l c h a r a c t e r i s - t i c s from s h o r t - p e r i o d P waves, Bull. Seism. Soc. Amer.. 3 58_, 1681-1700. Emura, K., 1960. Propagation of the d i s t u r b a n c e s i n the medium c o n s i s t i n g o f s e m i - i n f i n i t e l i q u i d and s o l i d , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 12, 63-100. Ewing, W. M., W. S. J a r d e t z k y , and F. Press, 1957. E l a s t i c waves i n l a y e r e d media, McGraw-Hill, New York. Fernandez, L. M. and Careaga, J . , 1968. The t h i c k n e s s of the c r u s t i n c e n t r a l U n i t e d S t a t e s and La Paz, B o l i v i a , from the spectrum of l o n g i t u d i n a l s e i s m i c waves, Bull. Seism. Soc. Amer., 58_, 711-741. Fuchs, K., 1966. S y n t h e t i c seismograms of P waves propaga- t i n g i n s o l i d wedges with f r e e boundaries, Geophysics, 3_1_, 524-535 . H a s k e l l , N. A., 1953. The d i s p e r s i o n of s u r f a c e waves i n m u l t i l a y e r e d media, Bull. Seism. Soc. Amer., 43, 17-34. H a s k e l l , N. A., 1960. C r u s t a l r e f l e c t i o n of plane SH waves, J. Geophys. Res., 65_, 4147-4150 . H a s k e l l , N. A. 1962. C r u s t a l r e f l e c t i o n of plane P and SV waves, J. Geophys. Res., 6J7 , 4751-4767 . Honda, H. and Nakamura, K., 1954. On the r e f l e c t i o n and r e f r a c t i o n of the e x p l o s i v e sounds at the ocean bottom I I , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 6, 70-84. Hudson, J . A., 196 3. SH waves i n a wedge - shaped medium, Geophys. J. R.A.S., 7, 517-546. 112 Hudson, "J. A. and Knopoff, L. , 1964. Transmission and r e f l e c t i o n of surface waves at a corner 2, Rayleigh waves, J. Geophys. Res., 69, 281-289. Ibrahim, A. B. , 1969. Determination of cr u s t a l thickness from spectral behavior of SH waves, Bull. Seism. Soo. Amer., S9_, 1247-1258. J e f f r e y s , H. and J e f f r e y s , B. S., 1956. Methods of mathe- matical physics, Cambridge Uriiv. Press, Cambridge, England. Kane, J . and Spence, J . , 1963. Rayleigh waves transmission on e l a s t i c wedges, Geophysics, 28_, 715-723 . Kane, J . , 1966. Teleseismic response of a uniform dipping crust (Part I of a series on cr u s t a l equalization of seismic arrays), Bull. Seism. Soo. Amer., 56, 841-859. K e l l e r , J . B., 1962. Geometrical theory of d i f f r a c t i o n , J. Aooust. Soo. Am., 5_2, 116-130. Lapwood, E. R., 1961. The transmission of a Rayleigh pulse round a c o r n e r , Geophys. J. R. Astr. Soo., 4_, 174-196. McGarr, A. and Alsop, L. E., 1967. Transmission and r e f l e c - t i o n of Rayleigh waves at v e r t i c a l , boundaries, J. Geophys. Res., 72_, 2169-2180. Nagumo, S. , 1961. E l a s t i c wave propagation in a l i q u i d layer overlying a sloping r i g i d bottom, J. Seism. Soo. Japan, 14_, 189-197. Nakamura, K., 1960. Normal mode waves i n an e l a s t i c plate ( 1 ) , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 12, 44-62. Phinney, R. A., 1964. Structure of the earth's crust from spectral behavior of long period body waves, J. Geophys. Res., 69, 2997-3017. Sato, R., 1963. D i f f r a c t i o n of SH waves at an obtuse-angled corner, J. Phys. Earth, 11_, 1-17. , 113 APPENDIX I ENERGY RELATIONS As a check on the amplitude r e l a t i o n s d e r i v e d i n the t e x t , the method used by Ewing et a l (1957) has been used to d e r i v e e x p r e s s i o n s f o r energy p a r t i t i o n between the i n c i d e n t , r e f l e c t e d and r e f r a c t e d waves. To c a l c u l a t e k i n e t i c e n e r g i e s , we note that the v e l o c i t i e s 1X are r e l a t e d to the displacements IA. by IL— LCOU. and hence may be obtained d i r e c t l y from equations (3.17) and (3.18). The energy f l u x f o r the waves can then be o b t a i n e d by m u l t i p l y i n g the k i n e t i c energy per u n i t volume, ~~^f(^r~^~ l^e') ^y the v e l o c i t y of pr o p a g a t i o n and the area o f wavefront i n v o l v e d . For a P wave i n c i d e n t on the s u r f a c e , i t s energy f l u x per u n i t area must be equal to the sum of the e n e r g i e s i n the r e f l e c t e d and r e f r a c t e d waves. We have X c*2 ctn \s\n(oLi~dd)\ ~ ~Z?2. Caz Crjt ^2. (SI n (oirsi- 0<0| + 2 ?z cti Drx cbx I s f n QO| (A-1.1) A f o l l o w i n g c o m p u t a t i o n a l l y more u s e f u l form i s ob t a i n e d 114 u s i n g (3.28) 1 ( C m / HvJlCiJ | S i n ( 6 k - o 6 ) | •+ >Y 1 1 A/Wf Yi -Ci/v^) ̂cos^Ce^-oO The c o r r e s p o n d i n g e q u a t i o n f o r S waves i s •'" W ^ W V P W | s i n ( 6 k - p ) | (A-1.2) + 4 § \ M I D i n / I Sln(6d -P)| For P and SV waves i n c i d e n t on the boundary from medium ( 1 ) , the r e l a t i o n s are r e s p e c t i v e l y 1 1 5 s i n ( e ^ - O L ) | , c ^ / C r * A v 7 1 - ^ £ C O S % d ( k - d 5 ( A - 1 . 4 ) B + S i Vb i / 1 B i n , ( A - 1 . 5 ) 116 APPENDIX II EXPRESSION OF A FREE RAYLEIGH WAVE USING COMPLEX ANGLES In the c a l c u l a t i o n o f the displ a c e m e n t s , complex angles have been used i n order that the cases o f t o t a l r e f l e c t i o n and i n c i d e n t angles g r e a t e r than the c r i t i c a l are i n v o l v e d i n the r e s u l t s . Although R a y l e i g h waves are not produced i n t h i s problem, the e x p r e s s i o n of R a y l e i g h waves i n terms of complex angles i s of i n t e r e s t . Consider an e l a s t i c h a l f - s p a c e with f r e e s u r f a c e 0 ~ O ( F i g u r e A - l ) The s o l u t i o n i n the medium can be w r i t t e n as (A-2.1) 0 > K - i V . c ^ T C O S ( ° - ^ The boundary c o n d i t i o n s at S — 0 are ee = o T0=O S u b s t i t u t i n g (A-2.1) i n t o the boundary c o n d i t i o n s u s i n g (3.15) and (3.16), we have ( l - ^ V b t c o s V O A ^ + BJ?SI n ^ c o s ^ = 0 (A-2.3) (A-2.2)  118 and V b [ C O S d ^ C O S p g . (A-2.4) From (A-2.3), we have ~ ^v^s\noi?<cosd,cis\n^cos$^o ( A ' 2 - 5 ) S u b s t i t u t i n g (A-2.4), and w r i t i n g % — COSbL^and V — t ^ i g i v e s | G ( i - v ) ? c 3 + ( i 6 - ^ ) ^ + - ^ ^ - i 7 3 = 0 ( A . 2 . 6 ) Assuming Poisson's r e l a t i o n , Z>~J^- , y i e l d s The r e a l root of t h i s e quation i s %=3. which co r - responds to COS olfi= ± 1,88^ and u s i n g (A-2.4), COS ± 1 . 0 88 • R e c a l l ing the r e l a t i o n s a.rccos(r B ) = J C - a r c c o s n arc cos p = C&rccoshp (p=rea\ > i ) we o b t a i n oCp.= 1.-2^-7 L o r j o - \ . 2 . < t 7 I (A-2.8) P a = o . ^ o 6 8 L o r jo™o . 4 -oG8o 119 If i n equations (A-2.1), we use c o s ( p ± 1%) = c o s p c o s h % : f " ls\n?s\nh% S i n ( p ± L ^ ) = S i n p c o s h S T icosps\r\]r\% we have O A ^ ± l ^ r C i . S 8 ^ c o s e ± i i . F ^ 7 s m a ) - o + L t b l r ( | . o 8 8 C o S 0 ± L o . ^ 7 3 s f n 6 ) (A-2.9) 60 _ 0.^2.78., %.aA (A-2.10) We see that the d i l a t a t i o n and rotation propagate with the v e l o c i t y 0.cJ{c|'j-Cfc>| which coincides with the v e l o c i t y of the free Rayleigh wave. As a re s u l t we see that for a Rayleigh wave written i n terms of complex angles, the re a l part of the angle indicates the propagation d i r e c t i o n and the imaginary part gives the decrease of amplitude with the two solutions of (A-2.8) representing waves propa- gating i n opposite directions ( 0 O^Hoi JC ) . 120 APPENDIX I I I EVALUATION OF THE SECOND SERIES TERMS OF THE INTEGRAL As a guide to computation of hi g h e r order terms i n the s e r i e s , a summary o f the e v a l u a t i o n procedures and r e s u l t s f o r the c o n t r i b u t i o n s by waves twice r e f l e c t e d from the boundary between the e l a s t i c media are e v a l u a t e d here. The second terms o f the s e r i e s have the form -LOO where ^ _ A S i n ( ^ + o 6 L ) - § y i - A " c o s ^ ^ o 6 L ) 1 A s i n( 0r+ou)+S/1 -tfcosXtf*-* ou) z A s i n ( 0 ^ oa)+S / 1 -tfcosXA+oil) and 1 = a n d 771 = + , — From equations (4.17) and (4.18) (A-3.1) (A-3.2) ( A - 3 . 3 ) and 0 ^ have been gi v e n by equations (4.29) to (4.32)*. 121 As the int e g r a n d s of (A-3.1) c o n t a i n the expres- s i o n s and (A- 3.4) which are both two-valued, a fo u r - s h e e t e d Riemann s u r f a c e i s r e q u i r e d f o r t h e i r r e p r e s e n t a t i o n . The branch c u t s , along which the four sheets c o a l e s c e are d e f i n e d by R . s (A ' s0 = O and (P^sz)= 0 • F ° r e v a l u a t i o n purposes the medium i s assumed to be very s l i g h t l y a b s o r p t i v e as b e f o r e . The sheets I, I I , I I I and IV are d e f i n e d c o r r e s p o n d i n g to the combinations „ „, (A-3.5, (Re(Asi)<o,Re(AS2)>o) , ( R e M > o , Re(AS2.)>o); (Re(ASi)>o,Re(Asa)<o) , (Re(A s l)<o,Re(A«a)<o). r e s p e c t i v e l y . The o r i g i n a l path of i n t e g r a t i o n can be s h i f t e d on any sheet o f the Riemann s u r f a c e f o r the f a c t o r 0 i&b\fi-2Si.COS(oLi, ^ ^ v a n i s h i n g along the path at a l a r g e d i s t a n c e from the o r i g i n . The o r i g i n a l path [_J i s taken on sheet II where the r e l a t i o n s Im(S\)T\oLl)<iO , I m (As i ) *C0 and h o l d along L . As an example, when QQ ̂ > Qc, 122 where (A- 3.6) Q 0 = ATCCOSO/A) the o r i g i n a l path can be s h i f t e d to Lis , ( L 3 } Ly- ) and ( b i , Ltjj, ) a s shown i n F i g u r e A-2. [ j S passes through the saddle p o i n t £> , and the contours ( L»3 ; LyO and ( L 1 ; ) go around the branch p o i n t s £ and 3 r e s p e c t i v e l y , each one of them being drawn along the path of s t e e p e s t descent g i v e n by cos(x-d?i)cosh*&= I (A-3.7) cosoc-e^coshty = cos(eo- eS) . (A-3.8) and C O S ( X - 0 j £ ) C O S h ^ = C O S ( 0 B - 0 ^ ) (A-3.9) where 123 n m Re X s , < o Re X S 2 >o Re X S I >o R e X S 2 >o Re X S ) >o Re X S 2 <o Re X s, <o Re X S 2 <o E i g . A-2. The oci-plane showing branch cuts and i n t e g r a l paths f o r evaluation of the second s e r i e s term of the i n t e g r a l s . Notation: B,C - branch points; S - saddle point; L - o r i g i n a l path of i n t e g r a t i o n ; L s - path of steepest descent through saddle point; and L{, ("i= 1,.2 ••••) - paths of branch l i n e i n t e g r a l * 124 I n t e g r a l Around C The contour i n t e g r a l s along ( L 3 , L » f ) can be ev a l u a t e d by the same procedure as b e f o r e . By n o t i n g the r e l a t i o n s CA-3.10) f o r the path Vu\. near C on sheet I, f o r a harmonic time v a r i a t i o n the c o n t r i b u t i o n to the displacement i s found to be where I n t e g r a l Around B The contour i n t e g r a l s along ( \jK } L^,) c a n a l s o be e v a l u a t e d by the same procedure as b e f o r e . By n o t i n g the 125 r e l a t i o n s H e ( X s O > 0 , I m ( A s O > 0 , (A-3.13) for the path L.z near B lying on sheet I I , for a harmonic time v a r i a t i o n the contribution to the displacement i s found to be : - : : ^ \ ' : f . - = A J / I J G S | i • (A-3.14) Integral through S The contour i n t e g r a l along L 5 can also be eva- luated by the same procedure as before. For a harmonic time v a r i a t i o n the contribution to the displacement is found to be u s-KJ^& AT(C) A : ( C ) o t f e M ! C i ^ l (A-3.15) 126 where f o r V* ^ B V' ^ c Al(C> A Sin (0r+ O - h J \ - t f c o s ^ + e%) A S in ($zT+ 65) + iJ\-£?cos*(0r+eZ) A s i n (ej ) .+ g / i - ^ C o s ^ ^ + eS.) (A-3.16) i f e B > e ^ > 0 c : At@£) = same as (A-3.16) (A-3.17) tanf,= Asinc^+e^) (A-3.18) i f 6& > 6c > dza t a n ^ A s i n ( ^ a ^ ) (A-3.19) (A-3.20) 127 A p e r i o d i c S o l u t i o n When the motions are a p e r i o d i c and vary as </>(t)=- t 4 c * A > 0 , O O the o p e r a t i o n 0 -<?o a p p l i e d to (A-3.11), (A-3.14) and (A-3.15) y i e l d s the f o l - lowing s o l u t i o n s : Head waves, jE{\-\/£?r* ( R - ) 3 A | S j n ( e c _ • * (A-3.21) X A Sin (f^+ ds)-Sj 1 - ^ 0 0 5 " ^ + 6s) Jl+pz___J"jl/f-A _sinC0r+aB)+§/i-A^ost$r-feB) (A-3.23) 128 where u T,„n = — — — & G b | R e f l e c t e d waves, ;7" ' f ' : fj . A A;(O-AM) V JT: ,.. f o r e B>0 27>0c i l = -A- 3 0 A 1 Y COS + ^ + 3ft 129 for eB 70c>e_jL . m n w - l ............ where R L ^ = / C b l (A-3. If the branch points are smaller than the saddle points head waves do not appear. The ray paths for a r r i v a l s which tr a v e l along part of the path as head waves are shown in Figure A-3. Hence, except for d i f f r a c t e d waves, we can formally obtain a complete synthetic seismogram in the case of a dip- ping layer by applying this procedure to the t h i r d and higher order series terms of the formal i n t e g r a l s o l u t i o n . 130 From branch point B From branch point C m = + i n (A-3.-1). PUBLICATIONS Nakamura, K. and I s h i i , H., 1965. Refraction of explosive sound waves from a line source i n a i r into water, S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 16, 90-107. Tohoku Univ. Aftershocks Observation Group, 1966. Observa- tion of aftershocks of an earthquake happened of f Oga-Peninsula on 7th, May, 1964, Tohoku Disaster Prevention Research Group Report, 85-101. I s h i i , H. and Takagi, A., 1967. Theoretical study on the c r u s t a l movements, Part I. The influence of surface topography (Two-dimensional SH torque source), S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 77-94. I s h i i , H. and Takagi, A., 1967. Theoretical study on the crustal movements, Part II. The influence of horizontal discontinuity, S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 95-106. I s h i i , H. and E l l i s , R. M., Multiple r e f l e c t i o n of plane SH waves by a dipping layer, B u l l . Seism. Soc. Amer. (accepted for publication).

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